The book could be a good companion for any graduate student in partial differential equations or in applied mathematics.

*389*
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H ANDBOOK OF D IFFERENTIAL E QUATIONS S TATIONARY PARTIAL D IFFERENTIAL E QUATIONS VOLUME I

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H ANDBOOK OF D IFFERENTIAL E QUATIONS S TATIONARY PARTIAL D IFFERENTIAL E QUATIONS Volume I

Edited by

M. CHIPOT Institute of Mathematics, University of Zurich, Zurich, Switzerland

P. QUITTNER Institute of Applied Mathematics, Comenius University, Bratislava, Slovak Republic

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© 2004 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-proﬁt educational classroom use. Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected] Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P OLP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made.

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ISBN: 0 444 51126 1 Set ISBN: 0 444 51743 x The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands

Preface This handbook is Volume I in a multi-volume series devoted to stationary partial differential equations. It is a collection of self contained, state-of-the-art surveys written by well-known experts in the ﬁeld. The authors have made an effort to achieve readability for mathematicians and scientists from other ﬁelds, and we hope that this series of handbooks will become a new reference for research, learning and teaching. Partial differential equations represent one of the most rapidly developing topics in mathematics. This is due to their numerous applications in science and engineering on one hand and to the challenge and beauty of associated mathematical problems on the other. This volume consists of eight chapters covering a variety of elliptic problems and explaining many useful ideas, techniques and results. Although the central theme is the mathematically rigorous analysis, many of the contributions are enriched by a plenty of ﬁgures originating in numerical simulations. We thank all the contributors for their clearly written and elegant articles, and Arjen Sevenster at Elsevier for efﬁcient collaboration. M. Chipot and P. Quittner

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List of Contributors Bandle, C., Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland (Ch. 1) Galdi, G.P., University of Pittsburgh, 15261 Pittsburgh, USA (Ch. 2) Ni, W.-M., University of Minnesota, Minneapolis, MN 55455, USA (Ch. 3) Pedregal, P., Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain (Ch. 4) Reichel, W., Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland (Ch. 1) Shafrir, I., Technion, Israel Institute of Technology, 32000 Haifa, Israel (Ch. 5) Takáˇc, P., Universität Rostock, D-18055 Rostock, Germany (Ch. 6) Tarantello, G., Università di Roma ‘Tor Vergata’, Dipartimento di Matematica, Via della Ricerca Scientiﬁca, 1, 00133 Rome, Italy (Ch. 7) Véron, L., Université de Tours, Parc de Grandmont, 37200 Tours, France (Ch. 8)

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Contents Preface List of Contributors

v vii

1. Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via Degree Theory C. Bandle and W. Reichel 2. Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain G.P. Galdi 3. Qualitative Properties of Solutions to Elliptic Problems W.-M. Ni 4. On Some Basic Aspects of the Relationship between the Calculus of Variations and Differential Equations P. Pedregal 5. On a Class of Singular Perturbation Problems I. Shafrir 6. Nonlinear Spectral Problems for Degenerate Elliptic Operators P. Takáˇc 7. Analytical Aspects of Liouville-Type Equations with Singular Sources G. Tarantello 8. Elliptic Equations Involving Measures L. Véron Author Index Subject Index

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CHAPTER 1

Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via Degree Theory

Catherine Bandle and Wolfgang Reichel Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland E-mail: {catherine.bandle;wolfgang.reichel}@unibas.ch

Contents 1. Degree theory . . . . . . . . . . . . . . . . . . . 1.1. Introduction . . . . . . . . . . . . . . . . 1.2. Brouwer degree in ﬁnite dimensions . . . 1.3. Leray–Schauder degree in Banach spaces 1.4. The index of an isolated solution . . . . . 1.5. Asymptotically linear equations . . . . . 1.6. Fixed point alternatives . . . . . . . . . . 1.7. Degree theory in unbounded domains . . 1.8. Degree theory in cones . . . . . . . . . . 1.9. Notes . . . . . . . . . . . . . . . . . . . . 2. Existence of solutions . . . . . . . . . . . . . . 2.1. Function spaces . . . . . . . . . . . . . . 2.2. Uniformly elliptic linear operators . . . . 2.3. Schauder estimates . . . . . . . . . . . . . 2.4. Lp -estimates . . . . . . . . . . . . . . . . 2.5. Applications to boundary value problems 2.6. Comparison principles . . . . . . . . . . . 2.7. Degree between sub- and supersolutions . 2.8. Emden–Fowler type equations . . . . . . 2.9. Multiplicity results . . . . . . . . . . . . . 2.10. Notes . . . . . . . . . . . . . . . . . . . . 3. Global continuation of solutions . . . . . . . . . 3.1. A global implicit function theorem . . . . 3.2. Applications – continuation of solutions . 3.3. Further applications . . . . . . . . . . . . 3.4. Notes . . . . . . . . . . . . . . . . . . . . 4. Bifurcation theory and related problems . . . .

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4.1. Bifurcation from the trivial solution 4.2. Bifurcation from inﬁnity . . . . . . 4.3. Perturbations at resonance . . . . . . Acknowledgments . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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1. Degree theory 1.1. Introduction In this chapter we shall develop a tool for proving the existence of solutions of nonlinear equations in a Banach space X of the form F (x) = y,

x ∈ Ω ⊂ X,

⊂ X → X is a continuous map. We want to study the solutions in the intewhere F : Ω rior of Ω knowing the restriction of F onto the boundary ∂Ω. This will be achieved by considering a topological invariant deﬁned on the triple (F, Ω, y). Such an invariant can easily be found for continuously differentiable functions F : [0, 1] → R with isolated solutions {xi }ki=1 of F (x) = y. Let us ﬁx F (0) and F (1). It clear that for given y ∈ / {F (0), F (1)} the number of solutions varies with F but is k (x ) is invariant under deformations of F which keep the endpoints ﬁxed, sign F i i=1 cf. Figures 1 and 2. More generally, F (0) and F (1) can also be deformed as long as they do not cross y. As soon as one of the endpoints coincides with y, the invariance under deformations is lost, cf. Figure 3. If the solutions are not isolated or if F (xi ) = 0 then a natural approach is to approximate F by functions Fn with isolated solutions, cf. Figure 4. Heuristically, the quantity described above seems to be stable if we pass to the limit Fn → F . For analytic functions F : Ω ⊂ C → C the argument principle can be employed to determine the number of solutions F (z) = w in a given domain. More precisely, if γ is a simple closed curve in Ω on which F is different from w then the number of solutions inside γ is F (z) 1 given by the boundary integral 2πi γ F (z)−w dz. Obviously this integral is invariant under “small” deformations of F on γ . In the subsequent sections these simple observations will be generalized to large classes of equations in ﬁnite and inﬁnite-dimensional spaces. The quantities ki=1 sign F (xi ) and F (z) 1 2πi γ F (z)−w dz will be replaced by a more general concept, namely the topological degree. In many cases it will be impossible to compute it directly. For the applications two properties will be crucial: 1. If the degree is different from zero then a solution of F (x) = y exists. 2. The degree is invariant under certain deformations. The deﬁnition and use of the degree goes back to Brouwer (1912) [16] and Leray and Schauder (1934) [49]. Since we are mainly interested in the degree theory as a tool for proving the existence of solutions to certain equations and less in its geometrical meaning we shall adopt an axiomatic approach common in analysis. It consists ﬁrst in listing the desired properties, then in proving that there is at most one quantity satisfying all these conditions, and ﬁnally in discussing one of several possible constructions of the degree. This text is intended for nonspecialists. The goal is to present a powerful tool for proving existence of solutions of linear and nonlinear second-order elliptic boundary value problems and to recount some of the most interesting properties and applications. Rather than describing more recent topological developments of the notion of degree and its properties

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Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

we discuss in some detail different classes of boundary value problems for which variational methods do not apply. The completeness of the proofs varies. Full details are given if the proofs are not available in the literature or if they contribute to a better understanding. The more difﬁcult technical proofs are only sketched and references are suggested.

1.2. Brouwer degree in ﬁnite dimensions In ﬁnite dimensions the notion of degree goes back to Brouwer [16]. The proofs of the next → RN two sections can be found in [26]. Let Ω ⊂ RN be a bounded open set and G : Ω N N be a continuous map. Let Id : R → R denote the identity map. / G(∂Ω). The degree is a mapping deg : (G, Ω, y) → Z D EFINITION 1.1. Suppose that y ∈ with the following properties: (d1) Normalization: deg(Id, Ω, y) = 1 if y ∈ Ω and deg(Id, Ω, y) = 0 if y ∈ / Ω.

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=Ω 1 ∪ Ω 2 with Ω1 , Ω2 open, disjoint and y ∈ (d2) Excision: if Ω / G(∂Ω1 ∪ ∂Ω2 ) then deg(G, Ω, y) = deg(G, Ω1 , y) + deg(G, Ω2 , y). → RN is continuous and y : [0, 1] → RN is (d3) Homotopy invariance: if h : [0, 1] × Ω continuous with y(t) ∈ / h(t, ∂Ω) for all t ∈ [0, 1] then deg h(t, ·), Ω, y(t) is independent of t. (d4) Existence: if deg(G, Ω, y) = 0 then G(x) = y has a solution x ∈ Ω. It can be shown that (d1)–(d3) imply (d4). Moreover, there is at most one function satisfying (d1)–(d3) (cf., e.g., Deimling [26]). One can show the following extension of the homotopy invariance (d3), cf. Amann [3] and Leray and Schauder [49]: (d3)g General homotopy invariance: let Θ ⊂ [0, 1] × RN be bounded and open in [0, 1] × RN and denote by Θt the slice at t, that is, Θt = x ∈ RN : (t, x) ∈ Θ . → RN is continuous and y : [0, 1] → RN is continuous with y(t) ∈ / If h : Θ h(t, ∂Θt ) for all t ∈ [0, 1] then deg h(t, ·), Θt , y(t) is independent of t. For the construction of the degree we proceed in several steps. and denote by G (x) its (I) Degree for regular values of C 1 -maps. Let G ∈ C 1 (Ω) Jacobian and by det G (x) the determinant of the Jacobian. Furthermore y ∈ RN is called a regular value of G if det G (x) = 0 for all x ∈ G−1 (y). Otherwise y is called a singular value. If y ∈ / G(∂Ω) is a regular value then we deﬁne

deg(G, Ω, y) :=

sign det G (x).

x∈G−1 (y)

It can be shown that for small ε > 0 the following integral representation holds

φε G(x) − y det G (x) dx,

deg(G, Ω, y) = Ω

where φε (x) = ε−N φ1 (x/ε) and φ1 ∈ C0∞ (RN ) with φ1 (0) > 0, RN φ1 (x) dx = 1. This integral representation plays a key role in the analytic approach to degree theory.

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For y ∈ (II) Degree for singular values of C 2 -maps. Let G ∈ C 2 (Ω). / G(∂Ω) let y1 be a regular value with |y1 − y| < dist(y, G(∂Ω)). By Sard’s lemma, which states that the set of singular values has N -dimensional Lebesgue measure 0, such a value always exists. Since it can be shown that deg(G, Ω, y1 ) is independent of the choice of y1 the following deﬁnition deg(G, Ω, y) := deg(G, Ω, y1) makes sense. The proof is done through the integral representation. E XAMPLE 1.1. Consider Ω = (−1, 1) and G(x) = x 3 . The value y = 0 is a singular value, but any neighboring value y1 = δ is regular. Then deg(G, Ω, y1 ) =√ sign G (δ 1/3 ) = 1. If √ 2 G(x) = x then similarly deg(G, Ω, y1 ) = sign G (− δ) + sign G ( δ) = 0. E XAMPLE 1.2. Consider Ω = {x12 + x22 < 1} and G(x1 , x2 ) = (x13 − x1 x22 , x23 ). The value y = (0, 0) is singular, and the neighboring value y1 = (0, δ 3 ) with δ > 0 is regular. The preimage G−1 (y1 ) consists of the three points (0, δ), (δ, δ) and (−δ, δ). In the ﬁrst point G has a negative and in the last two points a positive determinant. Hence deg(G, Ω, y) = 1. (III) Degree for continuous maps. An important fact of the degree is that it can be and y ∈ extended to maps which are merely continuous. Let G ∈ C(Ω) / G(∂Ω). Let be such that G−H ∞ < dist(y, G(∂Ω)). Then it turns out that deg(H, Ω, y) H ∈ C 2 (Ω) is independent of the choice of H . Therefore we can set deg(G, Ω, y) := deg(H, Ω, y). (IV) Degree in ﬁnite-dimensional spaces. The concept of degree is easily extended to arbitrary spaces of ﬁnite dimensions which are different from RN . Let (X, · ) be an and let N -dimensional normed space. Suppose Ω ⊂ X is an open, bounded set, G ∈ C(Ω) N y∈ / G(∂Ω). Let L : X → R be a linear homeomorphism. Then deg(G, Ω, y) := deg L ◦ G ◦ L−1 , LΩ, Ly is independent of the choice of L. A consequence of the elementary properties of degree theory is the following theorem. 1 (0) → T HEOREM 1.2 (Brouwer’s ﬁxed point theorem). Every continuous map F : B 1 (0), where B1 (0) is the open unit ball {x ∈ RN : x < 1} has a ﬁxed point. B P ROOF. If there is no ﬁxed point on the boundary of B1 (0) we consider the homotopy h(t, x) = Id −tF (x). There is no zero of h(t, ·) on ∂B1 (0), because for t = 1 this is excluded by assumption and for 0 t < 1 we have x − tF (x) 1 − t > 0 if x = 1. Thus deg(h(t, ·), B1 (0), 0) is well deﬁned. From the homotopy invariance (d3) we conclude that deg(h(t, ·), B1 (0), 0) = deg(Id, B1 (0), 0) = 1 which by (d4) establishes the assertion.

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1.3. Leray–Schauder degree in Banach spaces We wish to extend the previous results to inﬁnite-dimensional spaces. However, one needs to be careful: although Brouwer’s ﬁxed point theorem follows immediately from the elementary properties of the degree, its generalization to inﬁnite dimensions is false (cf. notes). A large class of nonlinear maps for which it is still valid is the class of continuous compact maps. And likewise the topological degree can be deﬁned for continuous compact perturbations of the identity. Suppose (X, ·) is a real Banach space. Let Ω = ∅ be an open, bounded set in X and let → X be compact which means that F is continuous and maps bounded closed sets F :Ω into compact sets. In contrast to the Brouwer degree, which is deﬁned for any continuous map, the Leray–Schauder degree is deﬁned only for compact perturbations of the identity, namely G = Id −F . T HEOREM 1.3. Let the above assumptions hold. If y ∈ / (Id −F )(∂Ω) then there exists a unique mapping deg : (Id −F, Ω, y) → Z for which the properties (d1), (d2) and (d4) of Deﬁnition 1.1 hold with G replaced by Id −F and for which (d3) holds in the following form: → X is compact in R × X and y : (d3) Homotopy invariance: if k : [0, 1] × Ω [0, 1] → X is continuous with y(t) ∈ / (Id −k(t, ·))(∂Ω) for all t ∈ [0, 1] then deg(Id −k(t, ·), Ω, y(t)) is independent on t. As for the Brouwer degree one can generalize (d3) : (d3)g General homotopy invariance: let Θ ⊂ [0, 1] × X be bounded and open in → X is compact and [0, 1] × X with Θt = {x ∈ X: (t, x) ∈ Θ}. If k : Θ y : [0, 1] → X is continuous with y(t) ∈ / (Id −k(t, ·))(∂Θt ) for all t ∈ [0, 1] then deg(Id −k(t, ·), Θt , y(t)) is independent of t. The class of maps Id −F , F compact is by no means the most general class for which the degree can be deﬁned. It is, however, sufﬁciently broad to include the applications discussed here. The fundamental idea in inﬁnite-dimensional degree theory goes back to Schauder. It consists of the following approximation of compact maps F deﬁned on bounded sets Ω: → Xε ⊂ X with ﬁnite-dimensional for every ε > 0 there exists a continuous map Fε : Ω . In general the approximation Fε is range Xε such that F (x) − Fε (x) < ε for all x ∈ Ω not unique. However, it turns out that the degree for Id −Fε on Ω ∩ Xε is well deﬁned, provided 0 < ε ε0 = dist(y, (Id −F )(∂Ω)). We then deﬁne deg(Id −F, Ω, y) := deg(Id −Fε , Ω ∩ Xε , y). This deﬁnition makes sense since the latter is independent of the choice of the Schauder approximation and independent of ε ∈ (0, ε0 ). 1.3.1. Retracts and Schauder’s ﬁxed point theorem D EFINITION 1.4. A subset R of a Banach space X is called a retract of X if there exists a continuous map r : X → R such that r|R = Id. The map r is called a retraction.

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E XAMPLES . (1). The closed unit ball is a retract. Consider the map r(x) = x/x2 if x > 1 and r(x) = x elsewhere. (2). Dugundji [27] proved that closed convex sets are retracts. T HEOREM 1.5 (Schauder). (i) Let X be a Banach space, C ⊂ X nonempty closed bounded and convex. If F : C → C is compact then F has a ﬁxed point. (ii) The same is true if C is homeomorphic to a closed bounded and convex set. P ROOF. (i) By Dugundji’s theorem C is a retract. Let r : X → C be the retraction. Consider the map F ◦ r : X → C. Any ﬁxed point of F ◦ r is a ﬁxed point of F . Let Bρ (0) be a large ball containing C. The map F ◦ r has no ﬁxed point on ∂Bρ (0). Consider the homotopy k(t, x) := tF (r(x)) for t ∈ [0, 1]. There is no ﬁxed point of k(t, ·) on ∂Bρ (0), because for t = 1 this has already been excluded, and for t < 1 we have k(t, x) < ρ if x ρ. By the homotopy invariance of the degree we get deg(Id −F ◦ r, Bρ (0), 0) = deg(Id, Bρ (0), 0) = 1, i.e., F ◦ r has a ﬁxed point in Bρ (0). This proves the theorem if C is closed bounded and convex. (ii) Suppose now that C = g(C0 ) where C0 is closed bounded and convex and g : C0 → C is a homeomorphism. Then g −1 ◦ F ◦ g : C0 → C0 has a ﬁxed point x ∈ C0 , i.e., g(x) ∈ C is a ﬁxed point of F . 1.3.2. Tools for calculating the degree T HEOREM 1.6 (Dimension reduction). Let (X, · ) be a Banach space and (X0 , · ) ⊂ X → X0 is compact. Let y ∈ X0 be such that be a closed subspace. Suppose F : Ω y∈ / (Id −F )(∂Ω). Then deg(Id −F, Ω, y) = deg(Id −F |X0 ∩Ω , X0 ∩ Ω, y). The property is ﬁrst established for maps with ﬁnite-dimensional range. Then it is used to show that the Leray–Schauder degree does not depend on the particular Schauder approximation. Finally the dimension reduction is proved for all compact perturbations of the identity. The basis for the general dimension reduction formula is illustrated next. E XAMPLE 1.3. Consider a linear map F : Rn → Rk Rn given by F (x) = Ax with an n × n matrix A. Since F maps into Rk with k < n the matrix A can be written as follows: B C A= , 0 0 where B, C are k × k and k × (n − k) matrices. The derivative of Id −F at x is Id −A given by −C Idk×k −B . Id −A = 0 Id(n−k)×(n−k) Therefore det(Id −A) = det(Id −B).

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→ X are compact and y ∈ L EMMA 1.7. Suppose F1 , F2 : Ω / (Id −F1 )(∂Ω). If F1 = F2 on ∂Ω then deg(Id −F1 , Ω, y) = deg(Id −F2 , Ω, y). P ROOF. We deﬁne the homotopy k(t, x) := tF1 (x)+(1−t)F2 (x) for t ∈ [0, 1]. On ∂Ω we have k(t, x) = F1 (x) = F2 (x). Therefore y ∈ / (Id −k(t, ·)(∂Ω)) and deg(Id −k(t, ·), Ω, y) is invariant for t ∈ [0, 1]. 1.3.3. Degree for linear maps L EMMA 1.8 (Product formula). (a) Let K, L : X → X be linear and compact with Id −K, Id −L injective and suppose 0 ∈ Ω. Then deg (Id −K) ◦ (Id −L), Ω, 0 = deg(Id −K, Ω, 0) · deg(Id −L, Ω, 0). (b) Let K : X → X be linear and compact with Id −K injective. Let also X = V ⊕ W with closed subspaces V , W such that K : V → V and K : W → W . Then deg Id −K, B1 (0), 0 = deg Id −K|V , B1 (0) ∩ V , 0 · deg Id −K|W , B1 (0) ∩ W, 0 . Part (a) reﬂects the multiplication rule for the determinant of products of matrices. Part (b) is best understood by an example: suppose the block-matrix A : Rn → Rn is given by A=

B 0

0 C

,

with a k × k-matrix B and an (n − k) × (n − k)-matrix C. Thus A maps the k-dimensional subspace V and the (n − k)-dimensional subspace W into itself. It is immediate that det(Id −A) = det(Id −B) · det(Id −C), and therefore Part (b) holds for this example. In order to state a degree formula for Id −K, where K is a compact linear operator, we recall the main facts from the classical Fredholm–Riesz–Schauder theory. Let 0 = λ ∈ R be an eigenvalue of a compact linear operator K. Its eigenspace is ﬁnite-dimensional, and the dimension of the eigenspace is called the geometric multiplicity of λ. For each n = 1, 2, . . . consider the operator (K − λ Id)n , its nullspace Nn and its range Rn . There exists an integer n0 = n0 (λ) 1 such that N1 N2 · · · Nn0 = Nn0 +1 = Nn0 +2 = · · · , R1 R2 · · · Rn0 = Rn0 +1 = Rn0 +2 = · · · . The set Nn0 (λ) is called the generalized nullspace of K − λ Id and m(λ) = dim Nn0 (λ) is called the algebraic multiplicity of the eigenvalue λ. The set Rn0 (λ) is called the generalized range.

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If λ is simple we have the well-known Fredholm alternative X = N1 ⊕ R1 . In the general case one has X = Nn0 (λ) ⊕ Rn0 (λ) . Moreover, K maps Nn0 (λ) to Nn0 (λ) , Rn0 (λ) to Rn0 (λ) and K − λ Id has a bounded inverse on Rn0 (λ) . L EMMA 1.9. Let K : X → X be linear and compact with Id −K injective and suppose 0 ∈ Ω. Then deg(Id −K, Ω, 0) = (−1)β , where β = m(λ). λ>1

The sum is taken over all eigenvalues λ > 1 of K and m(λ) is the algebraic multiplicity of λ. To understand the formula take a real matrix A in Jordan normal form. Calculating sign det(Id −A) amounts to counting the number of negative entries in the diagonal. Thus the contribution comes only from the eigenvalues of A larger than 1, each with its algebraic multiplicity. R EMARK . Observe that the degree formula in Lemma 1.9 remains valid for deg(Id − K − x0 , Ω, 0) provided (Id −K)−1 x0 ∈ Ω.

1.4. The index of an isolated solution Suppose the solution set of (Id −F )(x) = y with F compact consists of isolated points, and let x0 be such a solution. Then x0 is the only solution in some ball Bε0 (x0 ). Therefore deg(Id −F, Bε (x0 ), y) is independent of ε for 0 < ε < ε0 . We deﬁne the index of an isolated solution x0 by means of the degree as follows: ind(Id −F, x0 , y) = deg Id −F, Bε (x0 ), y for small ε. In general, it is difﬁcult to determine the index. We shall list some cases where this can be done. Recall that if F is compact and differentiable then its Fréchet derivative F (x0 ) is a compact linear operator. T HEOREM 1.10 (Leray–Schauder). Under the preceding assumptions and if Id −F (x0 ) is injective we have that ind(Id −F, x0 , y) = ±1. More precisely, ind(Id −F, x0 , y) = ind Id −F (x0 ), x0 , y m(λ). = (−1)β , β = λ>1

The sum is taken over all eigenvalues λ > 1 of F (x0 ) and m(λ) is the algebraic multiplicity of λ.

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P ROOF. Without loss of generality we may assume that y = 0 and that x0 = 0 is the isolated solution. Then for x near the origin we have (Id −F )(x) = (Id −F (0))x − ω(x), where ω(x)/x → 0 as x → 0. Hence deg(Id −F − tω, Bε (0), 0) is well deﬁned for all t ∈ [0, 1] and ε > 0 sufﬁciently small. Moreover it is independent of t. Consequently, we get that ind(Id −F, 0, 0) coincides with deg(Id −F (0), Bε (0), 0). Lemma 1.9 applies and proves the assertion. The condition that Id −F (x0 ) is injective in the previous theorem is necessary, as the following examples shows: E XAMPLE . Let F (x) = −x 2 + x for x ∈ R. The only solution of x − F (x) = 0 is x0 = 0. Then Id −F (x0 ) = 0. The index of x0 vanishes, cf. Example 1.1 in Section 1.2. In the next theorems we consider potential operator on a Hilbert space H. Let g : Bε (x0 ) → R be a C 1 -functional and let ∇g(x) be its gradient, i.e., the Riesz representation of its Fréchet derivative g (x). T HEOREM 1.11 (Rabinowitz [61]). Suppose that ∇g(x) = x − F (x) where F is compact. If x0 is an isolated local minimum of g then ind(∇g, x0 , 0) = 1. Rather than giving the proof we illustrate this result in the ﬁnite-dimensional case. Let g : RN → R. If 0 is a critical point of g then under suitable regularity assumptions we have for small |x| that g(x) = g(0) + 12 (g (0)x, x) + o(|x|2), where g (0) is the Hessian of g at 0. If 0 is a nondegenerate minimum all eigenvalues are positive and thus its index is 1. Notice that the index of a nondegenerate isolated maximum is (−1)N . It depends on the dimension N of the underlying space. The next example deals with saddle points, i.e., critical points which are neither local maxima nor minima. The index will depend on the type of saddle point as it is seen example. Consider the function g : RN → R given by g(x) = sin the 2 following N − i=1 ai xi + i=s+1 bi xi2 where ai > 0 and bi > 0. If s ∈ / {0, N} then 0 is a saddle point and ind(∇g, 0, 0) = (−1)s . The case s = 1 has received special attention. Its topological properties can be described in a more general setting as follows: Suppose U ⊂ X is a nonempty open set. For a C 1 -functional g : U → R and c ∈ R we deﬁne Mc := g −1 ((−∞, c)). The next deﬁnition is due to Hofer [39]. D EFINITION 1.12. Let 0 be a critical point of g with g(0) = c. The point 0 is said to be of mountain pass type if for all open neighborhoods W of 0 the set W ∩ Mc is nonempty and not path connected. This deﬁnition of a critical point of mountain pass type is satisﬁed by a mountain pass point in the sense of Ambrosetti and Rabinowitz. Notice that in the previous example 0 is of mountain pass type if and only if s = 1. Hofer [39] has extended Theorem 1.11 to critical points of mountain pass type. T HEOREM 1.13 (Hofer [39]). Let g be as in Theorem 1.11. Suppose in addition that it is in C 2 (U, R) for some open subset U ⊂ H. Suppose that 0 is an isolated critical point of

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C. Bandle and W. Reichel

mountain pass type. Assume also that if the smallest eigenvalue of g (x0 ) is zero, then it is simple. Then ind(∇g, x0 , 0) = −1.

1.5. Asymptotically linear equations A map G : X → X is called asymptotically linear if there exists a bounded linear operator A : X → X such that lim

x→∞

G(x) − Ax = 0. x

The linear operator A is uniquely determined and is therefore called the derivative of G at inﬁnity, written as G (∞). It can be shown that if G is compact then the same is true for G (∞). T HEOREM 1.14. Let G : X → X be asymptotically linear such that G (∞) is invertible. Assume also that G − G (∞) is compact. Then the nonlinear problem G(x) = y has a solution for every y ∈ X.

+ G (∞)x = y with G

= G − G (∞) has P ROOF. We have to show that the equation G(x) −1

a solution. Set F = −G ◦ [G (∞)] . Then the problem reduces to (Id −F )(z) = y, where F is compact and z = G (∞)x. By deﬁnition of the derivative at inﬁnity it follows that F (z)/z → 0 as z → ∞. For Ω = BR (0) we want to calculate deg(Id −tF, Ω, y) for t ∈ [0, 1]. For z ∈ ∂BR (0) we have z − tF (z) − y z 1 − t F (z) − y R − y, z 2 provided R is sufﬁciently large. If R is even bigger than 2y then we have that y ∈ / (Id −tF )(∂Ω) and by homotopy invariance of the degree we get deg(Id −F, Ω, y) = deg(Id, Ω, y) = 1. This completes the proof. C OROLLARY 1.15. It is sufﬁcient for Theorem 1.14 to have an invertible linear operator A such that lim supx→∞ G(x) − Ax/x < 1/A−1 . The following multiplicity result goes back to Amann [2], see also [3]. We present here the version given by Sattinger [64]. T HEOREM 1.16. Let F be compact and asymptotically linear. Suppose that Id −F (∞) is invertible. Assume that F has two different ﬁxed points x1 , x2 such that (Id −F (xi ))−1 exists for i = 1, 2. Then there exists a third ﬁxed point x3 . P ROOF. Since Id −F (∞) is invertible there exists a > 0 such that x − F (∞)x ax for all x ∈ X. Since F is asymptotically linear we can ﬁnd a positive number R0 such that

Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

13

F (x) − F (∞)x a2 x for all x R0 . Hence, for all x R0 and for all τ ∈ [0, 1], x − τ F (x) − (1 − τ )F (∞)x x − F (∞)x − τ F (x) − F (∞)x

a R0 . 2

Hence τ F + (1 − τ )F (∞) has no ﬁxed point outside of BR0 and as a consequence deg(Id −τ F − (1 − τ )F (∞), BR0 , 0) is well deﬁned and independent of τ . Thus setting τ = 0 and τ = 1 we get deg Id −F (∞), BR0 , 0 = deg(Id −F, BR0 , 0).

(1.1)

By Lemma 1.9 the left-hand side of (1.1) equals (−1)β = ±1 where β is related to the multiplicity of the eigenvalues of F (∞) larger than one. On the other hand if we assume that xi , i = 1, 2, are the only ﬁxed-points of F in BR0 then by the excision property (d2) the right-hand side of (1.1) is 2i=1 ind(Id −F, xi , 0) = 0 or ±2. This contradicts (1.1). Therefore at least one more ﬁxed point of F must exist. 1.6. Fixed point alternatives T HEOREM 1.17 (Leray–Schauder alternative). Let Ω ⊂ X be bounded, open and assume → X be compact. Then the following alternative holds: p ∈ Ω. Let furthermore F : Ω (i) F has a ﬁxed point in Ω or (ii) there exists λ ∈ (0, 1) and x ∈ ∂Ω such that x = λF (x) + (1 − λ)p. P ROOF. Suppose for contradiction that neither (i) nor (ii) holds. We want to show that deg(Id −tF, Ω, (1 − t)p) is well deﬁned. So suppose that for some t ∈ [0, 1] there is x ∈ ∂Ω with x − tF (x) = (1 − t)p. Since (i) does not hold the possibility t = 1 is excluded and since (ii) does not hold it is impossible that 0 < t < 1. And since p ∈ Ω also t = 0 is excluded. Hence, homotopy invariance applies and yields deg(Id −F, Ω, 0) = deg(Id, Ω, p) = 1 which shows that F has a ﬁxed point in Ω. This contradicts the assumption that (i) does not hold. T HEOREM 1.18 (Principle of a priori bounds). For t ∈ [0, 1] let F (t, ·) : X → X be a family of compact operators with F (0, ·) ≡ 0. Assume, moreover, that F (t, x) is continuous in t uniformly w.r.t. x in balls in X. Suppose that the set S = {x: ∃t ∈ [0, 1]: x = F (t, x)} is bounded. Then F (1, ·) has a ﬁxed point. P ROOF. Standard arguments show that the hypotheses imply that F : [0, 1] × X → X is compact. If BR (0) is such that all solutions of x = F (t, x) for t ∈ [0, 1] are a priori known to lie inside BR (0) then deg(Id −F (t, ·), BR (0), 0) is homotopy invariant. Hence deg(Id −F (1, ·), BR (0), 0) = deg(Id, BR (0), 0) = 1. This shows that F (1, ·) has a ﬁxed point.

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By taking F (t, x) = tF (x) we get the following result. C OROLLARY 1.19 (Schäfer’s theorem [65]). Let F : X → X be compact. Then the following alternative holds: (i) x − tF (x) = 0 has a solution for every t ∈ [0, 1] or (ii) S = {x : ∃t ∈ [0, 1]: x − tF (x) = 0} is unbounded.

1.7. Degree theory in unbounded domains Up to now the degree was deﬁned only in bounded domains. We indicate a generalization to unbounded domains which will be needed in the next section. Assume Ω ⊂ X is open and possibly unbounded. Let us consider the class of maps → X where (Id −F )−1 (y) is compact for every y ∈ F :Ω / (Id −F )(∂Ω). In order to deﬁne deg(Id −F, Ω, y) take any bounded open neighborhood V ⊂ Ω of (Id −F )−1 (y) and set deg(Id −F, Ω, y) =: deg(Id −F, V, y). This deﬁnition makes sense because the excision property (d2) implies that deg(Id −F, V, y) is the same for every bounded open neighborhood V of (Id −F )−1 (y). The following lemma is useful for practical purposes. → X be compact and assume that F (Ω) is bounded. Then L EMMA 1.20. Let F : Ω −1 (Id −F ) (y) is compact. P ROOF. Let {xn }n1 be a sequence of solutions to the equation x − F (x) = y. The se is bounded. Since F is compact quence is bounded because we have assumed that F (Ω) there exists a subsequence {xn }n 1 such that {F (xn )} converges. Hence xn converges to x, and from the continuity of F we conclude that the limit solves x − F (x) = y.

1.8. Degree theory in cones Krasnosel’skii derived a theorem to ﬁnd nontrivial ﬁxed points of cone preserving maps, cf. [44]. A cone C is a closed, convex subset of the Banach space X with the following properties: (i) if x, y ∈ C and α, β 0 then αx + βy ∈ C, (ii) if x ∈ C and x = 0 then −x ∈ / C. A cone induces a partial ordering x y in X whenever y − x ∈ C. The Leray–Schauder degree theory cannot be applied immediately to functions p F : C → C because many important cones such as L+ (D) = {x ∈ Lp (D): x 0 a.e.} for p 1 have empty interior. By Dugundji’s theorem [27] one knows that C is a retract. Hence it is possible to extend the degree to arbitrary cones in a natural way.

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D EFINITION 1.21. Let C ⊂ X be a cone, r : X → C be a retraction and let U ⊂ C be a bounded open set with respect to the relative topology of C. If F : C → C is compact and y ∈ C is such that (Id −F )x = y has no solution on the boundary of U (with respect to the relative topology of C) then we deﬁne deg(Id −F, U, y) := deg Id −F ◦ r, r −1 U, y . The deﬁnition makes sense because it is independent of the particular choice of the retraction. Moreover, the solution set A := {x ∈ U : x − F (x) = y} is the same as the solution set B := {x ∈ r −1 U : x − (F ◦ r)(x) = y} since for the latter, the fact that y, (F ◦ r)(x) ∈ C implies x ∈ C. In particular, the solution set A (= B) is compact since U is bounded. Note that even if r −1 U is unbounded the degree is nevertheless deﬁned by the arguments of the previous section. Many applications of degree theory in cones are due to Amann [3]. The next result is due to Krasnosel’skii, cf. [44]. It is also found in [9], Appendix 1. For 0 < r < R consider the sets S(r, R) := {x ∈ C: r < x < R} and S(R) = {x ∈ C: x < R}. Both sets are open in the relative topology of C. T HEOREM 1.22. Let C ⊂ X be a cone and F : C → C be compact. Assume there exist numbers 0 < r < R and a point 0 = v ∈ C such that (i) x = tF (x) for all 0 t 1 and x = r, (ii) x = F (x) + tv for t 0 and x = R. Then deg Id −F, S(r), 0 = 1, deg Id −F, S(R), 0 = 0 and deg Id −F, S(r, R), 0 = −1. In particular, F has a ﬁxed point in S(r, R). P ROOF. It follows from (i) that x − tF (x) = 0 on x = r for all t ∈ [0, 1]. By the homotopy invariance (d3) and by the normalization (d1) it follows that deg(Id −tF, S(r), 0) = deg(Id, S(r), 0) = 1 which establishes the ﬁrst assertion. By (ii) and the homotopy invariance we have deg(Id −F − tv, S(R), 0) = deg(Id −F, S(R), 0) for all t > 0. Suppose that deg(Id −F − tv, S(R), 0) = 0. Then by the existence property (d4) the equation F (x) + tv = x has always a solution xt in S(R). For large t we have the estimate R tv − F (xt ). This leads to a contradiction if t is too large and shows that deg(Id −F, S(R), 0) = 0. The last statement now follows from the excision property (d2). For later uses let us describe a result concerning the spectrum of compact linear operators in cones. For an elementary proof we refer to Takáˇc [69]. A proof using degree theory is given in Theorem 3.4 in Section 3.1. T HEOREM 1.23 (Krein–Rutman). Let X be a Banach space ordered with respect to a cone C. Suppose that Int(C) = ∅ and let T : X → X be a compact linear operator which is strongly positive in the sense that T (C \ {0}) ⊂ Int(C). Then:

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(1) the spectral radius r(T ) is a positive simple eigenvalue of T , (2) the eigenvector u ∈ X \ {0} associated with the eigenvalue r(T ) can be taken in Int(C), (3) if μ is in the spectrum of T , 0 = μ = r(T ) then μ is an eigenvalue of T satisfying |μ| < r(T ), (4) if μ is an eigenvalue of T associated with an eigenvector v ∈ C \ {0} then μ = r(T ).

1.9. Notes 1. There are several equivalent approaches to degree theory. Its origin dates back to Kronecker, Poincaré, Brouwer and Hopf. In 1869 Kronecker generalized the argument principle to higher dimensions. Hopf proposed a deﬁnition via homology groups. Another way based upon real analysis was carried out by Nagumo [51,53] and Heinz [34]. A definition of degree in terms of cohomology is found in [62] (Rado and Reichelderfer). One of the most complete texts is the celebrated book by Krasnosel’skii and Zabreiko [44] who have made important contributions to the theory and its applications. For an introduction to degree theory see also Deimling [26]. 2. Brezis [11,12] reviews degree theory for harmonic maps from SN to SN which are

critical points of the Dirichlet energy SN |∇u|2 dx. Similar to the Brouwer degree one can deﬁne for continuous maps SN → SN the degree deg(u, SN , y) for y ∈ SN . In contrast to degree theory on sets with boundary, the degree for functions u : SN → SN is independent of y ∈ SN . Hence, we set deg(u) = deg(u, SN , y) for every continuous map u : SN → SN . By Hopf’s result, if two continuous maps u, v : SN → SN have the same degree, then there exists a homotopy connecting u and v. Thus, the space C(SN , SN ) is decomposed into its connected components characterized by their degree. One can try to use this decomposition for ﬁnding harmonic maps in each connected component of C(SN , SN ). In order to apply direct methods of the calculus of variations one has to deﬁne the degree for maps in the Sobolev space H 1,2 (SN , SN ), and the question arises if the connected components remain the same when passing from C(SN , SN ) to H 1,2(SN , SN ). This is true in dimension N = 2, cf. Schoen and Uhlenbeck [67], but the problem of closedness of the components in the H 1,2 -topology still remains. These harmonic map problems are considered in [11,12] and Struwe [68]; see also the references given there. For dimensions N 3 however, the Schoen–Uhlenbeck approach only works for maps in H 1,N (SN , SN ), which poses again problems when minimizing the Dirichlet energy by the direct methods of the calculus of variations. Brezis and Nirenberg [13,14] consider the degree for maps in VMO(SN , SN ) (vanishing mean oscillation). 3. Counterexample to Brouwer’s ﬁxed point theorem in inﬁnite dimensions. Let X be the Banach space of real sequences x = (xn ) tending to zero with norm x = maxn |xn |. Let F : X → X be deﬁned by (F x)1 = (1 + x)/2

and (F x)n+1 = xn .

1 (0) into itself, but F has no ﬁxed point. F is continuous and maps B

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17

2. Existence of solutions In the theory of elliptic partial differential equations the Hölder and the Sobolev spaces play an outstanding role. Both are Banach spaces in which a complete theory for linear elliptic differential equations is available. For convenience we shall summarize the main results. More details and proofs are found in the reference texts of Miranda [50] and Gilbarg and Trudinger [32].

2.1. Function spaces Let D be an open set in RN and α ∈ [0, 1] be an arbitrary number. A function f : D → RN is said to be Hölder continuous with exponent α if |f (x) − f (y)| < ∞. |x − y|α x,y∈D

Mα (f ) = sup

For α = 1, f is Lipschitz continuous. Likewise we deﬁne by M0 (f ) the maximal modulus f ∞ . Let f α = f ∞ + Mα (f ). For each N -tuple k = (k1 , k2 , . . . , kN ) of nonnegative integers let Dk f =

∂ |k| f k

∂x1k1 ∂x2k2 · · · ∂xNN

with |k| =

N

kj .

j =1

Moreover, we set, for α ∈ [0, 1] and m ∈ N, Mm+α (f ) = sup Mα D k f , |k|=m

f m =

m

Mj (f ),

j =0

f m+α =

m

Mj (f ) + Mm+α (f ).

j =0

denote the space of functions with continuous mth order derivatives in D. With Let C m (D) this notation we can now deﬁne the Hölder spaces := f ∈ C m D : f m+α < ∞ . C m,α D and C m,α (D) consisting of functions with compact support in D The subspaces of C m (D) m,α m will be denoted by C0 (D) and C0 (D).

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C. Bandle and W. Reichel m,p

For bounded domains the Sobolev spaces H m,p (D), [H0 [C m (D)] with respect to the norm completion of C m (D), 0 f m,p =

m k p D f p L (D)

(D)] are obtained as the

1/p .

|k|=0

m,p

For m 1 every function f in H m,p or H0

has generalized derivatives up to order m.

2.2. Uniformly elliptic linear operators For the rest of this chapter let D ⊂ RN be a bounded domain. From now on we shall use 2 the summation convention and the abbreviations ∂i := ∂x∂ i and ∂ij2 := ∂x∂i ∂xj . The operator L := aij (x)∂ij2 + bi (x)∂i + c(x) is uniformly elliptic in D provided there exists a positive constant Λ such that aij (x)ξi ξj Λξi ξi

for all x ∈ D and ξ ∈ RN

and aij , bi , c ∈ L∞ (D). Associated to L is the operator L0 := aij (x)∂ij2 + bi (x)∂i . if We say that L satisﬁes the maximum principle for u ∈ C 2 (D) ∩ C(D) Lu 0 in D,

u 0 on ∂D

⇒

max u 0. D

(MP)

Sufﬁcient for the maximum principle is c 0. Moreover, under this condition and if Lu 0 in D the following strong maximum principle holds: (i) if u attains its nonnegative maximum in D then u ≡ const; (ii) if u attains its nonnegative maximum at a point x0 ∈ ∂D which lies on the boundary of a ball B ⊂ D and if u is continuous in D ∪ {x0 } and an outward directional ∂u derivative ∂u ∂ν (x0 ) exists then ∂ν (x0 ) > 0 unless u ≡ const. If the maximum principle (MP) holds then simple pointwise estimates for the classical solutions of the boundary value problem Lu = f

u=0

in D,

on ∂D

(2.1)

∩ C 2 (D) then can be derived. For instance, if c 0 and if u ∈ C(D) sup |u| C sup D

|f | , Λ

(2.2)

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where C depends only on the diameter of D and bi /Λ∞ , i = 1, . . . , N ; cf. Gilbarg and Trudinger [32]. The maximum principle holds for the class of operators deﬁned next. We assume that at every point x ∈ ∂D an outward normal ν(x) exists. D EFINITION 2.1. The operator −L is called strictly positive provided there exists a func ∩ C 2 (D) with φ > 0 in D and ∂φ (x0 ) < 0 at points x0 ∈ ∂D where tion φ ∈ C 1 (D) ∂ν φ(x0 ) = 0 such that −Lφ 0 but ≡ 0 in D. ∩ L EMMA 2.2. Suppose −L is strictly positive. Then (MP) holds in the class of C 1 (D) ∩ C 2 (D)-solution of (2.1), where the C 2 (D)-functions and (2.2) holds for every C 1 (D) constant C depends also on max{c, 0}∞ . ∩ C 2 (D) is such that Lu 0 in D, u 0 on ∂D but P ROOF. Suppose u ∈ C 1 (D) ∗ maxD u > 0. Let t > 0 be so large that t ∗ φ > u in D. Consider the smallest t¯ ∈ (0, t ∗ ) such that tφ u in D for all t ∈ (t¯, t ∗ ). Then we have v = t¯φ − u 0 in D and there exists with v(x0 ) = ∇v(x0 ) = 0. Let c− = min{c, 0}. The function v satisﬁes a point x0 in D − L0 v + c v Lv 0 in D. Since v attains its zero-inﬁmum at x0 with ∇v(x0 ) = 0 the strong form of the maximum principle implies v ≡ 0, i.e., u ≡ t¯φ. This is impossible, and proves (MP). The L∞ -estimate (2.2) carries over like in [32], Section 3.3. There are essentially two classical approaches concerning existence and estimates for solutions of (2.1): the Schauder theory for classical solutions in the Hölder spaces and the Lp -theory for strong solutions in the Sobolev spaces. Both are described in the next two sections. 2.3. Schauder estimates Consider the Dirichlet problem Lu = f

in D,

u = 0 on ∂D.

(2.3)

Moreover, let aij α , bi α , cα M. Let Assume ∂D ∈ C 2,α , aij , bi , c, f ∈ C α (D). 2,α u ∈ C (D) be a solution of (2.3). Then it satisﬁes the following Schauder boundary estimate u2+α C u∞ + f α , where C depends only on M, α, D and the ellipticity constant Λ. This estimate is true because u vanishes on the boundary. The Schauder interior estimates come into play if no information of the solutions of Lu = f in D on the boundary is available. They involve norms f m+α deﬁned as follows: j +α |f (x) − f (y)| , |x − y|α

Hj,α (f ) = sup dx,y x,y∈D

Hα (f ) = H0,α (f ),

20

C. Bandle and W. Reichel

where dx = dist(x, ∂D) and dx,y = min(dx , dy ). Then we deﬁne f α = f ∞ + Hα (f ), f m+α = supdx|k| D k f (x) + Hm,α D k f . |k|m D

|k|=m

Suppose f α < ∞. Furthermore let aij α , bi α , cα M. Then for any solution u ∈ C 2,α (D) ∩ L∞ (D) of Lu = f in D, we have u2+α C u∞ + f α , where C depends only on Λ, α, M and D. We consider the Dirichlet problem (2.3) where all data are α-Hölder continuous and ∂D ∈ C 2,α . Then Schauder’s famous result, obtained by means of a continuity argument (see, e.g., [32]) states: T HEOREM . Under the above conditions and if −L is strictly positive the Dirichlet prob and there exists a constant C such that lem (2.3) has a unique solution in C 2,α (D) u2+α Cf α . → C 2,α (D) for uniformly ellipThe result justiﬁes the use of the notation L−1 : C α (D) tic, strictly positive operators −L with α-Hölder continuous coefﬁcients. In this case L−1 → C k,β (D) provided k + β < 2 + α. is a compact linear operator from C α (D) 2.4. Lp -estimates Here the regularity assumptions on the data and the solutions are weaker. Assume bi , c ∈ L∞ (D), f ∈ Lp (D) and 1 < p < ∞. A strong solution ∂D ∈ C 1,1 , aij ∈ C(D), 1,p u ∈ H0 (D) ∩ H 2,p (D) of Lu = f in D satisﬁes the boundary Lp -estimate uH 2,p (D) C f Lp (D) + uLp (D) , where C depends only on the ellipticity constant, p, the domain D and the sup-norm of 2,p the coefﬁcients. For strong solutions u ∈ Hloc (D) ∩ Lp (D) the interior Lp -estimates are of the form uH 2,p (D ) C f Lp (D) + uLp (D) , where D ⊂ D is a compact subdomain and C depends as above only on the data and D . For the Dirichlet problem (2.3) we have the following existence theorem (see, for instance, [32]): T HEOREM . Let L satisfy the assumptions above and assume c(x) 0. If f ∈ Lp (D) then 1,p the Dirichlet problem (2.3) has a unique strong solution u ∈ H 2,p (D) ∩ H0 (D).

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21

More generally, let us suppose instead of c(x) 0 that whenever (2.3) has a solution 1,p then this solution is unique. Then the linear operator L−1 : Lp (D) → H 2,p ∩ H0 (D) is well deﬁned and bounded, i.e., uH 2,p (D) Cf Lp (D) with u = L−1 f . Moreover, L−1 : Lp (D) → H0 (D) is compact. For later purposes we shall need the fact that L−1 has a unique compact restriction, into C 1,α (D). Indeed if f is continuous then u = L−1 f denoted again by L−1 , from C(D) p p belongs to L (D) for all p > 1. The L -estimates imply that uH 2,p (D) Cf ∞ and by for p > N and a suitable α ∈ (0, 1 − N/p) the compact embedding H 2,p (D) → C 1,α (D) → C 1,α (D) is compact. it follows that u1+α Cf ∞ and L−1 : C(D) 1,p

2.5. Applications to boundary value problems 2.5.1. Asymptotically linear equations. Consider the boundary value problem Lu + λu + g(x, u) = 0 in D,

u = 0 on ∂D,

(2.4)

This problem can be written in the where g(x, s) = o(s) as |s| → ∞ uniformly for x ∈ D. form u + L−1 λu + g(x, u) = 0, or on X = C(D) depending on the regularity of where L−1 is either deﬁned on X = C α (D) the data. The operator Gu = u + L−1 (λu + g(x, u)) is asymptotically linear. Its derivative at inﬁnity is given by G (∞) = Id +λL−1 . From Theorem 1.14 we deduce: T HEOREM 2.3. Depending on the underlying space X assume that either uniformly w.r.t. s in bounded intervals and (i) g(x, s) is α-Hölder continuous in x ∈ D locally Lipschitz continuous in s uniformly w.r.t. x ∈ D or and s ∈ R. (ii) g(x, s) continuous in x ∈ D If λ is not an eigenvalue of −L then (2.4) possesses a solution. R EMARK . If λ is an eigenvalue of −L then (2.4) is discussed in Section 4.3.1. 2.5.2. Semilinear boundary value problems. Consider the problem Lu = g(x, u, ∇u)

in D,

u=0

on ∂D,

where L satisﬁes the conditions of Section 2.4 and L−1 : Lp (D) → H0 (D) exists and is × R × RN and subject to the condition compact. The nonlinearity is continuous in D g(x, u, ∇u) M 1 + |u| + |∇u| γ 1,p

22

C. Bandle and W. Reichel

for some positive γ < 1. T HEOREM 2.4. Let p > 1. Under the hypothesis above there exists a solution in 1,p H 2,p (D) ∩ H0 (D). P ROOF. Consider for t ∈ [0, 1] the problem Lu = tg(x, u, ∇u)

in D,

u=0

on ∂D.

(2.5) 1,p

By the Lp -estimates we have for any solution u ∈ H 2,p (D)∩H0 (D) of (2.5) the estimate uH 2,p (D) CM

γp 1 + |u| + |∇u| dx

1/p ,

D

where C is independent of t. Using subsequently the inequality (1 + s)γp c(ε) + εs p , s 0, together with Minkowski’s inequality we conclude for all t ∈ [0, 1] and any solution u of (2.5) that uH 2,p (D) C0 for some positive constant C0 . The same holds for the H0 -norm. The operator L−1 : Lp (D) → H0 (D) is compact. Likewise, 1,p 1,p L−1 G[u] : H0 (D) → H0 (D) is compact where G[u] := g(x, u, ∇u). Consequently Schäfer’s theorem, cf. Corollary 1.19, applies and shows the existence of a solution of (2.5) 1,p in H0 (D) for every t ∈ [0, 1] and in particular for t = 1. By a regularity step the solutions lie in H 2,p (D). This establishes the assertion. 1,p

1,p

2.5.3. Quasilinear boundary value problems. In this section we describe the Leray– Schauder method for solving the boundary value problem aij (x, u, ∇u) ∂ij2 u = f (x, u, ∇u) in D,

u = 0 on ∂D.

(2.6)

× R × RN ) and that a constant Λ > 0 exists Here we assume ∂D ∈ C 2,α , f, aij ∈ C α (D such that aij (x, z, χ)ξi ξj Λξi ξi for all x ∈ D, z ∈ R and χ, ξ ∈ RN . the linear problem For all z ∈ C 1,β (D) aij (x, z, ∇z) ∂ij2 U = f (x, z, ∇z) in D,

U = 0 on ∂D

Consider the solution operator F : C 1,β (D) → has a unique solution in U ∈ C 2,αβ (D). 2,αβ (D) mapping z → U (z). It is not difﬁcult to see that F is a compact operator from C into itself. The solutions of (2.6) can be interpreted as the ﬁxed points of F C 1,β (D) 1,β in C (D). For σ ∈ [0, 1] the equation u = σ F (u) is equivalent to the quasilinear problem aij (x, u, ∇u) ∂ij2 u = σf (x, u, ∇u)

in D,

u = 0 on ∂D.

(2.7)

The next theorem goes back to Leray and Schauder [49]. We present it in the version of Gilbarg and Trudinger [32]. It is an immediate consequence of Schäfer’s theorem (Corollary 1.19).

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23

T HEOREM 2.5. Under the above assumptions and if for some β > 0 there exists a constant M > 0 such that every solution of (2.7) satisﬁes u1+β M for any σ ∈ [0, 1] then (2.6) has a solution u ∈ C 2,α (D). This theorem reduces the solvability of quasilinear problems to ﬁnding a priori estimates. This step is by far the most difﬁcult problem in the application of the Leray– Schauder technique. Terminology: In the following L is always a uniformly elliptic operator with bounded coefﬁcients such that −L is strictly positive. 2.5.4. Eigenvalue problems. In this section we use the Krein–Rutman theorem, cf. Theorem 1.23, to show the existence of an eigenvalue and an eigenfunction for the problem Lψ + λmψ = 0 in D,

ψ = 0 on ∂D,

(2.8)

is a nonnegative weight m 0, m ≡ 0. where m ∈ C α (D) T HEOREM 2.6. Let the data be Hölder continuous. Then (2.8) has a smallest eigenvalue λ1 which is positive. The corresponding eigenspace is one-dimensional and the eigenfunction φ1 (x) may be taken positive in D. P ROOF. By the considerations at the end of Section 2.4 the differential operator has a → C 1,α (D). The application of the abstract Theorem 1.23 compact inverse L−1 : C(D) is not suitable requires a careful choice of the cone. The standard positive cone C0+ (D) because it has empty interior. We follow the proof given by Amann [3]. Let e be the solution of the boundary value problem Le + 1 = 0 in D, e = 0 on ∂D. By the maximum principle it follows that e > 0 ∂e ∃λ > 0 such that < 0 on ∂D. Consider the linear space Ce (D) = {v ∈ C0 (D): in D and ∂ν −λe v λe}. It is complete with respect to the norm ve := inf{λ 0: −λe v λe}. ∃λ > 0 such that 0 v λe} of nonnegative functions in The cone C = {v ∈ C0+ (D): Ce (D) has nonempty interior with respect to the · e -topology. → C 1,α (D) it follows that the operator L−1 Due to the compactness of L−1 : C(D) → Ce (D) deﬁned maps C(D) compactly into Ce (D). Moreover, the operator T : Ce (D) −1 as T u = −L (mu) is strongly positive with respect to C because of the strong version of the maximum principle. The theorem of Krein–Rutman applies to T and establishes the assertion.

2.6. Comparison principles Consider the boundary value problem Lu + f (x, u) = 0 in D,

u = 0 on ∂D.

(2.9)

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C. Bandle and W. Reichel

We assume that f (x, s) as a function D × R → R is locally uniformly w.r.t. s ∈ R, (H1) Hölder continuous in x ∈ D, (H2) locally Lipschitz continuous in s, uniformly w.r.t. x ∈ D. satisﬁes D EFINITION 2.7. Suppose a pair of functions (v, w) in C 2 (D) ∩ C 1 (D) Lv + f (x, v) 0,

Lw + f (x, w) 0 in D

with v 0 w on ∂D. Then (v, w) is called a pair of sub- and supersolutions for (2.9). L EMMA 2.8. Let (v, w) be a pair of sub- and supersolutions for (2.9). Then the following holds: (i) Strong comparison principle: Assume v w. Then either v ≡ w or v < w in D. In the second case suppose x ∈ ∂D is a point where v(x) = w(x). Then ∂v ∂ν (x) > ∂w (x). ∂ν (ii) Weak comparison principle: Suppose f (x, s) is nonincreasing in s. Then v w in D. The proof of (i) and (ii) follows from strong and weak versions of the maximum principle applied to u = v − w.

2.7. Degree between sub- and supersolutions A pair (v, w) of sub- and supersolutions is called strict if neither v nor w is a solution. L EMMA 2.9 (Monotone iterations). Let (v, w) be a pair of strict sub- and supersolutions of (2.9) and assume v < w. Then there exist a minimal solution u and a maximal solution : v < u < w in D}. u of (2.9) in V = {u ∈ C 1 (D) The idea is a follows: let σ > 0 be so large that g(x, s) = f (x, s) + σ s is increasing in where R max(v∞ , w∞ ). If M = L − σ Id then (2.9) is s ∈ [−R, R] for all x ∈ D, equivalent to Mu + g(x, u) = 0 in D,

u = 0 on ∂D.

→ C 1 (D) is compact and monotone increasing, The operator (−M)−1 ◦ g(x, ·) : C 1 (D) 1 satisfy u1 u2 then (−M)−1 (g(x, u1 )) (−M)−1 (g(x, u2 ). Morei.e., if u1 , u2 ∈ C (D) over, it maps V into itself. The sequence vn+1 = (−M)−1 (g(x, vn )), v0 = v is monotone increasing with u = limn→∞ vn . Likewise wn+1 = (−M)−1 (g(x, wn )), w0 = w is a monotone decreasing sequence with limn→∞ wn = u. ¯ T HEOREM 2.10. Let (v, w) be a pair of strict sub- and supersolutions for (2.9) with v < w

Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

25

and let : v < u < w in D and if x ∈ ∂D is such that v(x) = u(x) U = u ∈ C1 D ∂u ∂u ∂w ∂v (x) > (x), (x) > (x), resp. . or w(x) = u(x) then ∂ν ∂ν ∂ν ∂ν Choose h ∈ U . For sufﬁciently large R > 0 it follows that deg Id +L−1 ◦ f (x, ·), U ∩ BR (h), 0 = 1, centered at h and containing where BR (h) is the open norm ball of radius R in C 1 (D) v and w. Let P ROOF. The set U is open, however, it is unbounded in C 1 (D). ⎧ if s v(x), ⎪ ⎨ f x, v(x) if v(x) s w(x), f˜(x, s) := f (x, s) ⎪ ⎩ f x, w(x) if s w(x). Then u ∈ U is a solution of Lu + f (x, u) = 0 if and only if u solves Lu + f˜(x, u) = 0. Thus, by replacing f by f˜ we may suppose that f is bounded. Moreover, by choosing σ > 0 sufﬁciently large, we may suppose that f (x, s) + σ s is increasing in s ∈ R for all Let u be the minimal solution of (2.9) in U . Consider for t ∈ [0, 1] the following x ∈ D. one-parameter family of problems Lu − σ u + t f (x, u) + σ u + (1 − t) f (x, u ) + σ u = 0 in D

(2.10)

with u = 0 on ∂D. The pair (v, w) remains a pair of strict sub- and supersolutions for (2.10). By the strong comparison principle no solution of (2.10) lies on ∂U . Moreover, since we assumed boundedness of f , for all t ∈ [0, 1] every solution lies in the open where h ∈ U is arbitrary and R is sufﬁciently large – in parnorm ball BR (h) ⊂ C 1 (D), ticular large enough that v, w ∈ BR (h). Therefore the following topological degree is well deﬁned deg Id −σ L−1 + tL−1 ◦ f (x, ·) + σ Id + (1 − t)L−1 ◦ f (x, u ) + σ u , U ∩ BR (h), 0 and it is homotopy invariant in t. Hence the values at t = 1 and t = 0 coincide: deg Id +L−1 ◦ f (x, ·), U ∩ BR (h), 0 = deg Id −σ L−1 + L−1 f (x, u ) + σ u , U ∩ BR (h), 0 . =−u+σ L−1 u

(2.11)

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C. Bandle and W. Reichel

Since the unique solution of the linear problem, Lu − σ u = Lu − σ u

in D,

u=0

on ∂D,

is u and since L−1 has only negative eigenvalues the last degree in (2.11) equals 1 by Lemma 1.9. R EMARKS . (1) A close inspection of the proof shows that the radius R can be taken such that R Cw − v∞ for a constant C > 0 depending on L and f . (2) For practical use it is important to relax the regularity of the sub- and supersolutions and to allow them to be differentiable in a weak sense, see the notes for more details.

2.8. Emden–Fowler type equations 2N the critical Sobolev exponent in dimension N 3 and we We denote by 2∗ = N−2 ∗ set 2 = ∞ for N = 1, 2. Motivated by the example of the Emden–Fowler problem u + up = 0 in D with u = 0 on ∂D for 1 < p < 2∗ − 1, we look for solutions of

Lu + f (x, u) = 0,

u > 0 in D, u = 0 on ∂D,

(2.12)

for which u = 0 is a solution. In addition to hypotheses (H1) and (H2) we introduce: (H3) f (x, s) > λ1 s for large s > 0 where λ1 is the smallest eigenvalue of −L, (H4) lims→0 f (x, s)/s = 0 uniformly for x ∈ D. We also use an assumption on the solution set of (2.12) with f (x, s) replaced by f (x, s) + κ: (H5) for κ in bounded intervals there exists an upper bound M such that uC 1 < M for every solution of (2.12). Sufﬁcient conditions for (H5) are given, e.g., in the a priori bound principle of Gidas and Spruck [31] stated next; see also the notes for further results. This result is fundamental for a large number of applications. × R → R is continuous in x ∈ D and L EMMA 2.11 (Gidas and Spruck). Suppose f : D ∗ p there exists p ∈ (1, 2 − 1) and h ∈ C(D), h > 0, in D with lims→∞ f (x, s)/s = h(x) Then there exists a constant M > 0 such that every positive solution u uniformly for x ∈ D. of (2.12) satisﬁes uC 1 < M. R EMARK . In order to derive (H5) from Lemma 2.11 it is important to have a version which applies for positive solutions of the parameter-dependent problem Lu + f (x, u, λ) = 0 ×R×I → R is continuous both in D, u = 0 on ∂D, where λ ∈ I = [λa , λb ]. Suppose f : D × in x ∈ D and λ ∈ I and there exists p as above and a continuous, positive function h : D p I → R with lims→∞ f (x, s, λ)/s = h(x, λ) uniformly for x ∈ D and λ ∈ I . Then there exists a constant M > 0 such that every positive solution u for λ ∈ I satisﬁes uC 1 < M. Our result for the generalized Emden–Fowler problem (2.12) is as follows.

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27

× R → R satisﬁes (H1)–(H5). Then (2.12) has a positive T HEOREM 2.12. Suppose f : D solution. Before we can prove Theorem 2.12 by degree theory, we need the following result: × R → R satisﬁes (H1)–(H3). Then there exists a value L EMMA 2.13. Suppose f : D K > 0 such that Lu + f (x, u) + κ = 0

in D,

u=0

on ∂D

(2.13)

has no nonnegative solution if κ K. P ROOF. Let φ1 be the positive ﬁrst Dirichlet eigenfunction of −L with the positive eigenvalue λ1 . By (H3) there exists K > 0 such that f (x, s) > λ1 s − K for all s 0. Suppose κ K and assume u is a nonnegative solution of (2.13). Then Lu + λ1 u < 0 in D, and by the strong comparison principle of Lemma 2.8 we know u > 0 in D and ∂ν u < 0 on ∂D. The latter implies that {t > 0: u tφ1 in D} is nonempty. Hence we may deﬁne τ = sup{t > 0: u tφ1 in D}. Again by the strong comparison principle of Lemma 2.8 applied to u and τ φ1 we ﬁnd that either τ φ1 ≡ u or τ φ1 > u in D and τ ∂ν φ1 < ∂ν u on ∂D. The ﬁrst alternative can be excluded immediately and the second contradicts the deﬁnition of τ . Hence there is no nonnegative solution u of (2.13) for κ K. P ROOF OF T HEOREM 2.12. By (H4) we have f (x, 0) = 0. Since we are interested in positive solutions only we may assume that f (x, s) = 0 for all s < 0. Then every solution of (2.12) is positive or identically zero. By (H4) the function tφ1 is a strict supersolution to (2.12) for t > 0 small. Likewise, −tφ1 is a subsolution to (2.12) for t > 0 small. After rewriting (2.12) as u + L−1 f (x, u) = 0 we ﬁnd by Theorem 2.10 that deg Id +L−1 ◦ f (x, ·), U ∩ BR (0), 0 = 1, where U is the open set as in Theorem 2.10 spanned by (−tφ1 , tφ1 ). Let K > 0 be the constant from Lemma 2.13. By assumption (H5) we know that all solutions of Lu + f (x, u) + κ = 0

in D,

u=0

on ∂D,

(2.14)

for 0 κ K satisfy the bound uC 1 < M, where we may assume that M > R. Hence deg(Id +L−1 (f (x, ·) + κ), BM (0), 0) is well deﬁned and homotopy invariant with respect to κ, i.e., deg Id +L−1 ◦ f (x, ·), BM (0), 0 = deg Id +L−1 ◦ f (x, ·) + K , BM (0), 0 = 0

28

C. Bandle and W. Reichel

Fig. 5. Excision property of the degree.

since no solution of (2.14) exists for κ = K. By the excision property (d2) of the Leray– Schauder degree (see Figure 5), we know that next to the zero-solution a second solution exists in BM (0) \ U . This establishes the claim.

2.9. Multiplicity results In this section two examples for the existence of multiple solutions are given. The ﬁrst result of Amann [3] shows the existence of an additional solution between a pair of strict sub- and supersolutions (v, w) in case the maximal and the minimal solutions u, u are different. The arguments are based on the abstract result in Theorem 1.16. T HEOREM 2.14 (Amann [3]). Let (v, w) be a pair of strict sub- and supersolutions to (2.12). Assume that the maximal and the minimal solutions u¯ and u of (2.12) be¯ Id, tween v and w satisfy v < u < u¯ < w in D. Suppose that the operators L + fu (x, u) L + fu (x, u ) Id do not have the eigenvalue 0. Then there exists a third solution u such that u < u < u¯ in D. P ROOF. Deﬁne ⎧ ⎪ ⎨ f (x, u ) + u − t g(x, t) := f (x, t) ⎪ ⎩ f (x, u¯ ) + u¯ − t

if t u, ¯ if u t u, if t u, ¯

and consider the problem Lu + g(x, u) = 0

in D,

u = 0 on ∂D.

(2.15)

Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

29

We claim that every solution u of (2.15) also solves (2.12). Indeed if u u¯ in some subdomain D ⊂ D then L(u − u) ¯ − (u − u) ¯ = 0 in D ,

u = u¯

on ∂D .

By the maximum principle u = u¯ in D . In the same way we can exclude that u < u. The operator F = Problem (2.15) is equivalent to u + L−1 g(x, u) = 0 in C 1,α (D). −L−1 ◦ g(x, ·) is asymptotically linear with F (∞) = L−1 ◦ Id. In view of our assumptions Theorem 1.16 now applies and the conclusion follows. The second multiplicity result is due to P. Hess [37]. It combines topological degree methods with variational principles. Consider for λ > 0 the problem u + λf (u) = 0 in D,

u = 0 on ∂D,

(2.16)

where f is a sign-changing function. T HEOREM 2.15. Let f : R+ → R be a continuously differentiable function with f (0) > 0. Suppose (1) there exist numbers 0 < a1 < a2 < · · · < am such that f (ak ) = 0 for k = 1, 2, . . . , m, s (2) max{F (s): 0 s ak−1 } < F (ak ), k = 2, . . . , m, where F (s) := 0 f (t) dt. Then there exists a number λ¯ such that for all λ > λ¯ there are at least 2m − 1 positive solutions uˆ 1 , u2 , uˆ 2 , . . . , um , uˆ m of (2.16) with 0 < uˆ 1 ∞ < a1 , uˆ k ∞ < ak and uk ∞ > ak−1 for k = 2, . . . , m. Moreover, uˆ 1 < uˆ 2 < · · · < uˆ m and uk < uˆ k . R EMARK . The hypotheses imply that the graph of f has m positive humps and (m − 1) negative humps, each positive hump having greater area than the previous negative hump; ak is the right end point of the kth positive hump (Figure 6).

Fig. 6.

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C. Bandle and W. Reichel

P ROOF OF T HEOREM 2.15. Let us sketch the main steps. Consider the function ⎧ =0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 fk (s) = f (s) ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ =0

if s −1, if − 1 s 0, if 0 s ak , if ak s ak + 1, if ak + 1 s,

and denote by Fk (s) its primitive Jk (λ, v) =

1 2

s 0

fk (t) dt. Deﬁne the functional

|∇v|2 dx − λ D

Fk (v) dx D

for v ∈ H01,2 (D).

Let Kk (λ) be the set of critical points uk (λ) of Jk . They are solutions of the auxiliary problem with f replaced by fk . Since by the maximum principle they are positive and bounded from above by ak they correspond to positive solutions of the original problem. Moreover, Kk (λ) is not empty because there exists a minimizer vk (λ) in H01,2 (D). Clearly Kk (λ) ⊂ Kk+1 (λ). The Schauder theory implies that Kk (λ) is compact for ﬁxed λ and k. Next it is shown that for all positive λ no critical points uk (λ) lies on the boundary of a large ball BR in H01,2(D). Thus deg(Id +λ−1 fk , BR , 0) is well deﬁned and independent of λ. Hence, deg(Id +λ−1 fk , BR , 0) = 1. It is then shown that deg(Id +λ−1 fk , Uε (Kk−1 (λ)), 0) = 1 in a small neighborhood of Kk−1 (λ). Then condition (2) comes into play and guarantees that for large λ the minimizer / Kk−1 (λ). If vk (λ) is an isolated solution then by Theorem 1.11 of Rabinowitz vk (λ) ∈ ind(Id +λ−1 fk , vk (λ), 0) = 1. By the excision property of the degree one ﬁnds deg Id +λ−1 fk , BR \ Uε Kk−1 (λ) ∪ Bε (vk (λ), 0 = −1, which shows the existence of an additional solution in Kk (λ) (see Figure 7).

Fig. 7. Solution continua for (2.16).

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31

2.10. Notes Maximum principle 1. Protter and Weinberger [56] prove a generalized maximum principle for those op such that −Lφ 0 ∩ C 2 (D) exists with φ > 0 in D erators L where a function φ ∈ C(D) in D; compare with Deﬁnition 2.1. If Lu 0 in D then they obtain that u/φ cannot attain a nonnegative maximum in D unless u/φ ≡ const, with a corresponding statement about nonnegative boundary maxima. 2. For operators with principal part in divergence form Lu = ∂j (aij ∂i u) + bi ∂i u + cu with c 0 the maximum principle (MP) has a natural extension to weak solutions u ∈ H 1,2 (D) of the inequality Lu 0, cf. [32], Section 8. There also exists a weak analogue of (2.2): if the coefﬁcients of L are bounded and if f ∈ Lq (D) for some q > N2 then every of Lu = f with u = 0 on ∂D satisﬁes weak solution u ∈ H01,2(D) ∩ C(D) sup |u| C|Λ|−1 ∞ f Lq , D

where C = C(N, L, vol(D)). 2,N 3. Similarly, if c 0 then (MP) also holds for strong solutions u ∈ Hloc (D) ∩ C(D) with u = 0 on ∂D, cf. [32], Section 9. The analogue of (2.2) is sup |u| C|Λ|−1 ∞ f LN , D

where C = C(N, bi /ΛLN , diam(D)). 4. The strong form of the maximum principle for essentially bounded, nonclassical solutions of Lu 0 a.e. in D can be expressed as follows: If ess supD u is positive and if there exists x0 ∈ D and r0 > 0 such that ess supD u = ess supBr (x0 ) u for all r ∈ (0, r0 ) then u = const in D. Eigenvalue problem 1. Hess and Kato [38] have generalized Theorem 2.6 to the case where the weight m is positive somewhere in D, but may change sign. 2. If −L is not necessarily strictly positive one still obtains a smallest eigenvalue λ1 (which may not be positive) and a unique (up to multiples) ﬁrst eigenfunction of one sign. then the smallest eigenvalue λ1 is characterized by 3. If the weight m is positive in D λ1 =

−Lφ(x) , x∈D m(x)φ(x) φ>0 in D sup

inf

, φ ∈ C 2 (D) ∩ C D

cf. [56]. 4. Next we establish a criterion for −L to be strictly positive in the case c+ ≡ 0. Let + c = max{c, 0}, c− = min{c, 0} and let μ1 be the smallest eigenvalue of (L0 + c− )ϕ + μc+ ϕ = 0

in D,

ϕ = 0 on ∂D.

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The operator −L is strictly positive if and only if μ1 > 1. The proof is a consequence of Theorem 2.6 and Remark 3. In particular if λ1 > 0 denotes the ﬁrst Dirichlet eigenvalue of −L0 then c(x) λ1 , c(x) ≡ λ1 in D is sufﬁcient for −L to be strictly positive. Sub- and supersolutions 1. The method of sub- and supersolutions was known for a long time in the context of ordinary differential equations. The ﬁrst time it was used in partial differential equations seems to be by Nagumo [52] in 1954. Only much later after the seminal paper by Keller and Cohen [42] it became a standard tool. 2. The fact that the degree between an upper and a lower solution is +1 can be seen from a different point of view: let U be the C 1 -order interval as in Theorem 2.10. In a ﬁrst step the degree deg(Id +L−1 ◦ f, U ∩ BR (0), 0) is seen to be homotopy equivalent to deg(Id +M −1 ◦ g, U ∩ BR (0), 0) where M = L − λ Id, g(x, s) = f (x, s) + λs with λ so large that g(x, s) is increasing in s. In a second step one observes that Id +M −1 ◦ g maps U ∩ BR (0) into itself. The convexity of U ∩ BR (0) implies that its closure is a retract. Now the same proof as for Schauder’s ﬁxed point theorem (Theorem 1.5) shows that the degree equals +1. 3. Sattinger [63] raised the question of existence of a solution to (2.9) in the presence of sub- and supersolutions which are not ordered. The following example shows that in general further conditions are required. Consider the problem u + λm u + φm = 0 in D,

u = 0 on ∂D,

where λm is the mth eigenvalue of and φm the corresponding eigenfunction. For m 2 it is easy to see that for t− < 0 sufﬁciently small the function t− φ1 is a supersolution and for t+ > 0 sufﬁciently large the function t+ φ1 is a subsolution. Nevertheless the problem is not solvable. The ﬁrst existence result under the assumption that there exist a pair of nonwellordered sub- and supersolutions was established by Amann, Ambrosetti and Mancini [4] for nonlinearities of the form sup f (x, s) − λ1 s < ∞. D×R

For selfadjoint operators with Dirichlet or Neumann boundary conditions Gossez and Omari [33] established the existence of solutions in the presence of not necessarily ordered sub- and supersolutions under the assumptions lim inf |x|→∞

f (x, s) λ1 s

and

lim sup |x|→∞

f (x, s) γ (x), s

with γ (x) λ2 and strict inequality on a set of positive measure (in the result in [33] the nonlinearity was even allowed to depend on the gradient). Generalizations and extensions of these results based on a degree argument for nonwell-ordered sub- and supersolutions are contained in a paper by De Coster and Henrard [24] where many references to the his-

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torical developments are found. The set U considered in Theorem 2.10 has to be modiﬁed as follows : ∃x1 , x2 ∈ D : u(x1 ) < v(x1 ) and u(x2 ) > w(x2 ) . U = u ∈ C1 D Under suitable conditions it is shown that in contrast to the case of ordered upper and lower solutions deg Id +L−1 ◦ f (x, ·), U ∩ BR (0), 0 = −1. 4. Consider the elliptic operator L = ∂j (aij ∂i ) + bi ∂i + c with coefﬁcients in L∞ (D) and c 0. The associated bilinear form a(u, v) =

N D

aij (x) ∂i u ∂j v −

i,j =1

N

vbi ∂i u − c(x)uv dx

i=1

is well deﬁned in H 1,2 (D) × H 1,2 (D). Let f (x, s) be measurable in x ∈ D and continuous in s for almost all x ∈ D. The function w, resp. v in H 1,2(D) is called a weak supersolution of (2.9) if w 0 on ∂D in the sense of traces,

f (·, w) ∈ L2 (D)

and a(w, ξ )

f (x, w)ξ dx D

for all ξ ∈ H01,2 (D), ξ 0.

For a weak subsolution v the inequalities are reversed. According to a result of Hess [35] the following extension of Lemma 2.9 holds: L EMMA . Let (v, w) be a pair of weak sub- and supersolutions of (2.9) such that v w a.e. in D. Assume f (x, t)2 dx < ∞. sup D v(x)t w(x)

Then (2.9) admits a solution u ∈ H01,2 (D) with v(x) u(x) w(x) in D. A priori bounds Further conditions for the a priori boundedness of the solution set of (2.13), cf. condition (H5), are known in the literature. Brezis and Turner [15] studied the case where f (x, s) does not have exact power-growth at ∞. However, they need to assume that N+1 lims→∞ f (x, s)/s N−1 = 0. For L = De Figueiredo et al. [25] extended the result of Brezis and Turner to more general nonlinearities with subcritical growth. Also in [25]

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a version of Theorem 2.12 for L = is proved. Chen and Li [18] derived a priori bounds in dimension n = 2 for positive solution of u + f (x, u) = 0, where f (x, s) can have exponential growth in s. A new approach to a priori bounds is proposed by Quittner and Souplet [57]. There additional references can be found. An example for Krasnosel’skii’s ﬁxed point theorem in cones We give a variant for the proof of Theorem 2.12. We use Krasnosel’skii’s ﬁxed point theorem, cf. Theorem 1.22. As before, we set up the boundary value problem (2.12) as a where F (u) = −L−1 f (x, u). Let C be the cone ﬁxed point problem u = F (u) in C 1 (D) of nonnegative functions. Clearly F : C → C is compact. First, we verify the condition (i) u = tF (u)

for u = r and t ∈ [0, 1].

By (H4) the function f (x, s) = o(s) as s → 0. Hence, given ε > 0 we can choose r > 0 so small that tF (u) εL−1 r. Thus for sufﬁciently small ε condition (i) holds. Next we need to verify (ii) u = F (u) + tv

for u = R, all t 0 and some v ∈ C.

Let v = −L−1 K, where K is the constant from Lemma 2.13. For 0 t 1 the solutions of u = F (u) + tv are a priori bounded by (H5). Hence we may choose R strictly larger than this bound. Thus (ii) holds for 0 t 1. And for t 1 the problem u = F (u)+tv amounts to solving Lu + f (x, u) + tK = 0 in D with Dirichlet conditions on ∂D. By Lemma 2.13 no such solution exists for t 1. Hence condition (ii) holds, and Krasnosel’skii’s Theorem 1.22 shows the existence of a nontrivial ﬁxed point of F in C.

3. Global continuation of solutions 3.1. A global implicit function theorem Consider the problem x − F (λ, x) = 0

(3.1)

which depends on a parameter λ ∈ R. In the sequel R × X is equipped with the product norm. In the neighborhood of a known solution the solution set of (3.1) can be described by the implicit function theorem. L EMMA 3.1 (Implicit function theorem). Let X, Y, Z be Banach spaces and let G : Y × X → Z be continuous and continuously differentiable with respect to x in a neighborhood U of the point (y0 , x0 ). Suppose further that G(y0 , x0 ) = 0 and that Gx (y0 , x0 ) has a bounded inverse from Z → X. Then there exist neighborhoods Bδ (x0 ) ⊂ X, Bε (y0 ) ⊂ Y and a continuous mapping x : Bε (y0 ) → Bδ (x0 ) such that (i) G(y, x(y)) = 0,

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(ii) x(y0 ) = x0 , (iii) if G(y, x) is continuously differentiable with respect to both variables then x (y) = −1 ◦ Gy (y, x(y)). − Gx (y, x(y)) In the following we will describe how a solution of (3.1) for one value λ0 can be continued to a global continuum in λ. This can be considered as the global analogue of the implicit function theorem. In contrast to the implicit function theorem no differentiability is required. Instead the nondegeneracy of the known solution is expressed by a nonvanishing degree. Different versions of this continuation method are known in the literature. To illustrate the main idea we begin with an elementary result. L EMMA 3.2. Let F : R × X → X be such that for all λ ∈ R the map F (λ, ·) : X → X is compact and F (λ, x) is continuous in λ uniformly w.r.t. x in balls in X. Let (λ0 , x0 ) be a solution of (3.1). Then the solution (λ0 , x0 ) can only be isolated in [λ0 , ∞) × X or (−∞, λ0 ] × X if ind(Id −F (λ0 , ·), x0 , 0)) = 0. P ROOF. Suppose that there is a neighborhood O = [λ0 , λ0 + ε) × Br (x0 ) ⊂ [λ0 , ∞) × X is (λ0 , x0 ). Since in particular for λ ∈ [λ0 , λ0 + ε) such that the only solution of (3.1) in O there is no solution on ∂Br (x0 ) the homotopy invariance of the degree with respect to λ shows deg Id −F (λ0 , ·), Br (x0 ), 0 = deg Id −F (λ0 + ε, ·), Br (x0 ), 0 . The ﬁrst degree equals ind(Id −F (λ0 , ·), x0 , 0). The second degree is 0 since there is no r (x0 ) at λ0 + ε. This proves the result. solution of (3.1) in B T HEOREM 3.3. Let F : R × X → X be such that for all λ ∈ R the map F (λ, ·) : X → X is compact and F (λ, x) is continuous in λ uniformly w.r.t. x in balls in X. Let (λ0 , x0 ) be a solution of (3.1). Suppose U ⊂ X is an open, bounded set such that x0 ∈ U and (i) for ﬁxed λ0 there is no other solution in U, (ii) deg(Id −F (λ0 , ·), U, 0) = 0. Then there exist two connected and closed sets (= continua) C + ⊂ [λ0 , ∞) × X and C − ⊂ (−∞, λ0 ] × X of solutions of (3.1) with (λ0 , u0 ) ∈ C + , C − . For C + one of the following two alternatives holds: (a) C + is unbounded, cf. Figures 8(α)–(γ ), or (b) C + ∩ ({λ0 } × (X \ U)) = ∅, cf. Figure 8(δ). The same alternative holds for C − . R EMARKS . 1. Alternative (b) means that (3.1) has another solution at λ = λ0 outside U , i.e., that C + bends back to λ0 . 2. The condition (ii) of the theorem is necessary as seen by the following example: let F : R2 → R be given by F (λ, x) = x + x 2 + λ2 . The point (0, 0) is a solution of x − F (λ, x) = 0, which is isolated in R2 , and deg(Id −F (0, ·), Bρ (0), 0) = 0, see the Example in Section 1.4.

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(α)

(β)

(γ )

(δ)

Fig. 8. (α)–(γ ): C + unbounded, (δ): C + turns back.

P ROOF OF T HEOREM 3.3. The following arguments are based on Deimling [26] and Peitgen and Schmitt [55]. Let S = {(λ, x) ∈ [λ0 , ∞) × X: x − F (λ, x) = 0} be the set of all solutions and let C + be the connected component of S containing (λ0 , x0 ). Suppose for contradiction that C + is bounded and that C + ∩ ({λ0 } × X) only contains the element (λ0 , x0 ). Then one chooses a relatively open, bounded neighborhood O of C + in[λ0 , ∞) × X such that no solution of (3.1) lies on ∂O, cf. Figure 9. We write O = λ {λ} × Oλ where Oλ is the projection of O on X for ﬁxed λ. Then on ∂Oλ there is no solution u of (3.1). By the assumption that C + does not turn back and by the excision property (d2) we get deg Id −F (λ0 , ·), Oλ0 , 0 = deg Id −F (λ0 , ·), U, 0 = 0 (by (ii)).

(3.2)

Moreover, since O is bounded there exists λ∗ such that Oλ = ∅ for all λ λ∗ . The generalized homotopy invariance (d3)g of the degree in λ leads to deg Id −F (λ0 , ·), Oλ0 , 0 = deg Id −F (λ∗ , ·), Oλ∗ , 0 = 0. This contradicts (3.2). For the construction of O consider a δ-neighborhood Vδ of C + in the space [λ0 , ∞) × X. Clearly C + ∩ ∂Vδ = ∅. If also S ∩ ∂Vδ = ∅ then we have found such a neighborhood. If not, then we use the following result of Whyburn [71]: L EMMA . Let (K, d) be a compact metric space, M1 ⊂ K a connected component and M2 ⊂ K closed such that M1 ∩ M2 = ∅. Then there exist compact and disjoint sets A, B such that A ∪ B = K,

M1 ⊂ A,

M2 ⊂ B.

To apply this result consider the compact set K := V δ ∩ S. Since C + and ∂Vδ ∩ S are disjoint, nonempty compact subsets of K and since C + is a connected component of S (and hence of K) there exist two compact, disjoint sets A, B such that A ∪ B = K,

C + ⊂ A,

∂Vδ ∩ S ⊂ B.

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Fig. 9. The open neighborhood O of C + .

Since β = dist(A, B) > 0 we may take a β/2-neighborhood Wβ/2 (A) of A to deﬁne O = Vδ ∩ Wβ/2 (A). Then, O is a neighborhood of C + such that on ∂O there is no element of S. Notice that the construction excludes the case C + = {(λ0 , x0 )}, even if the exists a sequence of solutions (λn , xn ) converging to (λ0 , x0 ). As a ﬁrst application for global continuation let us use Theorem 3.3 to prove the Krein– Rutman theorem, cf. Theorem 1.23 in Chapter 1. This proof is attributed to Rabinowitz. T HEOREM 3.4. Let X be a Banach space ordered with respect to a cone C. Suppose that Int(C) = ∅ and let T : X → X be a compact linear operator which is strongly positive in the sense that T (C \ {0}) ⊂ Int(C). Then T has a positive eigenvalue with eigenvector in Int(C). P ROOF. Let w ∈ C \ {0} be arbitrary. Then there exists M > 0 such that MT w w, since if this is not the case then T w − M −1 w ∈ / C for all M > 0 and thus T w ∈ / Int(C) in contrary to the assumption on T . By Dugundji’s theorem there exist a retraction r : X → C. Let T : X → C be given by

T = T ◦ r. Let ε > 0 and consider the problem x − λT (x + εw) = 0.

(3.3)

For λ = 0 there is a unique solution x = 0. Hence the global continuation principle of Theorem 3.3 applies and shows the existence of an unbounded continuum Cε ⊂ [0, ∞) × X of solutions of (3.3). Notice that for λ > 0 the solutions are nontrivial and 0. Next we show that Cε is bounded in the positive λ-direction. If (λ, x) is solution of (3.3) then x ελT w ελM −1 w. Thus T x ελM −1 T w ελM −2 w. But x λT x, i.e., x ε(λM −1 )2 w. By induction we ﬁnd x ε(λM −1 )n w for all n ∈ N. If λ > M then we obtain w 0, i.e., w ∈ / C \ {0}, a contradiction. Hence λ M.

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By the unboundedness of Cε there exists a solution (λε , xε ) of (3.3) on Cε with xε = 1. Taking ε = 1/n and using the compactness of T one can extract a convergent subsequence such that xn → x 0, λn → λ with x = λT x and x = 1. Thus λ > 0. This completes the result.

3.2. Applications – continuation of solutions A famous example for the continuation of solutions is given by the problem u + λ(1 + u)p = 0, u > 0 in D and u = 0 on ∂D with λ 0 and p > 1, studied by Joseph and Lundgren [41]. For λ = 0 the only solution is u ≡ 0 with index +1. By Theorem 3.3 of (positive) solutions exists. Now we an unbounded continuum C + ⊂ [0, ∞) × C 1 (D) consider the more general problem Lu + λf (x, u) + g(x) = 0,

u > 0 in D, λ > 0, u = 0 on ∂D,

(3.4)

The function f (x, s) is subject to the following conditions with g ∈ C α (D). locally uniformly w.r.t. s ∈ R, Hölder continuous in x ∈ D, locally Lipschitz continuous in s, uniformly w.r.t. x ∈ D,

(H)

for all λ > 0, i.e., 0 is a strict subsolution of (3.4), (F1) λf (x, 0) + g(x) 0, ≡ 0 in D (F2) there exists σ > 0 such that f (x, s) σ s in D for all s 0. For some of our applications we will suppose that the solutions of (3.4) are C 1 -bounded uniformly in λ for λ bounded and bounded away from zero, i.e., (F3) for every n ∈ N there exists an upper bound Mn such that uC 1 Mn for every solution of (3.4) with λ ∈ [1/n, n]. By the remark following Lemma 2.11 (F3) is satisﬁed if ∃p ∈ (1, 2∗ − 1) and h ∈ C(D), with lims→∞ f (x, s)/s p = h(x) uniformly for x ∈ D. h > 0 in D Examples of functions λf (x, s) + g(x) satisfying (F1), (F2) are λ(s p + 1) and λs p + 1 for p 1. T HEOREM 3.5. Suppose f satisﬁes (H) and (F1). of solutions of (3.4) exists. (a) Then an unbounded continuum C + ⊂ [0, ∞) × C 1 (D) + (b) If f also satisﬁes (F2) then C is bounded in the λ-direction. (c) If (F2) and (F3) hold then there exists λ∗ > 0 such that (i) for 0 < λ < λ∗ there are at least two solutions on C + , (ii) for λ = λ∗ there is at least one solution on C + , (iii) for λ > λ∗ there is no solution of (3.4). P ROOF. We set f (x, s) = f (x, 0) for s 0. Then (F1) implies any that solution of (3.4) is positive. Problem (3.4) can be reformulated as u + L−1 λf (x, u) + g(x) = 0

. for u ∈ C 1 D

(3.5)

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Fig. 10. Regions of nonexistence.

Note that (3.5) has a unique solution for λ = 0. Therefore alternative (a) of Theorem 3.3 with λ0 = 0 applies. This shows the existence of an unbounded continuum C + of positive solutions of (3.5). Hence (a) is proven. To show (b) one needs to observe that under hypotheses (F2) problem (3.5) has no positive solution for sufﬁciently large λ, since then λf (x, s) + g(x) λ1 s for all s 0 and λ large. The proof is almost the same as the proof of Lemma 2.13. Hence C + is bounded in the λ-direction. It remains to show (c). We know from (b) that there exists a value Λ > 0 such that no solution of (3.5) exists for λ Λ. It follows from (F3) that C + bends back to λ = 0 and becomes unbounded as shown in Figure 10. Notice that C + is unbounded even in the larger space R × C(D). Note that 0 is a strict lower solution. Thus, whenever (3.4) has a solution for some λ, then it also has a minimal solution uλ . Let [0, λ∗ ] be the projection of C + onto the λ-axis. We claim: (α) for λ > λ∗ there is no solution of (3.4), (β) for all λ ∈ (0, λ∗ ) we have (λ, uλ ) ∈ C + , (γ ) for all λ ∈ (0, λ∗ ) there exist a solution v on C + such that v uλ∗ . Notice that the minimal solutions are strictly ordered, i.e., λ1 < λ2 implies uλ1 < uλ2 . In particular, uλ < uλ∗ for all λ ∈ (0, λ∗ ). Hence, (β) and (γ ) together imply that for every λ ∈ (0, λ∗ ) there are at least two solutions of (3.4) on C + . This means that part (c) of Theorem 3.5 is complete, provided (α), (β), (γ ) are proved. (α): Suppose there is a value λ0 > λ∗ such (3.4) has a solution and let uλ0 be the min 0 < u < uλ in D, 0 > ∂ν u > imal solution. Consider the set V = [0, λ0 ] × {u ∈ C 1 (D): 0 ∂ν uλ0 on ∂D}. Since f (x, s) > 0 for s > 0 by assumption (F2), notice that uλ0 is a strict upper solution to (3.4) for every λ ∈ [0, λ0 ). Observe that C + is connected and C + ∩ V = ∅. If C + meets ∂V then the strong comparison principle implies that this is only possible for (λ, u) = (0, v0 ), where v0 = −L−1 g. Hence C + stays inside V in contradiction to the un boundedness of C + in the space R × C(D).

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(β): Suppose for some λ0 we have (λ0 , uλ0 ) ∈ / C + . The same proof as for (α) with 0 < u < uλ in D, 0 > ∂ν u > ∂ν uλ on ∂D} shows that C + V = [0, λ0 ] × {u ∈ C 1 (D): 0 0 cannot cross ∂V except at (λ, u) = (0, v0 ). This contradicts the unboundedness of C + . (γ ): Assume for contradiction that there exists λ˜ ∈ (0, λ∗ ) such that all elements ˜ (λ, v) ∈ C + have the property v uλ∗ . Deﬁne the set W = [0, λ˜ ) × Z where : ∃x ∈ D : u(x) > uλ∗ (x) . Z = u ∈ C01 D Notice that C + ∩ W = ∅ since The set W is a relatively open subset of [0, ∞) × C01 (D). c c + + C becomes unbounded near λ = 0. Also C ∩ W = ∅ since (0, v0 ) ∈ C + ∩ W . Since c + + C is a connected set which intersects both W and W it follows that C ∩ ∂W = ∅. A contradiction will be reached if we can show C + ∩ ∂W = ∅. This is done next. First observe that ∂W = λ˜ × Z ∪ 0, λ˜ × ∂Z . =:A1

=:A2

By assumption we have C + ∩ A1 = ∅. To show the same for A2 we need to determine ∂Z. c We will do this via the observation ∂Z = ∂Z . First, one ﬁnds : ∃x ∈ D: u(x) uλ∗ or ∃x ∈ ∂D: ∂ν u(x) ∂ν uλ∗ (x) . Z = u ∈ C01 D Then c : u < uλ∗ in D and ∂ν u > ∂ν uλ∗ on ∂D . Z = u ∈ C01 D Finally, this leads to c : u uλ∗ in D and u, uλ∗ “touch” , ∂Z = ∂Z = u ∈ C01 D where two functions v, w “touch” if there exists x ∈ D with v(x) = w(x) or there exists x ∈ ∂D with ∂ν v(x) = ∂ν w(x). Now it is easy to see that C + ∩ ([0, λ˜ ] × ∂Z) = ∅, since the strong comparison principle of Lemma 2.8(i) implies that no solution of (3.4) for a value λ ∈ (0, λ∗ ) can be below the strict supersolution uλ∗ and “touch”. Hence we have obtained that C + ∩ ∂W = ∅, which is the desired contradiction. E XAMPLE 3.1. Consider for q > 0 the problem Lu + λ(1 + u)−q = 0, u > 0 in D with u = 0 on ∂D. Theorem 3.5(a) applies. Moreover, for any given λ > 0 the constant function 0 is a strict lower solution and if φ1 is the ﬁrst Dirichlet eigenfunction on D ⊃⊃ D then tφ1 is an upper solution for t sufﬁciently large. Thus, for any λ > 0 there is a solution, which is unique by the comparison principle of Lemma 2.8. Hence we are in the situation of Figure 8(α). In dimension N = 1 with q = 3 and L = d2 /dx 2 on D = (−1, 1) the unique solution is explicitly given by √ 1 − 1 + 4λ λ 2 . u(x) = −1 + x c − , c = c 2

Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

41

E XAMPLE 3.2. Next we consider the problem Lu + λu + 1 = 0, u > 0 in D with u = 0 on ∂D. Now Theorem 3.5(b) applies. Moreover, there is no positive solution for λ λ1 . This follows as in Lemma 2.13, since for any t > 0 the function tφ1 is a lower solution. However, for 0 < λ < λ1 there is a unique solution, which is positive. Hence the situation is as in the second picture of Figure 8. In dimension N = 1 with L = d2 /dx 2 on D = (−1, 1) the unique solution is given by √ 1 cos( λx) u(x) = − + . √ λ cos( λ)λ E XAMPLE 3.3. For 1 < p < 2∗ − 1 the problem Lu + λ(up + 1) = 0, u > 0 in D with u = 0 on ∂D satisﬁes Theorem 3.5(c), as depicted in the third picture of Figure 8. Again, this can be seen explicitly in dimension N = 1 with p = 2 and L = d2 /dx 2 on D = (−1, 1), since then the initial value u(0) = a must satisfy √

1 λ = F (a) = √ 2

a 0

ds a

+ a 3 /3 − s

− s 3 /3

.

Figure 11 shows a plot of F (a): for any λ ∈ (0, λ∗ ) with λ∗ ≈ 1.188 there exist exactly two values a1 , a2 = u(0), i.e., two solutions. As λ → 0 the L∞ -norm of the second solution blows up.

Fig. 11. Plot of F (a).

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3.3. Further applications To illustrate the wide use of Theorem 3.3 we give three further applications, where a global solution branch is obtained by ﬁrst solving a sequence of approximate problems with solutions branches Cn+ and then passing to the limit C + = limn→∞ Cn+ . This limit process is made precise in the following deﬁnition. D EFINITION 3.6 (Whyburn [70]). Let S be a topological space and let G be an inﬁnite collection of subsets of S. (a) The set of all points z ∈ S such that every neighborhood of z contains points of inﬁnitely many sets of G is called the superior limit of G and is written lim sup G. (b) The set of all points z ∈ S such that every neighborhood of z contains points of all but ﬁnitely many sets of G is called the inferior limit of G and is written lim inf G. L EMMA 3.7 (Whyburn [70]). Let S be a topological space and let {An }n∈N be an inﬁnite sequence of connected subsets of S such that n∈N An is relatively compact and lim inf{An } = ∅. Then lim sup{An } is connected. R EMARK . Let S = X ×R and let {An }n∈N be a familiy of continua such that (xn , λo ) ∈ An . If {xn }n∈N has an accumulation point x¯ then (x, ¯ λ0 ) ∈ lim inf{An }, see Figure 12. E XAMPLE 3.4. For 0 < q < 1, p > 1 and λ > 0 consider the problem Lu + λ uq + up = 0

in D,

u=0

on ∂D.

(3.6)

Notice that the nonlinearity is a sum of concave and a convex function. We have the following result:

Fig. 12. lim inf{An } = ∅ since all An converge at λ0 .

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T HEOREM 3.8. Suppose 1 < p < 2∗ − 1. Then there is an unbounded continuum C + ⊂ of positive solutions of (3.6) with the following properties: there exists [0, ∞) × C 1 (D) λ∗ > 0 such that (i) for 0 < λ < λ∗ there are at least two positive solutions on C + and C + becomes unbounded near λ = 0, (ii) for λ = λ∗ there is at least one positive solution on C + , (iii) for λ > λ∗ there is no positive solution of (3.6). P ROOF. In order to avoid difﬁculties with the trivial solution of (3.6), we consider for 0 < ε 1 the problem q p Lu + λ u+ + ε + u+ = 0 in D,

u = 0 on ∂D.

(3.7)

The solutions are positive for λ > 0. The boundedness of the solution set in the positive λ-direction is clear from the proof of Theorem 3.5(b). We may therefore assume 0 λ Λ, where Λ does not depend on ε. For λ ∈ [ n1 , Λ] the remark following Lemma 2.11 applies and shows that there exists a constant Mn > 0, which is independent of ε, such that every positive solution of (3.7) with λ ∈ [ n1 , Λ] satisﬁes u∞ Mn . We formulate (3.7) as the ﬁxed point problem u + L−1 f (u; λ, ε) = 0,

λ 0.

(3.8)

At λ = 0, (3.8) has the isolated solution u = 0 with degree +1. We can therefore apply Theorem 3.3(a) and obtain a continuum Cε+ of solutions of (3.8) starting at (0, 0), turning back to λ = 0 and becoming unbounded near λ = 0; see Figure 13. Let λ1 be the ﬁrst eigenvalue of −L with ﬁrst eigenfunction φ1 such that 0 < φ1 < 1 in D. For λ 0 the function tφ1 provides a subsolution to (3.7) if 0 t ( λλ1 )1/(1−q) . Hence, there is no positive solution of (3.7) with 0 u ( λλ1 )1/(1−q)φ1 . Thus for every nontrivial solution there exists at least one point x ∈ D with u(x) ( λλ1 )1/(1−q)φ1 (x), 1/(1−q) . This region is also i.e., there exists a constant c(D) such that uC 1 (D) c(D)λ depicted in Figure 13. + be the connected component of Cε+ ∩ ([0, ∞) × BL (0)) containing For L > 0 let Cε,L + + = ∅ since (0, 0) ∈ Cε,L for all ε > 0. Now we can apply (0, 0). Notice that lim inf Cε,L + Lemma 3.7 to Cε,L and deﬁne + CL+ = lim sup Cε,L ε>0

and C + =

!

CL+

L>0

which contains positive solutions of (3.6) for to obtain a continuum C + ⊂ [0, Λ) × C 1 (D) λ > 0. By possibly enlarging C + we may assume that C + is a set of solutions which is uC 1 (D) maximal connected and unbounded in the set P = {(λ, u) ∈ [0, Λ] × C 1 (D): 1/(1−q) ∗ c(D)λ }. The deﬁnition of λ and the multiplicity result are done as in Theorem 3.5(c). Note that the arguments based on the strong comparison principle of Lemma 2.8(i) also hold for the given nonlinearity λ(uq + up ).

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Fig. 13. The branch Cε+ .

R EMARKS . (1) The restriction 0 < q < 1 is crucial for the construction of lower solutions, which are small for λ positive and small, cf. Figure 13. (2) Note that u = 0 is a solution of (3.6). The linearization at u = 0 is not deﬁned because 0 < q < 1. This is the reason why more reﬁned topological arguments are needed for the proof of the theorem. For supercritical p we do not have a priori bounds available. Therefore we cannot expect the turning back of the continuum C + , i.e., the multiplicity of solutions is no longer available. The results is as follows. T HEOREM 3.9. Suppose p 2∗ − 1. Then there is an unbounded continuum C ⊂ (0, ∞) × of solutions u > 0 of (3.6) with the following properties: there exists λ∗ > 0 such C 1 (D) that (i) for 0 < λ < λ∗ there is at least one solution on C + , (ii) for λ > λ∗ there is no solution of (3.6). P ROOF. As before we consider the problem (3.7) for 0 < ε 1. By Theorem 3.3(a) a continuum Cε+ of solutions of (3.8) exists which is bounded in the λ-direction and with the 1/(1−q) . The same property that for every solution (λ, u) of (3.8) one has uC 1 (D) c(D)λ limit procedure ε → 0 as in Theorem 3.8 produces the claimed continuum C + . The deﬁnition of λ∗ and the fact that the minimal solution lies on C + follow as in Theorem 3.5(c). However, there is no claim of multiplicity. E XAMPLE 3.5. Let γ > 0 and p > 1 and consider the problem u + u−γ + λup = 0,

u > 0 in D, u = 0 on ∂D.

(3.9)

Due to the negative exponent the probwhere solutions are understood in C 2 (D) ∩ C0 (D). lem is singular near ∂D.

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T HEOREM 3.10. Let D be convex and suppose 1 < p < 2∗ − 1. Then there is an unbounded continuum C + ⊂ R × L∞ (D) with the following properties: there exists λ∗ > 0 such that (i) for 0 < λ < λ∗ there are at least two solutions on C + , (ii) for λ > λ∗ there is no solution of (3.9), (iii) for λ = λ∗ there is at least one function on C + , which satisﬁes the equation u + u−γ + λ∗ up = 0 in D. If p 2∗ − 1 the structure of the positive solutions is as in Theorem 3.9. To avoid the singularity at u = 0 one considers for ε > 0 the problem u + u−γ + λup = 0,

u > 0 in D, u = ε on ∂D.

(3.10)

Solutions are ε and the same argument as in Lemma 2.13 shows the existence of Λ > 0 such that (3.9), (3.10) has no solution for λ > Λ. A variant of Lemma 2.11 shows that for given 0 < λ < Λ there exists M = M( λ, Λ) > 0 independent of ε such that for λ ∈ [λ, Λ] every solution uελ of (3.10) satisﬁes uελ ∞ M. To obtain the uniformity in ε it is important to know that there exists a neighborhood of ∂D where no solution has a local maximum. By the convexity of D such a neighborhood was constructed by Bandle and Scarpellini [8]. It is found by applying the moving plane method of Aleksandrov; cf. [30]. The next lemma provides upper and lower bounds for the solutions of (3.10). L EMMA 3.11. Let uελ be the minimal solution of (3.10). Then 2

(i) there exists t0 > 0 such that (t0 φ1 ) γ +1 uελ for all ε > 0 and every solution uελ of (3.10); (ii) for every λ > 0 there exists a value Aλ such that uελ Aλ uελ for every ε > 0 and every solution uελ of (3.10). 2

P ROOF. (i) Following a computation of Lazer and McKenna [48] one shows that (tφ1 ) γ +1 is a subsolution to (3.10) for all sufﬁciently small t. (ii) Fix λ and let f (s) = s −γ + λs p . Let M = M(λ) be the a priori bound for the solutions uελ . Let N = N(λ) be so large that f (s) is decreasing for s ∈ (0, M/N). Notice that since f is decreasing near 0 we may choose N even larger such that Nf (s/N) f (s) for all 0 < s < M. Set v = uελ /N . As a result one ﬁnds Lv + f (v) =

1 ε 1 ε Luλ + Nf uελ /N Luλ + f uελ = 0. N N

Thus v is a subsolution, which attains values in the range [0, M/N], where f is decreasing. ε Consider the open set D = {x ∈ D: uελ < M N }. On ∂D we have v uλ . By applying the comparison principle of Lemma 2.8(ii) to the subsolution v and to the minimal solution uλε in the set D one obtains v uλε in D . The same relation holds trivially on D \ D . Hence v uελ in D which is equivalent to (ii).

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P ROOF OF T HEOREM 3.10. For λ = 0 problem (3.10) has a unique solution uε0 of index +1 obtained by the method of sub- and supersolutions, cf. [48]. Hence, standard arguments show for every ε > 0 the existence of a continuum Cε+ with the properties (i)–(iii). In particular, whenever (3.10) has a solution then the minimal solution uλε belongs to Cε+ , and moreover, a second solution vλε exists on Cε+ with vλε uελ∗ , i.e., ε

∃yλε ∈ D

with vλε yλε > uελ∗ε yλε .

We claim that the point yλε can be assumed to have distance δλ > 0 from ∂D uniformly for ε ∈ (0, 1]. To see this note that the nonlinearity s −γ + λs p is decreasing on the interval 1

γ p+γ ) . Hence we can assume that (0, Mλ ] with Mλ = ( λp

vλε yλε > Mλ , since otherwise the comparison principle of Lemma 2.8(ii) would imply vλε uελ∗ in D. ε Next recall from Lemma 3.11(ii) that ε

vλε Aλ uελ Aλ uλ0

in D for 0 < ε < ε0 . ε

For small ε0 there exists δλ > 0 such that Aλ uλ0 (x) Mλ for all x ∈ D with dist(x, ∂D) δλ . Therefore the point yλε must have at least distance δλ from ∂D. It remains to pass to the limit ε → 0. Let [0, λ∗ε ] be the projection of Cε+ onto the λ-axis. + be the connected component of Cε+ ∩ ([0, ∞) × BL (0)) containing (0, uε0 ). To pass Let Cε,L + as a subset of the complete metric space R × Cloc (D) to the limit ε → 0 we consider Cε,L with the metric on Cloc (D) given by d(f, g) =

∞ 1 f − g∞,Dn , 2n 1 + f − g∞,Dn n=1

+ where Dn are open with D1 ⊂ D2 ⊂ · · · ⊂⊂ D and D = ∞ n=1 Dn . Since solutions on Cε,L ∞ are in L and bounded away from zero on every Dn it is easy to see that bounded + C is relatively compact in [0, ∞) × Cloc (D). Moreover, at λ = 0 the unique soluε>0 ε,L + = ∅, and tions uε0 converge monotonically in ε to a solution u0 of (3.9). Thus lim inf C1/n,L + + −γ p CL = lim sup C1/n,L is a continuum of solutions of the equation u + u + λu = 0 in D. The same holds for C + = L CL+ . Let [0, λ∗ ] denote the projection of C + onto the λ-axis. Clearly λ∗ = supε>0 λ∗ε = limε→0 λ∗ε . It remains to show that for 0 < λ < λ∗ the solutions So ﬁx λ ∈ [0, λ∗ ). on C + attain Dirichlet boundary data, i.e., that they belong to C0 (D). Then uελ uλ monotonically as ε → 0. Together with the lower bound from Lemma 3.11 The bound uε Aλ uε for all ε > 0 from Lemma 3.11 implies this shows that uλ ∈ C0 (D). λ λ Thus we have shown that all solutions that every other solution uλ also belongs to C0 (D). on C + except for λ = λ∗ attain zero-boundary values on ∂D. By replacing C + by a maximally connected component of positive solutions of (3.9) all properties (i)–(iii) follow. To see the multiplicity result recall that for all λ ∈ (0, λ∗ε ) there exists a solution vλε on Cε+ with

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47

vλε (yλε ) > uελ∗ (yλε ) and dist(yλε , ∂D) δλ > 0. After passing to the limit this property says ε that for all λ ∈ (0, λ∗ ) there exists a solution vλ on C + with vλ (yλ ) uλ∗ (yλ ) at some interior point yλ ∈ D. In particular vλ is not the minimal solution. This shows the multiplicity result. Open problems. (i) Does there exists a solution of (3.9) at λ = λ∗ ? Note that on C + we have at λ = λ∗ a bounded solution of the differential equation u + u−γ + λ∗ up = 0, but we do not know if the Dirichlet conditions are attained. (ii) Can one generalize Theorem 3.10 to second order operators L instead of ? Note that the moving plane method used in the proof is currently the main obstruction. E XAMPLE 3.6. Let p > 1 and consider the problem u = λ(1 + |u|p )

in D,

u(x) → ∞

as dist(x, ∂D) → 0.

(3.11)

Solutions are supposed to be in C 2 (D). Let us introduce the weighted-norm uω = 2

supx∈D |u(x)| dist(x, ∂D) p−1 and consider the Banach space X = (C(D), · ω ). Existence of positive solutions for Example 3.6 was shown by Keller [43] and Osserman [54]. Parts of the following theorem were derived by Aftalion and Reichel [1]. T HEOREM 3.12. Let D be convex and suppose 1 < p < 2∗ − 1. Then there is an unbounded continuum C + ⊂ R × X with the following properties: there exists λ∗ > 0 such that (i) for 0 < λ < λ∗ there is at least one positive and one sign-changing solution on C + , (ii) for λ > λ∗ there is no solution of (3.11), (iii) for λ = λ∗ there is at least one positive solution on C + , (iv) uλ ω → ∞ as λ → 0 for any solution uλ of (3.11). If p 2∗ − 1 the structure of the positive solutions is as in Theorem 3.9. S KETCH OF THE PROOF. The problem with inﬁnite boundary values is replaced by u = λ 1 + |u|p in D,

u=c

on ∂D.

(3.12)

Solutions are c and by the further transformation v = c − u the problem becomes v + λ 1 + |c − v|p = 0 in D,

v=0

on ∂D.

(3.13)

Similar arguments to those used in Example 3.5 give that there is no solution λ > Λ. For given values 0 < λ < Λ there exists constants K, L independent of c such that for every λ ∈ [λ, Λ] and every solution ucλ of (3.12) the following holds: 2

(a) ucλ Kφ11−p (see [43] and [54]), (b) ucλ −L (see [1]).

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Fig. 14. Solution continua for vc , uc and u in respective norms.

A suitable translation of these bounds to solutions of (3.13) and standard arguments show of positive solutions that starting at λ = 0 and v = 0 a continuum Dc+ ⊂ [0, Λ] × C(D) of (3.13) exists with properties (i)–(iii) of the theorem. This translates back to a continuum Cc+ of solutions of (3.12). The minimal solution v cλ of (3.13) corresponds to the positive maximal solution u¯ cλ of (3.12). After the continuum Dc+ has turned back we have large positive solutions vλc which correspond to sign-changing solutions ucλ on Cc+ ; cf. Figure 14. It remains to pass to the limit c → ∞. Due to the estimates (a) and (b) this is done in the weighted space X = (C(D), · ω ). The veriﬁcation of the boundary conditions is clear for all maximal solutions u¯ λ = limc→∞ u¯ cλ , including the one at λ∗ , since they are monotone increasing in c. For the sign-changing solutions the boundary conditions are veriﬁed by the estimate ucλ ucλ − Aλ for ﬁxed λ and every c > 0, which is shown in a similar way as in Lemma 3.11(ii).

3.4. Notes 1. A theorem of two solutions in the spirit of Theorem 3.5(c) was discovered by Crandall and Rabinowitz [22] for problems with variational structure. The assumptions on the nonlinearity are weaker; on the other hand the variational methods do not provide continua of solutions. For more recent results and references concerning the problem u + λ(1 + u)p = 0 see Gazzola and Malchiodi [28]. 2. Minimal solutions are isolated. Here we recall that if f (x, s) > 0 is continuously differentiable and convex in s then for ﬁxed λ the minimal solution of Lu + λf (x, u) + g(x) = 0 in D with u = 0 on ∂D is isolated. Note ﬁrst by the positivity of f (x, s) that λ > λ0 implies uλ > uλ0 . By the convexity of f (x, s) in s we get 0 = L( uλ − uλ0 ) + λ0 f (x, uλ ) − f (x, uλ0 ) + (λ − λ0 )f (x, uλ ) >0

> L( uλ − uλ0 ) + λ0 ∂s f (x, uλ0 )( uλ − uλ0 ).

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Then the linearized operator −L − λ0 ∂s f (x, uλ0 ) is strictly positive, and hence invertible. By the inverse function theorem the minimal solution uλ0 of u + L−1 (λ0 f (x, u) + g(x)) = 0 is isolated. 3. Continuity of the minimal solution. Theorem 3.5 does not claim the continuity of the minimal solution uλ w.r.t. λ. However, continuity is known for problems Lu + λf (x, u) + g(x) = 0 in D with u = 0 on ∂D under the assumption that f (x, s) is nonnegative, convex and increasing in s and 0 is a strict subsolution, see also [3] for related conditions. Here we give a short proof: the nonnegativity of f (x, s) and g(x) implies that uλ is monotone increasing in λ since uμ serves as an upper solution for any λ < μ. Hence left-continuity of uλ follows. Right-continuity is more delicate: suppose limμλ uμ = u¯ uλ , but u¯ = uλ . Consider vn = 12 (uλ + uλ+ 2 ). Then n

1 1 2 −Lvn = λf (x, uλ ) + λ+ f (x, uλ+2/n ) + g(x) 2 2 n 1 1 1 f (x, uλ ) + f (x, uλ+2/n ) + g(x) = λ+ n 2 2 +

1 f (x, uλ+2/n ) − f (x, uλ ) 2n 0

1 f (x, vn ) + g(x) λ+ n by using convexity and monotonicity of f (x, s). Because vn is a supersolution and 0 is a strict subsolution the minimal solution uλ+1/n must satisfy vn uλ+1/n 0. Passing to ¯ u, ¯ which implies uλ u. ¯ Since uλ is the minimal the limit n → ∞ we obtain 12 (uλ + u) solution we obtain uλ = u. ¯ This ﬁnishes the proof of the right-continuity. 4. Discontinuity of the minimal solution. An example for discontinuity of the minimal solution can be found by adapting an example of Laetsch [45]: Consider u + λf (u) = 0 in (0, 1) with u(0) = u(1) = 0, where f (s) is positive on (0, ∞), convex and increasing and satisﬁes (F2) and (F3), e.g., f (s) = (1 + s p ) for p > 1. Let uλ be the minimal solution deﬁned for the maximal λ-interval (0, λ∗ ] and deﬁne ρ = uλ∗ ∞ . If we choose α > ρ and deﬁne " f (s) if 0 s α, h(s) = g(α)−g(s) f (α) + f (α) |g (α)| if s > α, where g is bounded, smooth and g < 0. Then the problem v + λh(v) = 0 in (0, 1) with v(0) = v(1) = 0 has a minimal solution v λ for every λ > 0. Moreover, for 0 < λ λ∗ we have v λ = uλ with v λ ∞ ρ. But for λ > λ∗ the minimal solution must attain values large then α, i.e., v λ ∞ α > ρ. This shows that the minimal solution v λ is discontinuous at λ∗ . Notice that h satisﬁes (F1) and (F3), but not (F2). 5. Nonexistence of three ordered solutions. Theorem 3.5(c) shows that in certain λ-ranges there are two ordered solutions: the minimal solution and a second positive, large

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solution. It is again a consequence of the strict convexity of f (x, s) in s that there can never be three ordered solutions u < v < w in D of the problem Lu + f (x, u) = 0 in D with u = 0 on ∂D. Suppose this were the case. Then f (v) − f (u) f (v) − f (w) (w − v) > (w − v), v−w v−u f (v) − f (u) (v − u). −L(v − u) = v−u

−L(w − v) =

(u) With d(x) = f (v)−f the ﬁrst equation implies that the function φ = w − v > 0 in D v−u satisﬁes −Lφ − d(x)φ > 0 in D. Hence the operator −L − d(x) is strictly positive, cf. Deﬁnition 2.1. With ψ = v − u > 0 in D the second equation shows that Lψ + d(x)ψ = 0. The maximum principle applied to L + d(x) shows that ψ ≡ 0. This contradiction shows that three ordered solutions are excluded for strictly convex nonlinearities. In contrast, Amann [3] obtained conditions on nonlinearities such that three ordered solutions do exist, see also Theorem 2.14. 6. Results similar to that of Example 3.4 with L = and the nonlinearity λuq + up were obtained by Ambrosetti, Brezis and Cerami [5] by variational methods. 7. The existence of solutions for problems with singularities of the type u−γ as in Example 3.5 was ﬁrst treated by Crandall, Rabinowitz and Tartar [23]. More recent results and references related to Example 3.5 are given by Ghergu and R˘adulescu [29].

4. Bifurcation theory and related problems 4.1. Bifurcation from the trivial solution 4.1.1. Abstract theory. Let F : R × X → X be a continuous map such that F (λ, 0) = 0 for all λ. The goal is to ﬁnd solutions (λ, x) of the nonlinear eigenvalue problem x − F (λ, x) = 0.

(4.1)

If F is differentiable and Id −Fx (λ0 , 0) is invertible the implicit function theorem implies that (λ, 0) is locally the unique branch of solutions near (λ0 , 0). In this chapter we analyze the situation where Id −Fx (λ0 , 0) fails to be invertible and in addition to the trivial branch at least one other branch emanates from (λ0 , 0). For this purpose let us introduce the notion of a bifurcation point. D EFINITION 4.1. The point (λ0 , 0) ∈ R × X is called a bifurcation point of (4.1) if there exists a sequence (λn , xn ) of solutions of (4.1) such that λn → λ0 and xn → 0 with xn = 0. It should be observed that the deﬁnition of a bifurcation point does not guarantee the existence of a continuous branch of solutions. A counterexample is given in [10]. The ﬁrst

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51

study of bifurcation points goes back to Krasnosel’skii, cf. [44]. It is based on the linearization of (4.1) at the bifurcation point (λ0 , 0). Large parts of the bifurcation theory including many references are in the textbook of Chow and Hale [19]. The next result is mainly due to Krasnosel’skii and Zabreiko [44], cf. also [64]. T HEOREM 4.2 (Krasnosel’skii). Let F (λ, x) = λAx + g(λ, x) where A is a compact linear operator and g is a compact map w.r.t. (λ, x) such that g(λ, x) = o(x) as x → 0 uniformly for λ in compact intervals. (i) Necessary condition: if (λ0 , 0) is a bifurcation point of F then λ−1 0 is an eigenvalue of A. (ii) Sufﬁcient condition: let 0 be an isolated solution of x − F (λ, x) = 0 at λ = λn and λ = λn where λn , λn → λ0 as n → ∞. Suppose further that ind Id −F λn , · , 0, 0 = ind Id −F λn , · , 0, 0 . Then (λ0 , 0) is a bifurcation point. P ROOF. (i) Let (λ0 , 0) be a bifurcation point. Then by deﬁnition there exists a sequence (λn , xn ), n = 1, 2 . . . , of solutions such that λn → λ0 and xn → 0, xn = 0 as n → ∞. We have xn − λ0 Axn = (λn − λ0 )Axn + g(λn , xn ). Dividing by xn and setting yn = xn /xn , we get yn − λ0 Ayn = (λn − λ0 )Ayn + g(λn , xn )/xn . The right-hand side converges to 0 as n → ∞. By the compactness of A there is a subsequence denoted again by yn such that Ayn converges. Hence yn → y as n → ∞, where y = 1 and y − λ0 Ay = 0. Consequently λ−1 0 is an eigenvalue of A. (ii) Assume that 0 is an isolated solution of x − F (λ0 , x) = 0 for otherwise there is nothing to prove. Hence there exists ρ > 0 such that x − F (λ0 , x) = 0 has no solution with x = ρ. Now we show that there is a δ > 0 such that (4.1) has no solution of norm ρ for |λ−λ0 | δ. Indeed suppose that the conclusion does not hold. Then there exists a sequence (xn , λn ) of solutions to (4.1) such that λn → λ0 and xn = ρ. Since F is compact in x and λ, there exists a subsequence, denoted again by (λn , xn ) such that g(λn , xn ) and λn Axn converge. It follows from (4.1) that xn converges as well. Taking n → ∞ we obtain a solution (λ0 , x) with x = ρ which contradicts the choice of ρ. For λ close to λ0 the deg(Id −F (λ, ·), Bρ , 0) is thus well deﬁned. In view of our assumptions we may select a new sequence λ¯ n out of the given sequences λn , λn such that ind Id −F λ¯ n , · , 0, 0 = ind Id −F (λ0 , ·), 0, 0 (4.2) and λ¯ n → λ0 as n → ∞. Choose 0 < ε < ρ sufﬁciently small such that x = 0 is the only ε . By the previous remark we may assume that x − solution of x − F (λ0 , x) = 0 in B F (λ, x) = 0 has no solution on ∂Bε for all λ between λ0 and λ¯ n if n is sufﬁciently large. By the homotopy invariance (d3) with respect to λ, we have deg Id −F (λ0 , ·), Bε , 0 = deg Id −F λ¯ n , · , Bε , 0 . (4.3)

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The left-hand side of (4.3) equals ind(Id −F (λ0 , ·), 0, 0). By (4.2) this is only possible if for every large n the equation x − F (λ¯ n , x) = 0 has an additional solution different from zero in Bε . Here we have used the excision property (d2). R EMARK . Simple examples show that not every eigenvalue λ−1 0 of A is a bifurcation point. Consider the equation #

1 0 0 1

1 −1 −λ 0 1

$ x 0 0 + = . y −x 3 0

The only eigenvalue of the corresponding linear operator A is λ = 1. It is easy to show that for λ near 1 the equation only possesses the trivial solution. Consequently (1, 0, 0) cannot be a bifurcation point. A SSUMPTION . In the sequel we shall always assume that F (λ, x) = λA + g(λ, x), where A is a compact linear operator and g is a compact map with respect to (λ, x) such that g(λ, x) = o(x) as x → 0 uniformly for λ in compact intervals. −1 T HEOREM 4.3. Let λ−1 0 be an eigenvalue of A. If the algebraic multiplicity of λ0 is odd then (λ0 , 0) is a bifurcation point for (4.1).

P ROOF. Suppose that (λ0 , 0) is not a bifurcation point. Therefore there exists ε > 0 such that (λ, 0) is the only solution of (4.1) with |λ0 − λ| ε and x ε. The homotopy Id −F (λ, ·) w.r.t. λ is thus well deﬁned in Bε = {x ∈ X: x < ε}. Since A is compact λ−1 0 is an isolated eigenvalue. By Theorem 1.10, (−1)β(λ0−0) = deg Id −F (λ0 − 0, ·), Bε , 0 = deg Id −F (λ0 + 0, ·), Bε , 0 = (−1)β(λ0+0) , where β(λ) is the sum of the algebraic multiplicity of all eigenvalues of A larger than 1/λ. By our assumption β(λ0 − 0) differs by an odd number from β(λ0 + 0) which leads to a contradiction. Notice that the counterexample in the above remark is in accordance with Theorem 4.3 for there the algebraic multiplicity of λ−1 0 is two. A different approach to the study of bifurcation points from the technical point of view is found in [7]. Artino attributes it to Ize. The basic idea is to look for solutions (λ, x) of a given norm x = ρ in the interval |λ − λ0 | δ0 by means of a degree argument. This approach not only clariﬁes the structure of the solutions near the bifurcation points, but it is also an example how to use the tools of Section 1.3.3. Let us start with some preliminary considerations which will be needed for the main result of this section. In the sequel we shall assume that R × X is equipped with the product norm and the topology is deﬁned with respect to this norm. Consider the problem x − λAx − g(λ, x) = 0, where as before g(λ, x) = o(x) as x → 0 uniformly for

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|λ − λ0 | ε. Suppose that λ−1 0 is an eigenvalue of A and let δ0 > 0 be so small that A has no other eigenvalues between (λ0 + δ0 )−1 and (λ0 − δ0 )−1 . Assume in addition that 0 is the only solution of x = F (λ0 , x) in the ball Bρ = {x ∈ X: x < ρ}. Since λ−1 0 is an isolated eigenvalue, 0 is an isolated solution of x = F (λ, x) for every ﬁxed λ ∈ [λ0 − δ0 , λ0 + δ0 ]. Consider the function G : X × (λ0 − δ0 , λ0 + δ0 ) → X × R deﬁned as follows: G(x, λ) = x − F (λ, x), x2 − ρ 2 . Under our assumptions the map G is a compact perturbation of the identity. L EMMA 4.4. Let the assumptions of Theorem 4.3 hold. Suppose that 0 is an isolated solution of x = F (λ0 , x). Let i± = ind(Id −(λ0 ± 0)A, 0, 0). For sufﬁciently small δ0 and ρ deg G, x2 + (λ − λ0 )2 < ρ 2 + δ02 , (0, 0) = i− − i+ .

t (x, λ) = (y, τ ) with P ROOF. Consider for 0 t 1 the homotopy G y = (Id −λA)x − tg(λ, x), τ = t x2 − ρ 2 + (1 − t) δ02 − (λ − λ0 )2 .

t (x, λ) = (0, 0) on the set Let δ0 be as above. Consider for t ∈ [0, 1] the solutions of G 2 2 2 2 {x + (λ − λ0 ) = ρ + δ0 }. Because of the second equation the solutions must satisfy λ = λ0 ± δ0 and x = ρ. From the ﬁrst equation we get −1 g(λ0 ± δ, x) x = t Id −(λ0 ± δ)A . ρ ρ From the behavior of g(λ, x) near x = 0 it follows that for sufﬁciently small ρ no solutions

t = 0 on {x2 + (λ − λ0 )2 = ρ 2 + δ 2 } exist for t ∈ [0, 1]. Hence ν := deg(G

t , {x2 + of G 0 2 2 2 (λ − λ0 ) < ρ + δ0 }, (0, 0)) is well deﬁned and independent of t. Consequently, ν = deg G0 , x2 + (λ − λ0 )2 < ρ 2 + δ02 , (0, 0) . The only solutions of G0 (x, λ) = ((Id −λA)x, δ02 − (λ − λ0 )2 ) = (0, 0) are (0, λ0 ± δ0 ). By the excision property (d2), ν = ind G0 , (0, λ0 + δ0 ), (0, 0) + ind G0 , (0, λ0 − δ0 ), (0, 0) . Theorem 1.10 and the product formula Lemma 1.8(b) complete the proof.

Provided λ−1 0 is an eigenvalue of A of odd multiplicity and provided 0 is an isolated solution of x − F (λ0 , x) = 0 we can apply the global continuation principle of Theorem 3.3 to the problem Gρ (x, λ) = (0, 0),

(4.4)

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C. Bandle and W. Reichel

Fig. 15. Rabinowitz alternative.

where we now emphasize the ρ-dependence of the map Gρ : Y → Y with Y = X × R. We continue the trivial solution (0, λ0 ) of G0 (x, λ) = (0, 0) with respect to ρ > 0. The result is a continuum C ⊂ [0, ∞) × Y of solutions (ρ, x, λ) ∈ R × Y of (4.4). Either the continuum turns back to ρ = 0 or it is unbounded. The more precise global behavior of C was studied by Rabinowitz [59]. T HEOREM 4.5 (Rabinowitz). Let the assumptions be the same as in Theorem 4.3. Then there exists a maximal connected continuum C ⊂ [0, ∞) × Y of solution of (4.4) emanating from ρ = 0 and (x, λ) = (0, λ0 ) such that the following alternative holds (see Figure 15): (i) C is unbounded in [0, ∞) × Y or (ii) C meets ρ = 0 at (x, λ) = (0, λj ) for j = 0 where λ−1 j is an eigenvalue of A. Furthermore the number of such points (0, λj ) belonging to C with λ−1 j having odd algebraic multiplicity (including the point (0, λ0 )) is even.

R EMARK . The projection of C onto Y = X × R provides the branch of nontrivial solutions for (4.1).

P ROOF OF T HEOREM 4.5. We present a variant of the proof given by Artino [7]. Assume that C + is bounded. Then by the compactness of A it contains at most a ﬁnite number of bifurcation points (0, λj ), j = 1, . . . , k. Let O ⊂ [0, ∞) × Y be a relatively open bounded set containing C and such that no solution (ρ, x, λ) of (4.4) lies on ∂O. As in Theorem 3.3 we write O = ρ0 {ρ} × Oρ . Then deg(Gρ , Oρ , (0, 0)) is well deﬁned and independent of ρ. By the boundedness of C there are no solutions for large ρ. Hence the degree is zero. On the other hand for small ρ the solutions are close to the bifurcation points (0, λj ), j = 1, . . . , k. The excision property together with Lemma 4.4 implies that for small ρ and δ0 0=

k 1

k deg Gρ , x2 + |λ − λj |2 < ρ 2 + δ02 , (0, 0) = i− (j ) − i+ (j ) , 1

where i± (j ) is as in Lemma 4.4 with λ0 replaced by λj . The conclusion now follows.

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55

4.1.2. Applications 1. Let D ⊂ RN be a bounded domain and assume that ∂D and L satisfy the regularity assumptions needed for the Schauder estimates. By the Krein–Rutman theorem (cf. Section 2.5.4) −L possesses a smallest eigenvalue λ1 > 0 with an eigenfunction φ > 0. Consider the boundary value problem Lu + λu + f (x, u, ∇u, λ) = 0

in D,

u=0

on ∂D,

(4.5)

where f : D × R × RN × R → R and f (x, s, ξ, λ) is Hölder continuous in x uniformly and λ in for (s, ξ, λ) in balls in RN+2 and locally Lipschitz in (s, ξ ) uniformly for x ∈ D compact intervals. In addition assume f (x, 0, 0, λ) = 0 for all x ∈ D and λ ∈ R. As already → C 1,α (D) is compact and λ−1 is a simple observed in Section 2.3 the map L−1 : C α (D) 1 −1 eigenvalue of −L . Hence (4.5) can be written as u + λL−1 u + L−1 f (x, u, ∇u, λ) = 0. This problem can be interpreted as an abstract bifurcation equation in the Banach space Assume |f (x, s, ξ, λ)| = o(s) as s → 0 uniformly w.r.t. (x, ξ ) ∈ D × RN X = C 1,α (D). and λ in compact intervals. Then Theorem 4.3 applies and λ1 is a bifurcation point of (4.5). Notice that if we use the Lp -theory the regularity assumptions of f can be loosened considerably. E.g., in the case where f is independent of the gradient continuity of f w.r.t. all variables sufﬁces. The behavior of the continuum of solutions C near the bifurcation point (λ1 , 0) depends on f . The following result is due to Sattinger [64]. T HEOREM 4.6. Assume f (x, s, ξ, λ) = g(x, s, ξ, λ)s with g(x, s, ξ, λ) < 0 in D × R × RN × R and g(x, s, ξ, λ) → 0 as s → 0 uniformly in x, ξ and λ. Then the continuum C of solutions of (4.5) emanating from λ1 lies to the right of λ1 . P ROOF. Assume that λ < λ1 . The sign assumption on g implies that tφ is a supersolution and −tφ is a subsolution for any t > 0. If there were a solution u = 0 for such a λ < λ1 then for large t 1 we have −tφ < u < tφ in D. By continuously decreasing t there would exists a ﬁrst value t0 > 0 where either t0 φ or −t0 φ touches u. By the strong comparison principle of Lemma 2.8 either t0 φ ≡ u or −t0 φ ≡ u. Both possibilities contradict the fact that g(x, s, ξ, λ)s = o(s) for s near 0, i.e. the equation does not support linear eigenfunctions as solutions. 2. The next example has ﬁrst been discussed by Rabinowitz [58]. Consider the quasilinear operator Lu = aij (x, u, ∇u) ∂ij2 u + bi (x, u, ∇u) ∂i u + c(x, u, ∇u)u, 2 N where c 0 and N i,j =1 aij (x, z, χ)ξi ξj Λ|ξ | in D for all x ∈ D, z ∈ R and ξ ∈ R . We shall assume the coefﬁcients and their derivatives with respect to z and χ to be Hölder

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continuous. Let m(x) m0 > 0 and let F be a positive function such that F (x, s, ξ, λ) = and on bounded λ-intervals. Our goal is to o((|s|2 + |ξ |2 )1/2 ) near (0, 0) uniformly on D establish a branch of positive solutions of the quasilinear boundary value problem Lu + λm(x)u + F (x, u, ∇u, λ) = 0 in D,

u=0

on ∂D.

(4.6)

Note that u = 0 is a solution for all λ. Put L := aij (x, 0, 0)∂ij2 + bi (x, 0, 0)∂i + c(x, 0, 0). By Krein–Rutman’s theorem the linear problem, Lφ + λm(x)φ = 0

in D,

φ=0

on ∂D,

has a principal eigenvalue λ1 with a eigenfunction φ of constant sign. T HEOREM 4.7. Under the above conditions problem (4.6) has an unbounded branch of positive solutions emanating from (λ1 , 0) and extending to the left of λ1 . We deﬁne a mapping G : R × X → X as follows: P ROOF. We shall take X = C 1,α (D). For (λ, u) ∈ R × X let G(λ, u) be the unique solution of aij (x, u, ∇u) ∂ij2 v + bi (x, u, ∇u) ∂i v + c(x, u, ∇u)v + λm(x)u + F (x, u, ∇u, λ) = 0

in D,

v=0

on ∂D.

Hence (4.6) is equivalent to u = G(λ, u). It is easily seen that H (λ, u) = G(λ, u) + λL−1 (mu) = o(u) as u → 0 uniformly in bounded λ-intervals. Thus, our problem can be stated in the following form u + λL−1 (mu) − H (λ, u) = 0.

(4.7)

By Theorem 4.5 a continuum of solutions emanates from (λ1 , 0). In the next step we use the Lyapunov–Schmidt reduction (see notes below) and decompose u = tφ + w, where X = span[φ] ⊕ W . As a result of the reduction procedure we obtain w = w(t, λ) = o(t) uniformly in λ. The projection P : X → span[φ] and division by t lead to 1 λ φ1 − P H λ, tφ + w(t, λ) = 0. 1− λ1 t Since the derivative of the function at the left-hand side with respect to λ evaluated at t = 0, λ = λ1 is different from zero the implicit function theorem yields that the solutions are of the form (λ(t), tφ + o(t)) for small t. Hence for small positive t the function u is positive. Thus from (λ1 , 0) a branch C of initially positive solutions emanates. The fact that C extends initially to the left of λ1 follows from the maximum principle. Moreover, as long as u stays positive the branch C lies to the left of λ1 , and reversely, as long as C lies to the

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left of λ1 the solutions on C stay positive, because if this were not the case we could ﬁnd a sequence (λn , un ) of positive solutions converging to (λ∗ , u∗ ) where u∗ vanishes at some ∗ ∗ inner point or ∂u ∂ν = 0 somewhere. By the maximum principle u ≡ 0 which is impossible by construction. Hence C stay globally to the left of λ1 and consists of positive solutions. By Rabinowitz’s alternative C escapes to inﬁnity, since to the left of λ1 there are no further eigenvalues. Observe that to the right of λ, there is locally a branch of negative solutions. 4.1.3. Notes 1. Krasnosels’kii and Zabreiko [44] were able to improve Theorem 4.2 considerably in the case of potential operators (see also [10]). They assume that X = H is a Hilbert space, A : H → H is a continuous, symmetric linear operator and g(λ, x) = ∇x G(λ, x) = o(x) as x → 0, where G : R × H → R is a C 1 -functional. If λ−1 0 is an isolated eigenvalue of A of ﬁnite multiplicity, then (λ0 , 0) is a bifurcation point for (Id −λA)x − g(λ, x) = 0. In addition there is a ρ0 > 0 such that (i) for each ρ ∈ (0, ρ0 ) at least two different solutions (xi (ρ), λi (ρ)), i = 1, 2, exist with xi = ρ and |λi − λ0 | small, (ii) (λi (ρ), xi (ρ)) → (λ0 , 0) as ρ → 0. The proof relies on variational techniques. It is based on the Lyapunov–Schmidt reduction which provides a general device for constructing solutions in the neighborhood of a bifurcation point. 2. Lyapunov–Schmidt reduction. We shall sketch the idea for the case where λ−1 0 is a simple eigenvalue of the compact linear operator A : X → X. Let N be the one-dimensional eigenspace N = {x: (Id −λ0 A)x = 0}, and let R be the range of Id −λ0 A. Then it follows from the Fredholm alternative that X = N ⊕ R, in the sense that every x can be written uniquely as x = xN + xR where xN and xR lie in N or R, respectively. With P and Q we denote the corresponding projection operators. We seek solutions of (Id −λA)x − g(λ, x) = 0 near (λ0 , 0). Applying successively the operators P and Q we obtain the pair of equations λ 1− xN − P g(λ, xN + xR ) = 0, λ0 (Id −λA)xR − Qg(λ, xN + xR ) = 0. We shall now apply the implicit function theorem (Lemma 3.1) to the second equation K(xR , xN , λ) := (Id −λA)xR − Qg(λ, xN + xR ) = 0. Since K(0, 0, λ) = 0 and since the Fréchet derivative Kx R (0, 0, λ0 ) = Id −λ0 A is invertible on R, there exists a solution xR = xR (xN , λ) for small xN and λ close to λ0 . Introducing this expression into the ﬁrst equation we obtain λ 1− xN − P g λ, xN + xR (xN , λ) = 0. λ0 This equation in the ﬁnite-dimensional space N for xN and λ is often called the bifurcation equation.

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The same reduction procedure applies to arbitrary (not necessarily simple) eigenvalues if N , R are replaced by the generalized nullspace and range. 3. Under additional regularity assumptions it can be shown that the continua emanating from a bifurcation point are continuous curves. The following result is taken from Artino [7]. It is generally attributed to Crandall and Rabinowitz [20]. T HEOREM . Let G(λ, x) be a C k map from a neighborhood of (λ0 , 0) in R × X to X with G(λ, 0) = 0 for all λ. Suppose (i) the null-space N of Gx (λ0 , 0) is one-dimensional spanned by φ, (ii) the range R of Gx (λ0 , 0) has co-dimension 1, (iii) Gλx (λ0 , 0)(1, φ) ∈ / R. k−2 Then there is a C curve Γ intersecting (λ0 , 0) which can be parametrized as follows λ(s), sφ + x(s) with x(0) = 0 and λ(0) = λ0 . / R is called the transversality condition. R EMARKS . (a) The condition Gλx (λ0 , 0)(1, φ) ∈ (b) Further results including conditions for the analyticity of bifurcating branches and their global description can be found in the book of Buffoni and Toland [17]. Let us check in the special case G(λ, x) = (Id −λA) − g(λ, x) with the usual assumption g(λ, x) = o(x) how the conditions (i)–(iii) can be satisﬁed: (i) if λ−1 0 is a simple eigenvalue of A with eigenvalue φ then Gx (λ0 , 0) = Id −λ0 A, (ii) if A is compact then by (i) and the Fredholm alternative the range R of Id −λ0 A has co-dimension 1, (iii) Gλx (λ0 , 0)(1, φ) = −Aφ = −λ−1 / R, since φ belongs to the null-space N . 0 φ∈ 4. Symmetry breaking bifurcation. The techniques of global bifurcation theory can be used to establish continua of symmetric solutions and the existence of points on these continua where symmetry is broken. At those points new continua of unsymmetric solutions bifurcate from the symmetric ones. For a survey on this topic see [40]. 4.2. Bifurcation from inﬁnity The concept of bifurcation from inﬁnity goes back to [44]. It is best explained by an example. Let A : X → X be a compact linear operator such that μ1 = A > 0 is an eigenvalue with eigenvector x1 . Consider problem (Id −λA)x = y. For |λ| < 1/μ1 the the linear 1 n An y. If, for example, y = x , then x = unique solution is given by x = ∞ λ 1 n=0 1−λμ1 x1 . As λ → 1/μ1 the norm of the solution tends to +∞. This phenomenon is called bifurcation from inﬁnity. In the following we study this phenomenon for problems of the type x − F (λ, x) = 0,

(4.8)

where F is asymptotically linear with respect to x. More precisely, we assume that a compact linear operator A : X → X exists such that lim

x→∞

F (λ, x) − λAx =0 x

(4.9)

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59

uniformly with respect to λ in bounded intervals. 4.2.1. Abstract theory D EFINITION 4.8. A point λ0 ∈ R is called a point of bifurcation from inﬁnity for (4.8) if there exists a sequence (λn , xn ) of solutions of (4.8) such that λn → λ0 and xn → ∞ as n → ∞. By the transformation y = x/x2 the original problem (4.8) is transformed into y − y2 F λ, y/y2 = 0.

(4.10)

Condition (4.9) means that λA is the linearization of y2 F (λ, y/y2 ) around y = 0; in particular y = 0 solves (4.10) for every λ ∈ R. Hence, the results of Section 4.1 can be applied to (4.10). Therefore the results for bifurcation from inﬁnity are very similar to those for bifurcation from the trivial solution. Still we present the theory independently, since a number of points will become more clear, cf. Section 4.2.2. T HEOREM 4.9. Let F : R × X → X be compact with respect to (λ, x) and assume (4.9) holds for a compact linear operator A. (i) Necessary condition: if λ0 is a point of bifurcation from inﬁnity for (4.8) then λ−1 0 is an eigenvalue of A. (ii) Sufﬁcient condition: if λ−1 0 is an eigenvalue of A of odd algebraic multiplicity then λ0 is a point of bifurcation from inﬁnity for (4.8). P ROOF. We set G(λ, x) = F (λ, x) − λAx. (i) Let (λn , xn ) be a sequence of solutions of (4.8) with λn → λ0 and xn → ∞. Deﬁne yn = xn /xn . Then yn − λn Ayn =

G(λn , xn ) →0 xn

as n → ∞.

By the compactness of A there is a subsequence again denoted by yn with yn → y and y = 1 such that y − λ0 Ay = 0. This shows that λ−1 0 is an eigenvalue of A. > 0; a similar proof works if λ−1 (ii) Suppose λ−1 0 0 < 0. For δ > 0 sufﬁciently small −1 there is no further eigenvalue of A in ((λ0 + δ) , (λ0 − δ)−1 ). Fix a value λ ∈ (λ0 − δ, λ0 + δ) with λ = λ0 . We claim that there exists a radius R(λ) > 0 such that for all R R(λ) deg Id −F (λ, ·), BR (0), 0 = (−1)β ,

where β =

m(μ).

μ>λ−1

To see this notice that for large R and t ∈ [0, 1] there is no solution of x − F (λ, x) + tG(λ, x) = 0

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on ∂BR (0), since otherwise the exists a sequence (tn , xn ) such that tn → t0 and xn → ∞ such that xn − F (λ, xn ) + tn G(λ, xn ) = 0 and by considering yn = xn /xn this implies that yn − λAyn → 0. Using the compactness of A one obtains a nontrivial solution of y − λAy = 0, which is impossible. Thus, deg(Id −F (λ, ·) + tG(λ, ·), BR (0), 0) is well deﬁned and homotopy invariant in t ∈ [0, 1]. This implies deg Id −F (λ, ·), BR (0), 0 = deg Id −λA, BR (0), 0 m(μ) = (−1)β , where β = μ>λ−1

as claimed. Now suppose for contradiction that there is no bifurcation from inﬁnity for (4.8). Then for λ ∈ (λ0 − δ, λ0 + δ) and R sufﬁciently large, deg(Id −F (λ, ·), BR (0), 0) is well deﬁned and homotopy invariant in λ. However, deg Id −F (λ0 − δ, ·), BR (0), 0 deg Id −F (λ0 + δ, ·), BR (0), 0 = −1 since λ−1 0 has odd algebraic multiplicity. This contradiction shows that λ0 is a point of bifurcation from inﬁnity. R EMARK . Let A : H → H be a continuous, symmetric linear operator on a Hilbert space H and let G(λ, x) = ∇x G(λ, x) such that G(λ, x)/x → 0 as x → ∞ uniformly for λ in compact intervals. If λ−1 0 is an isolated eigenvalue of A of ﬁnite multiplicity, then λ0 is a point of bifurcation from inﬁnity for (Id −λA)x − G(λ, x) = 0. The next result is a version of Theorem 4.9 with continuous solution branches bifurcating from inﬁnity. For the full analogue of Theorem 4.5, which is more delicate to state, see [60]. C OROLLARY 4.10. If λ−1 0 is an eigenvalue of A of odd algebraic multiplicity then there exists a continuum C of solutions of (4.8) bifurcating from inﬁnity at λ0 . 4.2.2. Applications. Consider the semilinear boundary value problem Lu + λf (x, u) + k(x) = 0 in D,

u=0

on ∂D,

(4.11)

and f (x, s) is α-Hölder continuous in x ∈ D locally uniformly in s ∈ R where k ∈ C α (D) with the condition of asymptotic and locally Lipschitz continuous in s uniformly in x ∈ D linearity lim

s→±∞

f (x, s) =1 s

uniformly for x ∈ D.

(AL)

T HEOREM 4.11. If f (x, s) satisﬁes (AL) then at every eigenvalue of L of odd algebraic which bifurcates from inﬁnity. Near the multiplicity there is a continuum C ⊂ R × C 1 (D) ﬁrst eigenvalue λ1 the solutions on C have one sign.

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61

P ROOF. Problem (4.11) is equivalent to . u + L−1 λf (x, u) + k(x) = 0 in C 1 D

(4.12)

=−F (λ,u)

It satisﬁes the asymptotic linearity conditions of the previous section. Hence Corollary 4.10 applies. If the bifurcation happens near λ1 then consider any sequence (λn , un ) ∈ C of solutions such that λn → λ1 and un → ∞. It is then easy to see that for a subsequence Hence, for large n, the function un has one sign. By the connectun /un → φ1 in C 1 (D). edness of C it follows that for large n either all un are positive or all un are negative. As a variant of (AL) we also consider the condition of asymptotic half-linearity lim

s→±∞

f (x, s) = 1 uniformly for x ∈ D. |s|

(AHL)

Such functions do not lead to asymptotically linear problems as in the previous section. Instead we have f (x, s) = |s| + g(x, s), where g(x, s)/s → 0 as |s| → ∞. If we deﬁne the half-linear operator A(u) = −L−1 |u| then (4.12) becomes , u − λA(u) − G(λ, u) = 0 in C 1 D where G(λ, u)/u → 0 as u → ∞. In order to have bifurcation from inﬁnity for such problems we ﬁrst show the following basic formula for the change of the degree for the half-linear operator A(u). L EMMA 4.12. Let A(u) = −L−1 |u|. For every R > 0 and δ > 0 sufﬁciently small, deg Id −λA, BR (0), 0 =

"

1

if 0 < λ < λ1 ,

0

if λ1 < λ < λ1 + δ.

R EMARK . The formula differs in an essential way from the corresponding one for linear Id +λL−1 since there the degree changes from 1 to −1 as λ passes through λ1 . P ROOF OF L EMMA 4.12. Let 0 < λ < λ1 . We use u+ = max{u, 0} and u− = min{u, 0}. Consider the homotopy At (u) = −L−1 (u+ + tu− ) for t ∈ [−1, 1]. Since the Lipschitz constant of the nonlinearity λ(u+ + tu− ) is strictly less than λ1 for all t ∈ [−1, 1] the maximum principle implies that (Id −λAt )u = 0 has only the trivial solution. Hence deg(Id −λAt , BR (0), 0) is well deﬁned. Homotopy invariance in t implies deg(Id − λA, BR (0), 0) = deg(Id +λL−1 , BR (0), 0) = 1. For λ1 < λ < λ√ 1 + δ we argue differently. We approximate the function |s| by the smooth function hε (s) = s 2 + ε2 . Consider the problem Lu + λhε (u) = 0 in D,

u = 0 on ∂D.

(4.13)

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Any solution of (4.13) has to be positive, and since any positive multiple of the ﬁrst eigenfunction φ1 > 0 is a subsolution it is easy to see that (4.13) has no solution. Moreover, and since the degree since A(u) + L−1 hε (u) → 0 as ε → 0 uniformly for u ∈ C 1 (D) deg(Id −λA, BR (0), 0) is well deﬁned, we ﬁnd that for small ε deg Id −λA, BR (0), 0 = deg Id +λL−1 hε , BR (0), 0 = 0.

This shows the claim.

T HEOREM 4.13. If f (x, s) (−f (x, s)) satisﬁes (AHL) then λ1 is a point of bifurcation from inﬁnity. The bifurcating continuum consists of positive (negative) solutions near λ1 . P ROOF. Suppose f (x, s) satisﬁes (AHL). We follow the proof of Theorem 4.9. Since Lu+ λ|u| = 0 in D with zero boundary data has no solution if λ ∈ (0, λ1 + δ) \ {λ1 } one can show that for large R deg Id −F (λ, ·) + tG(λ, ·), BR (0), 0 is well deﬁned and homotopy invariant in t ∈ [0, 1] and thus deg Id −F (λ, ·), BR (0), 0 = deg Id −λA, BR (0), 0 . By Lemma 4.12 the degree changes as λ passes through λ1 . Hence λ1 is a point of bifurcation from inﬁnity, and a continuum C bifurcating from inﬁnity exists. For a sequence (λn , un ) ∈ C of solutions with λn → λ1 and un → ∞ one can see that a subsequence vn = un /un converges to a nontrivial solution of Lv + λ1 |v| = 0 in D with v = 0 on ∂D. Clearly v is a positive multiple of φ1 and hence for large n, the function un is positive. E XAMPLE . Consider the problem Lu + λu + g(u) = 0,

u > 0 in D, u = 0 on ∂D,

where g(s) 0 and g(s)/s → 0 as s → ∞ and s → 0. We may assume that g(s) = 0 for s 0. By the results of Section 4.1 at λ = λ1 there is a continuum C1 of positive solutions bifurcating from 0. By Theorem 4.13 a continuum C2 of positive solution bifurcates from inﬁnity at λ = λ1 . Moreover, the assumption g 0 implies that both continua extend to the left of λ1 . Finally, for λ 0 there is no positive solution. Hence C1 has to become unbounded in the Banach space direction and C2 has to connect to the trivial solution at λ1 . By the local uniqueness of the bifurcating branch near λ1 we obtain C1 = C2 . As a consequence there exists δ > 0 such that for λ ∈ (λ1 − δ, λ1 ) there are at least two positive solutions, cf. Figure 16. The next result investigates conditions for at least three solutions as in Figure 17. It is taken from the lecture notes of Schmitt [66].

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Fig. 16. Bifurcation from 0 and from inﬁnity.

Fig. 17. Three solutions.

P ROPOSITION 4.14. Suppose that f (x, s) satisﬁes (AL) and assume that the solutions of (4.11) are a priori bounded uniformly for λ in compact intervals [−K, λ1 ]. Then there exists δ > 0 and three different solution continua C, C + , C − with the following properties: (a) for every λ < λ1 + δ there is a solution on C, bifurcates from inﬁnity at λ1 with positive/negative so(b) C ± ⊂ (λ1 , λ1 + δ) × C 1 (D) lutions. P ROOF. Deﬁne f˜(x, s) := f (x, |s|) and f¯(x, s) := f (x, −|s|). The functions f˜(x, s) and −f¯(x, s) satisfy (AHL). By Theorem 4.13 there are solution continua C + , C − for (4.11) with f (x, s) replaced by f˜(x, s), f¯(x, s), respectively. Since the solutions are positive, negative for λ near λ1 we obtain that C + and C − are solution continua for the original problem (4.11) for λ ∈ (λ1 , λ1 + δ). The third continuum C exists, since for λ = 0 there is a unique solution of index +1, which is continued for λ < 0 and λ > 0 by the global continuation result of Theorem 3.3. To the right and left of λ = 0 the continuum C has to

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be unbounded since it cannot turn back to λ = 0. By the assumption of boundedness of the solutions for λ λ1 the branch C must exist for values of λ ∈ (−∞, λ1 + δ). The previous proposition prepares the following result of the existence of three solutions. It is again taken essentially from [66]. −1 Let μ = λ−1 1 be the largest eigenvalue of −L . Consider the splitting of the space 1 −1 = N ⊕ R, where N = N(μ Id +L ) is the nullspace and R = R(μ Id +L−1 ) the C (D) range. Then μ Id +L−1 : R → R is bijective. × T HEOREM 4.15 (Existence of three solutions). Let f (x, s) = s + g(x, s) where g : D R → R is bounded and suppose, moreover, that g(x, s)s < 0 for s = 0 and all x ∈ D. −1 + − Assume that k ∈ R(μ Id +L ). Then three solution continua C, C , C as in Proposition 4.14 exist, cf. Figure 17. P ROOF. In order to apply Proposition 4.14 we need to show that the solutions of (4.11) are a priori bounded uniformly in λ λ1 for λ in compact intervals. According to the splitting of the space we write u = tφ1 + w with 0 < φ1 ∈ N(μ Id +L−1 ) and w ∈ R(μ Id +L−1 ). Let P : X → N, Q : X → R be the two projectors from X to null-space, range of μ Id +L−1 , respectively. Recall that L−1 : R → R. For later use let C = {u ∈ u 0} be the cone of nonnegative functions. Notice that R ∩ C = {0} since othC 1 (D): erwise by the Krein–Rutman theorem the strongly positive operator L−1 : R → R would have a ﬁrst eigenvalue with eigenvector in C. This is impossible. Hence R ∩ C = {0} and as a consequence, if u = tφ1 + w ∈ C \ {0} then necessarily t > 0. This shows that P maps C into itself. With the help of the two projectors (4.11) is equivalent to (4.14) w + λL−1 w + QL−1 λg(x, tφ1 + w) + k = 0, λ (4.15) + P L−1 λg(x, tφ1 + w) + k = 0. tφ1 1 − λ1 From (4.14) and the boundedness of the function g it follows that for λ ∈ [λ1 − K, λ1 + δ] there exists a bound M = M(K, δ) such that w M for any solution u = tφ1 + w. Now consider (4.15). Since L−1 k ∈ R the equation simpliﬁes to λ + λP L−1 g(x, tφ1 + w) = 0. tφ1 1 − (4.16) λ1 If λ is bounded away from λ1 then the boundedness of g implies the boundedness of t in (4.16). It remains to see what happens for λ ∈ [0, λ1 ]. Suppose tφ1 + w is a solution where t 1 is large. Then tφ1 + w > 0, g(x, tφ1 + w) < 0 and L−1 g(x, tφ + w) > 0. u 0} of nonnegative functions into itself we Since P maps the cone C = {u ∈ C 1 (D): get P L−1 g(x, tφ1 +w) > 0. Together with the λ ∈ [0, λ1 ] this contradicts (4.16). The same contradiction happens if tφ1 + w is a solution where t −1 is small. Altogether we ﬁnd that t is bounded if λ λ1 is in compact intervals. Together with the bound for w we obtain the desired a priori bound for u. Then we can apply Proposition 4.14.

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4.3. Perturbations at resonance This section deals with problems of the form Lu + λk u + g(x, u) = h(x)

in D,

u = 0 on ∂D,

(4.17)

where λk is a simple eigenvalue of L and the data satisfy the usual smoothness assump and g(x, s) is α-Hölder continuous in x ∈ D uniformly w.r.t. s in tions, h ∈ C α (D) bounded intervals and locally Lipschitz continuous in s uniformly w.r.t. x ∈ D. We present two prototypes of problems to which degree theory applies. The ﬁrst concerns bounded perturbations g and was ﬁrst studied by Landesman and Lazer [46] and the second deals with unbounded nonlinearities g and was ﬁrst considered by Ambrosetti and Prodi [6]. × R → R be bounded. Assume that the 4.3.1. Landesman–Lazer type result. Let g : D limiting functions g±∞ (x) := lims→±∞ g(x, s) exist and that the convergence is uniform This implies that the limit functions belong to L∞ (D). Denote by φ the eigenin x ∈ D. function corresponding to the eigenvalue λk . In addition suppose that the following integral inequality holds

D

+

g−∞ φ + g∞ φ

−

hφ dx

0 such that whenever u = tφ + w solves (4.18) then w M1 uniformly for s ∈ [0, 1]. Applying the projector Q : X → span[φ] yields

g(x, tφ + w)φ dx = s

s

hφ dx − (1 − s)t

D

D

φ 2 dx.

(4.19)

D

We will show that from here the boundedness of t follows. To see this we distinguish between two cases. (a) 0 s

1 2

and (b)

1 s 1. 2

In case (a) the boundedness of t follows immediately from (4.19) and the boundedness of g. In case (b) the Landesman–Lazer condition (LL) comes into play. Put Dδ+ = {x ∈ D: φ δ > 0} and Dδ− = {x ∈ D: φ −δ < 0}. On Dδ+ we have g(x, tφ + w)φ → g±∞ (x)φ as t → ±∞. Similar expressions hold on Dδ− . Hence

g(x, tφ + w)φ dx → D

D

g±∞ φ + + g∓∞ φ − dx + O(δ)

as t → ±∞.

This statement together with the Landesman–Lazer condition (LL) contradict (4.19) if |t| is too large and δ > 0 is chosen sufﬁciently small. Hence whenever u = tφ + w solves (4.18) then |t| M2 uniformly for s ∈ [0, 1]. This proves that a priori the solutions u of (4.18) lie inside a large ball BR (0) for all s ∈ [0, 1]. For large R we have by the homotopy invariance with respect to s ∈ [0, 1] that deg Id +λk L−1 + L−1 g(x, ·) − h , BR (0), 0 = deg Id +λk L−1 + L−1 Q, BR (0), 0 = 0, where the latter degree is nonzero, because the operator Id +λk L−1 + L−1 Q is injective and hence bijetive in L2 (D). Hence (4.17) has a solution. R EMARKS . (1) Landesman and Lazer [46] observed that (LL) with weak inequalities () is also necessary for the existence of a solution of (4.17) provided g−∞ (x) g(x, s) g∞ (x) for all x ∈ D and s ∈ R. (2) Theorem 4.16 holds if both inequalities in (LL) are reversed. For λ = λ1 such a situation was considered in Theorem 4.15. 4.3.2. Ambrosetti–Prodi type results. Consider the problem (4.17) with λk = λ1 and and g(x, s) is subject h(x) = f (x) − tr(x), where f and r 0, r ≡ 0, are in C α (D) to the conditions lim inf s→−∞

g(x, s) 0 s

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67

The following theorem is due to Hess [36]. A survey of uniformly with respect to x ∈ D. similar results is given by Lazer and McKenna [47]. T HEOREM 4.17. Under the above assumptions and if in addition there exists a positive number γ such that g(x, s) + λ1 s γ 1 + |s| ∀x ∈ D, s 0. Then there is a number t0 ∈ R such that (4.17) has no solution for all t > t0 and at least one solution for all t < t0 . P ROOF. The boundary value problem is equivalent to Lγ u + gγ (x, u) = h(x)

in D,

u=0

on ∂D,

(4.20)

where Lγ := L − γ and gγ (x, s) := g(x, s) + (γ + λ1 )s. The two main steps of the proof follow from clever a priori bounds which are based on the speciﬁc assumptions on g. More details are found in [36]. C LAIM 1. For given R > 0 there exists a value T = T (R) such that v + τ L−1 γ (gγ (·, v) − f + tr) = 0 for all v with v + = R, τ ∈ [0, 1] and t T (R). C LAIM 2. For ﬁxed t ∈ R there is ρ > 0 such that v + τ L−1 γ (gγ (·, v) − f + tr) = 0 for − all v with v = ρ, τ ∈ [0, 1]. Since for Consider next the open set SR,ρ := {v ∈ X: v + < R, v − < ρ} in C(D). −1 given t T (R), v +τ Lγ (gγ (·, v)−f +tr) = 0 for v ∈ ∂SR,ρ the Leray–Schauder degree is well deﬁned. By the homotopy invariance w.r.t. τ we conclude that deg Id +L−1 γ gγ (·, v) − f + tr , SR,ρ , 0 = deg(Id, SR,ρ , 0) = 1. This establishes the existence of a solution for all t T (R). Then it is shown by means of sub- and supersolutions that there exist also solutions in (t − ε, t + ε). Thus t0 := sup{t ∈ R: ∃ solution of (4.20)} is the number asserted in the theorem. The ﬁniteness of t0 follows from the asymptotic behavior of g. In fact let ψ > 0 be the eigenfunction corresponding to the principal eigenvalue λ1 of the adjoint boundary value problem L∗ ψ + λ1 ψ = 0 in D, ψ = 0 on ∂D and denote by ·, ·! the inner product in L2 . Multiplying (4.17) with ψ and integrating, we obtain %

& g(·, u), ψ = f, ψ! − t r, ψ!.

× R and since r, ψ! is positive the claim Since g(x, s) is bounded below on D follows.

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Acknowledgments We thank J. Horák, V. R˘adulescu and S. Stingelin for suggesting improvements of the text. We are particularly indebted to the referee who red the text very carefully, corrected many ﬂaws and helped us to eliminate some of the most egregious errors.

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CHAPTER 2

Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain Giovanni P. Galdi Department of Mechanical Engineering, University of Pittsburgh, 15261 Pittsburgh, USA E-mail: [email protected] Dedicated to Professor Salvatore Rionero on the occasion of his 70th birthday

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Analysis of some linearized problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The Stokes approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Some applications of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. The Oseen approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. The Oseen approximation in the limit of vanishing Reynolds number . . . . . . . . . . . . . . . . 1.5. A variant to the Oseen approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The nonlinear problem: Unique solvability for small Reynolds number and related results . . . . . . . 2.1. Unique solvability at small Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Limit of vanishing Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Perturbation theory at ﬁnite Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The nonlinear problem: On the solvability for arbitrary Reynolds number . . . . . . . . . . . . . . . . 3.1. Existence: Leray method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Existence: Fujita method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Some existence results when ξ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. On the pointwise asymptotic behavior of D-solutions . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Asymptotic structure of D-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The nonlinear problem: On the existence of symmetric solutions for arbitrary large Reynolds number 4.1. A remark about symmetric solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. A key result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Existence of symmetric solutions for arbitrary large Reynolds number . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF DIFFERENTIAL EQUATIONS Stationary Partial Differential Equations, volume 1 Edited by M. Chipot and P. Quittner © 2004 Elsevier B.V. All rights reserved 71

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Introduction As is well known, one of the most representative and fascinating issues in mathematical ﬂuid dynamics is the steady-state, plane, exterior boundary-value problem associated to the Navier–Stokes equations. The problem, in its classical formulation, consists in ﬁnding a vector function v = (v1 , v2 ) and a scalar function p, depending only on x = (x1 , x2 ) and satisfying the following system of equations v = λv · ∇v + ∇p div v = 0 v|∂Ω = v∗ ,

' in Ω, (1)

lim v(x) = ξ.

|x|→∞

In (1), Ω is the exterior of a two-dimensional compact, connected set B, ξ = (ξ1 , ξ2 ) is a ﬁxed constant vector, v∗ = (v∗1 , v∗2 ) is a prescribed vector function at the boundary ∂Ω of Ω, and λ is a given nonnegative real number. From the physical point of view, problem (1) is related to the stationary motion of a viscous, incompressible ﬂuid around a long, straight cylinder C, assuming that the ﬂuid is at rest at large distances from C. The vector −ξ is the (possibly zero) translational velocity of C, supposed to be orthogonal to its axis a. In a region of ﬂow sufﬁciently far from the two ends of C and including C, one can expect that the velocity ﬁeld of the ﬂuid is independent of the coordinate parallel to a and, moreover, that there is no motion in the direction of a. Under these hypotheses, the corresponding mathematical problem becomes two-dimensional and is described by (1), where v, scaled by a characteristic velocity V , is the dimensionless Eulerian velocity of the particle of the ﬂuid, p is the associated pressure ﬁeld, and Ω (the complement of B) is the relevant region of ﬂow. Moreover, v∗ represents the (dimensionless) velocity of the ﬂuid at the boundary of B (≡ ∂Ω). Finally, λ is a dimensionless parameter, the Reynolds number, that can be written as V d/ν, where d is a length scale (typically, the diameter of B), and ν (> 0) is the coefﬁcient of kinematical viscosity of the ﬂuid. The case v∗ = 0 and ξ = 0 in (1) deserves special attention. In fact, it describes the signiﬁcant physical situation of when the cylinder has impermeable, immobile walls and translates into the ﬂuid with constant velocity −ξ . Actually, it was just this problem that, in 1851, received the ﬁrst mathematical treatment ever, in the pioneering work of Sir George Gabriel Stokes on the motion of a pendulum in a viscous liquid [42]. In particular, in the wake of his successful study of the motion of a sphere in a viscous ﬂuid, Stokes looked for solutions to (1) with v∗ = 0 in the limiting case when the viscosity of the ﬂuid is much larger that the quantity V d, where V is taken as the magnitude of the velocity of the cylinder. This amounts to take λ = 0 in (1)1 and get a corresponding linearized problem that is nowadays called Stokes approximation. However, to his surprise, Stokes found that

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G.P. Galdi

this linearized problem, even in the simplest case when B is a circle, has no solution, and he concluded with the following statement [42], p. 63, It appears that the supposition of steady motion is inadmissible.

Such an observation constitutes what we currently refer to as Stokes paradox. This is deﬁnitely a very intriguing starting point for the mathematician who is interested in the resolution of the boundary-value problem (1). In fact, it appears that, if the problem admits a solution, the nonlinear term λv · ∇v has to play a key role in its determination. This latter fact was recognized, and made quantitative, by C.W. Oseen, more than half a century later, in 1910, in his fundamental paper [37]; see also [38], §15 and Chapter III. Speciﬁcally, Oseen proposed another approximation that takes into account, somehow, the nonlinear term by replacing it with λξ · ∇v. This procedure leads to a corresponding linearization of problem (1), called the Oseen approximation. The well-posedness of the boundary-value problem corresponding to the Oseen approximation was proved, in its full generality, almost one decade later by Faxén [13]. The ﬁrst mathematical study of the full nonlinear problem (1) in its complete generality is due to J. Leray in 1933 [33]. Actually, in [33] Leray investigated also the threedimensional counterpart of (1). By using topological degree theory (Leray–Schauder theorem) in conjunction with an a priori estimate for all possible solutions (in a given functional class) to (1) (see (2) below), Leray was able to show that for any λ there exists a smooth pair v, p that satisﬁes (1)1,2,3, provided v∗ and Ω are regular enough and the total ﬂux of v∗ through ∂Ω is zero, namely, ∂Ω

v∗ · n = 0,

where n is the unit outer normal to ∂Ω. The important question that Leray could not answer was whether or not the velocity v satisﬁes the prescription at inﬁnity (1)4 . Notice that this lack of information only appeared in the two-dimensional problem, while in the three-dimensional case he was able to show the validity of (1)1 uniformly pointwise, if ξ = 0 and in a generalized sense, if ξ = 0. The discrepancy between the two- and threedimensional cases is due to the following reason. The solution constructed by Leray veriﬁes the condition Ω

|∇v|2 M ≡ M(ξ, v∗ , Ω) < ∞.

Now, if Ω ⊂ R3 , Leray proved the following inequality [33], p. 47,

|v(x) − ξ |2 Ω

|x|2

|∇v|2

4 Ω

(2)

Stationary Navier–Stokes problem in a two-dimensional exterior domain

75

so that the method of construction he used, and (2) imply

|v(x) − ξ |2 Ω

|x|2

4M,

(3)

which, in turn, furnishes (1)4 in a generalized sense.1 If Ω ⊂ R2 , we only have the weaker inequality [33], pp. 54–55,

|v(x) − ξ |2 Ω

|x|2 log2 |x|

4

|∇v|2 .

(4)

Ω

Now, it is not hard to bring examples of plane solenoidal ﬁelds satisfying (4) (or (2)), vanishing at ∂Ω and growing as a power of log |x|, for sufﬁciently large |x|. As a consequence, in order to show the (possible) validity of (1)4 , in the two-dimensional case, the equations of motion (1)1,2 must play a fundamental role. It is worth emphasizing that Leray’s method applies to both situations ξ = 0 and ξ = 0 as well, furnishing the same partial answer in either case. Analogous conclusions were reached almost thirty years later, in 1961, by Fujita [17], who found essentially the same results as Leray’s, by a different method of constructing solutions, the so-called Galerkin method. The drawback of both Leray’s and Fujita’s solutions can be summarized by saying that the only information available on the asymptotic behavior of the solution, is that the velocity ﬁeld v has a ﬁnite Dirichlet integral which, by what we noticed, does not even ensure the boundedness of v. These solutions are called D-solutions. The above interlocutory results left open the worrisome possibility that a Stokes paradox could also hold for the fully nonlinear problem (1). If this chance turns out to be indeed true, it might cast serious doubts on the reliability of the Navier–Stokes ﬂuid model, in that it would not be able to catch the physics of such an elementary phenomenon. The possibility of a nonlinear Stokes paradox, was ruled out by R. Finn and D. Smith in a profound paper appeared in 1967 [16], where they show that if ξ = 0, and Ω and v∗ are sufﬁciently regular, then (1) has a solution, at least for “small” λ. Moreover, these solutions are Physically Reasonable, in the sense that they meet all the basic physical requirements, such as energy equation, and they show the presence of a wake in the direction ξ , opposite to the direction of the velocity of the cylinder. Finally, the solutions are unique in a ball (of “small” radius) of a suitable Banach space. The methods used by Finn and Smith are completely different than Leray’s, and are based on very accurate and detailed estimates of the Green’s tensor and of the fundamental solution associated to the Oseen approximation. Another approach to the problem addressed by Finn and Smith was given, more recently, by Galdi [18]. The approach, based on a suitable Lq -theory of the Oseen approximation, is very ﬂexible and in fact allowed the resolution of other, more complicated problems, such as the analogous boundary-value problem (1) for the case of a compressible (densityvarying) viscous ﬂuid [25]. 1 If ξ = 0, Leray further showed that (3) eventually implies (1) uniformly pointwise. 4

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G.P. Galdi

Though signiﬁcant, these results leave open several fundamental questions. The most important is, of course, that of whether problem (1) has a solution for all λ > 0, as it happens, in fact, in the analogous three-dimensional case. Another important question regards the solvability of (1) when ξ = 0. In such a circumstance, no result whatsoever is available, other than the incomplete one obtained by Leray and Fujita that we described before. As we already noticed, the solvability of problem (1) for arbitrary λ > 0 would be established, the moment we could show that the velocity ﬁeld of solutions constructed by Leray or Fujita (D-solutions) does tend to the prescribed value ξ . The question of the asymptotic behavior of the velocity ﬁeld of a D-solution was taken up in a series of remarkable papers by Gilbarg and Weinberger ﬁrst [29,30], and then by Amick [2,3]. Speciﬁcally, in the case when v∗ = 0, the above authors showed that (i) every solution to (1)1,2,3 that satisﬁes (2) is uniformly pointwise bounded; and (ii) for every solution to (1)1,2,3 that satisﬁes (2), there exists some vector ξ˜ such that the average over the angle of |v − ξ˜ |2 tends to 0 as |x| → ∞. If, in particular, B is symmetric around the direction of ξ (= 0), one can show that v tends to ξ˜ uniformly pointwise [2,22]. Notice that, in general, no information is available about ξ˜ . Actually, ξ˜ can, in principle, be zero, even though ξ = 0. Therefore, another spontaneous question arises: Does the vector ξ˜ coincide with the prescribed vector ξ ? Even though it is very probable that the answer is in the positive, at the present time, no answer is known. Of course, had this question have an afﬁrmative answer for all λ, problem (1) would admit a solution for all λ as well. Very recently, when B is symmetric around the direction of ξ = 0, Galdi has proved the following result concerning the solvability of (1) for arbitrary large λ, in the class of symmetric solutions [22]. Assuming ξ directed along the x1 -axis, such solutions (v = (v1 , v2 ), p), have v1 and p even in x2 and v2 odd in x2 . Let us denote by (1)0 problem (1) with ξ ≡ v∗ ≡ 0, and by C the class of symmetric solutions (v, p) to (1)0 with v having a ﬁnite Dirichlet integral. (Notice that, in the class C the velocity ﬁeld v tends uniformly pointwise to zero at inﬁnity.) It is clear that C contains the trivial solution v = 0, p = const. Now if the trivial solution is the only solution in C, then the set of λ for which (1) is solvable in the class of symmetric solutions corresponding to a prescribed ξ (= 0) and v∗ = 0 contains an unbounded set, M0 , of the positive real line. This result leaves open two interesting lines of investigation. The ﬁrst, and more important, is the proof of the validity of its assumption, and, the second, is the study of the property of the set M0 . Unfortunately, to date, no result is available in either direction. Objective of the present chapter is to furnish a complete, consistent and, as far as possible, self-contained, presentation of the state of the art of the unique solvability of problem (1). We shall also give some new results and point out several open questions, and, whenever is the case, we suggest possible ways of resolution.

Outline of the chapter The chapter is divided into four sections. Section 1 is dedicated to the mathematical analysis of some linearized versions of problem (1), including the Stokes and the Oseen approximations, while Sections 2–4 concern the full nonlinear problem.

Stationary Navier–Stokes problem in a two-dimensional exterior domain

77

Speciﬁcally, in Section 1.1, we study the well-posedness of the boundary-value problem, (1)S say, to which (1) reduces by taking formally λ = 0. Even though, as we noticed before, (1)S does not have a solution for an arbitrary choice of ξ and v∗ , nevertheless it is also known that there are physically interesting situation where (1)S furnishes results in a reasonable agreement with the experience. Our main objective is to investigate under which conditions on v∗ and ξ , problem (1)S admits one and only one regular solution. This objective is accomplished by showing that (1)S is uniquely solvable if and only if ξ and v∗ satisfy a “compatibility condition” (see (1.12)). In the case when B is a circle of radius one, this condition takes the following simple form 1 ξ= 2π

∂Ω

v∗ .

Applications of this result are furnished in Section 1.2. A noteworthy application is that given to the self-propelled motion of micro-organisms like Ciliata. In a commonly accepted model of Ciliata, the layer model, the motion of minuscule hair-like organelles (cilia) placed on the surface of the main body of the animal produces a distribution of velocity, which serves as a propeller [8–10,24]. Due to the small velocity and to the microscopic size of the micro-organism, the typical Reynolds number involved is λ ∼ 10−3 , and, therefore, the Stokes approximation is applicable. By using the results of Section 1.1, we give necessary and sufﬁcient conditions for self-propulsion, that contain, as a particular case, those furnished by other authors by different methods [8,9]. Section 1.3 is dedicated to the study of the basic mathematical properties of the Oseen approximation. As mentioned before, the associated boundary-value problem, (1)O , say, is obtained from (1) by replacing the nonlinear term λv · ∇v with λξ · ∇v. In particular, we present results of existence, uniqueness and corresponding estimates of solutions to (1)O . We also furnish results on the asymptotic behavior that show, among other things, the existence of a “wake” in the direction of ξ . In Section 1.4, we study the behavior of solutions to the Oseen problem as λ → 0 and show that they tend to solutions to the Stokes problem corresponding to the same data, if and only if the data satisfy the compatibility condition determined in Section 1.1. In the last subsection of Section 1, Section 1.5, we study the functional–analytic properties of a problem obtained by perturbing the Oseen problem by a suitable linear operator, furnishing, in particular, sufﬁcient conditions for the existence and uniqueness of solutions to the perturbed problem. In Section 2 we begin the study of the unique solvability of the full nonlinear problem (1) at “small” Reynolds number λ. Speciﬁcally, in Section 2.1, by using the results proved in Section 1.3, we show that, there is λ0 > 0 such that, for any 0 < λ < λ0 , (1) has at least one solution in a suitable Banach space, B say. Moreover, the solution is locally unique, in the sense that it is the only one within a ball of B of appropriately “small” radius. Whether or not these solutions are unique “in the large” remains an open question. This circumstance has an undesired consequence. In fact, even though solutions determined here and the solutions constructed by Finn and Smith [16] belong to the same functional class, we can not conclude that (for small λ) they coincide. In Section 2.2, we analyze the behavior of solutions previously found in the limit of λ → 0, and show that they tend to the solutions of the corresponding Stokes problem if and only if the data satisfy the compatibility

78

G.P. Galdi

conditions established in Section 1.1. An interesting issue obtained as a by-product of this result is that at small, nonzero Reynolds number the force exerted by the ﬂuid on B is independent of the shape of B, a fact ﬁrst discovered by Finn and Smith [16], and, more recently, reconsidered, with a completely different approach, in [43–45]. In Section 2.3 we are interested in the construction of a solution to (1) (v, p), corresponding to λ, in a neighborhood of another solution (v0 , p0 ), corresponding to λ0 . Using the results of Section 1.5, we give sufﬁcient conditions for the existence of (v, p) and show that, under these conditions, (v, p) is analytic in |λ − λ0 |. The remaining Sections 3 and 4 are dedicated to the existence of solutions to (1) for arbitrary Reynolds numbers. In the ﬁrst two Sections 3.1 and 3.2 we brieﬂy review the methods of construction of Leray [33] and that of Fujita [17] which provide existence of solutions to (1)1,2,3 with v having a ﬁnite Dirichlet integral (D-solutions), for any value of λ. The drawback with these solutions is two fold. On the one hand, the lack of information about the behavior of v at large distance and, as a consequence, the impossibility of checking the validity of condition (1)4 . On the other hand, in the case when v∗ ≡ 0, it is not excluded that such solutions are identically zero. Before investigating these two questions, in Section 3.3 we prove some existence results for problem (1) when ξ = 0. In particular, we show that if B has two orthogonal directions of symmetry and if the data satisfy suitable parity conditions, then, for any λ > 0, (1) has at least one D-solution. Notice that, unlike the case ξ = 0, in such a case we show that v satisﬁes also (1)4 . The remaining two subsections of Section 3 are devoted to the study of the asymptotic behavior of D-solutions. Speciﬁcally, in Section 3.4, we describe the results of Gilbarg and Weinberger [29,30], and of Amick [2,3], and show that, if v∗ = 0, for any D-solution there exists a vector ξ˜ such that lim

r→∞ 0

2π

v(r, θ ) − ξ˜ 2 dθ = 0.

Moreover, we also prove that, in fact, v tends to ξ˜ uniformly pointwise, a property known, so far in the literature, only for symmetric solutions [3]. Our proof is very simple and it is based on the vorticity equation in conjunction with a suitable pointwise estimate of the Sobolev type. However, since we do not know whether or not ξ˜ = ξ , the big question that remains open is whether or not (1) is solvable for arbitrary large λ. We shall return on this problem in Section 4. The asymptotic structure of D-solutions whose velocity ﬁeld v tends, uniformly pointwise, to a nonzero vector, ξ˜ , is the object of Section 3.5. Following the approach of Galdi and Sohr [28], ﬁnalized by the very recent results of Sazonov [39], we shall show that every such a solution is “physically reasonable” in the sense of Finn and Smith, and so, in particular, the velocity ﬁeld and the pressure ﬁeld behave, roughly speaking, as velocity and pressure ﬁelds of the corresponding Oseen problem. In this respect, we emphasize that, if ξ˜ = 0, the rate of decay of the velocity ﬁeld v of a D-solution is, in general, not predictable. Actually, as seen by means of counter-examples (see (2.5)), in general v is not representable, at large |x|, in terms of negative powers of |x|. The last Section 4 is based on the work of Galdi [22], and it is aimed at furnishing sufﬁcient conditions for the existence of symmetric, “physically reasonable” solutions to (1) in the case when B is symmetric around one direction. As explained previously, the basic

Stationary Navier–Stokes problem in a two-dimensional exterior domain

79

assumption, (H ) say, is that problem (1) with v∗ = ξ = 0 has only the trivial solution in the class of symmetric D-solutions. This result is based on a key lemma derived in Section 4.2, Theorem 4.1, which furnishes a positive lower bound, in terms of ξ 2 , for the Dirichlet integral of the velocity ﬁeld of a solution constructed by Leray method. This bound furnishes, in particular, that symmetric Leray solutions, corresponding to v∗ = 0 and ξ = 0, are not trivial, a fact ﬁrst discovered by Amick [2]. If this latter conclusion holds also for solutions that are not necessarily symmetric or for generic symmetric D-solutions is an open question. Using Theorem 4.1 and other preparatory results derived in Section 4.1, in Section 4.3 we then show that if the basic assumption (H ) is satisﬁed, the set of λ for which (1) with v∗ = 0 and ξ = 0 has a symmetric solution contains an unbounded set M0 . Proving or disproving (H ) remains an open question. If (H ) is proven to be true, the next step is to investigate the properties of M0 . A possible way is to use an analytic continuation argument, along the lines of the results proved in Section 2.3.

Notation In this paper we shall use the notation of [20]. However, for the reader’s convenience, we collect here the most frequently used symbols. N is the set of positive integers. Rn , n 1, is the Euclidean n-dimensional space and {e1 , e2 , . . . , en } ≡ {ei } is the associated canonical basis. Throughout this paper we shall essentially deal with the case n = 2. The coordinates (respectively, components) of a point x ∈ R2 (respectively, vector v) will be denoted by x1 and x2 (respectively, v1 and v2 ). We shall also use polar coordinates (r, θ ), where x1 = r cos θ and x2 = r sin θ . The corresponding components of the vector v will be denoted by vr and vθ , respectively. Given a second-order tensor A and a vector a of components {Aij } and {ai }, respectively, in the basis {ei }, by a · A (respectively, A · a), we mean the vector whose components are given by Aij ai (respectively, Aij aj ). Moreover, if B = {Bij } is another second-order tensor, by the symbol A · B we mean the second-order tensor whose components are given We also set A : B = trace(A · B T ), where the superscript “T ” denotes transpose, by Ail Blj . √ and |A| = A : A. For a > 0, we set Ba (x) = {y ∈ R2 : |y − x| < a}, and B a (x) = {y ∈ R2 : |y − x| > a}. If x = 0, we shall simply write Ba and B a , respectively. If A is a domain of R2 , we denote by δ(A) its diameter and by Ac its complement. If A is an exterior domain (the complement of the closure of a bounded domain) we shall take the origin of coordinates in the interior of Ac . Moreover, for a > δ(Ac ), we set Aa = A ∩ Ba , Aa = A ∩ B a and Aa,b = Aa ∩ Ab , a > b > δ(Ac ). With the Greek letter Ω we shall indicate the relevant “region of ﬂow” of the ﬂuid, that is, an exterior domain of R2 . If Ω = R2 , we shall assume, without loss, that Ω c ≡ B (the “cross-section of the cylinder”) is contained in B1 and contains B1/2 . Unless explicitly stated, all domains involved in this paper are contained in R2 . C k (A), k 0, Lq (A), W m,q (A), m 0, 1 < q < ∞, denote the usual space of functions of class C k on A, and Lebesgue and Sobolev spaces, respectively. Norms in Lq (A) and W m,q (A) are denoted by · q,A , · m,q,A . Unless confusion arises, we shall usually drop

80

G.P. Galdi

the subscript “A” in these norms. The trace space on ∂A for functions from W m,q (A) will be denoted by W m−1/q,q (∂A) and its norm by · m−1/q,q,∂ A . By D k,q (A), k 1, 1 < q < ∞, we indicate the homogeneous Sobolev space of order (m, q) on A, [20,40], that is, the class of functions u that are (Lebesgue) locally integrable in Ω and with D β u ∈ Lq (A), β = (β1 , β2 ), |β| = k, where Dβ =

∂ |β| β

β

∂x1 1 ∂x2 2

,

|β| = β1 + β2 .

For u ∈ D k,q (A), we set2 |u|k,q,A

|β|=k A

1/q |D β u|q

,

where, again, the subscript “A” will be generally omitted. k,q By D0 (A) we indicate the completion of C0∞ (A) in the norm |u|k,q,A , and denote by −k,q

−1,q

D0 (A), 1/q = 1 − 1/q, the dual space of D0 (A). The D0 –D0 duality pairing will be indicated by [·, ·]. Finally, D(A) denotes the subset of C0∞ (Ω) constituted by solenoidal vector functions, and D01,2 (A) is the completion of D(A) in the D 1,2 (Ω)-norm. k,q

1,q

1. Analysis of some linearized problems In this section we shall describe the most signiﬁcant mathematical properties related to several linearizations of problem (1). Speciﬁcally, in Section 1.1, we shall investigate the oldest linearization, namely, the so called Stokes approximation. This approximation, especially for plane ﬂow, is very well known because it leads to the famous Stokes paradox (see Theorem 1.2) according to which, roughly, a cylinder can not move by steady, translational motion in a ﬂuid of “very large” viscosity, since the corresponding boundary-value problem has no solution (in any “reasonable” class). We shall show, however, that such an approximation is valid in several other physically interesting problems and, in particular, we will furnish a characterization on the data in order that the boundary-value problem has one and only one solution; see Theorem 1.1. Applications of this result include selfpropelled motions of a body in a viscous liquid, and are presented in Section 1.2. In Section 1.3, we shall survey the relevant properties of another and, in a sense, more appropriate linearization, that is, the Oseen approximation. We shall collect the signiﬁcant results of the corresponding boundary-value problem which will be the cornerstone of the nonlinear theory developed in Sections 2–4. We shall also study in which sense the solutions of the Oseen boundary-value problem converge to those of the corresponding Stokes boundaryvalue problem; see Section 1.4. Finally, in Section 1.5, we shall analyze a variant to the Oseen linearization, obtained by adding to the Oseen operator a suitable linear operator 2 Typically, we shall omit in the integrals the inﬁnitesimal volume or surface of integration.

Stationary Navier–Stokes problem in a two-dimensional exterior domain

81

that, in Section 2, will play an important role in the nonlinear treatment of existence of solutions at ﬁnite Reynolds numbers.

1.1. The Stokes approximation If we formally take λ = 0 in (1) we obtain the so-called Stokes approximation: v0 = ∇p0

' in Ω,

div v0 = 0

(1.1)

v0 |∂Ω = v∗ , lim v0 (x) = ξ.

|x|→∞

In general, this boundary-value problem does not have a solution. In fact, as we mentioned in the Introduction, it certainly does not have a solution in the physically relevant circumstance when v∗ = 0 and ξ = 0, leading to the so-called Stokes paradox. However, there are also other well-known special cases where problem (1.1) has one and only one solution. For example, an elementary solution in a closed form can be constructed if Ω is the exterior of a circle, ξ = 0 and v∗ = ω × x, for some constant vector ω orthogonal to the plane of ﬂow; see, e.g., [7], p. 18. The investigation of the solvability of problem (1.1) has been the object of several researches; see, e.g., [1,4,5,11,12]. One of the main goals of this section is to establish a necessary and sufﬁcient condition on the data v∗ and ξ for the (unique) solvability of (1.1) in a suitable functional class; see (1.12). To this end, we recall some preliminary facts. The ﬁrst result concerns the asymptotic behavior of solutions to (1.1) possessing a very mild degree of regularity locally in Ω and at inﬁnity. To this end, we recall that the Stokes fundamental solution is a pair constituted by a tensor ﬁeld U = {Uij } and a vector ﬁeld q = {qj } deﬁned by (i, j = 1, 2) # $ (xi − yi )(xj − yj ) 1 1 Uij (x − y) = − δij log + , 4π |x − y| |x − y|2 1 (xj − yj ) qj (x − y) = . 2π |x − y|2 By direct inspection, one checks that U and q satisfy the following equations Uij (x − y) +

∂ qij (x − y) = 0 and ∂xi

∂ Uij (x − y) = 0 ∂xi

for x = y.

(1.2)

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Denote by T ≡ {Tij = Tij (u, p)} the stress tensor associated to the velocity ﬁeld u and associated scalar ﬁeld p, namely (i, j = 1, 2), Tij = −pδij + Dij (u), where ∂uj 1 ∂ui + 2 ∂xj ∂xi

Dij (u) =

is the stretching tensor. We have the following result, whose proof can be found in [20], Theorems V.1.1 and V.3.2. 1,q q 1 < q < ∞, be a pair of vector and scalar L EMMA 1.1. Let (v, p) ∈ Wloc (Ω) × Lloc (Ω), ﬁelds, respectively, solving (1.1)1,2 in the sense of distributions. Then, v, p ∈ C ∞ (Ω). Moreover, if at least one of the following conditions is satisﬁed (i) |v(x)| = o(|x|), all |x| r,

|v(x)|t (ii) |x|r (1+|x|) n+t dx = o(log r) for some r > 1 and some t ∈ (1, ∞), there exist vector and scalar constants v∞ , p∞ such that, as |x| → ∞ (j = 1, 2),

vj (x) = v∞j + mi Uij (x) + σj (x), (1.3) p(x) = p∞ − mi qi (x) + η(x), where mi = −

Ti (v, p)n ,

(1.4)

∂Ω

and, for all |α| 0, D α σ (x) = O |x|−1−|α| , D α η(x) = O |x|−2−|α| .

(1.5)

Our next result (see Lemma 1.3) concerns the structure of the nullspace of the problem (1.1)1,2,3, in the homogeneous Sobolev spaces D 1,q (Ω). For this reason, and also because these spaces will play an important role also in our approach to the mathematical theory of the nonlinear Navier–Stokes problem, we wish to collect here their most significant properties. Speciﬁcally, we have the following result, whose proof is given in [20], Theorem II.6.1.

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L EMMA 1.2. Let Ω be an exterior domain and let u ∈ D 1,q (Ω). Then: (i) If 1 q < 2, there exists a unique u0 ∈ R2 such that, for all sufﬁciently large r we have

2π

u(r, θ ) − u0 q dθ γ0 r q−2 |u|q

q,Ω r ,

0

where γ0 = [(q − 1)/(2 − q)]q−1 if q > 1 and γ0 = 1 if q = 1. (ii) If q = 2, we have 1 r→∞ log r

u(r, θ )2 dθ = 0.

2π

lim

0

This estimate is sharp, in the sense that there are functions u such that 1 lim inf r→∞ log r

u(r, θ )2 dθ = M > 0,

2π 0

and u ∈ / D 1,q (Ω), for all q ∈ [1, 2].3 (iii) If 2 < q < ∞, we have lim

1

r→∞ r q−2

u(r, θ )2 dθ = 0.

2π 0

Assume, moreover, that Ω is locally Lipschitzian, and let u ∈ D 1,q (Ω), 1 < q < 2. Then, 1,q u ∈ D0 (Ω), 1 < q < 2, if and only if u|∂Ω = 0 and the constant vector u0 in part (i) is 1,q zero. Finally, u ∈ D0 (Ω), q 2, if and only if u|∂Ω = 0. R EMARK 1.1. Even though D0 (Ω) is the completion of C0∞ (Ω) in the norm | · |1,q , 1,q functions from D0 (Ω) may grow at large spatial distances, if q 2. The ﬁelds in (1.9)1 below furnish an explicit example (see also Footnote 3). 1,q

In order to prove the mentioned characterization, we need to introduce some suitable “auxiliary ﬁelds”. These are particular solutions to the Stokes system (1.1)1,2, and they can be introduced in several different ways. Following [27], we introduce them as a basis of 1,q the null space of solutions to the Stokes system (1.1)1,2 in the space D0 (Ω). Speciﬁcally, we have the following lemma [27]. L EMMA 1.3. Let Ω be an exterior domain of class C 2 . Let Sq be the linear subspace 1,q of D0 (Ω) × Lq (Ω), 1 < q < ∞, constituted by the distributional solutions u to the 3 Take, for example, u = (log r)1/2 , r > 1.

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following problem: u = ∇π div u = 0

' in Ω, (1.6)

1,q (u, π) ∈ D0 (Ω) × Lq (Ω).

Then, if 1 < q 2, Sq = {0, 0}, while if 2 < q < ∞, dim Sq = 2. In this latter case, there exists a basis {h(i) , p(i) }i=1,2 in Sq satisfying the following properties. (i) For all 1 < q < ∞ and i = 1, 2, we have (i) (i) 2,q ∞ × W 1,q Ω ∩ C (Ω) × C ∞ (Ω) . h ,p ∈ Wloc Ω loc (i)

(i)

(ii) There exist h∞ ∈ R2 and p∞ ∈ R, i = 1, 2, such that, as |x| → ∞, the following representation holds −1 , h(i) (x) = h(i) ∞ − U (x) · ei + O |x| (1.7) (i) p(i) (x) = p∞ + q(x) · ei + O |x|−2 . (iii) For i = 1, 2, we have T h(i) , p(i) · n = ei .

(1.8)

∂Ω

R EMARK 1.2. In some special cases, the ﬁelds {h(i) , p(i) } are known in a closed form. For example, if Ω is the exterior of the unit circle, we have h(1) 1 = 2 log |x| + (1)

h2 = −2 p(1) = h(2) 1

2x22 (x12 − x22 ) + − 1, |x|2 |x|4

x1 x2 1 − |x|−2 , 2 |x|

x1 , |x|2

x1 x2 = −2 2 1 − |x|−2 , |x|

(2)

h2 = 2 log |x| + p(2) = 4

x2 . |x|2

2x12 (x22 − x12 ) + − 1, |x|2 |x|4

(1.9)

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Finally, we recall the following lemma on the solvability of a nonhomogeneous version of problem (1.6) [27]. −1,q

L EMMA 1.4. Let Ω be an exterior domain of class C 2 . Then, for every f ∈ D0 1 < q < 2, satisfying [f, hi ] = 0, i = 1, 2, the problem u = ∇π + f div u = 0

(Ω),

' in Ω

(1.10) 1,q

has one and only one solution (u, π) ∈ D0 (Ω) × Lq (Ω), in the sense of distributions. We are now in a position to furnish the desired characterization. T HEOREM 1.1. Let Ω be an exterior domain of class C 2 . Denote by F the class of 1,q q satisfying (1.1)1,2 in the sense of distributions, with pairs (v0 , p0 ) ∈ Wloc (Ω) × Lloc (Ω) v0 obeying (1.1)3 in the trace sense, and (1.1)4 in the following averaged sense lim

2π

r→∞ 0

v0 (r, θ ) − ξ dθ = 0.

(1.11)

Then, for any given v∗ ∈ W 1−1/q,q (∂Ω), 1 < q < ∞, the condition (v0 , p0 ) ∈ F implies that v∗ and ξ satisfy the following relation4 ξi = ∂Ω

v∗ · T h(i) , p(i) · n,

i = 1, 2.

(1.12)

Conversely, let v∗ ∈ W 1−1/q,q (∂), 1 < q < ∞, and ξ ∈ R2 satisfy (1.12). Then, there is a unique solution (v0 , p0 ) ∈ F . Moreover, v0 , p0 ∈ C ∞ (), and, as |x| → ∞, v0 (x) = ξ + ζ |x| ,

(1.13)

where D α ζ (x) = O |x|−1−|α| ,

all |α| 0.

P ROOF. Multiplying (1.1)1 by h(i) and integrating by parts over ΩR , we obtain

|x|=R

h(i) · T (v0 , p0 ) · n =

T (v0 , p0 ) : ∇h(i) . ΩR

4 Notice that, since (h(i) , p (i) ) ∈ W 2,q (Ω) × W 1,q (Ω) for all 1 < q < ∞ (Lemma 1.3(i)), it follows that loc loc ,q (i) (i) 1−1/q T (h , p ) · n|∂Ω is well deﬁned as an element of W (∂Ω), so that (1.12) makes sense.

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Equation (1.11) along with Lemma 1.1 implies that v0 , p0 have the asymptotic behavior given in (1.3) with v∞ ≡ ξ . Moreover, the vector m in (1.3) must vanish. Employing this information, recalling (1.5) and passing to the limit R → ∞ in the previous relation, we thus ﬁnd T (v0 , p0 ) : ∇h(i) = 0.

(1.14)

Ω

We next multiply (1.6) with u ≡ h(i) , π ≡ p(i) , by v0 − ξ , and integrate by parts over ΩR to obtain ∂Ω

(v∗ − ξ ) · T h(i) , p(i) · n +

|x|=R

(v0 − ξ ) · T h(i) , p(i) · n (1.15)

=

T h(i) , p

(i)

: ∇v0 .

ΩR

Observing that T h(i) , p(i) : ∇v0 = T (v0 , p0 ) : ∇h(i) , we may let R → ∞ in (1.15) and use (1.14), (1.3) and (1.7) to deduce (1.12). Conversely, assume that (1.12) holds. We look for a solution v0 , p0 to (1.1), with v0 of the form v0 = w + V + σ + ξ.

(1.16)

In this relation σ (x) =

1 Φ∇log|x|, 2π

Φ= ∂Ω

v∗ · n,

and V ∈ W 1,q (Ω) is a solenoidal extension in Ω of compact support of the ﬁeld v∗ (x) − σ (x) − ξ,

x ∈ ∂Ω.

(1.17)

The existence of such V is well known; see [20], Exercise II.3.4. Moreover, V 1,q cv∗ 1−1/q,q(∂Ω).

(1.18)

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Furthermore, w solves the following problem w = ∇p0 + f div w = 0

' in Ω, (1.19)

w|∂Ω = 0, lim w(x) = 0,

|x|→∞

where (in the sense of distributions) f = −V . −1,q¯

From (1.18) it follows that f ∈ D0 is the ﬁeld (1.17), we ﬁnd

(i)

f, h

(Ω) for some 1 < q¯ < 2. Also, recalling that V |∂Ω

= − V , h(i) = − (V + σ ), h(i) =

∂Ω

(v∗ − ξ ) · T h(i) , p(i) · n.

In view of the assumption (1.12) on the data, we obtain [f, h(i) ] = 0, i = 1, 2. So, from 1,q¯ Lemma 1.4, we deduce that problem (1.19) admits a unique solution (w, p0 ) ∈ D0 (Ω) × Lq¯ (Ω). By Lemma 1.2(i) we thus get, in particular, lim

r→∞ 0

w(r, θ ) dθ = 0.

2π

Combining this information with (1.16) we obtain that v0 , p0 is a solution to (1.1), where the condition (1.1)4 is attained in the following way lim

r→∞ 0

v0 (r, θ ) − ξ dθ = 0.

2π

Then, by Lemma 1.1(ii) we deduce that v0 has the asymptotic behavior given in (1.13). The theorem is completely proved. An important, immediate consequence of Theorem 1.1 is the following theorem. T HEOREM 1.2 (Stokes paradox). Let Ω and F be as in Theorem 1.1. Then, there is no solution to (1.1) in the class F with v∗ = 0 and ξ = 0. P ROOF. If v∗ = 0, the condition (1.12) is equivalent to ξ = 0, giving a contradiction.

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1.2. Some applications of Theorem 1.1 In this section we shall consider some application to physically relevant situations of the characterization proved in Theorem 1.1. 1.2.1. Symmetric domains. In the case when Ω is the exterior of the unit circle, the auxiliary ﬁelds {h(i) , p(i) } are given in (1.9). By a simple calculation, one shows that 1 δj i , Tj h(i) , p(i) n ∂Ω = 2π

j, i = 1, 2.

Therefore, the necessary and sufﬁcient condition (1.12) becomes 1 v∗ . ξ= 2π ∂Ω

(1.20)

(1.21)

This relation can be satisﬁed under several physically relevant assumptions on ξ and v∗ . For example, consider the case when B is a circular disk D uniformly rotating around an axis orthogonal to its plane,5 with angular velocity ω, in a ﬂuid that is rest at inﬁnity. We then have ξ = 0 and v∗ = ω × x, and it is at once veriﬁed that (1.21) is satisﬁed. Another example is furnished by the case when D is in a ﬂuid subject to a simple shear in the x1 -direction. In this circumstance we have ξ = 0 and v∗ = k x2 e1 , where k is a given constant. Again, it is readily seen that condition (1.21) is satisﬁed. Similar conclusions can be drawn in the more general case when B (≡ Ω c ) possesses two orthogonal straight lines of geometric symmetry. Speciﬁcally, assuming that these lines coincide with the x1 and x2 axes, respectively, we suppose (x1 , x2 ) ∈ ∂B →

(x1 , −x2 ) ∈ ∂B, (−x1 , x2 ) ∈ ∂B.

In such a case, the ﬁelds {h(i) , p(i) } possess the following symmetry properties (1) (1) h(1) 1 (x1 , x2 ) = h1 (−x1 , x2 ) = h1 (x1 , −x2 ), (1)

(1)

(1)

h2 (x1 , x2 ) = −h2 (−x1 , x2 ) = −h2 (x1 , −x2 ), p(1) (x1 , x2 ) = −p(1) (−x1 , x2 ) = p(1) (x1 , −x2 ), (2) (2) h(2) 1 (x1 , x2 ) = −h1 (−x1 , x2 ) = −h1 (x1 , −x2 ), (2) (2) h(2) 2 (x1 , x2 ) = h2 (−x1 , x2 ) = h2 (x1 , −x2 ),

p(2) (x1 , x2 ) = p(2) (−x1 , x2 ) = −p(2) (x1 , −x2 ). 5 As explained in the Introduction, we recall that the disk is the cross-section of a “long” cylinder in the plane of the ﬂow.

Stationary Navier–Stokes problem in a two-dimensional exterior domain

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Therefore, taking into account that the unit normal n satisﬁes n1 (x1 , x2 ) = −n1 (−x1 , x2 ) = n1 (x1 , −x2 ), n2 (x1 , x2 ) = n2 (−x1 , x2 ) = −n2 (x1 , −x2 ), by direct inspection we see that the right-hand side of (1.12) is zero, provided v∗ is chosen as above. 1.2.2. Self-propelled motions. A very interesting application of Theorem 1.1 when ξ = 0 is related to self-propulsion of a rigid body B [23] and [24], Part II. In such a situation, the ﬂuid is at rest at inﬁnity and B moves by constant motion. The motion of B is not due to external forces but, rather, to a suitable distribution of velocity v∗ at ∂B (≡ ∂Ω), that furnishes the needed “thrust”. This happens, for example, in modeling the motion of certain micro-organisms, such as Ciliata; see [8] and [24], Part II. One of the basic questions for this type of problems is the following one: in which ways can we choose the ﬁeld v∗ in order that B moves with a (constant) rigid motion velocity U ≡ −ξ − ω × x, where ξ = 0 (so that B does move)?6 Within the Stokes approximation, this amounts to ﬁnd u0 , π0 and U satisfying the following problem [23] u0 = ∇π0 ,

' in Ω,

div u0 = 0

u0 |∂Ω = v∗ − ξ − ω × x, lim u0 (x) = 0,

|x|→∞

(1.22)

T (u0 , π0 ) · n = 0, ∂Ω

x × T (u0 , π0 ) · n = 0. ∂Ω

The last two equations in (1.22) are consequences of Newton’s laws of conservation of linear and angular momentum, respectively, for the body B. They express the fact that total external force and torque acting on B are identically zero, that is, that B is self-propelled. In view of Theorem 1.1 we know that, given H0 ∈ R2 such that

x × T h(i) , p(i) · n,

H0i = e3 · ∂Ω

6 Of course, ω is orthogonal to the plane of motion.

i = 1, 2,

(1.23)

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there is one and only one solution H, P to the following problem H = ∇P in Ω, div H = 0 (1.24)

H |∂Ω = e3 × x, lim H (x) = H0 .

|x|→∞

The following properties are easily established. T (H, P ) · n = 0, ∂Ω

(1.25)

Q ≡ e3 ·

x × T (H, P ) · n = 0. ∂Ω

Actually (1.25)1 is a direct consequence of (1.24)4 and of Lemma 1.1. To establish (1.25)2, we multiply both sides of (1.24)1 by H and integrate by parts over ΩR . Using the asymptotic properties (1.3)1 for H we then let R → ∞ and obtain the following relation D(H )2 = Q, Ω

which shows (1.25)2. We are now in position to analyze the solvability of (1.22). We begin to observe that, as a consequence of Theorem 1.1, problem (1.22)1–4 has a solution (for sufﬁciently smooth Ω and v∗ ) if and only if ξi + ωH0i = Fi ,

i = 1, 2,

where H0i is deﬁned in (1.23), and Fi v∗ · T h(i) , p(i) · n,

(1.26)

i = 1, 2.

(1.27)

∂Ω

Again by Theorem 1.1 and by Lemma 1.1, condition (1.26) implies the vanishing of the total force; see (1.22)5. We next multiply (1.24)1 by u0 , and integrate by parts over ΩR . Using the asymptotic properties (1.3)1 for u0 we then let R → ∞ and obtain the following relation (v∗ − ξ − ω × x) · T (H, P ) · n = D(H ) : D(u0 ). (1.28) ∂Ω

Ω

Likewise, multiplying (1.22)1 by H , integrating by parts over ΩR , taking into account the asymptotic properties of H and u0 , and then letting R → ∞, we ﬁnd D(H ) : D(u0 ) = e3 · x × T (u0 , p0 ) · n. (1.29) Ω

∂Ω

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From (1.28), (1.29) and (1.25)1, we deduce that the vanishing of the torque, condition (1.22)6, is equivalent to the following one Qω = G,

(1.30)

where G= ∂Ω

v∗ · T (H, P ) · n.

(1.31)

Let si = T h(i) , p(i) · n|∂Ω ,

i = 1, 2,

s3 = T (H, P ) · n|∂Ω , and deﬁne the following three-dimensional subspaces of Lq (∂Ω) S(B) = u ∈ Lq (∂Ω): u = αi si for some α ∈ R3 .

(1.32)

Notice that S depends only on the geometric properties of B, like shape and symmetry. Denote by P the projection of Lq (∂Ω) onto T (B). The next theorem shows, among other things, that a sufﬁcient condition to self-propel B is that the boundary velocity (the “thrust”) has a nonzero projection on S(B). Moreover, the velocity of B is uniquely determined by this projection. Precisely, we have the following result. T HEOREM 1.3. Let Ω be as in Theorem 1.1. Then, for any v∗ ∈ W 1−1/q,q (∂Ω), 1 < q < ∞, satisfying P(v∗ ) = 0, there exists a solution {u0 , p0 , U ≡ −ξ − ω × x} to problem (1.22) with U ∈ R2 \ {0}. Moreover, the translational velocity ξ and the angular velocity ω are given by ξi = Fi − GH0i /Q,

i = 1, 2, (1.33)

ω = G/Q, where H0 , Q, F and G are given in (1.23), (1.25), (1.27) and (1.31), respectively. So, in particular, ξ = 0 if and only if F = GH0 /Q. Moreover, let v∗∗ ∈ W 1−1/q,q (∂Ω) be another boundary velocity with P(v∗ ) = P(v∗∗ )

≡ −ξ˜ − ω˜ × x} the corresponding solution. Then U

= U. and denote by {u˜ 0 , p˜ 0 , U P ROOF. Let v∗ be given as stated. We then choose ξ and ω as in (1.33) and solve problem (1.22)1–4. This is certainly possible by Theorem 1.1, because the data satisfy the compatibility condition expressed by (1.33)1. By what we have seen before, this implies that also (1.22)5 is satisﬁed. Moreover, the choice of ω in (1.33)2 ensures, again as shown earlier, that also condition (1.22)6 is satisﬁed. Finally, the last part of the theorem follows

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from (1.33), from the fact that if Pv∗ = 0, then F ≡ G = 0, and from the linearity of the problem. The result is completely proved. It might be of some interest to evaluate the “self-propelled” conditions (1.33) in the case when B is a disk (of radius 1, say), since this situation is of a certain relevance in the study of the motion of several micro-organisms [8]. We then take v∗ = f (θ )eθ , and would like to ﬁnd the conditions on f (θ ) for which P(v∗ ) = 0, so that v∗ is an appropriate “thrust”. From (1.27) and (1.20) we ﬁnd F1 = −

1 2π

2π

f (θ ) sin θ dθ, 0

F2 =

1 2π

2π

f (θ ) cos θ dθ. 0

Moreover, we have H = eθ /r [7], p. 18, so that H0 = 0 and, by a simple calculation, we also ﬁnd 2π Q = −4π, G = −2 f (θ ) dθ. 0

Therefore, with the speciﬁed choice of v∗ B will move with the following translational and angular velocities 2π 1 f (θ ) sin θ dθ, ξ1 = − 2π 0 1 2π ω= f (θ ) dθ. 2π 0

1 ξ2 = 2π

2π

f (θ ) cos θ dθ, 0

1.3. The Oseen approximation Even though the Stokes linearization may provide some insights and useful information in certain physically interesting problems (as illustrated in the previous section), it is not able to give any kind of information on one of the most important problems in ﬂuid dynamics, namely, the motion of a body of simple symmetric shape, such as a cylinder, steadily translating through a viscous ﬂuid with a velocity orthogonal to its major axis of symmetry. Actually, there are several other situations, also for three-dimensional ﬂows, where the Stokes linearization furnishes results that are at odds with the observation or, even, with the assumptions that are at the basis of the linearization itself. For instance, in the case of the motion of a sphere in a Navier–Stokes ﬂuid, the Stokes approximation does not show any “wake” behind the sphere. Moreover, the ratio of the inertial term (v ·∇v) to the viscous term (v) goes to inﬁnity as soon as we move f ar away from the boundary, contradicting the basic logic of the linearization, that assumes this ratio to be “small” everywhere in the region of ﬂow. Motivated by these considerations, Oseen proposed another type of linearization of (1.1) when ξ = 0; see [38]. Choosing, without loss, ξ = e1 , the linearization is obtained by setting v = u + e1 in (1.1) and by disregarding the nonlinear term u · ∇u. Such an assumption

Stationary Navier–Stokes problem in a two-dimensional exterior domain

93

leads to the following problem ∂u = ∇p + F u − λ ∂x 1

' in Ω,

div u = g

(1.34) u|∂Ω = u∗ , lim u(x) = 0,

|x|→∞

where, for future purposes, we have allowed for a nonzero “body force” −F acting on the ﬂuid, and a prescribed value g (not necessarily zero) for the divergence of u. Associated with problem (1.34), Oseen introduced the corresponding fundamental solution E, q deﬁned as follows (i, j = 1, 2) Eij (x − y) = δij −

∂2 Φ(x − y), ∂yi ∂yj (1.35)

∂ ∂ +λ Φ(x − y), qj (x − y) = − ∂yj ∂y1 where

y1 −x1

Φ(x − y) =

Ψ (τ, x2 − y2 ) dτ −

0

Ψ (x − y) =

1 4π

y2 −x2

(y2 − x2 − τ )K0

0

1 λ|x − y| −λ(x2 −y2 )/2 e log |x − y| + K0 2πλ 2

λ|τ | dτ, 2 (1.36)

and K0 is the modiﬁed Bessel function of the second kind of order zero. By a direct calculation we show ∂ ∂ +λ ej (x − y) and Eij (x − y) = ∂y1 ∂yi ∂ Ej (x − y) = 0 ∂y

for x = y,

and that qj (x − y) =

1 xj − yj . 2π |x − y|2

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G.P. Galdi

Moreover, from the property of K0 (z) for small and large values of |z|, we can show the following relations (see [20], Section VII.3, for details) Eij (x − y) = −

$ # (xi − yi )(xj − yj ) 1 1 + + o(1) δij log 4π λr r2

= Uij (x − y) −

1 1 δij log + o(1) as λr → 0, 4π λ

(1.37)

with r = |x − y|, U Stokes fundamental tensor (1.2)1 , and 1 + 3 cos ϕ e−s cos ϕ 1 − cos ϕ − + √ + R(λr) , 2πλr 4 λπr 4λr 3 e−s sin ϕ sin ϕ E12 (x − y) = E21 (x − y) = 1+ − √ + R(λr) , 2πλr 4λr 4 λπr

E11 (x − y) =

(1.38)

cos ϕ E22 (x − y) = − 4πλr 1 − 3 cos ϕ e−s + R(λr) as λr → ∞, s − + √ 8 2 π (λr)3/2

where ϕ is the angle made by a ray that starts from x and is directed toward y, with the direction of the positive x1 -axis, and 1 s = λr(1 + cos ϕ). 2 Finally, the “remainder” R(t) satisﬁes dk R = O t −2−k as t → ∞, k 0. dt k Using (1.38), one can show the following asymptotic estimates. If y is interior to the parabola |y|(1 + cos ϕ) = 1,

(1.39)

we have E11 (y)

c |y|1/2

as |y| → ∞,

(1.40)

while if (1 + cos ϕ) |y|−1+2σ

for some σ ∈ [0, 1/2],

(1.41)

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95

we have E11 (y)

c |y|1/2+σ

as |y| → ∞.

(1.42)

Moreover, Ei2 (y) c , |y|

i = 1, 2, as |y| → ∞.

(1.43)

R EMARK 1.3. The fact that the asymptotic decay of E11 (y) for large |y| is faster outside than inside the parabolic region deﬁned in (1.39), is representative of the existence of a “parabolic wake” in the positive y1 -direction. So far as the behavior of the ﬁrst derivatives of E is concerned, differentiating, we derive the following uniform bounds as |y| → ∞ ∂E11 (y) c , |y| ∂y 2 ∂E1i (y) c ∂y |y|3/2 , i

∂E12 (y) c , ∂y |y|2 1 ∂E22 (y) c ∂y |y|2 , i

(1.44) i = 1, 2.

It should be also observed that, as it can be easily proved, ∂E11/∂y2 and ∂E1i /∂yi , i = 1, 2, have a faster decay rate outside the wake than that given in (1.44). The next result is the Oseen counterpart of Lemma 1.1 given for the Stokes linearization. We refer to [20], Theorem VII.6.2, for a proof. × L EMMA 1.5. Let Ω be an exterior domain of class C 2 and let (u, p) ∈ Wloc (Ω) q Lloc (Ω), 1 < q < ∞, be a pair of vector and scalar ﬁelds, respectively, satisfying (1.34)1,2 with F ≡ g = 0, in the sense of distributions. Then, u, p ∈ C ∞ (Ω). Moreover, if u satisﬁes at least one of the conditions (i), (ii) of Lemma 1.1, then there exist vector and scalar constants u∞ , p∞ such that as |x| → ∞, 1,q

(1)

uj (x) = u∞j + Mi Eij (x) + σj (x), (1.45) p(x) = p∞ − Mi∗ qi (x) + η(x), where

Mi = −

Ti (u, p) − Rδ1 ui n ,

∂Ω

Mi∗ = −

∂Ω

Ti (u, p) − λ[δ1 ui − δ1i u ] n ,

(1.46)

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and, for all |α| 0, D α σ (x) = O |x|−(2+|α|)/2 , (1.47)

D η(x) = O |x|−2−|α| . α

R EMARK 1.4. It can be shown that, in fact, the “remnant” σ of the preceding lemma has the same asymptotic behavior as the corresponding ﬁrst derivatives of the tensor E; see (1.44). Therefore, in particular, σ decays faster outside than inside the wake. R EMARK 1.5. The previous lemma shows, in particular, that every solution to the Oseen problem satisfying the stated assumptions, behaves, asymptotically, as the Oseen fundamental tensor; see also the previous remark. In view of Remark 1.3, this implies, that these solutions show a wake structure in the positive x1 -direction, as expected on a physical ground. We shall next present a number of theorems that will play a fundamental role in further developments. In fact, on the one hand, they insure existence and uniqueness for the Oseen problem (1.34), and, on the other hand, they furnish key functional-analytic properties that will allow us to show, among other things, existence and uniqueness for the nonlinear problem (1.1), at least for “small” data. To this end, for u = (u1 , u2 ), we introduce the following notation: u!q = u2 2q/(2−q) + |u2 |1,q + u3q/(3−2q) ∂u + ∂x

+ |u|1,3q/(3−q),

(1.48) 1 < q < 3/2.

1 q

As usual, if we need to specify the domain A on which we take the norm ·!q , we will write ·!q,A . R EMARK 1.6. If Ω is an exterior domain, every u with u!q + |u|2,q < ∞ in Ω, satisﬁes the condition lim u(x) = 0,

|x|→∞

uniformly.

In fact, since u ∈ D 1,3q/(3−q) (Ω) ∩ D 2,q (Ω), from [20], Theorem II.5.1, we deduce u ∈ D 1,2q/(2−q)(Ω). Since 2q/(2 − q) > 2 and, also, u ∈ L3q/(3−2q)(Ω), the stated property follows from [20], Remark II.7.2. The following result holds. T HEOREM 1.4. Let Ω be an exterior domain of class C 2 . Given F ∈ Lq (Ω),

g ∈ W 1,q (Ω),

u∗ ∈ W 2−1/q,q (∂Ω),

1 < q < 3/2,

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there exists one and only one corresponding solution u, p to the Oseen problem (1.34) such that u ∈ Ls2 (Ω) ∩ D 1,s1 (Ω) ∩ D 2,q (Ω),

p ∈ D 1,q (Ω),

u2 ∈ L2q/(2−q)(Ω) ∩ D 1,q (Ω). with s1 =

3q 3−q , s2

=

3q 3−2q .

Moreover, u, p verify the following estimate

u!q + |u|2,q + |p|1,q c F q + u∗ 2−1/q,q(∂Ω) + g1,q ,

(1.49)

where the positive constant c depends on q, Ω and λ. P ROOF. A full proof of the theorem is given in [20], Theorem VII.7.1 and Exercise VII.7.1. Here, for reader’s convenience, we shall sketch a proof when Ω = R2 and g = 0. In such a case the proof is obtained by using Fourier transform in conjunction with elementary multipliers theory. For simplicity, we shall also set λ = 1. We look for a solution to (1.34) corresponding to F ∈ C0∞ (R2 ) of the form u(x) =

1 2π

R2

eix·ξ U (ξ ) dξ,

p(x) =

1 2π

R2

eix·ξ P (ξ ) dξ.

(1.50)

Replacing (1.50) into (1.34) furnishes the following algebraic system for U and P : 2 (m (ξ ), m = 1, 2, ξ + iξ1 Um (ξ ) + iξm P (ξ ) = F (1.51) iξi Ui (ξ ) = 0, where ((ξ ) = 1 F 2π

R2

e−ix·ξ F (x) dx

is the Fourier transform of F . Solving (1.51) for U and P delivers (k (ξ ), Um (ξ ) = Φmk (ξ )F

P (ξ ) = i

(k (ξ ) ξk F ξ2

,

where Φmk (ξ ) =

ξm ξk − ξ 2 δmk ξ 2 (ξ 2 + iξ1 )

.

We recall the following theorem of Lizorkin [35]. Given the integral transformation 1 Tf = eix·ξ Ψ (ξ )fˆ(ξ ) dξ, 2π R2

(1.52)

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G.P. Galdi

with Ψ (ξ ) : R2 → R2 continuous together with the derivatives ∂Ψ , ∂ξ1

∂Ψ , ∂ξ2

∂ 2Ψ ∂ξ1 ∂ξ2

for |ξ | > 0, then if, for some β ∈ [0, 1) and M > 0, κ +κ ∂ 1 2Ψ |ξ1 |κ1 +β |ξ2 |κ2 +β κ1 κ2 M, ∂ξ1 ∂ξ2

(1.53)

Tf is bounded from Lq (R2 ) into Lr (R2 ), 1 < q < ∞, 1/r = 1/q − β, and we have Tf r Cf q , where C = c(q, r)M, c(q, r) > 0. It is at once checked that the functions ξs ξk /ξ 2 , s, k = 1, 2, satisfy the assumption (1.53) with β = 0. Therefore, from (1.50) and (1.52), we ﬁnd |p|1,q cF q .

(1.54)

Likewise, the functions ξ1 Φmk (ξ ),

ξs ξr Φmk (ξ ),

s, r, m, k = 1, 2,

satisfy (1.53) with β = 0, and so, again from (1.50) and (1.52), we have ∂u cF q . |u|2,q + ∂x1 q

(1.55)

Moreover, by a simple calculation, we verify that, for any , m, k = 1, 2, Φmk veriﬁes (1.53) with β = 2/3, ξ Φmk with β = 1/3, Φ2,k with β = 1/2 and ξ Φ2k with β = 0. Thus, from (1.50) and (1.52), by Lizorkin’s theorem we ﬁnd u2 2q/(2−q) + |u2 |1,q + u3q/(3−2q) + |u|1,3q/(3−q) cF q ,

1 < q < 3/2.

(1.56)

The summability properties stated in the theorem along with the estimate (1.49) are then a consequence of (1.54)–(1.56). Notice that, as observed in Remark 1.6, the solution u satisﬁes the condition at inﬁnity (1.34)4, uniformly pointwise. R EMARK 1.7. If F and g are of compact support, the solutions determined in Theorem 1.4 have the asymptotic behavior described in Lemma 1.5.

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The theorem just proved has some simple but important consequences. To show them, for 1 < q < 3/2, we deﬁne the following Sobolev-like spaces X1,q (Ω) and X2,q (Ω) by X1,q (Ω) = u: div u = 0, u!q < ∞ , X2,q (Ω) = u ∈ X1,q (Ω): |u|2,q < ∞ .

(1.57)

We observe the validity of the embeddings Xm,q (Ω) !→ W m,q (ΩR ),

m = 1, 2, for all R > 1.

(1.58)

So, if Ω is locally Lipschitzian, a function u from X1,q (Ω) leaves a trace u|∂Ω on ∂Ω and the map u ∈ X1,q (Ω) → u|∂Ω ∈ W 1−1/q,q (∂Ω) is continuous. In particular, if Ω is locally Lipschitzian, the following space is well deﬁned 1,q X0 (Ω) = u ∈ X1,q (Ω): u|∂Ω = 0 .

(1.59)

The X-spaces are suitable spaces for velocity. We next introduce the appropriate space for the pressure Y 1,q (Ω) deﬁned by Y 1,q (Ω) = p ∈ L2q/(2−q)(Ω): |p|1,q < ∞ .

(1.60)

It is readily seen that the X, Y -spaces become Banach spaces when endowed with their “natural” norms uX1,q (Ω) ≡ u!q , uX2,q (Ω) ≡ u!q + |u|2,q , pY 1,q (Ω) ≡ p2q/(2−q) + |p|1,q . It is also shown, by standard methods, that they are reﬂexive and separable. We next introduce the Oseen operator Oλ (u, p) formally deﬁned as Oλ (u, p) = −u + λ

∂u + ∇p, ∂x1

(1.61)

where λ > 0.7 As an immediate corollary to Theorem 1.4 we obtain the following result. 7 More generally, we could take λ = 0, and all the properties stated below for the Oseen operator would continue to hold.

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T HEOREM 1.5. Let Ω be as in Theorem 1.4. The Oseen operator (1.61) is a linear iso1,q morphism from [X2,q (Ω) ∩ X0 (Ω)] × Y 1,q (Ω) into Lq (Ω). P ROOF. Clearly, the map 1,q Oλ : (u, p) ∈ X2,q (Ω) ∩ X0 (Ω) × Y 1,q (Ω) → Oλ (u, p) ∈ Lq (Ω) is linear and well deﬁned. It remains to show that the problem ∂u = ∇p + F u − λ ∂x 1

div u = 0

' in Ω,

u|∂Ω = 0, lim u(x) = 0

|x|→∞

1,q

has a unique solution (u, p) ∈ [X2,q (Ω) ∩ X0 (Ω)] × Y 1,q (Ω) for every F ∈ Lq (Ω). But this is exactly the statement of Theorem 1.4 with g ≡ u∗ = 0. The validity of an inequality of the type (1.49) with an explicit dependence of the constant c on λ ∈ (0, λ0 ], for some λ0 > 0, may be of fundamental importance for treating the nonlinear problem (1) when ξ = 0. Because of the Stokes paradox (see Theorem 1.4), one also expects that the constant c becomes unbounded as λ approaches zero. Now, if we restrict q in the interval 1 < q < 6/5, one can prove the validity of an inequality of the type (1.49), with a constant c which can be rendered independent of λ, for λ ranging in (0, λ0 ], but where the norm of u involves λ in an known way. To this end, for u = (u1 , u2 ), set u!λ,q = λ u2 2q/(2−q) + |u2 |1,q (1.62) + λ2/3 u3q/(3−2q) + λ1/3 |u|1,3q/(3−q). If we need to specify the domain A on which we take the norm ·!λ,q , we will write ·!λ,q,A . We then have the following result, for whose rather technically complicated proof we refer to [20], Theorem VII.5.1, and [21], Lemmas X.4.1 and X.4.2. T HEOREM 1.6. Let Ω be an exterior domain of class C 2 . Then, given F ∈ Lq (Ω),

u∗ ∈ W 2−1/q,q (∂Ω),

1 < q < 6/5,

there is a unique solution u, p to (1.34) with g ≡ 0, such that (u, p) ∈ X2,q (Ω) × Y 1,q (Ω). Moreover, there is λ0 > 0 such that for all 0 < λ λ0 , this solution satisﬁes the following

Stationary Navier–Stokes problem in a two-dimensional exterior domain

101

estimate |u|2,q + u!λ,q + |p|1,q c λ2(1−1/q)| log λ|−1 u∗ 2−1/q,q(∂Ω) + F q .

1.4. The Oseen approximation in the limit of vanishing Reynolds number A question that may spontaneously arise is the relation between the solutions to the Oseen problem (1.34) and those to the Stokes problem (1.1), in the limit of vanishing λ. In this section we present an interesting result relating the solutions to the Oseen problem (1.34) with F ≡ 0 and u∗ ≡ v∗ − e1 to the corresponding solutions to the Stokes problem (1.1), namely, u0 = ∇p0

'

div u0 = 0

in Ω, (1.63)

u0 |∂Ω = v∗ − e1 , lim u0 (x) = 0.

|x|→∞

Assume Ω and v∗ are prescribed as in Theorem 1.4. We begin to observe that, by the usual method based on a suitable solenoidal extension of the boundary data along with the Riesz representation theorem [20], Remark V.2.2, one can easily show the existence of a solution (u0 , p0 ) ∈ [D 1,2 (Ω) × L2 (Ω)] ∩ [C ∞ (Ω) × C ∞ (Ω)] satisfying (1.63)1–3. Of course, nothing in principle can be said about the attainability of the condition at inﬁnity (1.63)4, unless u0 |∂Ω satisﬁes the compatibility condition (1.12). However, despite this lack of information, this solution is unique in the class of solutions with velocity ﬁeld in D 1,2 (Ω). This is an immediate consequence of Lemmas 1.2 and 1.3. Actually, denoting by (w, φ) the difference between two such solutions, we have that w, φ satisfy (1.63)1–3 with v∗ − e1 ≡ 0. Thus, since w ∈ D 1,2 (Ω) and w|∂Ω = 0, from Lemma 1.2 it follows that w ∈ D01,2 (Ω). Therefore, by Lemma 1.3 we get w ≡ 0. The solution u0 admits the following representation [20], Theorem V.3.2, u0j (x) = u∞j +

∂Ω

v∗ (y) − e1 i Ti (Uj , qj )(x − y)

− Uij (x − y)Ti (u, p)(y) n (y) dσy ,

(1.64)

for some constant vector u∞ ∈ R2 , and where Uj = U · ej , j = 1, 2. The following theorem gives the answer to the question raised above. T HEOREM 1.7. Let Ω and v∗ be as in Theorem 1.4, and let u, p be the corresponding solution to (1.34) given in that theorem with F ≡ g = 0. Moreover, let (u0 , p0 ), u0 ∈

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G.P. Galdi

D 1,2 (Ω), be the uniquely determined solution to (1.63)1–3. Then, as λ → 0, (u, p) tends together with ﬁrst and second derivatives. to (u0 , p0 ), uniformly on compact subsets of Ω, Furthermore, lim m(u)| log λ| = 4πu∞ ,

λ→0

(1.65)

where u∞ is given in (1.64) and T (u, p) · n.

m(u) = − ∂Ω

Finally, the limit process preserves the prescription at inﬁnity, that is, u∞ = 0, if and only if v∗ satisﬁes condition (1.12), namely, ∂Ω

v∗ · T h(i) , p(i) · n = e1 .

(1.66)

P ROOF. We will sketch here only the proof of the second part, referring to [20], Section VII.8, for a complete proof of the theorem. The solution to the Oseen problem (1.34) with F ≡ g ≡ 0, given in Theorem 1.4 admits the following representation [20], Theorem VII.6.2, uj (x) = ∂Ω

(v∗i − e1 )i (y)Ti (Ej , ej )(x − y) − Eij (x − y)Ti (u, p)(y) + λ(v∗i − e1 )i (y)Eij (x − y)δ1 n dσy ,

(1.67)

where Ej = E · ej , j = 1, 2. From (1.37) and (1.66) we formally ﬁnd uj (x) =

1 1 mj (u) log 4π λ (v∗ − e1 )i (y)Ti (Uj , qj )(x − y) + ∂Ω

− Uij (x − y)Ti (u, p)(y) n (y) dσy + o(1) as λ|x − y| → 0. We now pass to the limit λ → 0 in this latter relation. Invoking the ﬁrst part of the theorem and using (1.64), we thus obtain (1.65). Finally, the validity of the characterization (1.66) is a consequence of Theorem 1.1.

Stationary Navier–Stokes problem in a two-dimensional exterior domain

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1.5. A variant to the Oseen approximation The objective of this section is to study some functional property of the following variant to the Oseen problem ' ∂u u − λ ∂x − λ(u0 · ∇u + u · ∇u0 ) = ∇p + F 1 in Ω, div u = 0 (1.68) u|∂Ω = 0, lim u(x) = 0,

|x|→∞

where u0 is a prescribed function from X1,q (Ω) (see (1.57)) and F ∈ Lq (Ω). We begin with a very simple but useful result. L EMMA 1.6. Let A be an arbitrary domain in R2 and let v, w be two divergence-free vectors in A for which the norm (1.62) with 1 < q 6/5, is ﬁnite. Then the following inequality holds for all λ > 0 v · ∇wq,A 4λ−1−2(1−1/q) v!λ,q,A w!λ,q,A . P ROOF. Taking into account that v and w are both divergence-free, we obtain ∂w2 ∂w1 ∂w2 ∂w2 v · ∇w = −v1 + v2 + v2 e1 + −v1 e2 ∂x2 ∂x2 ∂x1 ∂x2 and so, by the Hölder inequality and (1.62), v · ∇wq v1 3q/(3−2q)|w2 |1,3/2 + v2 3 |w|1,3q/(3−q) λ−2/3 |w2 |1,3/2 v!λ,q + λ−1/3 v2 3 w!λ,q .

(1.69)

From elementary Lq -interpolation inequalities we ﬁnd (with q = q/(q − 1)) that 3/q

1−3/q

|w2 |1,3/2 |w2 |1,q |w2 |1,3q/(3−q) λ−2/q −1/3 w!λ,q , 6/q

1−6/q

v2 3 v2 2q/(2−q)v3q/(3−2q) λ−2/q −2/3 v!λ,q , and the lemma becomes a consequence of this relation and (1.69).

For a given w ∈ X1,q (Ω), consider the operator Ku0 ,λ : u ∈ X2,q (Ω) → Ku0 ,λ (u) = λ(u0 · ∇u + u · ∇u0 ) ∈ Lq (Ω), 1 < q 6/5.

(1.70)

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In view of Lemma 1.6, the operator Ku0 ,λ is well deﬁned. Therefore, using Theorem 1.5 and recalling Remark 1.5, problem (1.68) can be re-written in the following functional form Oλ (u, p) − Ku0 ,λ (u) = F, 1,q (u, p) ∈ X2,q (Ω) ∩ X0 (Ω) × Y 1,q (Ω),

(1.71)

1,q

where Oλ (u, p) is the Oseen operator (1.61), and X0 (Ω), Y 1,q (Ω) are deﬁned in (1.59) and (1.60), respectively. The operator Ku0 ,λ enjoys the following important property. L EMMA 1.7. Ku0 ,λ is compact. P ROOF. Let {uk }k∈N ⊂ X2,q (Ω), with uk X2,q (Ω) = 1. Since X2,q (Ω) is reﬂexive, we may select a subsequence, which we continue to denote by {uk }k∈N that converges weakly to some u ∈ X2,q (Ω). Set Uk = uk − u. In view of the embedding (1.58) we get, in particular, Uk 2,q,ΩR 2

for all R > 1.

By Rellich theorem and by (1.58), we then deduce lim Uk !q,ΩR = 0 for all R > 1.

k→∞

(1.72)

From Lemma 1.6 and from the fact that Uk X2,q (Ω) 2, for all R > 1, we also ﬁnd Ku ,λ (Uk ) c(λ) w!q,Ω Uk !q,Ω + w! R Uk ! R q,Ω q,Ω R R 0 q c(λ) w!q,ΩR Uk !q,ΩR + 2 w!q,Ω R . This inequality together with (1.72) implies lim supKu0 ,λ (Uk )q 2c(λ) w!q,Ω R , k→∞

and the lemma follows from the fact that limR→∞ w!q,Ω R = 0.

From Theorem 1.5, Lemma 1.6 and well-known results on compact perturbations of isomorphisms, e.g., [32], Theorem IV.5.26, we then obtain the following theorem. T HEOREM 1.8. Let Ω be an exterior domain of class C 2 . Let Nλ,u0 be the linear sub1,q space of [X2,q (Ω) ∩ X0 (Ω)] × Y 1,q (Ω), 1 < q 6/5, constituted by the solutions of the problem Oλ (u, p) − Ku0 ,λ (u) = 0.

Stationary Navier–Stokes problem in a two-dimensional exterior domain

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Then, dimNλ,u0 < ∞. If, in particular, dimNλ,u0 = 0, then, for any F ∈ Lq (Ω), 1 < 1,q q < 6/5, problem (1.68) has a unique solution (u, p) ∈ [X2,q (Ω) ∩ X0 (Ω)] × Y 1,q (Ω), and this solution satisﬁes the following estimate u!q + |u|2,q + |p|1,q cF q .

2. The nonlinear problem: Unique solvability for small Reynolds number and related results The subject of Section 2 is to develop a perturbation theory for the boundary-value problem (1), when ξ = 0. Under this latter assumption, we can take, without loss, ξ = e1 . Thus, setting v = u + e1 , u∗ = v∗ − e1 , we at once obtain that (1) goes into the following equivalent boundary-value problem ' ∂u = λu · ∇u + ∇p u − λ ∂x 1 in Ω, div u = 0 (2.1) u|∂Ω = u∗ , lim u(x) = 0.

|x|→∞

One of the main goals of this section is to show that the nonlinear Navier–Stokes problem (2.1) possesses a solution if the Reynolds number λ is sufﬁciently small. However, the validity of this result is not so evident a priori, as we will know explain. A way of showing existence is to prove the existence of a ﬁxed point (in a subset S of an appropriate Banach space) of the mapping L : w ∈ S → L(w) = u ∈ S, where u solves the problem ∂u = λw · ∇w + ∇p u − λ ∂x 1

div u = 0

' in Ω,

u|∂Ω = u∗ ,

(2.2)

lim u(x) = 0.

|x|→∞

In the limit of λ → 0 there will be a competition between the linear term λ

∂u ∂x1

(2.3)

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G.P. Galdi

and the nonlinear one λw · ∇w.

(2.4)

If in the range of vanishing λ, the contribution of the former is negligible with respect to that of the latter, it would be very unlikely to prove existence, because the linear part in (2.2) would then approach the Stokes system for which, as we know from Section 1.1, solvability is established only under suitable compatibility conditions on the data. Fortunately, what happens is that (2.3) “prevails” on (2.4) and the machinery produces nonlinear existence. In fact, we shall show a stronger result, namely, that, provided λ is sufﬁciently small, a solution to (1) with ξ = 0 can be constructed in the form of a series, that is converging in a suitable Banach space X. Moreover, each coefﬁcient of the series can be evaluated as the solution to a suitable Oseen problem. Concerning uniqueness, we shall show that this solution is the only one that lies in a suitable ball of an appropriate Banach space. This type of solutions are called by R. Finn and D. Smith, who ﬁrst discovered their existence [16], Physically Reasonable (PR). The reason for such a name is because they satisfy all requirements expected on a physical ground such as uniqueness, validity of the energy equality (see Remark 2.2) and moreover, as we shall see in Section 3.5, they show the presence of a wake behind the body B (i.e., in the ξ ≡ e1 -direction). Another goal of this section is the construction of a perturbation theory at arbitrary Reynolds numbers. Speciﬁcally, we shall show that if λ0 is such that dim Nλ0 ,u0 = 0 (see Theorem 1.8), then we can construct a solution (u, p) to (2.1) that is (real) analytic in λ in a neighborhood of λ0 . We are thus able to obtain the solution to (2.1) by analytic continuation with respect to λ. This process will stop if, for some λ0 , either u!λ,q → ∞ as λ → λ− 0 , or dim Nλ0 ,u0 = 0. In this latter case, one can give sufﬁcient conditions for the existence of a bifurcating solution [26]. Let us now consider the solvability of problem (1) when ξ = 0. In this regard, we wish to emphasize that, to date, the existence of solutions to the nonlinear problem (1) when ξ = 0 for arbitrarily prescribed (sufﬁciently smooth) data v∗ is open, no matter what the magnitude of the Reynolds number λ. The major difﬁculty here is the choice of the function space where solutions should exist. Such a difﬁculty is mainly due to the fact that it is not clear what is the asymptotic spatial behavior that solutions a priori might have. Actually, this behavior cannot be, in general, of the type r −k (r = |x|) for some ﬁxed positive k, as the following example shows λ vr = − , r ω 1 1 − r −λ+2 , λ−2 r 1 dvr2 vθ2 − p = −λ . 2 dr r

vθ =

(2.5)

In the solution (2.5), due to G. Hamel [31], ω is an arbitrary constant and we assume λ = 2 and r > r0 > 0. Therefore, taking λ sufﬁciently close to 1, (2.5) provides an example of

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a solution that decays more slowly than any negative power of r. The ﬁelds in (2.5) show another undesired feature of problem (1) when ξ = 0. In fact, taking Ω = B 1 we see at once that the velocity ﬁeld v assumes the boundary data v∗r = −λ, v∗θ = 0, for all values of the constant ω. Consequently, solutions (2.5) also furnish an example of nonuniqueness to problem (1) with ξ = 0. In Part IV, we will consider the problem of uniqueness in relation to the solvability of problem (1) with ξ = 0 for arbitrary large λ. Coming back to the question of existence, it is very probable that a solution to (1) with ξ = 0 does not exist unless the data v∗ satisfy certain compatibility conditions. This guess is strongly suggested by the results presented in Section 1.1 for the Stokes approximation; see, in particular, Theorem 1.1. In fact, in Section 3.3, we shall show that problem (1) with ξ = 0 has at least one solution, provided B and v∗ satisfy certain symmetry conditions. Such a solution exists for all Reynolds number. 2.1. Unique solvability at small Reynolds number In this section we shall prove that problem (2.1) has one and only one solution in a ball of a suitable Banach space, provided λ is positive and “sufﬁciently small”. This solution can be expressed in the form of a series. To reach this goal, we propose a very simple result on the convergence of certain power series. L EMMA 2.1. Let {ak }k∈N , a0 > 0, be a sequence of positive real numbers satisfying the condition: an+1 C

n

ak an−k ,

n 0,

(2.6)

k=0

where C is a positive constant independent of n. Then the power series g(x) ≡

∞ an x n ,

x > 0,

n=0

is convergent provided 4a0 Cx < 1, and we have g(x) 2a0 .

(2.7)

P ROOF. Consider the sequence of positive numbers {Ak }k∈N deﬁned as follows A0 = a0 ,

An+1 = C

n

Ak An−k ,

n 0.

(2.8)

k=0

From (2.6) and (2.8) we ﬁnd that an An

for all n 0.

(2.9)

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If we multiply (2.8) by x n , sum from 0 to ∞ and use Cauchy’s product formula for series, from (2.8) we formally obtain −a0 + Φ(x) = CxΦ 2 (x),

(2.10)

where ∞ An x n . Φ(x) = n=0

The solution to (2.10) that reduces to a0 at x = 0 is given by Φ(x) =

1 1− 2Cx

1 − 4a0Cx ,

which has an analytic branch provided 4a0Cx < 1. The lemma then follows from this fact, from (2.9) and from the inequality 1−

1 − y y,

0 < y 1.

We are in a position to show the main result of this section. T HEOREM 2.1. Let Ω be an exterior domain of class C 2 and let u∗ ∈ W 2−1/q,q (∂Ω),

1 < q < 6/5.

There exists a positive constant λ0 > 0 such that, if for some λ ∈ (0, λ0 ], | log λ|−1 u∗ 2−1/q,q(∂Ω) < 1/16c2,

(2.11)

with c given in Theorem 1.6, then problem (2.1) has at least one solution (u, p) ∈ X2,q (Ω) × Y 1,q (Ω). This solution can be written in the form of a series u(x) =

∞

λn un (x, λ),

n=0

p(x) =

∞ λn pn (x, λ),

(2.12)

n=0

where (u0 , p0 ) is the solution to the Oseen problem (1.34) with F ≡ g ≡ 0, and, for n 0, un+1 − λ ∂u∂xn+1 = 1 div un+1 = 0 un+1 |∂Ω = 0, lim un+1 (x) = 0.

|x|→∞

n

k=0 uk

· ∇un−k + ∇pn+1

' in Ω, (2.13)

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The two series in (2.14) are converging in X2,q (Ω) and Y 1,q (Ω), respectively. Furthermore, the solution satisﬁes the estimate u!λ,q + |u|2,q + |p|1,q 2cλ2(1−1/q)| log λ|−1 u∗ 2−1/q,q(∂Ω).

(2.14)

Finally, if (u1 , p1 ) ∈ X2,q (Ω) × Y 1,q (Ω) is another solution corresponding to the same data and such that λ−2(1−1/q) u1 !λ,q < 1/8c,

(2.15)

then u = u1 and p = p1 . P ROOF. Let us temporarily set ε = λ on the right-hand side of (2.1) and consider ε as a positive small parameter. We then look for a solution to (2.1) of the form u(x) =

∞ εn un (x, λ),

p(x) =

n=0

∞

εn pn (x, λ),

(2.16)

n=0

where the coefﬁcients u0 , p0 and un+1 , pn+1 , n 0, satisfy the conditions stated in the theorem. We shall show that (2.16) are converging in X2,q (Ω) × Y 1,q (Ω) also for ε = λ. Applying the results of Theorem 1.6 to problem (2.13), and taking into account Lemma 1.6 we ﬁnd that, for some λ0 > 0 and all λ ∈ (0, λ0 ], Un+1 4cλ−1−2(1−1/q)

n Uk Un−k ,

n 0,

(2.17)

k=0

where Un = un !λ,q + |un |2,q + |pn |1,q ,

n 0.

Moreover, again by Theorem 1.6, we have U0 cλ2(1−1/q)| log λ|−1 u∗ 2−1/q,q(∂Ω).

(2.18)

We thus obtain that the sequence {Un }n∈N veriﬁes the assumptions of Lemma 2.1 with C = 4c λ−1−2(1−1/q). From (2.18) it then follows that the condition 4U0 Cε < 1 is satisﬁed if 16c2 λ−1 ε| log λ|−1 u∗ 2−1/q,q(∂Ω) < 1. Thus, the series (2.16) will converge for ε = λ if condition (2.11) holds. Moreover, in view of (2.7) and (2.18), we also recover the estimate (2.14). The existence proof is thus completed. It remains to show uniqueness. Denote by (u1 , p1 ) ∈ X2,q (Ω) × Y 1,q (Ω) another

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solution corresponding to the same data, and set w = u − u1 , π = p − p1 . We then have that w, π satisfy the following problem ∂w w − λ ∂x = ∇π + F 1

'

div w = 0

in Ω,

w|∂Ω = 0, lim w(x) = 0,

|x|→∞

where F = λ(w · ∇u1 + u · ∇w). From Theorem 1.6 and Lemma 1.6 it follows that w!λ,q 4cλ−2(1−1/q) w!λ,q u1 !λ,q + u!λ,q .

(2.19)

By a direct computation that uses (2.11) and (2.14), we ﬁnd λ−2(1−1/q) u!λ,q 1/8c, and so, from this inequality and from (2.15), we obtain 4cλ−2(1−1/q) u1 !λ,q + u!λ,q < 1, so that (2.19) implies w ≡ 0, thus completing the proof of the theorem.

R EMARK 2.1. An important question that the previous theorem leaves open is that of whether or not the solution there constructed is unique in the class of solutions (u1 , p1 ) merely belonging to X2,q (Ω) × Y 1,q (Ω), but not necessarily satisfying the smallness condition (2.15). R EMARK 2.2. It is veriﬁed at once that the solutions of Theorem 2.1 satisfy the energy equality: 2 Ω

D(v)2 =

∂Ω

(v∗ − ξ ) · T (v, p) · n.

This is immediately established by multiplying (2.2)1 by u ≡ v −ξ , integrating by parts and using the asymptotic properties following from the fact that (u, p) ∈ X2,q (Ω) ∩ Y q (Ω).

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2.2. Limit of vanishing Reynolds number In this section we collect some results related to the behavior of solutions determined in Theorem 2.1 in the limit λ → 0. In fact, these results are quite similar to those obtained for the Oseen linearization in Section 1.3. Speciﬁcally, we have the following theorem, for whose proof we refer to [21], Theorem X.7.1. T HEOREM 2.2. Let the assumptions of Theorem 2.1 hold and let u, p be the solution constructed in that theorem. Moreover, let (u0 , p0 ), u0 ∈ D 1,2 (Ω), be the uniquely determined solution to the Stokes problem (1.63)1–3; see Section 1.3. Then, as λ → 0, (u, p) tends to (u0 , p0 ), uniformly on compact sets, together with their ﬁrst and second derivatives. Furthermore, there is a u∞ ∈ R2 such that lim u0 (x) = u∞ ,

|x|→∞

(2.20)

and we have lim m(u)| log λ| = 4πu∞ ,

λ→0

(2.21)

where m(u) =

T (u, p) · n. ∂Ω

Finally, the limit process preserves the prescription at inﬁnity, i.e., u∞ = 0 if and only if the data satisfy condition (1.66). An interesting consequence of this theorem is the derivation of an asymptotic formula (in the limit of vanishing Reynolds number) for the force F ≡ −m(u) exerted by the ﬂuid on a body moving in it with constant velocity e1 . Speciﬁcally, taking u∗ ≡ −e1 , from the results of the ﬁrst part of Section 1.3, we have that the limit solution u0 is identically equal to −e1 , and so from (2.21) it follows, in the limit λ → 0, that F = 4πe1 + o(1) | log λ|−1 , (2.22) where o(1) denotes a vector quantity tending to zero with λ. This formula shows that in the limit of vanishingly small Reynolds number, the total force exerted from the ﬂuid on the body is determined entirely by the velocity at inﬁnity e1 and that it is directed along the line of this vector (only “drag” and no “lift”). Surprisingly enough, it does not depend on the shape of the body. This type of problem has been addressed also in [43–45]. 2.3. Perturbation theory at ﬁnite Reynolds number Let us suppose that we know the existence of a solution (u0 , p0 ) to (2.1) in the class X2,q (Ω) × Y 1,q (Ω) corresponding to a certain λ0 (> 0). Our objective in this section

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is to investigate the existence of a solution to (2.1) corresponding to a Reynolds number in a neighborhood of λ0 . Therefore, writing λ = λ0 + ε, |ε| ε0 , ε0 > 0, we look for a solution to (2.1) of the form u = u0 + w, p = p0 + φ, where w and φ satisfy the following boundary-value problem ⎫ ∂w w − λ0 ∂x − λ0 (u0 · ∇w + w · ∇u0 ) ⎪ ⎪ 1 ⎪ ⎪ ∂u0 ⎪ ∂w = ε ∂x1 + ∂x1 + u0 · ∇w + w · ∇u0 + u0 · ∇u0 ⎬ ⎪ ⎪ ⎪ + (λ0 + ε)w · ∇w + ∇φ ⎪ ⎪ ⎭ div w = 0

in Ω, (2.23)

w|∂Ω = 0, lim w(x) = 0.

|x|→∞

We have the following theorem. T HEOREM 2.3. Let Ω be an exterior domain of class C 2 and let (u0 , p0 ) be a solution to (2.1) with u0 ∈ X1,q (Ω), 1 < q < 6/5. Then, if dimNλ0 ,u0 = 0 (see Theorem 1.8), there exists ε0 > 0 such that problem (2.23) has at least one solution (w, φ) ∈ 1,q [X2,q (Ω) ∩ X0 (Ω)] × Y 1,q (Ω) for all −ε0 ε ε0 . Moreover, this solution can be 1,q expressed as power series in ε, converging in the space [X2,q (Ω) ∩ X0 (Ω)] × Y 1,q (Ω). P ROOF. We look for a solution in the form w(x) =

∞

εk wk (x),

k=0

φ(x) =

∞

εk φk (x).

(2.24)

k=0

Formally replacing these expressions in (2.23) and equating to zero the terms of equal power in ε, we ﬁnd, for all k 1, ∂w wk − λ0 ∂x − λ0 (u0 · ∇wk + wk · ∇u0 ) = Fk + ∇φk 1

div wk = 0

' in Ω, (2.25)

wk |∂Ω = 0, lim wk (x) = 0,

|x|→∞

where ∂wk−1 Fk = + λ0 wi · ∇wk−i + wi · ∇wk−1−i , ∂x1 k−1

k−1

i=1

i=0

k 1.

(2.26)

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113

From the assumptions, Theorem 1.8, and (2.25) and (2.26), we obtain, for k = 1, |w1 |2,q + w1 !λ,q + |φ1 |1,q M,

(2.27)

where the positive constant M depends only on u0 and λ0 . We want to show that there is a (sufﬁciently large) constant C such that |wk |2,q + wk !λ,q + |φk |1,q M C k−1 k −2

for all k 1.

(2.28)

We will use an induction argument that we have learned from [6]. Clearly, in view of (2.27), condition (2.28) is true for k = 1. Thus, assuming that Wi ≡ |wi |2,q + wi !λ,q + |φi |1,q MC i−1 i −2

(2.29)

for all 1 i k − 1, k 2, we have to prove that (2.29) holds also for i = k. From (2.29), Theorem 1.8 and Lemma 1.6, we obtain , Wk M1 MC k−2 (k − 1)−2 k−1 + M2 C i−1 C k−i−1 i −2 (k − i)−2 i=1 k−1 + C i−1 C k−i−2 i −2 (k − 1 − i)−2

(2.30)

,

i=0

where M1 depends only on u0 and λ0 . Observing that (k 2) k

2

k−1

i

−2

(k − i)

−2

i=1

k−1 −2 −2 + + i (k − 1 − i) i=0

k2 (k − 1)2

with a constant c0 independent of k, from (2.30) we ﬁnd MM1 MM1 k−1 −2 M1 c0 + c0 + Wk MC k c0 . C C C2

c0 ,

(2.31)

Recalling that M, M1 depend only on u0 and λ0 , we can choose C so large that the quantity in brackets in (2.31) is less than 1. In this way we obtain (2.29) also for i = k, and the induction proof is completed. From the estimate (2.29) for the coefﬁcients of the se1,q ries (2.24), we deduce that these series in [X2,q (Ω) ∩ X0 (Ω)] and Y 1,q (Ω), respectively, are both bounded from above by the numerical series ∞

k M C|ε| k −2 . C k=0

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Thus, they both converge if C|ε| 1. The proof of the theorem is completed.

R EMARK 2.3. According to the theorem just proved, we are able to obtain the solution (u, p) to the boundary-value problem (2.1) by analytic continuation with respect to the Reynolds number λ. The process will break if, for a certain λ0 , either u!λ,q → ∞ as λ → λ− 0 , or the problem (1.68) has a nonzero solution. In this latter case, bifurcation may occur. Sufﬁcient conditions for the occurrence of bifurcation are given in [26], Section 7.

3. The nonlinear problem: On the solvability for arbitrary Reynolds number Since the appearance of the seminal paper of J. Leray in 1933 [33], it is known that the system of equations (1)1,2,3 possesses at least one solution, provided the boundary value v∗ satisﬁes the zero-outﬂow condition Φ≡ ∂Ω

v∗ · n = 0.

(3.1)

This solution presents two important properties: (i) it is smooth in Ω (v, p ∈ C ∞ (Ω)) and (ii) it exists for all values of the Reynolds number λ; see Section 3.1. The main, basic question that Leray left open (see [33], pp. 54–55) was the proof of whether or not this solution satisﬁes the condition at inﬁnity (1)4 . In fact, concerning the asymptotic behavior of the solution, he was only able to prove that the velocity ﬁeld is in D 1,2 (Ω), that is, ∇v : ∇v M,

(3.2)

Ω

where M is a constant depending only on Ω, v∗ and λ. As we know from Lemma 1.2, this property alone is not enough to control the behavior at inﬁnity of the velocity ﬁeld. Apparently, Leray’s problem did not catch the attention of mathematicians for more than forty years, till when, in a series of remarkable papers, Gilbarg and Weinberger ﬁrst [29,30], and then Amick [2,3] investigated in great detail if and when a Leray’s solution (and, more generally, a solution with velocity ﬁelds in D 1,2 (Ω)) satisﬁes the prescription at inﬁnity (1)4 . Speciﬁcally, in the case when v∗ = 0, the above authors showed the validity of the following assertions; see Section 3.2. (i) Every solution to (1)1–3 that satisﬁes (3.2) (and so, in particular, every Leray’s solution) is uniformly pointwise bounded. (ii) For every solution to (1)1–3 that satisﬁes (3.2), there exists ξ˜ ∈ R2 such that lim

|x|→∞ 0

v |x|, θ − ξ˜ 2 dθ = 0.

2π

More information about ξ˜ can be obtained if B (≡ Ω c ) is symmetric around the direction of ξ (e1 , say). This means that (x1 , x2 ) ∈ ∂B implies (x1 , −x2) ∈ ∂B. In such a case, one

Stationary Navier–Stokes problem in a two-dimensional exterior domain

115

can show the existence of symmetric solutions v = (v1 , v2 ), p, that is, v1 (x1 , x2 ) = v1 (x1 , −x2 ), v2 (x1 , x2 ) = −v2 (x1 , −x2 ),

(3.3)

p(x1 , x2 ) = p(x1 , −x2 ), provided v∗ veriﬁes the same parity properties as v does. For symmetric solutions one then proves that ξ˜ = αξ , for some α ∈ [0, 1] [22], and that lim v(x) = ξ˜ ,

|x|→∞

uniformly;

(3.4)

see [2]. Actually, in Section 3.2, we shall furnish a new (and simpler) proof of (3.4) that extends to ﬂows that are not necessarily symmetric. Even though the above results represent a signiﬁcant contribution to the original achievement of Leray, the fundamental, outstanding question remains still open: Does v satisfy the condition at inﬁnity (1)4 ? Or, in other words, can we show that ξ˜ = ξ ? Notice that the possibility that ξ˜ = 0 is not excluded. A positive answer to this question would imply that (1) has a solution for arbitrary large Reynolds numbers. In this connection, we observe that, recently, Galdi has given another contribution to the problem, in the case of symmetric solutions [22]. Speciﬁcally, he has shown that if the problem (1) with v∗ = ξ = 0, has only the zero solution in the class of solutions satisfying (3.2) and (3.3), then problem (1) has at least one symmetric solution in a range of Reynolds numbers belonging to an unbounded set M of the positive real axis. This result will be described in detail in Section 4. Interestingly enough, we are able to give some results of existence for all Reynolds numbers, if ξ = 0. These results require that B is symmetric with respect to two orthogonal directions, and that the boundary data v∗ satisfy suitable parity conditions. We shall give a simple, self-contained proof of this fact in Section 3.3. The proof would also follow from much more elaborated arguments presented in Section 3.4. These results are interesting in that, as we emphasized in the introduction to Section 3, the case ξ = 0 is a completely unexplored territory, even for “small” (nonvanishing) Reynolds number.

3.1. Existence: Leray method As mentioned in the previous section, Leray was the ﬁrst to show existence of regular solutions to the Navier–Stokes system (1)1–3 for arbitrary values of the Reynolds number λ [33]. In this section we shall brieﬂy describe Leray’s method of constructing solutions, and recall some of their properties that we shall use later on. In the rest of this article, with the exception of Section 3.3, we will be concerned with the physically relevant case when ξ = 0. With this in mind, we ﬁnd it convenient to rewrite (1) in a different form, that is obtained by introducing the new velocity ﬁeld u = λ v. If we do

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this, and continue to denote by v the rescaled velocity ﬁeld, and moreover we take, without loss, ξ = e1 , we then obtain that (1) can be re-written as follows ' v = v · ∇v + ∇p in Ω, div v = 0 (3.5) v|∂Ω = v∗ along with the condition at inﬁnity lim v(x) = λe1 .

(3.6)

|x|→∞

A solution to (3.5) and (3.6) was sought by Leray [33] by means of the following procedure of “invading domains”. Let {Rk }k∈N be an unbounded, increasing sequence of positive numbers, with Rk > 1. For each k, consider the sequence of problems: vk = vk · ∇vk + ∇pk

'

div vk = 0

(3.7)

vk |∂Ω = v∗ , vk (x) = λe1

in ΩRk ,

at |x| = Rk .

Leray’s proof is based on the observation that every solution to (3.7) formally obeys the following a priori estimate ∇vk : ∇vk M, (3.8) ΩRk

where M depends only on Ω, v∗ and λ, but not on k. Such a uniform bound, along with Odqvist estimates for the Green’s tensor of the Stokes problem in bounded domains [36], and what we call nowadays the Leray–Schauder theorem [34] allowed Leray to prove existence of a regular solution to (3.7) for all k ∈ N, provided ∂Ω and v∗ have a suitable degree of smoothness. Letting k → ∞ and using the uniform bound (3.8), one can then show the existence of a regular solution to (3.5), whose velocity ﬁeld has a ﬁnite Dirichlet integral. If B is symmetric around the x1 -axis, this method delivers symmetric solutions, in the sense of (3.3). If we use this procedure along with well-known regularity theory for the classical Stokes problem in a bounded domain (see, e.g., [21]), we can reformulate the original result of Leray in the following convenient form. T HEOREM 3.1. Assume that Ω is of class C 3 and that v∗ ∈ W 3−1/s,s (∂Ω), s > 3. Then, there exist a subsequence of {vk , pk }k∈N – that we still denote by {vk , pk }k∈N – and two ﬁelds v = (v1 , v2 ) and p such that (i) ΩR |∇vk |2 M, for some M depending only on Ω, v∗ and λ; k

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117

), p ∈ C ∞ (Ω) ∩ C 1 (Ω ), for all bounded subdomains Ω ; (ii) v ∈C ∞ (Ω) ∩ C 2 (Ω (iii) vk − vC 2 (Ω ) + pk − pC 1 (Ω ) → 0 as k → ∞; (iv) v, p satisfy (3.5); (v) v ∈ D 1,2 (Ω) and the following energy inequality holds 2 D(v) (v∗ − λe1 ) · T (v, p) · n. 2 Ω

(3.9)

∂Ω

Finally, if B (≡ Ω c ) is symmetric around the x1 -axis, in addition to the above properties we have also that v, p satisfy (3.3). R EMARK 3.1. One fundamental issue that comes with the method of Leray of invading domains (and with any other method we are aware of, like Fujita’s; see next section) is related to the physically remarkable case when v∗ = 0, and is the following one (cf. [14], p. 88): Is the solution (v, p) nontrivial? Actually, we are not assured, a priori that v is nonidentically zero. As a matter of fact, Leray’s construction in the linear case would lead to an identically vanishing solution, as a consequence of the Stokes paradox. To see this, let us disregard in (3.7) the nonlinear term vk · ∇vk for each k ∈ N, and take v∗ = 0 as well. Applying Leray’s procedure, we then obtain that the limit ﬁeld, v (s) say, solves the Stokes problem with zero boundary data and that, in view of property (i), v (s) has a ﬁnite Dirichlet integral. Therefore, by Lemma 1.3 we infer v (s) ≡ 0. In the general nonlinear case, the answer to the question is still unknown. However, in Section 4, we shall prove that v is nontrivial at least for symmetric ﬂow, a fact ﬁrst discovered by Amick [2], §4.2. R EMARK 3.2. equality 2 Ωk

The (approximating) solutions vk , pk to (3.7), satisfy the following energy D(vk )2 =

∂Ω

(v∗ − λe1 ) · T (vk , pk ) · n.

This is at once established by multiplying both sides of (3.7)1 by vk − λ e1 , and integrating by parts over ΩRk . Notice that in the limit k → ∞, the energy equality is lost and we only obtain an energy inequality; see (3.9). This loss is essentially due to the little (uniform) information that the approximating solutions bring about their behavior at inﬁnity. Should one be able to show that the limit solution satisﬁes the energy equality, the problem of existence for arbitrary Reynolds numbers would be “almost” solved, at least in the class of symmetric solutions; see Remark 4.1.

3.2. Existence: Fujita method An alternative method of constructing solutions to (3.5), based on the so called “Galerkin approximation” was introduced by H. Fujita [17] in 1961. We will sketch it in the following, referring to [21], Chapter X, for details. Throughout this section, we shall denote by (·, ·) the duality pairing in L2 (Ω).

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Assuming ∂Ω locally Lipschitzian, given an arbitrary v∗ ∈ W 1/2,2 (∂Ω) satisfying (3.1), we can ﬁnd V ∈ W 1,2 (ΩR ) ∩ D 1,2 (Ω), all R > 1, which equals v∗ at ∂Ω and λe1 at |x| = R0 , for some R0 > 1; see [21], Lemma IX.4.1 and Remark IX.4.2. Moreover, cf. [21] loc. cit., for a given γ > 0, the ﬁeld V can be chosen in such a way as to verify the following inequality (u · ∇V , u) γ |u|2

1,2

for all u ∈ D01,2 (Ω).

(3.10)

A sequence of approximating solutions to (3.5), vm = um + V is then searched in the form um =

m

ckm ψk ,

k=1

(∇um , ∇ψk ) + (um · ∇um , ψk ) + (um · ∇V , ψk ) + (V · ∇um , ψk ) = −(∇V , ∇ψk ) − (V · ∇V , ψk ),

(3.11)

k = 1, 2, . . . , m,

where {ψk }k∈N ⊂ D(Ω) is a basis of D01,2 (Ω) that is orthonormal in L2 (Ω). For each m ∈ N we may establish existence to the nonlinear system (3.10), provided we show a uniform bound for |um |1,2 in terms of V ; see [21], Lemma VIII.3.2. Multiplying (3.10)2 by ckm , summing over k from 1 to m and observing that (V · ∇um , um ) = (um · ∇um , um ) = 0

for all m ∈ N,

we obtain |um |21,2 + (um · ∇V , um ) = −(∇V , ∇um ) − (V · ∇V , um ).

(3.12)

From (3.12), using Hölder’s inequality, (3.10), and the fact that the support of ∇V is bounded, one can show the following estimate |um |1,2 C(V ),

(3.13)

where C(V ) is a positive constant depending only on V . This latter inequality, on the one hand, proves that the nonlinear system (3.11) has at least one solution ([21], Lemma VIII.3.2) and, on the other hand, it implies that there exists u ∈ D01,2 (Ω) and a subsequence, that we continue to denote by {um }m∈N such that um → u

weakly in D01,2 (Ω),

um → u

strongly in L2 Ω

(3.14)

. Passing to the limit m → ∞ in (3.11)2 and employing (3.14), for any compact Ω ⊂ Ω we easily obtain that v ≡ u + V satisﬁes the following relation (∇v, ∇ψk ) + (v · ∇v, ψk ) = 0

for all k ∈ N.

(3.15)

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119

However, given any ϕ ∈ D(Ω), both ϕ and ∇ϕ can be approximated in the uniform norm by a ﬁnite linear combination of ψk ; see [20], Lemma VII.2.1. Thus, from (3.15), we infer (∇v, ∇ϕ) + (v · ∇v, ϕ) = 0 for all ϕ ∈ D(Ω).

(3.16)

Since v ∈ W 1,2 (ΩR ) for all R > 1, from (3.16) and well-known regularity results for the Navier–Stokes equations (see, e.g., [21], Corollary VIII.5.1) we have that v ∈ C ∞ (Ω) and that there exists p ∈ C ∞ (Ω) such that (v, p) satisﬁes (3.5)1,2. Moreover, v assumes the boundary data v∗ in the sense of trace. Also, in view of (3.13), we ﬁnd that v ∈ D 1,2 (Ω), which, as in Leray’s method, is the only information that Fujita’s method provides about the asymptotic behavior of the solution. Finally, using (3.11), we can show that v satisﬁes the energy inequality (3.9). The same type of argument would lead to a symmetric solution, in case when B is symmetric around the x1 -axis. This is achieved by using, instead of D01,2 (Ω), its subspace constituted by vector ﬁelds satisfying the parity condition (3.3). R EMARK 3.3. As in the case of Leray’s construction, the solution v just constructed with the Galerkin approximation may reduce, when v∗ = 0 to the trivial one v ≡ 0. In fact, we recall that v is of the form u + V , with V an extension of λ e1 and u ∈ D01,2 (Ω). In dimension 2 the ﬁeld V belongs to D01,2 (Ω), since D01,2 (Ω) contains also the functions that are constant in a neighborhood of inﬁnity; see Lemma 1.2. Thus, the possibility u = −V can not be ruled out, which would give v ≡ 0. While, as mentioned in Remark 3.1, one can show that symmetric solutions constructed with the method of Leray are nontrivial (Section 4), it is not known if the same conclusion can be drawn for the same type of solutions constructed via the Galerkin approximation. 3.3. Some existence results when ξ = 0 As we mentioned in the introduction to Section 2, it is not known if (1) possesses a solution if ξ = 0, in the case when v∗ is arbitrarily (sufﬁciently smooth) prescribed. However, it is not difﬁcult to show that if B is symmetric with respect to two orthogonal directions and v∗ satisﬁes suitable parity conditions, (1) with ξ = 0 has at least one solution for every value of λ. Speciﬁcally, assuming that these directions coincide with the x1 and x2 axes, respectively, we suppose (x1 , −x2 ) ∈ ∂B, (x1 , x2 ) ∈ ∂B → (3.17) (−x1 , x2 ) ∈ ∂B and that v∗1 (x1 , x2 ) = −v∗1 (−x1 , x2 ) = v∗1 (x1 , −x2 ), (3.18) v∗2 (x1 , x2 ) = v∗2 (−x1 , x2 ) = −v∗2 (x1 , −x2 ). In order to prove the existence result, we need two simple lemmas.

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L EMMA 3.1. Let Ω be a locally Lipschitzian, exterior domain, and let u ∈ D01,2 (Ω) satisfying either (i) u(x1 , x2 ) = −u(x1, −x2 ) or (ii) u(x1 , x2 ) = −u(−x1 , x2 ). Then, there exists c = c(Ω) > 0 such that Ω

u2 |x|

2

c

|∇u|2 .

(3.19)

Ω

P ROOF. We ﬁrst assume Ω = R2 and condition (i). The proof under condition (ii) is exactly the same, with the change x1 → x2 . Let ψ be a nondecreasing function that is zero in a neighborhood of ∂Ω and is one for sufﬁciently large |x|, and set w = ψu. Since, by hypothesis u(x1 , 0) = 0 for all x1 ∈ Ω, we ﬁnd that w(x1 , 0) = 0 for all x1 ∈ R. Thus, by the Hardy inequality, it follows that

w2 |x|2

x2 >0

w2

x22

x2 >0

|∇w|2 .

c x2 >0

By the properties of ψ, we ﬁnd that

|u|2 ,

|∇w|2 c

|∇u|2 +

x2 >0

Ω

K

where K is a bounded subset containing the support of ∇ψ. Thus, from the Hardy inequality, we obtain

w2 |x|2

x2 >0

|u|2 .

c

|∇u|2 + Ω

K

Since an analogous inequality holds for the half-plane {x2 < 0}, and since u(x) = w(x), for all sufﬁciently large |x|, |x| > R, say, we conclude

u2

x2 >R

|x|2

|u| .

2

c

2

|∇u| + Ω

(3.20)

K

Recalling that u|∂Ω = 0 we obtain that u obeys the following Poincaré inequality, for all ρ>1

|∇u| ,

2

2

u c Ωρ

Ωρ

where c = c(ρ) > 0; see, e.g., [20], Exercise II.4.10. If Ω = R2 , the lemma then follows from this latter inequality and (3.20). If Ω = R2 , the proof goes exactly as before, without the use of the function ψ.

Stationary Navier–Stokes problem in a two-dimensional exterior domain

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R EMARK 3.4. As we shall show in the following lemma, functions satisfying (3.19) tend to zero at inﬁnity in a suitable sense. If u merely belongs to D01,2 (Ω) and does not necessarily satisfy the parity conditions of Lemma 3.1, the following weaker inequality holds

u2 Ω

|x|2 log2 (|x|)

|∇u|2

c Ω

if Ω c = R2 , which does not prevent u from growing logarithmically fast at large distances; see Lemma 1.2. L EMMA 3.2. Let the assumptions of Lemma 3.1 be satisﬁed. Then

u(r, θ )2 dθ = 0.

2π

lim

r→∞ 0

(3.21)

P ROOF. We shall again assume condition (i) of Lemma 3.1, since the proof goes exactly the same way if (ii) is assumed instead. From the fact that u ∈ D 1,2 (Ω), we have

2k+1 2π 2k

0

1 ∂u(rθ ) 2 dθ dr → 0 as k → ∞. r ∂θ

Moreover, by the mean value theorem, there is rk ∈ (2k , 2k+1 ) such that

0

2π ∂u(r θ ) 2 k 1 ∂θ dθ log 2

2k+1 2π 2k

0

1 ∂u(rθ ) 2 dθ dr. r ∂θ

Therefore,

0

2π ∂u(r θ ) 2 k ∂θ dθ

→0

as k → ∞.

(3.22)

However, by condition (i) in Lemma 3.1, for all sufﬁciently large r we have u(r, 0) = 0, and so 2π ∂u(rk θ ) 2 2 max u(rk , θ ) c ∂θ dθ , θ∈[0,2π] 0 which, in turn, by (3.22) furnishes max u(rk , θ ) → 0 as k → ∞. θ∈[0,2π]

Set χ(r) = 0

2π ∂u(r θ ) 2 k

∂θ

dθ.

(3.23)

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G.P. Galdi

For any r ∈ (rk , rk+1 ), we have χ(r) χ(rk ) +

dr , dr

rk+1 dχ rk

χ(rk ) → 0 as k → ∞.

(3.24)

Using Cauchy inequality, we get 2π 2 2π 2 2π dχ ∂u |u| ∂u √ u r dθ 2 dθ + √ r dθ dr ∂r r r ∂r 0 0 0 and so, from Lemma 3.1, we deduce χ ∈ L1 (r0 , ∞). The lemma then follows from this property and from (3.24). With Lemma 3.1 in hands, we can then prove the following existence result. T HEOREM 3.2. Let Ω and v∗ satisfy the assumptions (3.17) and (3.18). Assume, moreover, that Ω is locally Lipschitzian and that v∗ ∈ W 1/2,2 (∂Ω). Then, for any λ = 0,8 problem (1) has at least one solution (v, p) ∈ C ∞ (Ω) × C ∞ (Ω) that satisﬁes (1)3 in the trace sense and (1)4 in the following sense lim

r→∞ 0

v(r, θ )2 dθ = 0.

2π

(3.25)

P ROOF. Using, for instance, Fujita method restricted to the subspace of D01,2 (Ω) constituted by vector ﬁeld satisfying parity properties similar to (3.18), we can ﬁnd a pair (v, p) that solves (1)1,2,3 in the sense speciﬁed in the theorem. Moreover, since v1 satisﬁes condition (i) of Lemma 3.1 and v2 satisﬁes condition (ii) of the same lemma, from Lemma 3.2 we deduce the validity of (3.25), and the result follows. R EMARK 3.5. A simple example where the symmetry assumptions of Theorem 3.2 are satisﬁed, is given by the case when B is the unit disk and v∗ = (x1 f (θ ), x2 g(θ )) where f and g are even functions of θ . Notice that the solutions of Hamel given in (2.5) satisfy all these requirements.

3.4. On the pointwise asymptotic behavior of D-solutions We now draw attention to the behavior at inﬁnity of a solution (v, p) to (3.5). The only assumption we shall make a priori on (v, p), is that v ∈ D 1,2 (Ω). Usually, these solutions are referred to as D-solutions. Solutions constructed by Leray in Theorem 3.1 and by Fujita in Section 3.2 are D-solutions. Notice that D-solutions are inﬁnitely differentiable in Ω. We shall mainly focus on the behavior of the velocity ﬁeld itself, referring to Remark 3.5 and to Lemma 3.3 for information regarding the behavior of the derivatives of v, and of p and its derivatives. 8 In fact, the result continues to hold also for λ = 0; see Section 1.2.1.

Stationary Navier–Stokes problem in a two-dimensional exterior domain

123

Our study will be done through a number of intermediate steps due, mostly, to Gilbarg and Weinberger [30] and to Amick [2]. We shall give here only the main ideas, referring the reader to those papers and to [21], Section X.3, for full details. The ﬁrst result concerns the pointwise convergence of the pressure ﬁeld p at large distances. L EMMA 3.3. Let (v, p) be a D-solution. Then, there exists p0 ∈ R such that lim p(x) = p0 .

|x|→∞

P ROOF. See [30], §4, and [21], Theorem X.3.3.

In order to investigate the behavior at inﬁnity of the velocity ﬁeld v, we begin to prove that v is uniformly bounded. To this end, we notice that, deﬁning the total head pressure as 1 Φ = p + |v|2 , 2 and the vorticity as ω=

∂v1 ∂v2 − , ∂x2 ∂x1

(3.26)

by a simple calculation based on (3.5)1,2, we show that Φ − v · ∇Φ = ω2 .

(3.27)

Consider now (3.27) in Ωρ1 ,ρ2 , for arbitrary ρ1 , ρ2 , with ρ2 > ρ1 > ρ0 , ρ0 sufﬁciently large. We may then apply Hopf’s maximum principle and obtain that Φ can not attain a maximum in Ωρ1 ,ρ2 , unless it is a constant. It also follows that max Φ(r, θ )

θ∈[0,2π]

has no maximum. We thus deduce that # $ 2 1 lim max p(r, θ ) + v(r, θ ) ≡A r→∞ θ∈[0,2π] 2 exists, implying, by Lemma 3.3 that9 lim

√ max v(r, θ ) = 2A ≡ L.

r→∞ θ∈[0,2π]

9 We assume, without loss, that p = 0. 0

(3.28)

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G.P. Galdi

However, we do not know if L is ﬁnite or inﬁnite. As a consequence, the maximum principle is not enough to obtain the boundedness of v, and we need more information about the function Φ. In the particular case v∗ ≡ 0 this is achieved through the following profound result, due to C.J. Amick [2], Theorem 11, that we shall state without proof. L EMMA 3.4. Let (v, p) be a D-solution to (3.5) corresponding to v∗ ≡ 0. Then there exists a Jordan arc, ρ , γ : t ∈ [0, 1) → γ (t) ∈ Ω such that (i) γ (0) ∈ ∂Ω ρ ; (ii) |γ (t)| → ∞ as t → 1. In addition, the function Φ is monotonically decreasing along γ , namely, Φ γ (t) < Φ γ (s)

for all s, t ∈ [0, 1), s < t.

(3.29)

With this result in hand, we can show the following one. L EMMA 3.5. Let v and v∗ be as in Lemma 3.4. Then v ∈ L∞ Ω ρ , and there is an L ∈ [0, ∞) such that lim max v(x) = L, uniformly.

(3.30)

P ROOF. Since p(x) tends to zero for large |x|, by (3.29) we deduce that v γ (t) c for all t ∈ [0, 1),

(3.31)

|x|→∞ θ∈[0,2π)

with c independent of t. Using the assumption that v ∈ D 1,2 (Ω), we have that 2k+1 2π

2k

0

1 ∂v 2 dθ dr → 0 as k → ∞, r ∂θ

implying 0

2π ∂v(r , θ ) 2 k

∂θ

dθ → 0

as k → ∞,

(3.32)

for some sequence {rk } with rk ∈ (2k , 2k+1 ). Since γ is connected and extends to inﬁnity, for any k ∈ N we can ﬁnd at least one tk ∈ [0, 1) such that γ (tk ) = (rk , θk ),

Stationary Navier–Stokes problem in a two-dimensional exterior domain

125

for some θk ∈ [0, 2π). Thus, in view of (3.31), it follows that v(rk , θk ) c

for all k ∈ N.

(3.33)

From the identity

θk

v(rk , θ ) = v(rk , θk ) −

∂v(rk , τ ) dτ, ∂τ

θ

and from (3.32) and (3.33), we ﬁnd max v(x) c1

x∈∂Brk

for all k ∈ N,

(3.34)

with c1 independent of k. We next apply the maximum principle to (3.27) in the annulus Ωrk ,rk+1 to ﬁnd max

x∈Ωrk ,rk+1

Φ(x) ≡

max

x∈Ωrk ,rk+1

2 1 Φ(x). p(x) + v(x) max x∈∂Brk ∪∂Brk+1 2

(3.35)

However, by (3.34) and Lemma 3.3 we deduce max

x∈∂Brk ∪∂Brk+1

Φ(x) c2

for all k ∈ N,

so that, again Lemma 3.3 and (3.35) deliver 2 max v(x) c3

x∈Ωrk ,rk+1

for all k ∈ N,

with a constant c3 independent of k. Therefore, v ∈ L∞ (Ω ρ ). Since the second part of the lemma is an immediate consequence of the ﬁrst and (3.28), the proof is complete. A conclusion similar to that of Lemma 3.5 can be reached by a more elementary proof that does not use Lemma 3.4, in the case when (v, p) is a solution constructed with Leray’s method. Actually, we have the following result whose proof can be found in [29], §2. L EMMA 3.6. Let (v, p) be a solution to (3.5) in the sense of Theorem 3.1. Then, there exists a constant C0 > 0 independent of k ∈ N, such that vk (x) C0

for all x ∈ Ω3Rk /4 .

Thus, from Theorem 3.1(iii) we ﬁnd, in particular, v ∈ L∞ (Ω). We shall next investigate if v approaches some vector ξ˜ at inﬁnity. We have the following two possibilities:

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G.P. Galdi

(i) the number L in (3.28) is zero; (ii) the number L in (3.28) belongs to (0, ∞].10 In case (i) we have lim v(x) = 0

|x|→∞

uniformly,

and we deduce at once that ξ˜ = 0. On the other hand, if L > 0, using the ideas of Gilbarg and Weinberger [30], §5, we proceed as follows. We set 1 f¯ = 2π

2π

f (r, θ ) dθ,

(3.36)

0

and recall the Wirtinger inequality

f (r, θ ) − f¯(r)2 dθ

2π

0

dθ.

2π ∂f (r, θ ) 2 0

∂θ

(3.37)

We need two preliminary lemmas. L EMMA 3.7. Let v and v∗ be as in Lemma 3.4. Then

2π 2 dθ = 0, ¯ (i) limr→∞ 0 |v(r, θ ) − v(r)| ¯ = L, (ii) limr→∞ |v(r)| where L is deﬁned in (3.28). P ROOF. By the Wirtinger inequality (3.37) and the Cauchy inequality, we have 2π 2π 2 d ∂v v(r, θ ) − v(r) ¯ dθ = 2 (v − v) ¯ · dθ dr ∂r 0 0 2π #

r|∇v|2 +

0

2π

c

r|∇v|2 dθ ;

0

therefore, lim

2 v(r, θ ) − v(r) ¯ = ∈ [0, ∞).

2π

r→∞ 0

However, again by (3.37), we have

∞ ρ

1 r

$ |v − v| ¯2 r dθ r2

2 v(r, θ ) − v(r) ¯ dθ dr < ∞,

2π 0

10 Of course, by Lemma 3.5, if v ≡ 0 it follows that L < ∞. ∗

Stationary Navier–Stokes problem in a two-dimensional exterior domain

127

which implies = 0, and (i) is proved. To show (ii), we observe that (3.30) implies that, given any sequence {rk } ⊂ R+ with rk → ∞, there is a corresponding sequence {θk } ⊂ [0, 2π) such that (3.38) lim v(rk , θk ) = L. rk →∞

However, as in the proof of Lemma 3.5, we show the existence of a sequence rk ∈ (2k , 2k+1 ) such that (3.32) holds. Since v(rk , θ ) − v(rk , θk )2 2π

2π ∂v(r , τ ) 2 k

0

dτ,

∂τ

by (3.38) and the triangle inequality, we ﬁnd that lim v(rk , θ ) = L uniformly in θ.

(3.39)

rk →∞

Moreover, since

2π

v(rk , θ ) − v(r ¯ k ) dθ = 0 for all k ∈ N,

0

it follows that 2 v(rk , θ ) − v(r ¯ k ) 2π

2π ∂v(r , τ ) 2 k

0

∂τ

dτ,

which together with (3.32) and (3.39) allows us to conclude that ¯ k ) = L. lim v(r

(3.40)

k→∞

Now, for r ∈ (rk , rk+1 ) we have 2 1 r v(r) ¯ − v(r ¯ k ) = 2π

2π

rk 0

1 (2π)

2

2 ∂v dr dθ ∂r

rk+1 2π rk

0

1 dr dθ r

Ω rk

|∇v|2

1 log 2|v|21,2,Ω rk , 2π

and so, in view of (3.40), the property (ii) follows by letting k → ∞ in this inequality. L EMMA 3.8. Let v be as in Lemma 3.4 and assume that the number L in (3.28) is ﬁnite. Then r 1/2 ∇ω ∈ L2 Ω ρ .

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G.P. Galdi

P ROOF. From (3.5)1 we ﬁnd ω − v · ∇ω = 0.

(3.41)

Let ψR = ψR (x) be a “cut-off” function that is 1 for R > |x| and is 0 for |x| > 2R, and satisﬁes |D α ψR | C/R |α| , |α| = 1, 2, for a constant C independent of R. Setting ηR = rψR , multiplying (3.41) by ηR ω and integrating by parts over Ω ρ we deduce that ηR |∇ω|2 Ωρ

=

1 2

Ωρ

ω2 (ηR + v · ∇ηR ) +

1 2

∂Bρ

∂ω2 − v · nω2 . ∂n

(3.42)

By the properties of ψR , it follows that |∇ηR | + |v · ∇ηR | + |ηR | c

(3.43)

for some constant c independent of R, so that by (3.43), identity (3.42) gives ηR |∇ω|2 C Ωρ

for a constant C independent of R. Letting R → ∞ and using the monotone convergence theorem completes the proof. We are now in a position to prove a ﬁrst result on the behavior of the velocity ﬁeld of a D-solution at inﬁnity. T HEOREM 3.3. Let (v, p) be a D-solution to (3.5) and let L ∈ [0, ∞] be the number deﬁned in (3.28). Then, if L < ∞ (this certainly happens whenever v∗ ≡ 0), there is ξ˜ ∈ R2 such that 2π v(r, θ ) − ξ˜ 2 dθ = 0. lim (3.44) r→∞ 0

If L = ∞, lim

r→∞ 0

v(r, θ )2 dθ = ∞.

2π

P ROOF. Let ψ = ψ(r) be the argument of v(r), ¯ that is, ¯ cos ψ(r), v¯1 (r) = v(r) ¯ sin ψ(r), and ψ ∈ [0, 2π). v¯2 (r) = v(r)

(3.45)

(3.46)

Stationary Navier–Stokes problem in a two-dimensional exterior domain

129

Clearly, we have ψ (r) =

v¯1 v¯2 − v¯1 v¯2 , |v|2

(3.47)

where the prime means differentiation. Multiplying the ﬁrst component of (3.5)1 by sin θ , the second component by cos θ , and adding up, for sufﬁciently large |x| we ﬁnd that ∂v2 ∂v1 1 ∂p ∂ω + v1 − v2 + = 0. ∂r ∂r ∂r r ∂θ

(3.48)

We take the average over θ of both sides of (3.48) to deduce that ∂ω + v¯1 v¯2 − v¯1 v¯2 ∂r 2π ∂v2 (r, θ ) ∂v1 (r, θ ) 1 dθ − v2 (r, θ ) − v¯2 (r) + v1 (r, θ ) − v¯1 (r) 2π 0 ∂r ∂r = 0.

(3.49)

From Lemma 3.7 we know that |v(r)| ¯ converges to L 0. Assume, for a while, that L > 0. Then we may ﬁnd ρ¯ > δ(Ω c ) such that v(r) ¯ > L/2,

for all r > ρ. ¯

(3.50)

2 and integrate over θ ∈ [0, 2π) and over We then divide both sides of (3.49) by |v(r)| ¯ r ∈ (r1 , r2 ), r2 > r1 > ρ, ¯ to obtain

ψ(r2 ) − ψ(r1 ) 1 =− 2π

r2 2π

1

0

2 |v(r)| ¯

r1

#

$ ∂ω ∂v2 ∂v1 dr dθ. + (v1 − v¯1 ) − (v2 − v¯2 ) ∂r ∂r ∂r

Using (3.50) and the Schwarz and the Wirtinger inequalities we see that ψ(r2 ) − ψ(r1 )

2 r 1/2 ω + |v|1,2,Ωr1 ,r2 2,Ωr1 ,r2 πL2 r2 2π 1/2 $ # dr × |v|1,2,Ωr1 ,r2 + r2 r1 0

and, therefore, letting r1 , r2 → ∞ and recalling Lemma 3.8, we obtain lim ψ(r) = ψ0

r→∞

(3.51)

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G.P. Galdi

for some ψ0 ∈ [0, 2π]. For L 0 we deﬁne the vector ξ˜ = (L cos ψ0 , L sin ψ0 ). If L ∈ (0, ∞), from Lemma 3.7(ii), (3.46) and (3.51), we conclude that lim v(r) ¯ = ξ˜ ,

r→∞

which along with Lemma 3.7(i) implies (3.44). If L = 0, we have ξ˜ = 0 and (3.44) follows from (3.28) even in a stronger, pointwise sense. Finally, if L = ∞, (3.45) follows directly from Lemma 3.7. Notice that, in view of Lemma 3.5, this latter circumstance cannot occur if v∗ ≡ 0. The theorem is proved. Our next task is to show that, in fact, v tends uniformly pointwise to ξ˜ . To this end we need two preliminary results. L EMMA 3.9. Let (v, p) be a D-solution and let ρ > 1. Then v ∈ D 2,2 (Ω ρ ) ∩ D 1,q (Ω ρ ) for any 2 q < ∞. P ROOF. From the identity v1 =

∂ω , ∂x2

we obtain that w = F

in R2 ,

(3.52)

where w = φρ v1 ,

F = φρ

∂ω − 2∇φρ · ∇v1 − v1 φρ , ∂x2

and φρ = 1 − ψρ , with ψρ the “cut-off” function introduced in the proof of Lemma 3.8. By Lemma 3.8 we deduce, in particular, that ω ∈ D 1,2 (Ω ρ ), and since v1 ∈ C ∞ (Ω), it follows that F ∈ L2 (R2 ). By well-known results of existence and uniqueness for the Poisson equation in the plane, we deduce w ∈ D 2,2 (R2 ) which, by the properties of φρ and the regularity of v in turn implies v1 ∈ D 2,2 (Ω 2ρ ). Since v1 ∈ D 1,2 (Ω) ∩ C ∞ (Ω), we obtain ∇v1 ∈ W 1,2 (Ω ρ ). By the Sobolev embedding theorem we thus ﬁnd ∇v1 ∈ Lq (Ω ρ ), 2 q < ∞. Since v2 = −

∂ω , ∂x1

by a similar reasoning we show ∇v2 ∈ Lq (Ω ρ ), 2 q < ∞. The proof of the lemma is, therefore, accomplished.

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131

L EMMA 3.10. Let ρ > 2, and assume u ∈ D 1,q (Ω ρ ) for some q > 2. Then, there exists a constant c depending only on q such that u(x) c

2π

u |x|, θ dθ + |u|1,q,B (x) 1

0

for all x ∈ Ω ρ .

P ROOF. The proof follows standard arguments. Let x = (r, θ ) and let (r , θ ) be a polar coordinate system with the origin at x. We have u r , θ = u(x) +

r 0

∂u(ρ, θ ) dρ. ∂ρ

(3.53)

Thus

u r , θ dθ 2π u(x) +

2π 1

2π 0

0

ρ −q /q dρ dθ

1/q

0

|u|1,q,B1(x)

q − 1 1/q 2π u(x) + 2π |u|1,q,B1 (x). q −2

(3.54)

Multiplying both sides of this inequality by r and integrating over r ∈ [0, 1] and θ ∈ [0, 2π], we get u1,B1 (x)

1 2

u(r, θ ) dθ + c1 |u|1,q,B

2π 0

1 (x)

,

(3.55)

where c1 = c1 (q) > 0. We now go back to (3.53) and, by the same arguments leading to (3.54), we obtain 2π u(x)

2π 0

u r , θ dθ + c2 |u|1,q,B (x), 1

with c2 = c2 (q) > 0. Multiplying by r and integrating over r ∈ [0, 1], we ﬁnd u(x) 1 u1,B (x) + c3 |u|1,q,B (x), 1 1 π with c3 = c3 (q) > 0. The lemma then follows from this latter inequality and (3.55).

Coupling the results of Theorem 3.3, Lemmas 3.9 and 3.10 we obtain the following. T HEOREM 3.4. Let (v, p) be a D-solution to (3.5). Then, there exists ξ˜ ∈ R2 such that lim v(x) = ξ˜ ,

|x|→∞

uniformly.

(3.56)

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G.P. Galdi

P ROOF. In view of Lemma 3.9 and Theorem 3.3, given ε > 0 there exists R0 > 1 such that |v|1,q,Ω R0 <

ε , 2c

v(r, θ ) − ξ˜ dθ < ε 2c

2π 0

for all r > R0 ,

where c = c(q) is the constant entering the inequality in Lemma 3.10. Applying Lemma 3.10 with u ≡ v − ξ˜ and ρ = R0 + 1 we thus conclude v(x) − ξ˜ < ε

for all x such that |x| > R0 + 1.

The theorem is proved.

R EMARK 3.6. The preceding theorem asserts that every solution to (3.5) with v∗ = 0, with corresponding velocity ﬁeld having a ﬁnite Dirichlet integral tends uniformly pointwise to some vector ξ˜ . The fundamental question that remains open is whether or not ξ˜ = λ e1 , so that also condition (3.6) may be satisﬁed. Actually, the vector ξ˜ can, in principle, even be zero. So, the question of solvability of (3.5)–(3.6) for “large” values of λ is still open. However, using the result of Theorem 3.4, in Section 4 we shall show that if Ω is symmetric around the x1 -axis and if a certain homogeneous problem related to (3.5)–(3.6) has only the zero solution, then problem (3.5)–(3.6) is solvable for arbitrarily large λ in the class of symmetric solutions. A crucial step in getting this result is the knowledge of a detailed asymptotic behavior of D-solutions that satisfy (3.56), which will be the object of Section 3.5. R EMARK 3.7. We would like to collect here the main results concerning the behavior at inﬁnity of the derivatives of v and p, when (v, p) is a D-solution. We begin to observe that, by Lemma 3.9, it follows that ∇v converges uniformly pointwise to zero. By using arguments similar to those employed in Lemma 3.8 and Lemma 3.9, it is possible to show that lim D α v(x) = 0,

|x|→∞

uniformly for any |α| 1;

see [21], Theorem X.3.2. Using this property along with (1)1 , one can also prove that lim D α p(x) = 0,

|x|→∞

uniformly for any |α| 1;

see [21], Theorem X.3.2. These results are silent about the rate of decay of v and p and their derivatives at large distances. If ξ˜ = 0, then, as we shall see in the next section, the ﬁelds v and p present the same asymptotic structure of the Oseen fundamental tensor. In the general case, very little can be said, and the available results concern only the ﬁrst derivatives of the velocity ﬁeld. Precisely, using Lemma 3.8 and the fact that ω, by (3.41),

Stationary Navier–Stokes problem in a two-dimensional exterior domain

133

satisﬁes the maximum principle in Ωρ1 , ρ2 , one can show that if v is bounded (as it happens when v∗ ≡ 0) then lim |x|3/4ω(x) = 0

|x|→∞

uniformly;

(3.57)

cf. [30], Theorem 5, and that, moreover, |x|3/4 ∇v(x) = 0 |x|→∞ log |x| lim

uniformly;

see [30], Theorem 7. The proof of (3.57) goes as follows. From Lemma 3.8 and from the assumption that v ∈ D 1,2 (Ω), we have that

2k+1

1 r

2k

2π

r ω + 2r 2 2

0

∂ω ω ∂θ dr dθ < ∞

3/2

for all k ∈ N,

which implies the existence of rk ∈ (2k , 2k+1 ) such that ∂ω(rk , θ ) 3/2 rk2 ω2 (rk , θ ) + 2rk ω(rk , θ ) dθ → 0 ∂θ

2π

0

as k → ∞.

(3.58)

However, ω2 (rk , θ )

1 2π

2π

ω2 (rk , ϕ) dϕ +

0

1 π

ω(rk , ϕ) ∂ω(rk , ϕ) dϕ ∂ϕ

2π 0

which, by (3.58), implies 3/2 rk ω2 (rk , θ ) → 0 as k → ∞, uniformly in θ. The result then follows by applying the maximum principle to (3.41) in the annuli Ωrk ,rk+1 .

3.5. Asymptotic structure of D-solutions The objective of this section is to show that every D-solution satisfying (3.56) for some ξ˜ = 0 admits an asymptotic expansion, for large |x|, whose dominating term is the Oseen fundamental solution (E, q). In particular, v − ξ˜ and p satisfy all the summability properties at large distance possessed by the fundamental solution. This result is by no means obvious and, for symmetric solution, is due to Amick [3], while in the general case is obtained from the work of Galdi and Sohr [28] and Sazonov [39].

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G.P. Galdi

We wish to emphasize that, if ξ˜ = 0, the structure of a D-solution at large distances is an open question. In this regard, it should be noted that, when ξ˜ = 0, D-solutions that are regular in a neighborhood of inﬁnity need not be represented there by an expansion in negative powers of r (≡ |x|) with coefﬁcients independent of r. Actually, the ﬁelds given in (2.5) for λ ∈ (1, 2) provide examples of D-solutions that decay to zero at inﬁnity more slowly than any negative power of r. Thus, assuming ξ˜ = 0, by means of an orthogonal transformation, we can always bring ξ˜ into the vector μe1 , for some μ = 0. As a matter of fact, the speciﬁc value of μ plays no role at all in our proof, so that, for simplicity, we shall put μ = 1. Furthermore, even though some of the results continue to hold also when v∗ = 0, for simplicity we take v∗ = 0. We are then lead to the investigation of the asymptotic structure of D-solutions to the following problem v = v · ∇v + ∇p div v = 0

' in Ω, (3.59)

v|∂Ω = 0, lim v(x) = e1 .

|x|→∞

To reach our objective, we begin to recall, without proof, the following result due to Smith [41] (part(i)) and to Galdi [19], Lemma X.5.1 (part (ii)). L EMMA 3.11. Let (v, p) be a regular solution to (3.59) satisfying either of the following conditions for all large |x|, and for some R > 1, (i) v − e1 = O(|x|−1/4−ε ) for some ε > 0;

(ii) Ω R |v − e1 |q < ∞ for some 1 q < 6. Then, the following asymptotic representations hold as |x| → ∞, v(x) = e1 + m · E(x) + V(x), ∂v ∂E(x) (x) = m · + Gk (x), ∂xk ∂xk

k = 1, 2,

(3.60)

p(x) = p0 − m · q(x) + P(x), where p0 is a constant, m is deﬁned in (1.4), and V, Gk , k = 1, 2, and P are “remnants” satisfying the following asymptotic estimates V(x) = O |x|−1 log2 |x| , Gk (x) = O |x|−αk log2 |x| , P(x) = O |x|−1 log |x| ,

(3.61)

Stationary Navier–Stokes problem in a two-dimensional exterior domain

135

where α1 = 3/2 and α2 = 1. In particular, the following estimate holds, uniformly in x, |v(x) − e1 | c|x|−1/2, |∇v(x)| c|x|−1 log2 |x|,

(3.62)

|p(x)| c|x|−1 log |x|, with c independent of x. Following Finn and Smith [16], we shall call solutions satisfying (3.60)–(3.62), Physically Reasonable Solutions. R EMARK 3.8. The results of Lemma 3.11 imply that the solutions constructed in Theorem 2.1 are physically reasonable. The following result, due to Galdi and Sohr [28], shows that if the component v2 of a D-solution (v, p) to (3.59) is in Ls (Ω) for some 1 < s < ∞, then (v, p) is physically reasonable. L EMMA 3.12. Let (v, p) be a solution to (3.59)1–3 such that v ∈ D 1,2 (Ω),

lim v1 (x) = 1.

|x|→∞

Assume, further, that there exists ρ > 1 such that v2 ∈ Ls Ω ρ for some s ∈ [1, ∞).

(3.63)

Then (v, p) is physically reasonable. P ROOF. In view of the preceding lemma, it is enough to show that v enjoys suitable summability properties in a neighborhood of inﬁnity. To reach this goal, we begin to notice that, by Lemma 3.9, ∇v ∈ W 1,2 Ω ρ .

(3.64)

Thus, from (3.63) and Sobolev-like inequalities one shows that lim v(x) = e1 .

|x|→∞

(3.65)

From the assumptions, (3.63), (3.64) and (3.59)1, we also have ∇p ∈ L2 Ω ρ .

(3.66)

136

G.P. Galdi

For R ρ > 2, let ψR be a smooth “cut-off” function deﬁned by ψR (x) =

0 if |x| < R/2, 1 if |x| R.

Setting ¯ u = ψR (v − e1 ) ≡ ψR v,

π = ψR p,

from (3.59) we deduce that u, π satisfy the following system in R2 u −

∂u ∂u =a + Au2 + ∇π + F, ∂x1 ∂x1

(3.67)

div u = g, where F = 2∇ψR · ∇v + ψR v¯ −

∂ψR ∂ψR v¯ − v¯1 v − p∇ψR , ∂x1 ∂x1

g = v¯ · ∇ψR , a = (ψR/2 v¯1 ), A = ψR/2

∂v . ∂x2

Clearly, we have F ∈ Lq R2 ,

g ∈ W 1,q R2 for all q ∈ (1, 2].

Moreover, we observe that in view of (3.64) and (3.65), by taking R sufﬁciently large, the quantities a∞

and A2

can be made less than any prescribed constant. For q ∈ (1, 3/2), we set 8u8q = u2,q + u!q , where ·, · !q is deﬁned in (1.48). Since for all q ∈ (1, 2), by the Hölder inequality, we have that ∂u a a∞ ∂u + A2 u2q/(2−q), + Au (3.68) 2 ∂x ∂x 1 1 q q

Stationary Navier–Stokes problem in a two-dimensional exterior domain

137

from Theorem 1.4 we deduce that 8u8q c1 a∞ + A2 8u8q +F q + g1,q

(3.69)

for some c1 = c1 (q) > 0. Thus, assuming, for instance, a∞ + A2

0: Φ|x|−1/2−ε/2 ∈ L2 (Ω).

(3.73)

It is well known that, by using a simple argument based on elliptic regularity, the function Φ decays (pointwise) exponentially fast outside a sector containing the positive x1 -axis (see [2], pp. 106–107). Thus, taking into account that within the sector it is |x2 | b x1 , for some b > 0, in order to show (3.73), it is enough to prove, for sufﬁciently large a > 0, that Πa

Φ2 x11+ε

< ∞,

(3.74)

where Πa = {x ∈ R2 : x1 > a}. The property (3.74) is established in [39], p. 205, by combining the fact that Φ obeys a maximum principle (see (3.27)) with a weight-function technique. Once (3.73) has been established, by means of an integral formula, based on the complex variable, relating Φ to v, in [39], Lemma 4, it is shown that Ω

v2 |x|1+ε

< ∞.

(3.75)

We next recall the well-known representation formula for v in terms of ω 1 v(x) = 2πρ

1 v(y) dy + 2π Bρ (x)

ω(y)(x − y)⊥ Bρ (x)

(x − y)2

dy,

where (x − y)⊥ = (−(x2 − y2 ), x1 − y1 ). Integrating the previous inequality over ρ from R/2 to R and using (3.75) and Schwarz inequality, we readily ﬁnd that v(x) c

BR

|v(y)| R2

+

. / |ω(y)| dy c |x|(1+ε)/2R −1 + max ω(y)R . y∈BR |x − y|

Using the decay property (3.57), from the preceding inequality we get v(x) c |x|(1+ε)/2R −1 + |x|−3/4R . Taking |x| > R/2 and minimizing this latter inequality with respect to R, we ﬁnally obtain |v(x)| c|x|(−1+2ε)/8, which furnishes v ∈ Ls (Ω ρ ), s > 16. Coupling this information with Lemma 3.12 we conclude with the following result. T HEOREM 3.5. Let (v, p) be a solution to (3.59), with v ∈ D 1,2 (Ω). Then (v, p) is physically reasonable.

Stationary Navier–Stokes problem in a two-dimensional exterior domain

139

4. The nonlinear problem: On the existence of symmetric solutions for arbitrary large Reynolds number In Section 3 we have shown that the velocity ﬁeld of a D-solution to problem (3.5) with v∗ = 0, necessarily tends to some ξ˜ uniformly pointwise, and, in fact, such solutions are even physically reasonable if ξ˜ = 0. However, we are not able to relate ξ˜ with the prescribed value λe1 ; see (3.6). As a result, we still do not know if the problem ' v = v · ∇v + ∇p in Ω, div v = 0 v|∂Ω = 0,

(4.1)

lim v(x) = λe1

|x|→∞

has a solution for “large” Reynolds number λ. The objective of this section is to give a contribution along this direction. Speciﬁcally, let us denote by (NS)0 the problem (4.1) with λ = 0. Clearly, the zero solution v = 0, p = const is a solution to (NS)0 . Furthermore, assume B symmetric around the x1 -axis (say) and denote by C the class of symmetric D-solutions v, p, that is, D-solutions satisfying (3.3). Then we shall prove that if the zero solution is the only solution to (NS)0 in the class C, then problem (4.1) is solvable in C, for arbitrary large Reynolds numbers and the corresponding solutions are physically reasonable. In particular, denoting by M the set of λ for which (4.1) has at least one symmetric solution associated to a given λe1 , we show that M contains an unbounded set, M0 , of the positive real axis.11 The crucial point in the proof of this result is to show a bound from below for the D 1,2 -norm of v in terms of λ (Section 4.2). To our knowledge, this is the ﬁrst contribution relating the solution (v, p) to its prescribed value at inﬁnity. The important question with this theorem is, of course, to verify the validity of its assumption. Stated in a different way, for our result to be true it is sufﬁcient that every symmetric solution to the homogeneous problem (NS)0 with v having a ﬁnite Dirichlet integral is identically zero. Actually, even a weaker version of this statement would be enough; see Remark 4.7. We wish to emphasize that the problem here is not related to local regularity of solutions to (NS)0 (they are of class C ∞ , and even real-analytic in Ω) but, rather, to their behavior at large distances. Actually, we know that any D-solution (v, p) to (NS)0 tends to zero uniformly pointwise together with all its derivatives of arbitrary order (see Remark 3.5), but this is not enough to make the “classical” energy method for uniqueness to work (see Remark 4.1). It should be said that if Ω ≡ R2 (unfortunately, a case of no interest in the present situation), then our assumption is easily shown to be satisﬁed; see [30] and Remark 4.2. However, we wish also to mention that, if λ = 0, Ω ≡ R2 , and v at ∂Ω is not zero, example of nonuniqueness are well known [31]. Proving or disproving the assumption of the theorem will certainly shed new light on this long-standing problem. I regret I was not able to get any result in this direction, and I leave it to the interested mathematician as a challenging open question. 11 Without loss of generality, we may take λ > 0.

140

G.P. Galdi

Another interesting problem that we leave open is the study of the properties of the set M0 , like topological or measure-theoretical ones. The properties of the set M0 can be studied, for instance, by means of the results of Theorem 2.3. In particular, one may try a continuation argument to show that M0 coincides with the whole positive real axis. This requires, on the one hand, that solutions (u0 , p0 ) with Reynolds number λ0 ∈ M0 must have the velocity ﬁeld in the space X1,q (Ω), for some 1 < q < 6/5, and, on the other hand, that they do not belong to the nullspace Nu0 ,λ0 of the operator Oλ0 − Ku0 ,λ0 ; see Theorem 1.8. While the ﬁrst issue ﬁnds a positive answer as a result of Theorem 3.5, we can not exclude, a priori, that solutions with λ0 ∈ M0 belong to Nλ0 ,u0 . As a ﬁnal remark, we believe that our result, as is stands, can be in principle extended to the case of nonsymmetric solutions. However, this would require a substantial technical effort in generalizing the result of Amick used in the proof of Lemma 4.5 to nonsymmetric ﬂow.

4.1. A remark about symmetric solutions In this section we shall show how the results proved in Theorem 3.4 specialize to the case of symmetric solutions. In fact, by a very simple observation, we can relate ξ˜ to λe1 . Speciﬁcally, have the following result. L EMMA 4.1. Let ξ˜ be as in Theorem 3.4. Then, ξ˜ = μe1 , where μ = αλ for some α ∈ [0, 1]. P ROOF. We claim that ξ˜ = μe1 for some μ ∈ R. In fact, since the component w of v along the x2 -axis satisﬁes w(x1 , x2 ) = −w(x1 , −x2 ), we ﬁnd that w(|x|, 0) = 0 for all |x| > 1. Therefore, our claim is a consequence of Theorem 3.4. We next multiply (4.1)1 by v − μe1 and integrate by parts over ΩR to get

|∇v|2 = −μe1 ·

T (v, p) · n +

ΩR

∂Ω

|x|=R

(v − μe1 ) · T (v, p) · n.

If μ = 0, we use the asymptotic properties of Theorem 3.5 and let R → ∞ in the previous relation. We then ﬁnd that v, p obey the following energy equality

|∇v|2 = −μ e1 · Ω

T (v, p) · n.

(4.2)

∂Ω

From (3.9) and (4.2) we deduce (μ − λ)e1 ·

T (v, p) · n 0. ∂Ω

(4.3)

Stationary Navier–Stokes problem in a two-dimensional exterior domain

In view of the symmetry properties of v, p, and of (3.9) it follows that T (v, p) · n = ηe

141

(4.4)

∂Ω

for some η < 0. Therefore, from (4.2) and (4.3) we ﬁnd μ = αλ for some nonzero α ∈ (−∞, 1], and ﬁnally, from (4.2) and (4.4) we ﬁnd −αλη > 0, which implies α > 0. The lemma is proved. R EMARK 4.1. From the proof of the previous lemma it follows that if we could construct D-solutions satisfying the energy equality and if α = 0, then ξ˜ = λe1 and we would show existence for arbitrary λ. The possibility of being α = 0 would be ruled out if we could show that the problem (NS)0 has only the zero solution in the class C; see the introduction to Section 4.

4.2. A key result In this section we prove a fundamental inequality for a symmetric Leray solution; see Theorem 4.1. By this nomenclature, we mean a symmetric D-solution to (4.1)1–3 that has been constructed by the method of Leray described in Section 3.1, for a given λ > 0, provided Ω is of class C 3 , which will be assumed throughout. We recall that these solutions satisfy the properties stated in Theorem 3.1 and Lemma 3.6. Everywhere in this section, we denote by {vk = (uk , wk ), pk } a symmetric solution to (3.7) and by ωk ≡ ∂uk /∂x2 − ∂wk /∂x1 the corresponding vorticity. We also set Dk ≡ |∇vk |2 ΩRk

and

1/4

δ≡

|∇v|2

,

Ω3

where v, p is a symmetric Leray solution, and 8f 8m ≡ f C m (Ω2 ) = max maxD α f (x). 0|α|m Ω2

Furthermore, we indicate by c a constant depending at most on ∂Ω, and whose numerical value is not essential to our aims. In particular, c may have several different values in a single computation. Finally, in order to avoid cumbersome notation, we shall denote the components v1 and v2 of the velocity ﬁeld v, by u and w, respectively. We shall also set x1 = x and x2 = y. We recall that B ≡ Ω c is contained in B1 . The main objective of this section is to prove the following key result.

142

G.P. Galdi

T HEOREM 4.1. Let v, p be a symmetric Leray solution corresponding to a given λ. Then, there exists a polynomial P = P (δ) with coefﬁcients depending only on Ω, such that P (0) = 0 and λ2 P (δ). R EMARK 4.2. The proof of this theorem will be achieved through several intermediate steps. Before doing this, however, we wish to point out a particular, immediate consequence of our result, namely, that a symmetric Leray solution corresponding to λ > 0 can never be trivial, i.e, v ≡ 0, p = const. This was proved for the ﬁrst time by Amick [2], Theorem 29. It is not known if the same result is true for nonsymmetric solutions, or for symmetric solutions constructed by a method different than Leray’s (like the Fujita method). L EMMA 4.2. The following inequality holds, for all ρk ∈ (Rk /2, 3Rk /4), λ c Dk +

v(ρk , θ )2 dθ .

2π

2

0

P ROOF. From the identity λe1 ≡ vk (Rk , θ ) = vk (ρk , θ ) +

Rk ρk

∂vk dr, ∂r

we obtain λ vk (ρk , θ ) +

Rk ρk

dr r

1/2

Rk

1/2 |∇vk |2

.

ρk

The lemma then follows after squaring both sides of this inequality and integrating over θ ∈ [0, 2π). L EMMA 4.3. The following inequality holds |∇ωk |2 c 8vk 822 + 8vk 832 +Rk−1 M(C0 + 1) ≡ Ak , Ω3Rk /4

where M and C0 are the constants introduced in Theorem 3.1(i) and Lemma 3.6, respectively. P ROOF. We recall that the vorticity ωk satisﬁes the following equation ωk − v · ∇ωk = 0.

(4.5)

Let ψRk (x) be a smooth, nonincreasing function which is 1 for |x| < 3Rk /4 and is 0 outside ΩRk , and let ζ (x) be a smooth, nonincreasing function which is 0 for |x| < 1 and is 1

Stationary Navier–Stokes problem in a two-dimensional exterior domain

143

for |x| > 2. We may take |∇ψRk (x)| cRk−1 . Setting η = ψRk ζ , multiplying (4.5) by η2 ωk and integrating over ΩRk , we ﬁnd

η2 |∇ωk |2 = −2 ΩRk

ηωk ∇ωk · ∇η + ΩRk

η|ωk |2 v · ∇η.

(4.6)

ΩRk

From Theorem 3.1(i), Lemma 3.6 and the properties of η, we get

−1 η|ωk | vk · ∇η c C0 Rk

ΩRk

|ωk | +

2

|vk ||ωk |

2

ΩRk

2

Ω2

c C0 Rk−1 M + 8vk 832 . By a similar argument and by Cauchy inequality, −2

ηωk ∇ωk · ∇η ΩRk

1 2 1 2

η2 |∇ωk |2 + 2

ΩRk

ΩRk

|∇η|2 |ωk |2 ΩRk

η2 |∇ωk |2 + c MRk−1 + 8vk 822 .

Replacing these two last displayed inequalities into (4.6), and recalling that η(x) ≡ 1, |x| ∈ (2, 3Rk /4), we obtain |∇ωk |2 c 8 vk 822 + 8 vk 832 +Rk−1 M(C0 + 1) , 2 0 and let v, p be a solution to (4.27) corresponding to μ ∈ (0, μ0 ]. Then, there exists a positive constant κ, depending only on Ω and μ0 , such that |∇v|2 κ. Ω

P ROOF. For a given ε > 0, let ψε = ψε (x) be a Hopf “cut-off” function, namely, a smooth function satisfying the following properties: (i) |ψε (x)| 1 for all x ∈ Ω; (ii) ψε (x) = 1 if δ(x) < γ 2 /2; (iii) ψε (x) = 0 if δ(x) 2γ (ε); (iv) |∇ψε (x)| ε/δ(x) for all x ∈ Ω, where δ(x) is the distance of x from ∂Ω and γ (ε) = exp(−ε−1 ); see, e.g., [20], Lemma III.6.2. We then deﬁne ∂(ψε x1 ) ∂(ψε x1 ) ,− Vε = μ ∂x2 ∂x1 and set u = v − Vε − μ. From (4.27) we thus get that u is a solution to the following problem: u = u · ∇u + u · ∇Vε + (Vε + μ) · ∇u + (Vε + μ) · ∇Vε − Vε + ∇p, ∇ · u = 0, (4.30) u|∂Ω = 0, lim u(x) = 0.

|x|→∞

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We now multiply both sides of (4.30)1 by u, integrate by parts over ΩR , let R → ∞ and use Theorem 3.5. We then obtain the following relation u · ∇u · Vε − (Vε + μ) · ∇u · Vε + ∇Vε : ∇u . |∇u|2 = − (4.31) Ω

Ω

Employing the properties of the function ψε , and noticing that the support of ψε is contained in Ω2γ ⊂ Ω2 , we easily obtain uVε 2 μ ψε u2 + 2|u| |∇ψε |2,Ω 2

μ u4,Ω2 ψε 4,Ω2γ + cεu/δ2,Ω2 .

(4.32)

Using the Sobolev inequality (4.19) and the following Hardy inequality, u/δ2,Ω2 c∇u2,Ω2 , into (4.32), we ﬁnd uVε 2 cμ∇u2,Ω2 ψε 4,Ω2γ + ε 2cμε∇u2,Ω2 .

(4.33)

Therefore, recalling that μ μ0 and that Vε is of compact support, from (4.31), (4.33) and Schwarz inequality, it follows that

|∇u|2 cμ0 ε Ω

Ω

1/2 |∇u|2 + c1 (ε) μ0 + μ20 |∇u|2 ,

(4.34)

Ω

where c1 (ε) = Vε 24 + ∇Vε 2 . We now choose ε = 1/2cμ0 , so that (4.34) furnishes |∇u|2 c(μ0 ).

(4.35)

Ω

Since

|∇v|2 Ω

|∇u|2 +

Ω

|∇Vε |2 , Ω

the lemma follows from (4.35) and this latter inequality.

P ROOF OF T HEOREM 4.2. Let us denote by M the set of those μ > 0 for which problem (4.27) has a corresponding solution v, p. From the results of Section 2 (see Theorem 2.1), we know that M ⊃ [0, c] for some positive c = c(Ω). We shall now show that M ⊃ M0 where M0 enjoys the properties:

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(i) M0 ⊂ (0, ∞); (ii) M0 is unbounded. Actually, let M0 be deﬁned as follows: μ ∈ M0

if and only if

lim v(x) = μe uniformly,

|x|→∞

where v, p is a symmetric Leray solution corresponding to a given λ > 0. Clearly, M0 ⊂ M. Also, by Lemma 2.2, M0 = ∅. Furthermore, M0 ⊂ (0, ∞). In fact, by Lemma 4.1, M0 ⊂ [0, ∞). However, 0 ∈ / M0 , because, otherwise, v, p satisfy the homogeneous equation (4.26), and this, by assumption, would imply v ≡ 0. So, by Theorem 4.1, we would conclude λ = 0, which leads to a contradiction. This proves property (i) of M0 . Let us show (ii). Assuming M0 bounded means 0 < μ μ0 < ∞ for some μ0 > 0. Thus, from Lemma 4.9, we have |∇v|2 c(μ0 ) (4.36) Ω

for all symmetric Leray solutions corresponding to arbitrary λ > 0. Using (4.36) into Theorem 4.2, we obtain λ c1 (μ0 ) for arbitrary λ > 0, which gives a contradiction. The theorem is, therefore, completely proved. R EMARK 4.7. The assumption of Theorem 4.1 can be somehow weakened, with some interesting consequences. Actually, to prove that M0 is unbounded what we really need is the existence of at least one diverging sequence {λm } for which the corresponding symmetric Leray solutions {vm , pm } satisfy lim vm (x) = μm e = 0.

|x|→∞

It is worth of emphasizing that if this property is not true, symmetric Leray solutions would present a very anomalous behavior, namely, there would exist a positive λ¯ , such that the velocity ﬁeld of such solutions corresponding to any λ > λ¯ would tend to zero as |x| → ∞, uniformly pointwise.

Acknowledgments Good part of this work was accomplished while I was a Deutsche Forschungsgemeinschaft (DFG) Mercator Professor at the University of Paderborn in the period May–August 2003. I wish to express my warm thanks to Professor Hermann Sohr for his hospitality and for several stimulating conversations.

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References [1] G. Allaire, Homogenization of the Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math. 44 (1991), 605–641. [2] C.J. Amick, On Leray’s problem of steady Navier–Stokes ﬂow past a body in the plane, Acta Math. 161 (1988), 71–130. [3] C.J. Amick, On the asymptotic form of Navier–Stokes ﬂow past a body in the plane, J. Differential Equations 91 (1991), 149–167. [4] A. Avudainayagam, B. Jothiram and J. Ramakrishna, A necessary condition for the existence of a class of plane Stokes ﬂows, Quart. J. Mech. Appl. Math. 39 (1986), 425–434. [5] A. Avudainayagam and J. Geetha, A necessary condition for the existence of plane Stokes’ ﬂows around an ellipse, Canad. Appl. Math. Quart. 3 (1995), 237–251. [6] K.I. Babenko, Theory of perturbations of stationary ﬂows of viscous incompressible ﬂuids at small Reynolds numbers, Selecta Math. Soviet 3 (1983/84), 111–149. [7] R. Berker, Intégration des équations du mouvement d’un ﬂuide visqueux incompressible, Handbuch der Physik, Band VIII/2, S. Flügge ed., Springer-Verlag, Berlin (1963), 1–384. [8] J.R. Blake, A ﬁnite model for ciliated micro-organisms, J. Biomech. 6 (1973), 133–140. [9] J.R. Blake and S.R. Otto, Ciliary propulsion, chaotic ﬁltration and a ‘blinking’ stokeslet, J. Engrg. Math. 30 (1996), 151–168. [10] C. Brennen and H. Winet, Fluid Mechanics of Propulsion by Cilia and Flagella, Annu. Rev. Fluid Mech. 9 (1977), 339–398. [11] I.-D. Chang and R. Finn, On the solutions of a class of equations occurring in continuum mechanics, with application to the Stokes paradox, Arch. Ration. Mech. Anal. 7 (1961), 388–401. [12] C. Christov, A note on the paper by K. Shulev: “A solution of the Stokes problem for a circular cylinder”, Teoret. Prilozhna Mekh. 19 (1988), 97–98. [13] H. Faxén, Fredholm’sche Integraleichungen zu der Hydrodynamik Zäher Flüssigkeiten, Ark. Mat. Astron. Fys. 21 (14) (1928/1929), 1–20. [14] R. Finn, The exterior problem for the Navier–Stokes equations, Actes, Congrès Intern. Math. 3 (1970), 85–94. [15] R. Finn and D.R. Smith, On the linearized hydrodynamical equations in two dimensions, Arch. Ration. Mech. Anal. 25 (1967), 1–25. [16] R. Finn and D.R. Smith, On the stationary solution of the Navier–Stokes equations in two dimensions, Arch. Ration. Mech. Anal. 25 (1967), 26–39. [17] H. Fujita, On the existence and regularity of the steady-state solutions of the Navier–Stokes theorem. J. Fac. Sci. Univ. Tokyo Sect. I 9 (1961), 59–102. [18] G.P. Galdi, Existence and uniqueness at low Reynolds number of stationary plane ﬂow of a viscous ﬂuid in exterior domains, Recent Developments in Theoretical Fluid Mechanics (Paseky, 1992), Pitman Res. Notes Math. Ser., Vol. 291, Longman, Harlow (1993), 1–33. [19] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Nonlinear Steady problems, 1st Edition, Springer Tracts in Natural Philosophy, Vol. 39, Springer-Verlag, New York (1994). [20] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Linearized Steady Problems, Rev. Edition, Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, New York (1998). [21] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes equations: Nonlinear Steady Problems, Rev. Edition, Springer Tracts in Natural Philosophy, Vol. 39, Springer-Verlag, New York (1998). [22] G.P. Galdi, On the existence of symmetric steady-state solutions to the plane exterior Navier–Stokes problem for arbitrary large Reynolds number, Advances in Fluid Dynamics, Quad. Mat. 4 (1999), 1–25. [23] G.P. Galdi, On the steady self-propelled motion of a body in a viscous incompressible ﬂuid, Arch. Ration. Mech. Anal. 148 (1999), 53–88. [24] G.P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam (2002), 653–791. [25] G.P. Galdi, A. Novotný and M. Padula, On the two-dimensional steady-state problem of a viscous gas in an exterior domain, Paciﬁc J. Math. 179 (1997), 65–100. [26] G.P. Galdi and P.J. Rabier, Functional properties of the Navier–Stokes operator and bifurcation of stationary solutions: Planar exterior domains, Progr. Nonlinear Differential Equations Appl. 35 (1999), 273–303.

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[27] G.P. Galdi and C.G. Simader, Existence, uniqueness and Lq -estimates for the Stokes problem in an exterior domain, Arch. Ration. Mech. Anal. 112 (1990), 291–318. [28] G.P. Galdi and H. Sohr, On the asymptotic structure of plane steady ﬂow of a viscous ﬂuid in exterior domains, Arch. Ration. Mech. Anal. 131 (1995), 101–119. [29] D. Gilbarg and H.F. Weinberger, Asymptotic properties of Leray’s solution of the stationary twodimensional Navier–Stokes equations, Russian Math. Surveys 29 (1974), 109–123. [30] D. Gilbarg and H.F. Weinberger, Asymptotic properties of steady plane solutions of the Navier–Stokes equations with bounded Dirichlet integral, Ann. Sc. Norm. Super. Pisa 5 (1978), 381–404. [31] G. Hamel, Spiralförmige bewegungen Zäher Flüssigkeiten, Jahresber. Deutsch. Math.-Verein. 25 (1916), 34–60; English transl.: NACA Tech. Memo. (1953) 1342. [32] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin (1995). [33] J. Leray, Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pures Appl. 12 (1933), 1–82. [34] J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–78. [35] P.I. Lizorkin, (Lp , Lq )-multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 152 (1963), 808–811 (in Russian). [36] F.K.G. Odqvist, Über die Randwertaufgaben der Hydrodynamik Zäher Flüssigkeiten, Math. Z. 32 (1930), 329–375. [37] C.W. Oseen, Über die Stokessche Formel und über eine verwandte Aufgabe in der Hydrodynamik, Ark. Mat. Astron. Fys. 6 (29) (1910), 1–20. [38] C.W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Akad. Verlagsgesellschaft M.B.H., Leipzig (1927). [39] L.I. Sazonov, On the asymptotic behavior of the solution of the two-dimensional stationary problem of the ﬂow past a body far from it, Mat. Zametki 65 (1999), 246–253 (in Russian); English transl.: Math. Notes 65 (1999), 202–207. [40] C.G. Simader and H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. A New Approach to Weak, Strong and (2 + k)-Solutions in Sobolev-Type Spaces, Pitman Research Notes in Mathematics Series, Vol. 360, Longman, Harlow (1996). [41] D.R. Smith, Estimates at inﬁnity for stationary solutions of the Navier–Stokes equations in two dimensions, Arch. Ration. Mech. Anal. 20 (1965), 341–372. [42] G.G. Stokes, On the effect of the internal friction of ﬂuids on the motion of pendulums, Trans. Cambridge Phil. Soc. 9 (1851), 8–106. [43] G. van Baalen, Stationary solutions of the Navier–Stokes equations in a half-plane downstream of an obstacle: “Universality” of the wake, Nonlinearity 15 (2002), 315–366. [44] P. Wittwer, On the structure of stationary solutions of the Navier–Stokes equations, Comm. Math. Phys. 234 (2003), 557–565. [45] P. Wittwer, On the structure of stationary solutions of the Navier–Stokes equations, Comm. Math. Phys. 226 (2002), 455–474.

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CHAPTER 3

Qualitative Properties of Solutions to Elliptic Problems Wei-Ming Ni School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: ni@math.umn.edu

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Concentrations of solutions: Single equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Spike-layer solutions in elliptic boundary-value problems . . . . . . . . . . . . . . . . . . . . 1.2. Multi-peak spike-layer solutions in elliptic boundary-value problems . . . . . . . . . . . . . . 1.3. Solutions with multidimensional concentration sets . . . . . . . . . . . . . . . . . . . . . . . 1.4. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Concentrations of solutions: Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The activator–inhibitor system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. A Lotka–Volterra competition system with cross-diffusion . . . . . . . . . . . . . . . . . . . 2.3. A chemotaxis system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Stability of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Single equations with Neumann boundary conditions . . . . . . . . . . . . . . . . . . . . . . 3.2. Single equations with Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 3.3. Shadow systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Diffusion systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Symmetry and related properties of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Symmetry of semilinear elliptic equations in a ball . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Symmetry of semilinear elliptic equations in an annulus . . . . . . . . . . . . . . . . . . . . . 4.3. Symmetry of semilinear elliptic equations in entire space . . . . . . . . . . . . . . . . . . . . 4.4. Related monotonicity properties, level sets and more general domains . . . . . . . . . . . . . 4.5. Generalizations and other types of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Symmetry of nonlinear elliptic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Miscellaneous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Graphics and visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Mountain-pass and scaling algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Visualization of solutions of singularly perturbed semilinear elliptic boundary value problems 5.3. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HANDBOOK OF DIFFERENTIAL EQUATIONS Stationary Partial Differential Equations, volume 1 Edited by M. Chipot and P. Quittner © 2004 Elsevier B.V. All rights reserved 157

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159 163 163 170 173 177 177 178 181 185 187 189 190 194 196 203 204 206 208 208 211 214 215 217 218 220 222 228

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Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

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Introduction Qualitative properties of solutions to elliptic equations can be interpreted in an extremely broad sense to include virtually every property of solutions. In this chapter, however, I shall focus more on concrete and/or geometric properties of solutions. In particular, I shall emphasize the following two properties of solutions: the “shape” of solutions and the stability of solutions. Naturally, the connections between them will also be discussed. Boundary conditions clearly play important roles in the qualitative behavior of solutions. One feature of this survey is the inclusion of comparisons of the different, sometimes opposite effects of Dirichlet and Neumann boundary conditions whenever possible. Qualitative properties of solutions are closely related to the existence of solution; in fact, it seems obvious that existence of solutions provides the basis for the study of qualitative properties. On the other hand, searching for solutions with particular properties in mind (often reﬂected in the phenomena for which the equations are modeling) could provide clues for existence. Therefore, in this chapter, we shall also talk about the existence of solutions, especially those solutions with “desired” properties, whenever is necessary or appropriate. Systematic studies of qualitative properties of solutions to general nonlinear elliptic equations or systems essentially began in the late 1970s, although some nonlinear elliptic equations (such as the Lane–Emden equation in astrophysics [Ch]) actually go back to the 19th century. It should be noted, however, that earlier works in this direction on linear elliptic equations, such as symmetrization or nodal properties of eigenfunctions, have had their consequences in nonlinear equations. Symmetry remains an important topic in modern theory of nonlinear partial differential equations. In particular, it is now understood how different boundary conditions may inﬂuence the symmetry properties of positive solutions in domains with symmetries. First, solutions of boundary-value problems are very different from solutions on entire space. Moreover, solutions to Neumann boundary-value problems exhibit drastically different behavior to their Dirichlet counterparts. For instance, it is known [GNN1] that all positive solutions of the following Dirichlet problem

where =

ε2 u + f (u) = 0

in BR (0),

u=0

on ∂BR (0),

n

∂2 i=1 ∂x 2 i

is the usual Laplace operator in Rn , ε > 0 is a constant, f is a

C 1 -function and BR (0) is the ball of radius R centered at the origin 0, must be radially symmetric. On the other hand, it has been proved [NT2] that for ε sufﬁciently small the following Neumann problem "

ε2 u − u + up = 0 in BR (0), ∂u on ∂BR (0), ∂ν = 0

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where ν is the unit outer normal to ∂BR (0) and 1 < p < n+2 n−2 (= ∞ if n = 2), possesses a positive solution uε with a unique maximum point located on the boundary ∂BR (0). Thus, this solution uε cannot possibly be radially symmetric. In fact, the number of positive nonradial solutions of the Neumann problem above tends to ∞ as ε tends to 0. Furthermore, while it has been known for decades that symmetrization reduces the “energy” of positive solutions for Dirichlet problems, it can be shown that symmetrization actually increases the “energy” of the solution uε above. (Here, by “energy” we mean the variational integral #

BR (0)

$ 1 1 2 2 p+1 |∇u| + u − u . 2 p+1

Note that, since symmetrization does not alter integrals involving u, only the Dirichlet integral |∇u|2 BR (0)

gets changed after symmetrization.) In other words, the most “stable” solutions to the Neumann problem above must not be radially symmetric – a remarkable difference between Neumann and Dirichlet boundary conditions. In fact, solutions to Neumann problems also possess some restricted symmetry properties – they seem to be more subtle. (See Section 4.1.) Generally speaking, Dirichlet boundary conditions are far more rigid and imposing than Neumann boundary conditions, as is already indicated by the above discussions. This is also true for general bounded smooth domains Ω in Rn . Symmetry properties of solutions to elliptic equations on entire space (or unbounded domains) clearly require appropriate conditions at ∞. It seems that the simplest, most general result in this direction is that, all positive solutions of the following problem

u + f (u) = 0 u→0

in Rn , at ∞,

must be radially symmetric (up to a translation) provided that f (0) < 0. (See [LiN] or Theorem 4.3.) The case f (0) = 0 turns out to be far more complicated. Roughly speaking, to guarantee radial symmetry in this case, additional hypothesis on suitable decay of solutions are needed. Symmetries and related properties, such as monotonicity, are discussed in Section 4. In a different but very important direction, signiﬁcant progress has been made in the past ﬁfteen years in understanding the “shape” of solutions; in particular, the “concentration” behavior of solutions to nonlinear elliptic equations and systems. More precisely, positive solutions concentrating near isolated points, i.e., spike-layer solutions (or, singleand multi-peak solutions), and the locations of these points (determined by the geometry of the underlying domains) have been obtained for both Dirichlet and Neumann boundaryvalue problems. For instance, as we shall see, in Section 1 that, for ε small, a “least-energy

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solution” of the Neumann problem ⎧ 2 p ⎪ ⎨ ε u − u + u = 0 in Ω, u>0 in Ω, ⎪ ⎩ ∂u = 0 on ∂Ω, ∂ν

(N)

where 1 < p < n+2 n−2 (= ∞ if n = 2), must have its only (local and thus global) maximum located on ∂Ω and near the most “curved ” part of ∂Ω. (See [NT2,NT3] or point (in Ω) Theorem 1.1. Here, the “curvedness” is measured by the mean curvature of ∂Ω.) On the other hand, a “least-energy solution” of the Dirichlet problem ⎧ 2 ⎨ ε u − u + up = 0 in Ω, u>0 in Ω, ⎩ u=0 on ∂Ω,

(D)

where 1 < p < n+2 n−2 (= ∞ if n = 2), must have its only (local and thus global) maximum point located near a “center” of the domain Ω. (See [NW] or Theorem 1.1. Here a “center” is deﬁned as a point in Ω which is most distant from ∂Ω.) Furthermore, the “proﬁles” of these least-energy solutions for both (N) and (D) have been determined in [NT2,NT3] and [NW]. There has been a huge amount of literature on those spike-layer solutions published since the papers [NT2,NT3] ﬁrst appeared in the early 1990s, and many interesting and excellent results have been obtained. For example, the locations of multiple interior peaks to a solution of (N), for ε small, are determined by the “spherepacking” property of the domain Ω. (See [GW1] or Section 1.2.) Those solutions often represent pattern formation in various branches of science. In Sections 1 and 2, we shall describe the recent progress in this direction as well as some models leading to those solutions. (We will, for instance, include the Gierer–Meinhardt system, based on Turing’s idea of “diffusion-driven instability”, in modeling the regeneration phenomenon of hydra.) Furthermore, positive solutions concentrating on multidimensional subsets (instead of isolated points which are 0-dimensional) of the underlying domains will also be discussed in Section 1, although advance in this direction has been rather limited so far. It is interesting to note that the multidimensional concentration sets in all results obtained thus far are located either on or near the boundary of the underlying domain; in particular, no multidimensional concentration set in the interior of the underlying domain has been found. The “shape” of solutions of elliptic equations or systems turns out to be closely related to the stability properties of those solutions. It seems clear that stability properties are crucial to our understanding of the entire dynamics of the original evolution problems. In Section 3 we include a brief discussion on this important and fascinating matter. Roughly speaking, the “general principle” here seems to be that the “simpler the shape” of a solution, the “more stable” it tends to be. For solutions of single (autonomous) elliptic equations under homogeneous Neumann boundary conditions

u + f (u) = 0 in Ω, ∂u on ∂Ω, ∂ν = 0

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it was established in 1979 that the only stable solutions are constant solutions, if Ω is convex. (See [CH,Ma1].) Thus, stability implies triviality. When we come to 2 × 2 elliptic systems ⎧ ⎨ d1 u + f (u, v) = 0 d2 v + g(u, v) = 0 ⎩ ∂u ∂v ∂ν = ∂ν = 0

in Ω, in Ω, on ∂Ω,

(S)

the situation becomes much more complicated and, up to this date, no general result has been established. However, as an intermediate step between single equations and 2 × 2 systems, progress has been made for the “shadow” system obtained by letting d2 → ∞ in (S) and formally replacing v by a constant ξ ⎧ ⎪ ⎨ d1 u + f (u, ξ ) = 0 Ω g(u, ξ ) dx = 0, ⎪ ⎩ ∂u = 0 ∂ν

in Ω, on ∂Ω.

(See Section 3.3.) It was proved in [NPY] that if n = 1 (i.e., Ω = (0, 1)) then stable solutions of shadow systems must be monotone. That is, stability implies monotonicity. In fact, similar results have been established in [NPY] for time-dependent solutions of the corresponding parabolic “shadow” systems as well. However, it remains an outstanding open problem when n > 1. One of the most direct ways to understand the qualitative behavior of solutions is to be able to “view” the graphs of solutions. With the help of modern computing power, it is now possible to graph solutions of nonlinear elliptic problems quite accurately by numerical simulations and thereby “visualize” the shape of solutions, even those exhibiting strikingly singular behavior. In Section 5, a brief illustration of this approach is presented. Again, particular attention has been paid to comparing the effects of different boundary conditions on the shapes of solutions. For the sake of simplicity, we have only included illustrations involving two-dimensional domains, although three-dimensional domains can be handled as well. This section is mainly written in collaboration with Goong Chen, Alain Perronnett and Jianxin Zhou, to whom I am grateful. Throughout this entire paper I have focused only on properties of positive solutions. Sign-changing solutions and their nodal domain properties are extremely interesting, although relatively unexplored. In this regard, I mention the very nice work of Castro et al. [CCN] in which a solution to a nonlinear Dirichlet problem that changes sign exactly once is established. Many other topics of elliptic equations have been omitted here; for instance, the Morse index of solutions; the topological degree of solutions; equations involving critical Sobolev exponents; and others. Many of those are quite important and interesting. Fortunately, some of them are covered by other articles in this volume. The selection of topics in this survey does not imply any value judgment on the topics but rather reﬂects my own taste. In particular, the Ginzburg–Landau system has received enormous attention in recent years. Although originally I intended to include it in this survey, I soon realized that it deserves

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a separate paper of its own. Here, we simply refer the interested readers to [BBH] and the more recent papers [Lin,LR]. For superconductivity with magnetic ﬁeld, the readers should consult [Pan] and the references therein. Many colleagues offered generous help in the writing of this survey-expository paper. Besides these already mentioned above, I would like to thank Changfeng Gui, Yi Li and Juncheng Wei, as I have learned from them on various topics included here. I am particularly grateful to Fang-Hua Lin, who explained, among other things, the Ginzburg–Landau vortices to me.

1. Concentrations of solutions: Single equations One of the greatest advances in the theory of partial differential equations is the recent progress on the studies of concentration behaviors of solutions to elliptic equations and systems. It is remarkable to see that similar, and in many cases, independent results have been obtained simultaneously concerning these striking behaviors in various models from different areas of science. These include activator–inhibitor systems in modeling the regeneration phenomenon of hydra, Ginzburg–Landau systems in superconductivity, nonlinear Schrödinger equations, the Gray–Scott model, the Lotka–Volterra competition system with cross-diffusions, and others. In this and next sections, we shall include descriptions of some of these systems from their backgrounds to the signiﬁcance of the mathematical results obtained. Comparisons of results under different boundary conditions also will be made to illustrate the importance of boundary effect on the behaviors of solutions.

1.1. Spike-layer solutions in elliptic boundary-value problems We have indicated in the Introduction that Neumann boundary condition is far less restrictive than Dirichlet boundary condition. Consequently, Neumann boundary-value problems tend to allow more solutions with more interesting behaviors than their Dirichlet counterparts. However, it is interesting to note that systematic studies of nonlinear Neumann problems seem to have a much shorter history. Studies of nonlinear Neumann problems are often motivated by models in pattern formation in mathematical biology. One of the more well-known examples is the Turing’s “diffusion-driven instability”. The regeneration phenomenon of hydra, ﬁrst discovered by A. Trembley [Tr] in 1744, is one of the earliest examples in morphogenesis. Hydra, an animal of a few millimeters in length, is made up of approximately 100,000 cells of about 15 different types. It consists of a “head” region located at one end along its length. Typical experiments on hydra involve removing part of its “head” region and transplanting it to other parts of the body column. Then, a new “head” will form if and only if the transplanted area is sufﬁciently far from the (old) “head”. These observations have led to the assumption of the existence of two chemical substances – a slowly diffusing (short-range) activator and a rapidly diffusing (long-range) inhibitor. In 1952, A. Turing [Tu] argued, although diffusion is a smoothing

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and trivializing process in a system of a single chemical, for systems of two or more chemicals, different diffusion rates could force the uniform steady states to become unstable and lead to nonhomogeneous distributions of such reactants. This is now known as the “diffusion-driven instability”. Exploring this idea further, in 1972, Gierer and Meinhardt [GM] proposed the following activator–inhibitor system (already normalized) to model the above regeneration phenomenon of hydra: ⎧ Up ⎪ ⎨ Ut = d1 U − U + V q r τ Vt = d2 V − V + U Vs ⎪ ⎩ ∂U ∂V ∂ν = ∂ν = 0

in Ω × [0, T ), in Ω × [0, T ),

(1.1)

on ∂Ω × [0, T ),

where, as before, is the usual Laplacian, Ω is a bounded smooth domain in Rn , ν denotes the outward unit normal to ∂Ω, T ∞, and the constants τ , d1 , d2 , p, q, r are all positive, s 0 and 0

0 in Ω, ⎪ ⎩ ∂u = 0 on ∂Ω. ∂ν

(1.4)

In the case n = 1, a lot of work has been done by I. Takagi [T]. For n 2, the situation becomes far more interesting. The pioneering work [NT1–NT3], [LNT] have produced a single-peak spike-layer solution uε of (1.4) in 1993. Furthermore, steady states of the shadow system (1.3) as well as the original system (1.1) have been constructed from uε – at least for small d1 and large d2 , and their stability properties have been investigated [NT4,NTY1,NTY2]. It seems illuminating to “solve” (1.4) as well as its Dirichlet counterpart side-by-side: ⎧ 2 ⎨ ε u − u + up = 0 in Ω, u>0 in Ω, ⎩ u=0 on ∂Ω,

(1.5)

and compare the qualitative properties of the solutions. I shall ﬁrst describe how the existence of a single-peak spike-layer solution is established, and then discuss the location and the proﬁle of this single peak. Since the equation in (1.4) and (1.5) is “autonomous” (i.e., no explicit spatial dependence in the equation), the location of the spike must be determined by the geometry of Ω. I would like to call the reader’s attention to see how exactly the geometry of Ω enters the picture in each of the problems (1.4) and (1.5) separately, and to compare the effects of different boundary conditions on the location of the peak. For ε small, (1.4) and (1.5) are singular perturbation problems. However, the traditional method in applied mathematics, using inner and outer expansions, simply does not apply here, because a spike-layer solution of (1.4) or (1.5) is exponentially small away from its peaks. To solve (1.4) or (1.5), it is important to note that although (1.1) does not admit a variational structure, there is a natural “energy” functional for (1.4) or (1.5). In the rest of this section, we will always assume that 1 < p < n+2 n−2 if n 3, and 1 < p < ∞ if n = 1, 2. We ﬁrst deﬁne the “energy” functional in H 1 (Ω) 1 Jε,N (u) = 2

2 ε |∇u|2 + u2 − Ω

1 p+1

p+1

Ω

u+ ,

(1.6)

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where u+ = max{u, 0}. It is standard to check that a critical point corresponding to a positive critical value of Jε,N is a classical solution of (1.4). Similarly, we deﬁne the “energy” functional in H01 (Ω) 1 Jε,D (u) = 2

2 ε |∇u|2 + u2 − Ω

1 p+1

p+1

Ω

u+

(1.7)

and observe that a critical point corresponding to a positive critical value of Jε,D is a classical solution of (1.5). Our ﬁrst step appears to be nothing unusual; namely, we shall use the well-known Mountain–Pass lemma to guarantee that each of Jε,N and Jε,D has a positive critical value. However, in order to use this variational formulation to obtain useful information later, our formulation of the Mountain–Pass lemma deviates from the usual one (see Section 1.4). More precisely, setting 0 1 cε,N = inf max Jε,N (tv) v 0, ≡ 0 in H 1 (Ω)

(1.8)

0 1 cε,D = inf max Jε,D (tv) v 0, ≡ 0 in H01 (Ω) ,

(1.9)

t 0

and t 0

we show that cε,N is a positive critical value of Jε,N , thus gives rise to a solution uε,N of (1.4); and similarly that cε,D is a positive critical value of Jε,D , thus gives rise to a solution uε,D of (1.5). Our main task here is to prove that both uε,N and uε,D exhibit a single spike-layer behavior, and we are going to determine the location as well as the proﬁle of the spike-layer of uε,N and uε,D separately. Roughly speaking, both uε,N and uε,D can have only one “peak” (i.e., a local maximum denoted by Pε,N and Pε,D , respectively, and must tend to 0 everywhere else. point in Ω), Moreover, Pε,N must lie on the boundary ∂Ω and tend to the “most-curved” part of ∂Ω, while Pε,D must tend to the “most-centered” part of Ω as ε tends to 0. As for the proﬁles of uε,N and uε,D , again, roughly speaking, uε,D is approximately a “scaled” version of w near Pε,D where w is the unique solution of ⎧ p n ⎨ w − w + w = 0 in R , n w → 0 at ∞, w > 0 in R , ⎩ w(0) = max w,

(1.10)

while uε,N is approximately a “scaled” and “deformed” version of “half” of w. To make those descriptions precise, we start with uε,N . T HEOREM 1.1. For each ε sufﬁciently small, the solution uε,N has exactly one local (thus and it is achieved at exactly one point Pε,N in Ω . Moreover, global) maximum in Ω uε,N has the following properties: (i) As ε → 0 the translated solution uε (· + Pε,N ) → 0 except at 0, and uε,N (Pε,N ) → w(0), where w is the unique solution of (1.10).

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(ii) Pε,N ∈ ∂Ω and H (Pε,N ) → maxP ∈∂Ω H (P ) as ε → 0, where H denotes the mean curvature of ∂Ω. (iii) Through rotation and translation we may suppose that Pε,N is the origin and near Pε,N the boundary ∂Ω = {(x , xn ) | xn = ψε (x )} and Ω = {(x , xn ) | xn > ψε (x )}, where x = (x1 , . . . , xn−1 ), and ψε (0) = 0, ∇ψε (0) = 0. Then the diffeo˜ = (Φε,1 (x), ˜ . . . , Φε,n (x)) ˜ deﬁned by morphism x = Φε (x) Φε,j x˜ =

"

ε for j = 1, . . . , n − 1, x˜j − x˜n ∂ψ ∂xj x˜ x˜n + ψε x˜ for j = n,

ﬂattens the boundary ∂Ω near Pε,N , and uε,N Φε (εy) = w(y) + εφ(y) + o(ε),

(1.11)

where φ is the unique solution of ⎧ φ − φ + pwp−1 φ ⎪ ⎪ ⎨ 2w ∂w +2|yn | ni,j =1 ψε,ij ∂y∂i ∂y − αε (sgn yn ) ∂y = 0 in Rn , n j ⎪

⎪ ∂w ⎩ φ(y) → 0 as y → ∞, and = 0 for j = 1, . . . , n, nφ R

with ψε,ij =

∂ 2 ψε ∂xi ∂xj

(1.12)

∂yj

(0), αε = ψε (0).

Note that (1.10) gives the proﬁle of uε,N up to the second order, where it can be proved that φ actually decays exponentially near ∞. The detailed proof of Theorem 1.1 may be found in [NT2,NT3]. We now turn to the Dirichlet case. To describe our results, ﬁrst we need to introduce

in Rn , we let PΩ w be the solution of the some notation. For a bounded smooth domain Ω linear problem

v − v + wp = 0 v=0

in Ω,

on ∂ Ω,

(1.13)

where w is the unique solution of (1.10). Now, set z=

x − Pε,D ε

and Ωε = {z ∈ Rn | x = Pε,D + εz ∈ Ω}, where Pε,D is the unique peak of uε,D as is stated in Theorem 1.2. Since eventually we will show the scaled version of PΩε w is a very good approximation of uε , we need to study the difference between w and PΩε w, that is, the function ϕε ≡ w − PΩε w, which satisﬁes

ϕε − ϕε = 0 in Ωε , ϕε = w on ∂Ωε .

(1.14)

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The quantity ϕε is extremely small and it turns out that the “correct” order of the difference w − PΩε w (for our purposes) is the logarithmic of ϕε−ε ; i.e., the function δε (x) = −ε log ϕε (z), which satisﬁes a nonlinear equation instead "

in Ω, εδε − |∇δε |2 + 1 = 0 x−Pε,D δε (x) = −ε log w on ∂Ω. ε

(1.15)

1

Finally, we enlarge ϕε to Vε (z) = e ε δε (Pε,D ) ϕε (z). It is clear that Vε satisﬁes

Vε − Vε = 0 Vε (0) = 1.

in Ωε ,

We are now ready to state our main results for the Dirichlet problem (1.5). T HEOREM 1.2. For each ε sufﬁciently small, the solution uε,D has exactly one local (thus global) maximum in Ω and it is achieved at exactly one point Pε,D in Ω. Moreover, uε,D has the following properties: (i) As ε → 0 the translated solution uε,D (·+Pε,D ) → 0 except at 0, and uε,D (Pε,D ) → w(0), where w is the unique solution of (1.10). (ii) d(Pε,D , ∂Ω) → maxP ∈Ω d(P , ∂Ω) as ε → 0, where d denotes the usual distance function. (iii) For every sequence εk → 0, there is a subsequence εki → 0 such that for ε = εki it holds that 1 1 uε,D (x) = PΩε w(z) + e− ε δε (Pε,D ) φε (z) + o e− ε δε (Pε,D ) ,

(1.16)

where δε (Pε,D ) → 2 maxP ∈Ω d(P , ∂Ω), e−μ|z| (φε − φ0 )L∞ (Rn ) → 0 with 1 > μ > max{0, 2 − p}, φ0 being a solution of − 1 + pwp−1 φ0 = pwp−1 V0

in Rn ,

and V0 being the pointwise limit of Vε , ε = εki . Several remarks are in order. First, comparing Theorems 1.1 and 1.2, we see that part (i) of Theorems 1.1 and 1.2, respectively, shows that each of the solutions uε,N and uε,D possesses a single-peak spike-layer structure, and, part (ii) of Theorems 1.1 and 1.2, also respectively, locates the peak of uε,N and of uε,D . It is interesting to note that although x−P intuitively, by the exponential decay of w, a scaled w (i.e., w( ε ε,D )) which is truncated near ∂Ω seems to be an excellent approximation for uε,D , part (iii) of Theorem 1.2 indix−P cates that the function PΩε w( ε ε,D ) is actually a better approximation for uε,D . This is quite delicate since the error terms induced by these two approximations are both of exponentially small order and, are very close. In fact, this observation turns out to be crucial in pushing our method through for the Dirichlet case (1.5).

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We now describe our method of proofs. In both cases (1.4) and (1.5), the most important idea is to obtain an estimate for cε,N and cε,D , respectively, which is sufﬁciently accurate to reﬂect the inﬂuence of the geometry of the domain Ω. More precisely, both the zerothorder approximations for cε,N and cε,D depend only on the unique solution w (and its energy) of (1.10). The geometry of the domain Ω, namely, the boundary mean curvature H (Pε,N ) at Pε,N in the Neumann case and the distance d(Pε,D , ∂Ω) in the Dirichlet case, enters the ﬁrst-order approximation of cε,N and cε,D . To be explicit, we have cε,N = ε

n

1 I (w) − (n − 1)γN H (Pε,N )ε + o(ε) 2

(1.17)

and 1 1 cε,D = εn I (w) + e− ε δε (Pε,D ) γD + o e− ε δε (Pε,D ) ,

(1.18)

where γN and γD are positive constants independent of ε and I (w) =

1 2

Rn

|∇w|2 + w2 −

1 p+1

Rn

wp+1 .

(1.19)

Observe that part (ii) of Theorems 1.1 and 1.2 follows from (1.17) and (1.18), respectively, together with some useful upper bounds of cε,N and cε,D . However, to obtain (1.17) and (1.18), one must ﬁrst establish part (iii) of Theorems 1.1 and 1.2, respectively, namely, the proﬁles of uε,N and uε,D (i.e., (1.11) and (1.16)). This turns out to rely heavily on some preliminary versions of (1.17) and (1.18). The proofs are indeed very involved and we shall refer the reader to the papers [NT2,NT3] and [NW] for the full details. One interesting component in our proof of Theorem 1.2 is that the limit of δε turns out to be a “viscosity” solution of the Hamilton–Jacobi equation |∇δ| = 1

in Ω

which gives rise to the distance d(Pε,D , ∂Ω). This also seems to indicate that although the function ϕε (or Vε ) satisﬁes a simple linear elliptic equation while δε satisﬁes a nonlinear one, the “correct” order of the error (w − PΩε w) is far more important than the form of the equation it satisﬁes. It turns out that the Neumann problem (1.4) also has a single-interior-peak spike-layer solution which is very close to the solution obtained in Theorem 1.2 (for the Dirichlet case (1.5)). We refer the interested reader to [W2] or the next section. Theorems 1.1 and 1.2 establish the existence of single-peak spike-layer solutions of (1.4) and (1.5), respectively. One natural question is that: Are there other single-peak spikelayer solutions of (1.4) or (1.5)? And, if there are, where are the locations of their peaks? For boundary spikes of the Neumann problem (1.4), Wei showed that if uε is a solution of (1.4) which has a single boundary peak Pε , then, as ε → 0, by passing to a subsequence if necessary, Pε must tend to a critical point of the boundary mean curvature. On the other hand, for each nondegenerate critical point P0 of the boundary mean curvature, one can

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always construct, for every ε > 0 small, a solution of (1.4) which has exactly one peak at Pε ∈ ∂Ω such that Pε → P0 as ε → 0. (See [W1] for details). For interior spikes of the Neumann problem (1.4), it is established in [GPW] that (by passing to a subsequence if necessary) the single peak Pε of a solution of (1.4) must tend to a critical point of the distance function d(P , ∂Ω). Conversely, again with additional hypotheses on Ω and a nondegeneracy condition on a critical point P0 ∈ Ω of d(P , ∂Ω), one can construct, for every ε > 0 small, a solution of (1.4) which has exactly one peak at Pε ∈ Ω such that Pε → P0 as ε → 0. (See [W2] for details). The counterparts of the above results for the Dirichlet case (1.5) are, however, not settled. Progress has been made in [W4].

1.2. Multi-peak spike-layer solutions in elliptic boundary-value problems A vast amount of literature on (1.4) has been produced since the publication of [NT3] in 1993. Much progress has been made and fascinating results concerning multi-peak spike-layer solutions have been obtained. We will only include the most recent and complete results here. Again, in this section we shall always assume that 1 < p < n+2 n−2 in (1.4). An “ideal” result for multi-peak spike-layer solutions to (1.4) would read as follows: C ONJECTURE . For any given nonnegative integers k and , (1.4) always possesses a multi-peak spike-layer solution with exactly k interior-peaks and boundary-peaks, provided that ε is sufﬁciently small. This conjecture almost has been proved in this generality. In [GW2], this conjecture is established with some minor conditions imposed on the domain Ω. The main difﬁculty here comes from the fact the “error” in the boundary-peak case is algebraic (as shown in (1.11) or (1.17)) while the “error” in the interior-peak case is transcendental (as indicated in (1.16) or (1.18)). To overcome this, a delicate argument was devised in [GW2] to handle the gap in the error, but only under additional technical assumptions on Ω. On the other hand, if we are to treat interior-peaks and boundary-peaks separately, deﬁnitive results are possible. For the case of interior-peaks, the following result was obtained in [GW1]. T HEOREM 1.3. Given any positive integer k, for every ε sufﬁciently small, (1.4) always possesses a multi-peak spike-layer solution with exactly k interior peaks. Furthermore, as ε → 0 the k peaks converge to a maximum point of the function 1 ϕ P 1 , . . . , P k = min d P i , ∂Ω , P − P m i, , m = 1, . . . , k , 2

(1.20)

where P 1 , . . . , P k ∈ Ω. Intuitively speaking, a maximum point (Q1 , . . . , Qk ) of the function ϕ in (1.20) corresponds to the centers of k disjoint balls of equal size packed in Ω (i.e., contained in Ω)

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with the largest possible diameter. Such a maximum point certainly exists, although may not be unique. The method of proof is still variational; however, with the help of Lyapunov–Schmidt reduction, the original “global” variational approach has now evolved to a powerful “localized” version. To illustrate the basic idea involved here, it seems best that we only treat the simplest case k = 1. This “localized” energy method is semiconstructive. The strategy is simple: First, we construct an approximate solution in the sense of Lyapunov–Schmidt with its peak located at a prescribed point P ∈ Ω. Then we perturb this point P and ﬁnd a critical point of the corresponding “energy” of this approximate solution, which gives rise to an interior-peak spike-layer solution of (1.4). To carry out this strategy, we let w be the solution of (1.10), as before, and, for any given point P = (P1 , . . . , Pn ) in Ω, let Pε,P w be the solution of

v − v + wp = 0 ∂v ∂ν = 0

in Ωε,P , on ∂Ωε,P ,

where Ωε,P = {z ∈ Rn | x = P + εz ∈ Ω}. Now solve uε,P = Pε,P w + ψε,P

(1.21)

⊥ , where with ψε,P ∈ Kε,P

Kε,P

∂Pε,P w = span j = 1, . . . , n , ∂Pj

(1.22)

p

and that ψε,P is C 1 in P , uε,P − uε,P + uε,P ∈ Kε,P and $ # C ψε,P H 2 (Ωε,P ) C exp − d(P , ∂Ω) . ε

(1.23)

Next, we deﬁne Φε (P ) = Jε (uε,P ) = Jε (Pε,P w + ψε,P ).

(1.24)

It is not hard to see that a maximum point Pε of Φε , i.e., Φε (Pε ) = max Φε (P )

(1.25)

P ∈Ω

gives a solution uε = Pε,Pε w + ψε,Pε of (1.4). For uε − uε + upε =

n j =1

αj

∂Pε,Pε w , ∂Pj

(1.26)

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for some α1 , . . . , αn ∈ R. From (1.25) and (1.26) it follows that ∂uε ∂Φε (P ) = Jε (uε ) 0= ∂Pk P =Pε ∂Pk n n ∂Pε,Pε w ∂(Pε,Pε w + ψε,Pε ) = αj = akj αj , ∂Pj ∂Pk Ωε,Pε j =1

(1.27)

j =1

where the matrix (akj ) is deﬁned by the last equality. Due to the cancellation property of the integral Ωε,Pε

∂Pε,Pε w ∂Pε,Pε w , ∂Pj ∂Pk

k = j,

(1.28)

and (1.23), we see that akk is much larger than akj , k = j . Consequently, det(akj ) = 0. Therefore, αj = 0, j = 1, . . . , n, and the proof is complete. This method generalizes to arbitrary k > 1. For the case of boundary-peaks, the situation becomes more interesting. The following result is obtained in [GWW]. T HEOREM 1.4. Given any positive integer k, (1.4) always possesses a multipeak spikelayer solution with exactly k boundary peaks, provided that ε is sufﬁciently small. Furthermore, as ε → 0, the k peaks Qε1 , . . . , Qεk have the following property: H Qεj → min H (P ), P ∈∂Ω

(1.29)

where H denotes the mean curvature of ∂Ω. Comparing the above result to Theorem 1.1, we see a very interesting difference in the location of the peaks: Theorem 1.1 guarantees the existence of a single boundary peak near a maximum of the boundary mean curvature, while Theorem 1.4 implies the existence of an arbitrary number of boundary peaks near a minimum of the boundary mean curvature. Whether (1.4) has a spike-layer solution with exactly k boundary peaks near a maximum of the boundary mean curvature, for a prescribed positive integer k, remains an interesting open question. The proof uses basically the same approach as that of Theorem 1.3. The detailed computations are, of course, quite different. We refer the interested readers to [GWW] for details. In [GW1,GPW], it is also proved that if Pε1 , . . . , Pεk in Ω are the locations of the k (interior) peaks of a solution to (1.4), then, by passing to subsequence if necessary, Pε1 , . . . , Pεk must tend to a critical point of ϕ in (1.20). We see that we have acquired a fairly good understanding of spike-layer solutions of (1.4) although there are still major questions left open. On the other hand, our knowledge of solutions of (1.4) with multidimensional concentration sets is very limited at this time. In the next section we shall report our progress in that direction.

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To conclude this section we remark that the existence of multi-peak spike-layer solutions for the Dirichlet problem (1.5) is, in general, not possible. For instance, when Ω is a ball, [GNN1] implies that solutions of (1.5) must be radially symmetric and thus (1.5) can only have single-peak solutions. This result is extended to strictly convex domains by Wei [W2]. Therefore, the existence of multi-peak spike-layer solutions for the Dirichlet problem (1.5) is drastically different from its counterpart of the Neumann case (1.4), and generally depends on the geometry of Ω.

1.3. Solutions with multidimensional concentration sets

A spike-layer solution (discussed in previous sections) has the property that its “energy” which are 0-dimensional. or “mass” concentrates near isolated points (i.e., its peaks) in Ω, Therefore we view a spike-layer solution as a solution with zero-dimensional concentration set. Similarly, solutions which are small everywhere except near a curve or curves are regarded as solutions with one-dimensional concentration sets. Generalizing in this manner, we can deﬁne solutions with k-dimensional concentration sets. The following conjecture has been around for quite some time: C ONJECTURE . Given any integer 0 k n − 1, there exists pk ∈ (1, ∞] such that for 1 < p < pk , (1.4) possesses a solution with k-dimensional concentration set, provided that ε is sufﬁciently small. (See, for instance, [N2].) Progress in this direction has only been made very recently. In [MM1,MM2] the above conjecture was established for a sequence of ε → 0 in the case k = n − 1 with the boundary ∂Ω (or, part of ∂Ω) being the concentration set. T HEOREM 1.5. Let Ω ⊆ Rn be a bounded smooth domain and p > 1. Then, for any component Γ of ∂Ω, there exists a sequence εm → 0 such that (1.4) possesses a solution uεm for ε = εm and uεm concentrates at Γ ; i.e., uεm → 0 away from Γ and uεm (εm (x − x0 )) → w(x · ν0 ) near x0 ∈ Γ where ν0 is the unit inner normal to Γ at x0 and w is the solution of (1.10) with n = 1. Note that in Theorem 1.5, no upper bound is imposed on p. The proof is very technical, and we shall only give a very brief outline. The ﬁrst crucial step is to construct a “good” approximate solution u˜ ε . Then, a detailed analysis of the second differential Jε (u˜ ε ), where Jε is deﬁned in (1.6), is essential for using the contraction mapping argument to obtain the solution uε near u˜ ε . In the second step it is observed that the Morse index of u˜ ε tends to ∞ as ε → 0, and thus the invertibility of Jε (u˜ ε ) is only guaranteed along a sequence εm → 0. Similar to the proofs of spike-layer solutions in previous sections, the construction of approximate solutions here is crucial, and is delicate and interesting. Roughly speaking, it is natural to use the one-dimensional solution of (1.10) as a candidate for approximate

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solutions. This turns out to be inadequate. In [MM1] (1.10) was replaced by

w − w + wp = −λw in [0, ∞), w (0) = w(∞) = 0 and w > 0,

(1.30)

where λ is related to the mean curvature of the boundary of Ωε = 1ε Ω (which tends to 0 as ε → 0). A natural question would be that whether (1.4) admits any solutions which concentrate on “interior” curves, and if there are such solutions, how the locations of these “interior” curves are determined. Even at the formal level, these are difﬁcult questions. To this date, progress has been made only in radially symmetric cases. In [AMN3], the following result was established. T HEOREM 1.6. Let Ω be the unit ball B1 in Rn . Then, for every p > 1 and ε small, (1.4) possesses a radial solution uε concentrating at |x| = rε for which 1 − rε ∼ ε| log ε|. (Here we use the notation “f ∼ g” to denote that as ε → 0, the quotient f/g is bounded from above and below by two positive constants.) Note that again we do not impose any upper bound on p here. We remark that the solution guaranteed by Theorem 1.6 is different from the one in Theorem 1.5, as the maximum of the solution in Theorem 1.5 takes place on the boundary, while the maximum of the solution in Theorem 1.6 lies in the interior of Ω. In fact, it is possible to construct a solution of (1.4) which concentrates on a cluster of spheres. T HEOREM 1.7 [MNW]. Let Ω be the unit ball B1 in Rn and N be a given positive integer. Then for every p > 1 and ε small, (1.4) possesses a radial solution uε concentrating on N spheres |x| = rε,j , j = 1, . . . , N , where 1 − rε,1 ∼ ε| log ε| and rε,j − rε,j +1 ∼ ε| log ε| for j = 1, . . . , N − 1. Since the basic ideas involved in Theorems 1.6 and 1.7 are similar, we shall conﬁne our discussions in the rest of this section to Theorem 1.6 for the sake of simplicity. It is obvious that (1.5) – the Dirichlet counterpart of (1.4) – does not admit any solutions other than a single-peak solution in case Ω is a ball, as is guaranteed by Gidas et al. [GNN1]. Nevertheless, Theorem 1.6 gives us a good opportunity to compare Dirichlet and Neumann boundary conditions. To illustrate the ideas involved, it seems best to discuss a slightly more general equation ε2 u − V |x| u + up = 0 and u > 0

in B1 ,

(1.31)

under the boundary conditions ∂u =0 ∂ν

on ∂Ω

(1.32)

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or u=0

on ∂Ω,

(1.33)

where V is a radial potential bounded by two positive constants. In fact, the relevant quantity turns out to be M(r) = r n−1 V θ (r),

θ=

p+1 1 − , p−1 2

(1.34)

and Theorem 1.6 is a special case of the following result. T HEOREM 1.8 [AMN3]. (i) If M (1) > 0 then for every p > 1 and ε small, the problem (1.31), (1.32) possesses a solution uε concentrating at |x| = rε , where 1 − rε ∼ ε| log ε|. (ii) If M (1) < 0 then for every p > 1 and ε small, the problem (1.31), (1.33) possesses a solution uε concentrating at |x| = rε , where 1 − rε ∼ ε| log ε|. R EMARK . In addition to the solutions in Theorem 1.8, the problems (1.31), (1.32) and (1.31), (1.33) also have solutions concentrating near |x| = r¯ , where (and if ) r¯ is a local extreme point of M. This particular solution also exists for the nonlinear Schrödinger equations in Rn . (See [AMN2].) The proof relies upon a ﬁnite-dimensional Lyapunov–Schmidt reduction and a “localized” energy method. Again, the ﬁrst crucial step, for both (i) and (ii), is to ﬁnd a good apε ε for (i) and zρ,D for (ii), where ρ is a parameter between 0 and 1, proximate solution zρ,N denoting the radius of concentration and will be determined eventually. Observe that a solution to the problem (1.31) and (1.32) is a critical point of the (rescaled) functional on Hr1 (B1/ε ) 1 J ε,N (u) = 2

|∇u|2 + V ε|x| u2 −

B1/ε

1 p+1

p+1

B1/ε

u+ ,

(1.35)

where Hr1 is the space of all radial H 1 functions on B1/ε . The general abstract procedure (zε establishing J ε,N ρ,N + w) = 0 is equivalent to ε ε )⊥ such that P J

(zε + w) = 0, and (a) ﬁnding w = wρ,N ∈ (Tz ZN ε,N ρ,N (b) ﬁnding a stationary point of ε ε + wρ,N , Ψε,N (ρ) = J ε,N zρ,N

(1.36)

ε ⊥ ε ) , Tz Z N is the tangent where P denotes the orthogonal projection from Hr1 onto (Tz ZN ε ε ε space of ZN at z and ZN is the family of the approximate solutions zρ,N . To ﬁnd nontrivial critical value of Ψε,N , we have the following expansion

1 εn−1 Ψε,N (ρ) = M(ερ) α − βe−2λ( ε −ρ) + higher-order terms,

(1.37)

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where α, β are two positive constants and λ2 = V (ερ). Now, differentiating (1.37) with respect to ρ and setting the leading term to 0, we obtain 1 εM (ερ) α − βe−2λ( ε −ρ) # $ 1 1 + 2βM(ερ) ελ (ερ) − ρ − λ(ερ) e−2λ( ε −ρ) = 0. ε If

1 ε

− ρ ∼ | log ε|, then, as ε → 0 1

e−2λ( ε −ρ) → 0, and therefore εαM (ερ) ∼ 2βM(ερ)λ(ερ)e−2λ( ε −ρ) 1

(1.38)

which can be achieved if M (1) > 0 (since ερ → 1 and 1ε − ρ ∼ | log ε|). For the Dirichlet case (1.31) and (1.33) we deﬁne similarly the functionals J ε,D and Ψε,D and the expansion corresponding to (1.37) now reads as follows: 1 εn−1 Ψε,D (ρ) = M(ερ) α + βe−2λ( ε −ρ) + higher-order terms.

(1.39)

Comparing (1.39) to (1.37), we see that only the sign for the term βe−2λ( ε −ρ) is different, which reﬂects the different or, opposite effects of Dirichlet and Neumann boundary conditions. Heuristically, when V ≡ 1, the ﬁrst term in (1.37) and (1.39) is due to the volume 1 energy which always has a tendency to “shrink”, while the second term ±βe−2λ( ε −ρ) in (1.39) and (1.37), respectively, indicates that in the Dirichlet case the boundary “pushes” the mass of the solution away from the boundary (therefore only single-peak solutions are possible), but in the Neumann case the boundary “pulls” the mass of the solution and thereby reaches a balance at ρ = rε ∼ 1 − ε| log ε| creating an extra solution. We remark that the method described above also applies to the annulus case and yields the following interesting results for V ≡ 1, which illustrates the opposite effects between Dirichlet and Neumann boundary conditions most vividly. 1

T HEOREM 1.9 [AMN3]. (i) For every p > 1 and ε small, the Neumann problem (1.4) with Ω = {x ∈ Rn | 0 < a < |x| < b} possesses a solution concentrating at |x| = rε , where b − rε ∼ ε| log ε|, near the outer boundary |x| = b. (ii) For every p > 1 and ε small, the Dirichlet problem (1.5) with Ω = {x ∈ Rn | 0 < a < |x| < b} possesses a solution concentrating at |x| = rε , where rε − a ∼ ε| log ε|, near the inner boundary |x| = a. Observe that from the “moving plane” method [GNN1] it follows easily that the Dirichlet problem (1.5) does not have a solution concentrating on a sphere near the outer boundary |x| = b.

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In conclusion, we mention that the method of Theorem 1.8 can be extended to produce solutions with k-dimensional concentration sets but again, some symmetry assumptions are needed. The conjecture stated at the beginning of this section remains as a major open problem. 1.4. Remarks In this section, we have considered the various concentration phenomena for essentially just one equation, namely, ε2 u − u + up = 0

(1.40)

in a bounded domain Ω under either Dirichlet or Neumann boundary conditions in (1.5) or (1.4), respectively. However, since the equation (1.40) is quite basic, similar phenomena could be expected for a more general class of equations. As a side remark, perhaps we ought to mention that, as ε becomes large, (1.4) will eventually lose all of its solutions except the trivial one uε ≡ 1 [LNT]. The methods involved in handling (1.40) are basically variational; more precisely, via the mountain-pass lemma. However, the mountain-pass approach we have used here in establishing Theorem 1.1 is due to Ding and Ni [DN] in 1983, which deviates from the original one due to Ambrosetti and Rabinowitz [AR] and is less general but more constructive. As a result, it is proved in [N1] that this approach yields the same critical value as the constrained minimization approach due to Nehari [Ne] in 1960. In studying multipeak solutions and other concentration sets, this approach has been modiﬁed; namely, it is also coupled with the Lyapunov–Schmidt ﬁnite-dimensional reduction, and becomes “local” in nature. This “localized energy method” is a major achievement, and is due to Gui and Wei [GW1]. It is interesting to note that the concentration sets of solutions to (1.4) we have discussed so far have dimensions ranging from 0 (i.e., peaks) to n − 1 (spheres in Rn ). A natural question arises: Does (1.4) possess solutions with n-dimensional concentration sets? In general, this question remains open although the answer is probably negative. Solutions with n-dimensional concentration sets (often referred to as internal transition layers) appear in phase transitions. This problem has been studied extensively in the past 30 years by many authors, including Alikakos, Bates, Xinfu Chen, del Pino, Fife, Fusco, Hale, Mimura and others. We refer the interested readers to the monograph [F] for further details. Finally, we remark that in Section 5, some of the single-peak spike-layer solutions of (1.40) are graphed numerically under homogeneous Dirichlet or Neumann boundary conditions. 2. Concentrations of solutions: Systems In Section 2.1, we shall return to the activator–inhibitor system (1.1) discussed in Section 1. Our ﬁrst goal is to construct stationary solutions of (1.1) for large d2 given the

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knowledge of the single-peak spike-layer solutions of (1.4) in Section 1. It turns out that this is accomplished only under additional assumptions. Next, we shall discuss cross-diffusion systems. Unlike (1.1) which is coupled in the reaction terms, these systems are coupled also in the highest-order terms. These systems typically appear in population dynamics with the environmental inﬂuences on the movement of individuals taken into consideration. That is, we no longer assume that individuals move around randomly. Instead, “directed movements” seem more reasonable. Thus, a basic equation in population dynamics (without the reaction term for the time being) is ut = ∇ · d∇u ± u∇ψ E(x, t) ,

(2.1)

where d > 0, ψ is increasing and E represents environmental inﬂuences that could also depend on u. Note that the ﬁrst term in (2.1) is diffusion, while the second term there represents the “directed movement” or the “taxis”. Examples for ψ include ψ(E) = kE, k log E or kE m /(1 + aE m ), where k > 0 and m > 0. When the positive sign in (2.1) is used, we refer to the movement as “negative taxis”, as in the classical Lotka–Volterra competition system with cross-diffusion proposed by Shigesada, Kawasaki and Teramoto in 1979, which will be studied in Section 2.2. When the negative sign in (2.1) is adopted, we have “positive taxis”, as in the Keller–Segel system in modeling the chemotactic aggregation stage of cellular slime molds (amoebae), which will be described in Section 2.3. As we shall see, in all these examples, solutions to the single equation (1.4) play fundamental roles. Another common feature in those three examples is that they all do not have variational structures – that is, none of the three systems is the Euler–Lagrange equation of a variational functional. In many ways, this makes them more interesting and challenging. In Section 2.4 we include two more systems: the CIMA reaction and the Gray–Scott model. Both present extremely rich and interesting phenomena in pattern formations.

2.1. The activator–inhibitor system In the one-dimensional case, much is known due to the work of Takagi [T]. We shall therefore focus on the case n 2 in this section. The existence question for nontrivial stationary solutions to the activator–inhibitor system (1.1) under the condition (1.2) for general domain Ω remains open. Progress has been made and there are two approaches to this problem. The ﬁrst one is via the shadow system. The best result in this direction so far seems to be [NT4] in which the domain Ω is assumed to be axially symmetric and multipeak spike-layer steady states are constructed. Here we are going to give a brief description of this result. The steady states of (1.1) satisfy the following elliptic system ⎧ p d1 U − U + U ⎪ q =0 V ⎪ ⎪ ⎨ Ur d2 V − V + V s = 0 ⎪ U > 0 and V > 0 ⎪ ⎪ ⎩ ∂U ∂V ∂ν = 0 = ∂ν

in Ω, in Ω, in Ω, on ∂Ω,

(2.2)

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179

where p, q, r are positive, s 0 and 0

0. p−1

(2.6)

Now, suppose that the xn -axis is the axis of symmetry for Ω and that P1 , . . . , P2N are the points at which ∂Ω intersects the xn -axis. The following result is proved in [NT4]. T HEOREM 2.1. Under the assumption (2.6), given any m distinct points Pj1 , . . . , Pjm in {P1 , P2 , . . . , P2N }, there are two constants D1 and D2 such that for every 0 < d1 < D1 and d2 > D2 the system (2.2) has a spike-layer solution with exactly m peaks at Pj1 , . . . , Pjm . To illustrate our approach, we shall only treat the case m = 1, as the general case m > 1 requires no new ideas or techniques. First, we introduce a diffeomorphism to ﬂatten a boundary portion around P ∈ {P1 , . . . , P2N }, as follows. Assuming P is the origin, we see that there is a smooth function ψ ∈ C ∞ ([−δ, δ]) with ψ(0) = ψ (0) = 0 such that, near P , ∂Ω is represented by {(x , ψ(|x |) | |x | < δ}. Setting " y yj − yn ψ y |yj | , j = 1, . . . , n − 1, (2.7) Φj (y) = j = n, y n + ψ y , we see that x = Φ(y) = (Φ1 (y), . . . , Φn (y)) is a diffeomorphism from an open set contain3κ , where κ > 0 is small, onto a neighborhood of P with DΦ(0) = I , ing the closed ball B

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W.-M. Ni

the identity map. Observe that x = Φ(y) maps the hyperplane {yn = 0} into ∂Ω. We write Ψ = Φ −1 . Now we can write u(x) = χ κ −1 Ψ (x) w ε−1 Ψ (x) + εφ ≡ u˜ ε + εφ,

(2.8)

where w is the solution of (1.10) and χ ∈ C0∞ (R) is a cut-off function such that (i) 0 χ(s) 1, (ii) χ(s) = 1 if |s| 1, and (iii) χ(s) = 0 if |s| 2. Note that u˜ is an approximate solution of the equation in (2.5) and the equation in (2.5) now takes the following form Lε φ + gε + Mε [φ] = 0, where Lε = ε2 − 1 + pu˜ p−1 , ε 1 2 ε u˜ ε − u˜ ε + u˜ pε , ε p 1 u˜ ε + εφ − u˜ pε − εpu˜ p−1 φ . Mε [φ] = ε ε

gε =

It turns out that Mε [φ] is small and gε is bounded. While Lε is not invertible in general, it actually has a bounded inverse when restricted to axially symmetric functions. This enables us to solve the equation in (2.5) with a solution of the form (2.8). Then we simply deﬁne ξ∞ = |Ω|

1/α

u

r

−1/α

Ω

q/(p−1)

and U∞ (x) = ξ∞ u(x), we obtain a solution (U, ξ ) of the shadow system (2.4). The original system (2.2) with d2 large turns out to be a regular perturbation of the shadow system (2.4). If we write δ = d2−1 and deﬁne the operator 1 Pu = u − |Ω|

u, Ω

then we can convert solving the system (2.2) to ﬁnding a zero (U, ξ, φ) of the map F = (F1 , F2 , F3 ) for δ > 0 but small, where F1 (U, ξ, φ; δ) = ε2 U − U +

Up , (ξ + φ)q

# F2 (U, ξ, φ; δ) = −(ξ + φ) +

$ Ur , (ξ + φ)s Ω $ # Ur , F3 (U, ξ, φ; δ) = φ + δ −φ + P (ξ + φ)s

Qualitative properties of solutions to elliptic problems

181

near (U∞ , ξ∞ , 0) (the solution for (2.4), corresponding to the case δ = 0). Notice that we have decomposed the second equation in (2.2) into two equations so that the linearization of the map F at (U∞ , ξ∞ , 0; 0) is invertible in suitable function spaces and thereby (2.2) can be solved by the implicit function theorem. The second approach is due to Wei and his collaborators. In this approach, d2 needs not to be very large, although there are other restrictions; in particular, this approach only works for planar domains, i.e., n = 2. To illustrate the basic idea involved here, we take the special case s = 0 in (2.2). The ﬁrst step here is to solve the second equation in (2.2)

d2 V − V + U r = 0 in Ω, ∂V on ∂Ω. ∂ν = 0

(2.9)

Then, writing V = T [U r ] and substituting into the ﬁrst equation in (2.2), we have "

d1 U − U + ∂U ∂ν

Up (T [U r ])q

=0

=0

in Ω, on ∂Ω.

(2.10)

It is observed that, under suitable scalings, (2.9) will have a solution close to a large constant, namely, V ∼ ξε 1 + O

1 | log ε|

where d1 = ε2 is small and ξε → ∞ as ε → 0. In this approach, the asymptotic behavior of the Green’s function G − G + δP = 0 in Ω, ∂G on ∂Ω, ∂ν = 0 where δP denotes the Dirac δ-function at the point P , is essential, which limits this approach to n = 2 only. (See [WW1,WW2].) 2.2. A Lotka–Volterra competition system with cross-diffusion Although diffusion is generally regarded as a trivializing process in single equations (see Section 1), we have seen how different diffusion rates could produce patterns strikingly different from trivial ones for 2×2 systems. However, for that to happen, reaction terms are essential as well: for some systems, no matter what the diffusion rates are, no nonconstant steady state could possibly exist. For example, the classical Lotka–Volterra competitiondiffusion system takes the following form: ⎧ ⎨ ut = d1 u + u(a1 − b1 u − c1 v) vt = d2 v + v(a2 − b2 u − c2 v) ⎩ ∂u ∂v ∂ν = 0 = ∂ν

in Ω × (0, ∞), in Ω × (0, ∞), on ∂Ω × (0, ∞),

(2.11)

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W.-M. Ni

where all the constants ai , bi , ci , di , i = 1, 2, are positive, and u, v are nonnegative. Here, as is explained in [Wm], u and v represent the population densities of two competing species. (A nice and thorough reference for (2.11) is the recent monograph by Cantrell and Cosner [CC].) For convenience, we set A = aa12 , B = bb12 , and C = cc12 . It is well known that in the “weak competition” case, i.e., (2.12)

B > A > C,

2 c1 b1 a2 −b2 a1 the constant steady state (u∗ , v∗ ) ≡ ( ab11 cc22 −a −b2 c1 , b1 c2 −b2 c1 ) is globally asymptotically stable regardless of the diffusion rates d1 and d2 . This implies, in particular, that no nonconstant steady state can exist for any diffusion rates d1 , d2 . On the other hand, it seems not entirely reasonable to add just diffusions to models in population dynamics, since individuals do not move around completely randomly. In particular, while modeling segregation phenomena for two competing species one must take into account the population pressures created by the competitors. In an attempt to model segregation phenomena between two competing species, Shigesada, Kawasaki and Teramoto [SKT] proposed in 1979 the following cross-diffusion model

⎧ ⎪ ⎨ ut = (d1 + ρ12 v)u + u(a1 − b1 u − c1 v) vt = (d2 + ρ21 u)v + v(a1 − b2 u − c2 v) ⎪ ⎩ ∂u ∂v ∂ν = 0 = ∂ν

in Ω × (0, T ), in Ω × (0, T ),

(2.13)

on ∂Ω × (0, T ),

where ρ12 and ρ21 represent the cross-diffusion pressures and are nonnegative. (In fact, the model in [SKT] also includes “self-diffusion” pressures that turn out to be not so different from the usual diffusion as is shown in [LN1]. Here, for simplicity, we shall discuss only (2.13).) Considerable work has been done concerning the global existence of solutions to the system (2.13) under various hypotheses. However, it is worth noting that even the local existence question for (2.13) is highly nontrivial and was resolved in a series of long papers by Amann [A1,A2] about ten years ago. We ﬁrst focus on the effect of cross-diffusion on steady states. To illustrate the significance of cross-diffusions, we again go to the weak competition case (i.e., B > A > C) since in this case (2.13) has no nonconstant steady states if both ρ12 = ρ21 = 0. (We refer to [Hu] for some interesting discussions on the ecological signiﬁcance of coexistence, “competition–exclusion”, and weak/strong competitions. One point of view is that whether “competition–exclusion” holds in nature is a matter of interpretation. See [Wm].) Recent work of Lou and myself [LN1,LN2] show that, indeed, if one of the two cross-diffusion rates, say ρ12 , is large, then (2.13) will have nonconstant steady states provided that d2 belongs to a proper range. On the other hand, if both ρ12 and ρ21 are small, then (2.13) will have no nonconstant steady states under the condition (2.12). This shows the introduction of cross-diffusion does seem to help create patterns. In the “strong competition” case, i.e., B < A < C, even the situation of steady states solutions of (2.11) becomes more interesting and complicated, and is not completely understood. Nonetheless, cross-diffusions still have similar effects in help creating nontrivial patterns of (2.13). We refer the interested readers to [LN1,LN2] for details.

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183

So far in this section, we have only touched upon the existence and nonexistence of nonconstant steady states. It seems natural and important to ask if we can derive any qualitative properties (such as the spike-layers in the previous section) of those steady states. Our ﬁrst step in this direction is to classify all the possible (limiting) steady states as one of the cross-diffusion pressures tends to inﬁnity. T HEOREM 2.2 [LN2]. Suppose for simplicity that ρ21 = 0. Suppose further that B = A = C, n 3, and ad22 = λk for all k, where λk is the kth eigenvalue of − on Ω with zero Neumann boundary data. Let (uj , vj ) be a nonconstant steady state solution of (2.13) with ρ12 = ρ12,j . Then by passing to a subsequence if necessary, either (i) or (ii) holds as ρ12,j → ∞, where ρ (i) (uj , 12,j d1 vj ) → (u, v) uniformly, u > 0, v > 0, and ⎧ ⎪ ⎨ d1 (1 + v)u + u(a1 − b1 u) = 0 in Ω, d2 v + v(a2 − b2 u) = 0 in Ω, ⎪ ⎩ ∂u = 0 = ∂v on ∂Ω; ∂ν ∂ν

(2.14)

and (ii) (uj , vj ) → ( wζ , w) uniformly, ζ is a positive constant, w > 0, and ⎧ ⎪ ⎨ d2 w + w(a2 − c2 w) − b2 ζ = 0 in Ω, ∂w on ∂Ω, ∂ν = 0

⎪ ⎩ b2 ζ 1 a − − c w = 0. Ω w

1

w

(2.15)

1

The proof is quite lengthy. The most important step in the proof is to obtain a priori bounds on steady states of (2.13) that are independent of ρ12 . We ought to remark that both systems (2.14) and (2.15) possess spike-layer solutions. For instance, using a suitable change of variables, the equation in (2.15) may be transformed into (1.4) with p = 2. Thus our results in Section 1 apply. Perhaps we ought to point out that in fact, what is important is the ratio of cross-diffusion versus diffusion ρ12 /d1 in which d1 can also vary. A deeper classiﬁcation result is obtained in [LN2] as ρ12 → ∞ in (2.13) in terms of various possibilities of ρ12 /d1 and d1 . To see how (1.4) turns up in (2.15), at least heuristically, we proceed as follows. Formally, setting ζ = u∗ v∗

and w = v ∗ − ϕ,

(2.16)

we have d2 ϕ − (c2 v∗ − b2 u∗ )ϕ + c2 ϕ 2 = 0.

(2.17)

Rescaling (2.17) we obtain (1.4) provided that c2 v∗ − b2 u∗ > 0,

(2.18)

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which is equivalent to "

1 2 (B 1 2 (B

+ C) > A if B > A > C, + C) < A if B < A < C.

(2.19)

Note that in (2.16) we need w > 0, or, v ∗ > ϕ. In n = 1 this is guaranteed by 1 A > (B + 3C). 4

(2.20)

Under these conditions, our results in Section 1 imply that (2.17) has spike-layer solutions for d2 small. Observe that those solutions tend to 0 as d2 → 0 except at isolated points. Let ϕ be, e.g., the solution of (1.4) guaranteed by Theorem 1.1. Then the pair (w, u∗ v∗ ) satisﬁes the differential equation with the homogeneous Neumann boundary condition in (2.15), and it almost satisﬁes the integral constraint in (2.15) since w is close to v∗ a.e. for d2 small. It is then not hard to ﬁnd a solution, for d2 small, near the pair (w, u∗ v∗ ) by the implicit function theorem, as was done in [LN2]. Although (2.14) is still an elliptic system, it is a bit easier to analyze than the original one. We refer the interested reader to [LN2], Section 5, for details. It turns out that both alternatives (i) and (ii) in Theorem 2.2 occur under suitable conditions. Therefore, to understand the steady states of (2.13) a good model would be (2.14) or (2.15), at least when ρ12 is large. In the recent work of Lou, Yotsutani and myself [LNY], we were able to achieve an almost complete understanding of the “shadow” system (2.15) for n = 1 (and Ω is an interval, say, (0, 1)). To illustrate our results, we include the following. T HEOREM 2.3. Suppose B < C. Then (2.15) does not have any nonconstant solution if either one of the following two conditions hold: (i) d2 a2 /π 2 , (ii) A B. T HEOREM 2.4. Suppose B < C. Then (2.15) has a nonconstant solution if d2 < a2 /π 2 and A (B + C)/2. The case d2 < a2 /π 2 and B < A < (B + C)/2 is more delicate – existence holds for d2 closer to a2 /π 2 while nonexistence holds when d2 is near 0. The behavior of solutions is also obtained for d2 close to one of the two endpoints, 0 or a2 /π 2 . T HEOREM 2.5. (i) As d2 → a2 /π 2 , (w, ζ ) → (0, 0) in such a way that a2 (1 + μ) ζ → w 2[μ + (1 − μ) sin2 (πx/2)] uniformly on [0, 1] where μ = (2A/B) − 1 − 2 (A/B)2 − (A/B) ∈ (0, 1].

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185

(ii) As d2 → 0 we have (α) if A < B+3C 4 , then ζ→

a22 (B − A)(A − C) , b 2 c2 (B − C)2

w(0) → 2 w(·) → (β) if A

a2 A − (B + 3C)/4 , c2 B −C

a2 B − A c2 B − C B+3C 4 ,

on (0, 1],

then ζ →

2 3 a2 16 b2 c2 ,

w(0) → 0, and w →

3a2 4c2

on (0, 1].

It seems interesting to note that the limits in (β) above are independent of a1 , b1 , c1 . Our method of proof here is a bit unusual: we convert the problem of solving (w, ζ ) of (2.15) to a problem of solving its “representation” in a different parameter space. This is done ﬁrst without the integral constraint in (2.15). Then we use the integral constraint to ﬁnd the “solution curve” in the new parameter space as the diffusion rate d2 varies. This method turns out to be very powerful as it gives fairly precise information about the solution. Of course, our ultimate goal is to be able to obtain the steady states of (2.13) from our knowledge of the simpler limiting systems (2.14) or (2.15). This turns out to be possible, at least in the one-dimensional case Ω = [0, 1], as the next two results show. (For simplicity, we shall assume that ρ21 = 0 in the next two theorems.) T HEOREM 2.6 [LN2]. Suppose that A > B. There exists a small d ∗ > 0 such that for any d2 ∈ (0, d ∗ ), we can ﬁnd a large d˜ > 0 such that if d1 d˜ is ﬁxed, then there exists a large α > 0 such that if ρ12 > α, (2.13) has a nonconstant positive steady state (u, v), with ¯ v) ¯ uniformly in [0,1] as ρ12 → ∞, where (u, ¯ v) ¯ is a nonconstant positive (u, ρ12 v) → (u, solution of (2.14). T HEOREM 2.7 [LN2]. Suppose that d1 > 0 is ﬁxed and that either A ∈ ( 12 (B + C), ( 14 B + 3 1 3 1 ∗ ∗ 4 C)) or A ∈ (( 4 B + 4 C), 2 (B + C)). There exists a small d > 0 such that for d2 ∈ (0, d ) we can ﬁnd a large α > 0 such that if ρ12 > α, (2.13) has a nonconstant positive steady state (u, v) with (u, v) → ( wζ , w) as ρ12 → ∞ where w > 0, nonconstant and (w, ζ ) is a solution of (2.15). The proofs of Theorems 2.6 and 2.7 involve careful analysis of the linearized systems of (2.14) and (2.15) at their nonconstant positive solutions.

2.3. A chemotaxis system Chemotaxis is the oriented movement of cells in response to chemicals in their environment. Cellular slime molds (amoebae) release a certain chemical, c-AMP, move toward its

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higher concentration, and eventually form aggregates. Letting u(x, t) be the population of amoebae at place x and at time t, and v(x, t) be the concentration of this chemical, Keller and Segel [KS] proposed the following model to describe the chemotactic aggregation stage of amoebae: ⎧ ut = d1 u − χ∇ · u∇ψ(v) in Ω × (0, T ), ⎪ ⎪ ⎨ vt = d2 v − av + bu in Ω × (0, T ), ∂u ∂v ⎪ = 0 = on ∂Ω × (0, T ), ⎪ ∂ν ⎩ ∂ν u(x, 0) = u0 (x), v(x, 0) = v0 (x) in Ω,

(2.21)

where the constants χ, a and b are positive. Comparing the ﬁrst equation in (2.21) to (2.1) we see that (2.21) is indeed an example for “positive taxis”. Popular examples for the “sensitivity function” ψ include ψ(v) = kv, k log v or kv 2 /(1 + v 2 ), where k > 0 is a constant. A large amount of work has focused on the linear case ψ(v) = kv, and much is known in this case, at least for the low spatial dimensions, n = 1 or 2. The mathematical phenomena exhibited here are rich, from nontrivial steady states to blow-up dynamics. A nice article due to Horstmann [Ho] contains a thorough survey, from a derivation of the Keller–Segel model to the descriptions of many signiﬁcant results, on this linear sensitivity case. It also includes some discussions on the (biological) consequences of those mathematical results. We will therefore refer the readers to [Ho] and the references therein for the linear case and concentrate on the other cases here. For the logarithmic case ψ(v) = k log v, Nagai and Senba [NS] recently proved global existence for a modiﬁed parabolic–elliptic system in case n = 2. Observe that in (2.21) the total population is always conserved; that is, for all t > 0 we have

u(x, t) dx ≡ Ω

u0 (x) dx. Ω

Therefore to study the steady states of (2.21) for the case ψ(v) = log v we consider the following elliptic system ⎧ d1 u − χ∇ · (u∇ log v) = 0 ⎪ ⎪ ⎪ ⎨ d2 v − av + bu = 0 ∂u ∂v ⎪ ∂ν = 0 = ∂ν ⎪

⎪ ⎩ 1 u(x) dx = u¯ |Ω| Ω

in Ω, in Ω, on ∂Ω,

(2.22)

(prescribed).

With p = χ/d1 , it is not hard to show that u = λv p for some constant λ > 0. Thus, setting ε2 = d2 /a, μ = (bλ/a)1/(p−1), and w = μv, we see that w satisﬁes (1.4), i.e.,

ε2 w − w + wp = 0 in Ω, ∂w on ∂Ω, ∂ν = 0

(2.23)

and our previous results for (1.4) apply. To obtain a solution pair for (2.22) from a solution

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187

of (2.23), simply set p u|Ω|w ¯ u=

p Ωw

v|Ω|w ¯ and v =

Ωw

with v¯ = b u/a. ¯ In this way, we obtain spike-layer steady states for the chemotaxis system (2.22) when d2 /a is small and 1 < χ/d1 < n+2 n−2 (∞ if n = 1, 2). Although many believe the particular steady state corresponding to the “least-energy” solution of (1.4) is stable, its proof has thus far eluded us. 2.4. Other systems The 1952 paper of Turing [Tu], in which the novel notion of “diffusion-driven instability” was ﬁrst posed in an attempt to model the regeneration phenomenon of hydra, is one of the most important papers in theoretical biology in the last century. However, the two chemicals, activator and inhibitor in Turing’s theory, are yet to be identiﬁed in hydra. The ﬁrst experimental evidence of Turing pattern was observed in 1990, nearly 40 years after Turing’s prediction, by the Bordeaux group in France on the chlorite–iodide– malonic acid–starch (CIMA) reaction in an open unstirred gel reactor [CDBD]. In their scheme, the two sides of the gel strip loaded with starch indicator are, respectively, in con− tact with solutions of chlorite (ClO− 2 ) and iodide (I ) ions on one side, and malonic acid (MA) on the other side, which are fed through two continuously-ﬂow stirred tank reactors. These reactants diffuse into the gel, encountering each other at signiﬁcant concentrations in a region near the middle of the gel, where the Turing patterns of lines of periodic spots can be observed. This observation represents a signiﬁcant breakthrough for one of the most fundamental ideas in morphogenesis and biological pattern formation. The Brandeis group later found that, after a relatively brief initial period, it is really the simpler chlorine dioxide ClO2 –I2 –MA (CDIMA) reaction that governs the formation of the patterns [LE1,LE2]. The CDIMA reaction can be described in a ﬁve-variable model consists of three component processes. However, observing that three of the ﬁve concentrations remain nearly constants in the reaction, Lengyel and Epstein [LE1,LE2] simpliﬁed the model to a 2 × 2 system: Let u = u(x, t) and v = v(x, t) denote the chemical concentrations (rescaled) of iodide (I− ) and chlorite (ClO− 2 ), respectively, at time t and x ∈ Ω, where Ω is a smooth, bounded domain in Rn . Then the Lengyel and Epstein model takes the form ⎧ 4uv ⎪ u = u + a − u − 1+u in Ω × [0, T ), ⎪ 2 ⎨ t uv in Ω × [0, T ), vt = σ cv + b u − 1+u2 (2.24) ⎪ ⎪ ⎩ ∂u = ∂v = 0 on ∂Ω × [0, T ), ∂ν

∂ν

where a and b are parameters related to the feed concentrations; c is the ratio of the diffusion coefﬁcients; σ > 1 is a rescaling parameter depending on the concentration of the

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starch, enlarging the effective diffusion ratio to σ c. All constants a, b, c, and σ are assumed to be positive. It was established in [NTa] that solutions of (2.24) must eventually enter the region Ra = (0, a) × (0, 1 + a 2 ) for t large, regardless of the initial values u(x, 0), v(x, 0). Furthermore, the existence and nonexistence of steady states of (2.24) are also investigated in [NTa]. Results there show that, roughly speaking, if any one of the following three quantities (i) the parameter a (related to the feed concentrations), (ii) the size of the reactor Ω (reﬂected by its ﬁrst eigenvalue), (iii) the “effective” diffusion rate d = c/b, is not large enough, then the system (2.24) has no nonconstant steady states. On the other hand, it was also established in [NTa] that if a lies in a suitable range, then (2.24) possesses nonconstant steady states for large d. The proof of the existence uses a degree-theoretical approach combined with the a priori bounds. However, such an approach does not provide much information about the shape of the solution. In the case n = 1, a better description for the structure of the set of nonconstant steady states to (2.24) is given in [JNT]; namely, a global bifurcation theorem which gives the existence of nonconstant steady states to (2.24) for all d suitably large (under a rather natural condition) is obtained. Moreover, the corresponding shadow system (as d → ∞) is also solved in [JNT]. There are various experimental and numerical studies on the system (2.24), see, e.g., [CK,JS] and the references therein. However, the qualitative properties of solutions to (2.24) largely remain open. Another system supporting many interesting spatio-temporal patterns is the Gray–Scott model [GS]. It models an irreversible autocatalytic chemical reaction involving two reactants in a gel reactor, where the reactor is maintained in contact with a reservoir of one of the two chemicals in the reaction. In dimensionless units it can be written as ⎧ 2 ⎪ ⎨ Ut = DU U + F (1 − U ) − U V Vt = DV V − (F + k)V + U V 2 ⎪ ⎩ ∂U = ∂V = 0 ∂ν ∂ν

in Ω × [0, T ), in Ω × [0, T ), on ∂Ω × [0, T ),

(2.25)

where the unknowns U = U (x, t) and V = V (x, t) represent the concentrations of the two chemicals at a point x ∈ Ω ⊂ Rn , n 3 and at a time t > 0, respectively; DU , DV are the diffusion coefﬁcients of U and V , respectively. F denotes the rate at which U is fed from the reservoir into the reactor, and k is a reaction-time constant. For various ranges of these parameters, (2.25) is expected to admit a rich solution structure involving pulses or spots, rings, stripes, self-replication spots, and spatio-temporal chaos. See [Pe] and [LMPS] for numerical simulations and experimental observations.

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In one-dimensional case n = 1, stationary one-pulse solution in the entire real line (i.e., Ω = R1 and no boundary condition in (2.25)) is studied in [DGK]. In case n = 2, “spotty” solutions are investigated in [W3] and [WW3]. Many other patterns here remain to be established with mathematical rigor.

3. Stability of solutions The stability/instability properties of solutions to elliptic problems, viewed as steady states of the appropriate corresponding evolution equations, are perhaps among the most important aspects if we are to understand the entire dynamics of the original evolution problems. For instance, in Section 1 we have obtained many solutions exhibiting various striking concentration patterns, to the semilinear Neumann problem (1.4). Which ones are stable? More precisely, from which ones can we construct stable steady states to the shadow system (1.3) or to the original system (1.1)? An ultimate question would be: For given initial data U (x, 0) and V (x, 0) in (1.1) how do we determine the large time behavior of U (x, t) and V (x, t)? To answer that, one important intermediate step would be to study the stability/instability properties of each and every steady state of (1.1). Therefore, we emphasize that it is not just the stability properties of the spike-layer solutions obtained in Theorems 1.1 and 1.3–1.6 in the context of the single equation (1.4) that we need to understand, what we are really after is the stability properties of the corresponding spike-layer steady states obtained in Section 2.1, in the context of the original system (1.1). It turns out to be a general principle that the stability properties of a steady state are closely related to the “shape” of the steady state. Roughly speaking, the more complicated the shape of the steady state, the less stable the steady state is. For example, in Section 3.1 we will show that for a solution of a single equation with homogeneous Neumann boundary condition to be stable, it must be a constant if the domain is convex – a nice result due to Matano [Ma1]. This may be regarded as “stability implies triviality” for single equations. In Section 3.3 we will show that, in one space dimension and under homogeneous Neumann boundary condition, for a (time-dependent) solution of a “shadow system” (i.e., a reaction–diffusion equation coupled with a nonlocal ordinary differential equation) to be stable, it must be eventually monotone in space. In short, “stability implies monotonicity” holds for shadow systems – a recent result due to [NPY]. For 2 × 2 systems, the situation can be very complicated and will be illustrated via the example (1.1) in Section 3.4. Stability properties of solutions to homogeneous Dirichlet problems, even in the single equation case, are not well understood. A discussion is included in Section 3.2. For equations in entire space Rn , stability properties could be extremely interesting and sometimes even counter-intuitive. For instance, positive solutions of the simple-looking equation u + up = 0

in Rn ,

for n 3 and p n+2 n−2 , exhibit surprisingly sophisticated stability and instability behaviors. (See [Wa,GNW1,GNW2] and [PY] for details.)

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3.1. Single equations with Neumann boundary conditions We start our discussion on the stability analysis of solutions to single equations with homogeneous Neumann boundary conditions

u + f (u) = 0 in Ω, ∂u on ∂Ω, ∂ν = 0

(3.1)

where f ∈ C 1 (R), Ω is a bounded smooth domain in Rn , ν is the unit outer normal to ∂Ω. In order to discuss the notion of stability in an intuitive way, it is best to introduce the corresponding parabolic initial-boundary problem ⎧ ⎨ vt = v + f (v) ∂v =0 ⎩ ∂ν v(x, 0) = v0 (x)

in Ω × R+ , on ∂Ω × R+ , in Ω.

(3.2)

A solution of (3.1) is said to be a steady state of (3.2), and a solution of u of (3.1) is said to be stable if for every ε > 0, there exists a δ > 0 such that v(·, t) − u(·)L∞ (Ω) < ε for all t > 0 provided that v0 − uL∞ (Ω) < δ. A steady state u is said to be asymptotically stable if there exists δ > 0 such that v(·, t) − u(·)L∞ (Ω) → 0 as t → ∞ provided that v0 − uL∞ (Ω) < δ. Naturally we say that u is unstable if it is not stable. It is also possible to discuss the stability of a solution u to (3.1) without going into its parabolic counterpart (3.2). This may be done via the “linearized stability”. Standard arguments show that (see, e.g., [Ma1], Theorem 3.3, p. 423) if u is stable, then H(ϕ) =

|Dϕ|2 − f (u)ϕ 2 0

(3.3)

Ω

for all ϕ ∈ H 1 (Ω). Putting this in a different way, we look at the linearized problem of (3.1) at this particular solution u

ϕ + f (u)ϕ + λϕ = 0 ∂ϕ ∂ν = 0

in Ω, on ∂Ω.

(3.4)

Denoting the ﬁrst eigenvalue by λ1 , we have λ1 = min H(ϕ) ϕ ∈ H 1 (Ω), ϕL2 (Ω) = 1 and, the assertion (3.3) follows from the following proposition. P ROPOSITION 3.1. If λ1 < 0, then u is unstable.

(3.5)

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191

P ROOF. Let ϕ1 be an eigenfunction (normalized, ϕ1 L2 (Ω) = 1) corresponding to λ1 . Then λ1 is simple and ϕ1 > 0 (or < 0) in Ω by the Krein–Rutman theory. Next, since λ1 < 0, there exists ε0 > 0 such that, for every 0 < ε ε0 , λ1 f (u(x) + ε) − f (u(x)) f u(x) + ε 2

(3.6)

for all x ∈ Ω. Suppose that u is stable. Then, in particular, there exists v0 close to u with v0 > u and v(·, t) − u(·)L∞ (Ω) < ε0 for all t > 0. (Note that v(x, t) > u(x) for all x ∈ Ω and for all t > 0 by the usual maximum principle.) Now setting

g(t) =

v(x, t) − u(x) ϕ1 (x) dx,

Ω

we have g(t) ε0 |Ω|1/2 for all t 0, and g (t) =

vt (x, t)ϕ1 (x) dx

Ω

Ω

Ω

= =

v + f (v) ϕ1

(v − u) + f (v) − f (u) ϕ1

= Ω

(v − u)ϕ1 + f (v) − f (u) ϕ1

f (v) − f (u) − f (u) − λ1 ϕ1 = (v − u) v−u Ω λ1 (v − u)ϕ1 − 2 Ω

by (3.6), i.e., we have obtained g (t) + Then

λ1 2 g(t) 0

for t 0.

d λ1 t /2 e g(t) 0 dt for t 0, which implies that eλ1 t /2g(t) is increasing and eλ1 t /2g(t) g(0) > 0 for all t > 0 which, in turn, implies that g(t) e−λ1 t /2 g(0) → ∞

as t → ∞,

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a contradiction. Therefore u must be unstable. It is generally believed that the diffusion process is a “smoothing” and “trivializing” process. Thus in a closed system it seems reasonable to expect that the only stable steady states are constants (i.e., spatially homogeneous). It turns out that this is indeed the case for single equations (3.1) or (3.2) provided that the domain Ω is nice, e.g., convex. (For systems of equations with different diffusion coefﬁcients, this is generally not true and we shall discuss this later.) This result was proved by Matano [Ma1] in 1979. (See [CH] also.) Matano also showed that this result also holds for other domains such as annuli {x ∈ Rn | a < |x| < b}, and gave a counterexample showing that for certain nonconvex domains, nontrivial stable steady states of (3.1) or (3.2) do exist. Following Matano’s proof, we see that the role of convexity is contained in the following lemma. L EMMA 3.2. Let Ω be a bounded smooth convex domain in Rn . Suppose that v ∈ C 3 (Ω) ∂v with ∂ν = 0 on ∂Ω. Then ∂ |Dv|2 0 ∂ν

on ∂Ω.

The main result in this section may be stated as follows. T HEOREM 3.3. If Ω is convex, then the only stable solutions of (3.1) are constants. P ROOF. The approach is to show that if u is a nonconstant solution of (3.1), then λ1 (given by (3.5)) must be negative. We shall achieve this by choosing appropriate test functions in (3.3). However, it is natural to question it a priori whether this approach would work. For, it seems that if f < 0 on R, then H(ϕ) is always positive for all ϕ ≡ 0 in H 1 (Ω). It turns out that if f < 0 on R, then (3.1) has no nonconstant solutions. To prove this, we

let u be a solution of (3.1). Integrating the equation yields Ω f (u(x)) dx = 0 and thus there exists a unique a such that f (a) = 0 (since f is monotonically decreasing). Without loss of generality, we may assume that a = 0, i.e., f (0) = 0. (For example, we may set v ≡ u − a, ˜ ˜ then v + f˜(v) = 0 and ∂v ∂ν = 0 on ∂Ω, where f (v) = f (v + a). Thus f (0) = f (a) = 0.) Assume u ≡ 0, then {x ∈ Ω | u(x) > 0} and {x ∈ Ω | u(x) < 0} are both nonempty. Let u(P ) = maxΩ u. Then u(P ) > 0 and we have two cases: (i) P ∈ Ω. Since f (u(P )) < 0 (f < 0 on R+ ) we have u(P ) > 0. On the other hand, u assumes its maximum at P , so u(P ) 0, a contradiction. Then (ii) P ∈ ∂Ω. Choose a ball B ⊆ Ω which is tangent to ∂Ω at P with u > 0 on B. and u(x) > 0 on B with u(P ) = max u. By Hopf’s boundary point f (u(x)) < 0 on B, B ∂u lemma, ∂u ∂ν > 0 at P , which contradicts the boundary condition ∂ν = 0 on ∂Ω. Coming back to the proof of the theorem, we choose ϕ = ui = ∂u/∂xi . Then differentiating the equation in (3.1) gives ui + f (u)ui = 0, and |Dui |2 − f (u)u2i H(ui ) = i

i

=

Ω

i

|Dui |2 + ui ui Ω

Qualitative properties of solutions to elliptic problems

=

i

=

1 2

|Dui |2 − |Dui |2 +

Ω

∂Ω

ui ∂Ω

∂ui ∂ν

193

∂ |Du|2 ∂ν

0 by Lemma 3.2. If one of the H(ui ), i = 1, 2, . . . , n, is negative, then, we are done since ui ∈ H 1 (Ω). Therefore we only have to deal with the case that H(ui ) = 0, i = 1, 2, . . . , n, and λ1 = 0. We shall derive a contradiction. First of all, we note that under this assumption each ui is an eigenfunction of λ1 . Since λ1 is simple we see that for each i, there exists ci such that ui = ci ϕ1 where ϕ1 > 0 is the normalized eigenfunction corresponding to λ1 (i.e., ϕ1 L2 (Ω) = 1). Thus Du = cϕ1 where c = (c1 , . . . , cn ). This implies that u is constant when restricted to hyperplanes which are perpendicular to c (by mean-value theorem); i.e., u is a function of one variable only. If we rotate the coordinate system so that the new coordinate system, denoted by x , has its x1 -axis pointing in the direction of c, then u is a function of x1 only. Since everything involved here are invariant under rotation, from now on, we shall be working with the new coordinate system x . To keep the notations simple, we shall still denote this new coordinate by x, the domain by Ω. So we have u(x) = u(x1 ) where x = (x1 , . . . , xn ) and Du(x) = (cϕ1(x), 0, . . . , 0) where c = |c|. Thus, for some a < b, we have

u + f (u) = 0 in (a, b), u (a) = u (b) = 0.

Recall that ui = ci ϕ1 , this implies that in particular ui satisﬁes the homogeneous Neumann boundary condition ∂ui =0 ∂ν on ∂Ω, which in turn implies that u11 (a) = 0, i.e., u (a) = 0. Now we have

u + f (u) = 0 in (a, b), u (a) = u (a) = 0.

Thus at x1 = a, f (u(a)) = 0 and u ≡ u(a) is a solution of this problem. By the uniqueness of solutions of ordinary differential equations, u ≡ u(a), a constant. This contradicts our assumption on u. Thus λ1 < 0, and our proof is complete. In [Ma1], an example is given to illustrate the importance of the convexity of Ω in the above theorem; namely, a stable nonconstant solution for (3.1) on a dumbbell-shaped domain Ω was constructed with a bistable nonlinearity f (u). Further research in this direction has been conducted by many authors, see [HV,JM] and the references therein.

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3.2. Single equations with Dirichlet boundary conditions It is clear that we can deﬁne the notions of stability, asymptotic stability, linearized stability and instability for solutions to single equations under homogeneous Dirichlet boundary conditions u + f (u) = 0 in Ω, (3.7) u=0 on ∂Ω, in a similar fashion as we did in Section 3.1. Attempts have been made to obtain the counterpart of Theorem 3.3 in Section 3.1. However, the situation here is more complicated. In [LinN] (see [Sw]) the following result was established: P ROPOSITION 3.4. Let Ω be a ball or an annulus. Then a stable solution of (3.7) must not change sign in Ω. We ought to remark that in the case n = 1, more general boundary condition than u = 0 in (3.7) was studied by [Mg]. However, in general, even if Ω is convex, a stable solution of (3.7) is not necessarily of one sign. Such an example was constructed in [Ma2] and [Sw]. P ROOF OF P ROPOSITION 3.4. To prove Proposition 3.4, we proceed as follows. Note that an interesting intermediate step in the proof is that stability implies radial symmetry. Let u be a stable solution (3.7). As the ﬁrst step, we claim that u must be radial. To this end, we set Tij = xi

∂ ∂ − xj , ∂xj ∂xi

i, j = 1, . . . , n,

where x = (x1 , . . . , xn ) ∈ Rn . A straightforward computation shows that Tij = Tij . Applying Tij to (3.7), we have

(Tij u) + f (u)Tij u = 0 Tij u = 0

in Ω, on ∂Ω.

(Recall that Ω has rotational symmetry.) Since (3.3) holds for all ϕ ∈ H01 (Ω), Tij u is the ﬁrst eigenfunction of the linearized operator + f (u) if Tij u ≡ 0. Tij u must then have only one sign in Ω, which is impossible. Hence Tij u ≡ 0 for all 1 i, j n and our assertion is proved. We now divide the rest of the proof into two cases. Case 1. Ω = Bb (i.e., Ω is the ball of radius b centered at the origin). Since u is radial, it satisﬁes urr + n−1 r ur + f (u) = 0 in 0 r b, u(b) = 0.

Qualitative properties of solutions to elliptic problems

195

Suppose that u changes sign in (0, b). Then there exists an r0 ∈ (0, b) such that ur (r0 ) = 0. Differentiating the above equation with respect to r, we obtain (ur )rr +

n−1 n−1 (ur )r + f (u)ur − 2 ur = 0. r r

Multiplying the above equation by r n−1 ur and integrating over (0, r0 ), we have, by (3.3), that u2r 2 dx = |∇u | dx − f (u)|ur |2 dx 0 −(n − 1) r 2 Br0 r Br0 Br0 since the function ur (r) ϕ(r) = 0

if r r0 , if r0 r b,

belongs to H01 (Ω). Therefore, ur ≡ 0 in (0, r0 ) which implies that u is a constant in Br0 and thus in Ω, which is a contradiction. Case 2. Ω = {x ∈ Rn | a < |x| < b} where 0 < a < b < ∞. Now u satisﬁes urr + n−1 r ur + f (u) = 0 in (a, b), u(a) = u(b) = 0. Suppose that u changes sign, there exist r0 , r1 such that a < r0 < r1 < b and ur (r0 ) = ur (r1 ) = 0. Differentiating the above equation and repeating the same arguments as in Case 1, we obtain that ur ≡ 0 in (r0 , r1 ). (In the present case, the “test function” is chosen to be ur (r) if r0 r r1 , ϕ(r) = 0 if r r1 or r r0 , which clearly belongs to H01 (Ω).) Thus u is a constant in (r0 , r1 ) which again implies u is a constant in Ω, a contradiction, and the proof of Proposition 3.4 is complete. For general nonlinearity f (u), even positive solutions of (3.7) are often unstable. To guarantee stability for positive solutions, we need to restrict ourselves to special classes of nonlinearities. P ROPOSITION 3.5. Let u be a positive solution of (3.7) where f satisﬁes the following condition: f (z) is decreasing in z > 0. z Then u must be the only positive solution of (3.7) and is stable.

(3.8)

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Well-known examples include the case f (u) = e−u . The sublinear case f (u) = uγ , 0 < γ < 1, although not C 1 in R, can be handled by exactly the same arguments below. The proof makes use of the well-known “monotone method”. Let u1 and u2 be two positive solutions of (3.7). Then, observe that for 0 < λ < 1, λu1 is a subsolution of (3.7). On the other hand, since f (0) 0 (guaranteed by (3.8)), we have by Hopf Boundary Point lemma that ∂ui /∂ν < 0 on ∂ for i = 1, 2, and consequently λu1 < u2 for sufﬁciently small λ > 0. Therefore, if we apply the monotone iteration scheme (see, e.g., [Sa], Theorem 2.1) to λu1 , for some λ small, eventually we obtain a solution u3 . Since u1 and u2 are both supersolutions of (3.7) and λu1 < u1 and λu1 < u2 , we have u3 u1 and u3 u2 . Since u1 = u2 , we must have u3 < u1 or u3 < u2 . Assume that u3 < u2 . Applying the Green’s identity, we derive from (3.8) that 0=

#

(u2 u3 − u3 u2 ) =

Ω

u2 u3 Ω

$ f (u2 ) f (u3 ) − < 0, u2 u3

a contradiction. Thus (3.7) has at most one positive solution under the hypothesis (3.8). The stability of a positive solution u to (3.7), if exists, also follows from the monotone method. For, if u is a positive solution of (3.7), then λu is a supersolution for every λ > 1, and λu is a subsolution of (3.7) for every 0 < λ < 1. Our conclusion now follows from Theorem 3.3 in [Sa].

3.3. Shadow systems From Theorem 3.3 in Section 3.1, it seems clear that single equations with homogeneous Neumann boundary conditions are simply inadequate in modeling nontrivial pattern in reality. Therefore we need to go to systems, and it seems that 2 × 2 systems already admit many stable steady state solutions with highly nontrivial patterns. As a ﬁrst step in understanding 2 × 2 systems, we shall ﬁrst study the shadow systems which, in some sense, lie between single equations and 2 × 2 systems (as we have seen in Section 1). For a 2 × 2 system ⎧ ⎨ ut = d1 u + f (u, v) vt = d2 v + g(u, v) ⎩ ∂u ∂v ∂ν = ∂ν = 0

in Ω × [0, T ), in Ω × [0, T ), on ∂Ω × [0, T ),

(3.9)

it has been known for quite some time that when both the diffusion coefﬁcients d1 and d2 are large, the dynamics of (3.9) is essentially determined by the corresponding system of ordinary differential equations, at least in many important cases. It has also been understood that when one of the diffusion coefﬁcients, say, d2 is large, the dynamics of (3.9) is essentially determined by the following shadow system ⎧ f (u, ξ ) in Ω × [0, T ), ⎪ ⎨ ut = d1 u +

−1 ξt = |Ω| (3.10) Ω g(u, ξ ) dx in [0, T ), ⎪ ⎩ ∂u = 0 on ∂Ω × [0, T ), ∂ν

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again in many important cases. (See [HS].) Note that the equation for v in (3.9) is replaced by an ordinary differential equation for ξ with nonlocal effects. In [NPY] it is established that any bounded (not necessarily stationary) stable solution of (3.10) in n = 1 must be either asymptotically homogeneous or eventually monotone in x. In particular, the fact that “stability implies monotonicity” for the shadow system (3.10) we discussed at the beginning of this chapter holds. To make the basic ideas involved here transparent, we ﬁrst treat the steady state case. P ROPOSITION 3.6 [NPY]. Suppose that f (u, v) and g(u, v) are of class C 1 . Then any spatially inhomogeneous nonmonotone steady state of ⎧ ⎨ ut = uxx + f (u, ξ ) ux (0, t) = 0 = ux (1, t),

1 ⎩ ξt = 0 g(u, ξ ) dx,

in (0, 1) × [0, ∞), t > 0,

(3.11)

t > 0,

is unstable. The proof relies heavily on symmetry properties of the domain Ω = (0, 1) and thus is strictly one-dimensional. We begin with the notion of k-symmetry. We say that a function u(x) is k-symmetric i+1 in [0, 1], k 2, if the restriction u(x), x ∈ [ i−1 k , k ], is (even) symmetric with respect to the point x = i/k for all i = 1, 2, . . . , k − 1, that is, u(x) = u

2i −x k

# for all x ∈

$ i −1 i +1 . , k k

We call a solution (u, ξ ) of (3.11) k-symmetric if u(x, t) is k-symmetric for every t. Let (u(x), ξ ) be a stationary solution of (3.11), that is, (u(x), ξ ) satisﬁes ⎧ ⎪ ⎨ u + f (u, ξ ) = 0, x ∈ (0, 1), u (0) = 0 = u (1), (3.12) ⎪ ⎩ 1 g u(x), ξ dx = 0. 0 Clearly, if (u(x), ξ ) is a nonconstant nonmonotone solution of (3.12), then u(x) is k-symmetric with some k 2 and monotone in [0, 1/k]. Let us consider the following eigenvalue problem associated with the linearized operator at u(x): ϕ(x) = ϕ (x) + fu u(x), ξ ϕ(x), x ∈ (0, 1), (3.13) ϕ (0) = 0 = ϕ (1). According to the Sturm–Liouville theory, the eigenvalues of (3.13) are real numbers 0 > 1 > 2 > · · · → −∞, and the corresponding eigenfunctions ϕ0 , ϕ1 , ϕ2 , . . . , are characterized by the property that ϕj has exactly j zeros in (0, 1). We assume that these eigenfunctions are normalized in L2 (0, 1).

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Next, let us consider the eigenvalue problem

˜ϕ(x) ˜ = ϕ˜ (x) + fu u(x), ξ ϕ(x), ˜ x ∈ (0, 1/k), ϕ˜ (0) = 0 = ϕ˜ (1/k).

(3.14)

We denote by ˜j and ϕ˜j the j th eigenvalue and corresponding eigenfunction of (3.14), respectively. We assume that the eigenfunctions are normalized in L2 (0, 1/k). Since ϕ˜j has exactly j zeros in (0, 1/k), it follows from reﬂection and the number of zeros that ˜j = j k ,

ϕ˜j (x) ≡

√ kϕj k (x) on [0, 1/k],

for all j = 0, 1, 2, . . . . L EMMA 3.7. Let w(x) be any k-symmetric function on [0, 1]. Then

1

w(x)ϕj (x) dx = 0,

j = 0, k, 2k, . . . .

0

P ROOF. Let ·, ·!L2 (a,b) denote the L2 -inner product on (a, b). By reﬂection, we have for x ∈ (0, 1/k) w=

∞ % j =0

=

∞ j =0

=

∞

w, ϕ˜j

&

ϕ˜ L2 (0,1/ k) j

k w, ϕj k !L2

(0,1/k)

ϕj k

w, ϕj k !L2 (0,1)ϕj k .

j =0

Hence, again by reﬂection, we obtain w=

∞

w, ϕj k !L2 (0,1)ϕj k

on [0, 1].

j =0

On the other hand, we can expand w as w=

∞

w, ϕ!L2 (0,1)ϕj

on [0, 1].

j =0

Comparing these two expansions termwise, we obtain the conclusion.

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L EMMA 3.8. If u(x) is k-symmetric, then the eigenvalues of (3.13) satisfy 0 > 1 > · · · > k−1 > 0. P ROOF. Differentiating (3.12), we obtain

u (x)

+ fu u(x), ξ u (x) = 0,

x ∈ (0, 1).

We also have u (0) = u (1) = 0. Clearly u (x) has k − 1 zeros in (0, 1) and ϕj (x) has exactly j zeros in (0, 1). Then it follows from the Sturm comparison theorem (see, e.g., [CoL]) that k−1 > 0. We now give a proof of Proposition 3.6. P ROOF OF P ROPOSITION 3.6. Let (u(x), ξ ) be any spatially inhomogeneous nonmonotone solution of (3.12), and consider the eigenvalue problem ⎧ ⎪ ⎨ λΦ(x) = Φ (x) + fu u(x), ξ Φ(x) + fv u(x), ξ η,

1 λη = 0 gu u(x), ξ Φ(x) + gv u(x), ξ η dx, ⎪ ⎩ Φ (0) = 0 = Φ (1).

x ∈ (0, 1), (3.15)

Since gu (u(x), ξ ) is k-symmetric with some k 2, it follows from Lemma 3.7 that

1

gu u(x), ξ ϕj (x) dx = 0 for j = 0, k, 2k, . . . .

0

Hence, (λ, Φ, η) = (j , ϕj , 0) satisﬁes (3.15) if j = 0, k, 2k, . . . . This implies that

η) (Φ, ˜ = (ej t ϕj (x), 0) satisﬁes the linearized system for (3.11) ⎧

t = Φ

xx + fu (u, ξ )Φ

+ fv (u, ξ )η, ⎪ ˜ 0 < x < 1, t > 0, ⎨Φ

1

η˜ t = 0 gu (u, ξ )Φ + gv (u, ξ )η˜ dx, t > 0, ⎪ ⎩

x (1, t), t > 0, Φx (0, t) = 0 = Φ if j = 0, k, 2k, . . . . Since j > 0 for j = 1, 2, . . . , k − 1, by Lemma 3.8, the steady state (u, ξ ) is unstable. The proof of the “parabolic” version of Proposition 3.6 is more involved, and we refer the interested readers to [NPY] for details. Among major problems left open concerning (3.10) is perhaps the multidimensional analogue of Proposition 3.6 for, say, convex domains. Very little is known so far in this generality. On the other hand, with more speciﬁc shadow systems, for instance, the one derived from the 2 × 2 activator–inhibitor system in Section 1, namely, (1.3), we do have results concerning its stability and instability properties in multidimensions. To describe some of the existing results we ﬁrst introduce the notion of weak stability. We say that a steady

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state solution (Uε , ξε ) of (1.3), with d1 = ε2 , is weakly stable if all the eigenvalues of the linearized operator Lε,τ =

p−1

q

ε2 − 1 + pUε /ξε

r r−1 dx/ξ s ε τ |Ω| Ω Uε

p

q+1

−qUε /ξε

− τ1 + s Ω Uεr dx/ τ |Ω|ξεs+1

are contained in the set {λ ∈ C | Re λ < 0 or λ = 0}. By standard theory we know that if all the eigenvalues of Lε,τ are contained in the left half plane {λ ∈ C | Re λ < 0}, then (Uε , ξε ) is asymptotically stable; while it is unstable if Lε,τ has an eigenvalue with positive real part. The weak stability allows 0 to be an eigenvalue. Next, we assume for the rest of this section that 1 < p < n+2 n−2 . Then we say that (Uε , ξε ) is a least-energy pattern if Uε (x) = ξεq/(p−1) uε (x) and ξε =

1 |Ω|

Ω

urε dx

−1/α ,

(3.16)

where α is deﬁned in (2.6) and uε is the least-energy solution uε,N of (1.4) guaranteed by Theorem 1.1. (It is easy to see that (3.16) gives a steady state of (1.3).) We are now ready to state some results concerning the stability and instability properties of (Uε , ξε ). (See [NTY2] for details.) T HEOREM 3.9 (Instability). For each ε sufﬁciently small, there is a τ0 0, depending on p, q, r, s and ε, such that (Uε , ξε ) is unstable if τ > τ0 . T HEOREM 3.10 (Stability). Suppose that r = p + 1. Then, as long as α does not belong to qr − 1), there exist positive a certain ﬁnite ( possibly empty) subset C of the interval (0, p−1 numbers τ1 > τ2 > · · · > τ2m−1 > 0, depending on p, q, s, ε, for which the following hold: (i) (Uε , ξε ) is weakly stable if τ ∈ (0, τ2m−1 ) ∪ (τ2m−2 , τ2m−3 ) ∪ · · · ∪ (τ2 , τ1 ); (ii) (Uε , ξε ) is unstable if τ ∈ (τ2m−1 , τ2m−2 ) ∪ · · · ∪ (τ3 , τ2 ) ∪ (τ1 , ∞), provided that ε is sufﬁciently small. Furthermore, if C is empty, then m = 1. /C T HEOREM 3.11 (Hopf bifurcation). Under the hypothesis of the above theorem, if α ∈ and ε is sufﬁciently small, then in each small neighborhood of τj , j = 1, . . . , 2m − 1, (1.3) has a one-parameter family of periodic solutions (Uε (x, t; μ), ξε (t; μ)) for τ = τ (μ) deﬁned for μ ∈ (−μ0 , μ0 ) bifurcating from (Uε , ξε ) at τ = τj , i.e., Uε (x, t; μ) = Uε (x) + O(μ), ξε (t; μ) = ξε + O(μ), τ (μ) = τj + O(μ) as μ → 0. Intuitively speaking, if τ is small, the inhibitor responds quickly to the change, thus one may expect (Uε , ξε ) to be stabilized. On the other hand, if τ is large, then the response of the inhibitor to changes is slow, suggesting that (Uε , ξε ) be unstable. Our results support these intuitions. It is clear that if the domain Ω is a ball or an annulus, then the linearized operator Lε,τ of the least-energy pattern (Uε , ξε ) always has 0 as an eigenvalue, as rotations generate a continuum of steady states of (1.3) since the single-peak of uε is assumed on the boundary. Therefore, in general one could expect at most the weak stability. However, in case Ω is

Qualitative properties of solutions to elliptic problems

201

a ball or an annulus, the weak stability does imply the nonlinear stability. (See [NTY2], Section 4.) Although the proofs in [NTY2] rely heavily on the assumption r = p + 1, the method used in [NTY2] is quite general. We will brieﬂy sketch the main ideas involved. First, observe that, with a suitable scaling argument, Lε,τ

q/(p−1)

ξε

φ

ξε η

ξ q/(p−1) φ =λ ξε η

if and only if ⎧ p−1 p 2 ⎪ puε φ − quε η = λφ ⎨ ε φ − φ + 2

r r Ω ur−1 ε φ Ω uε − (s + 1)η = τ λη, ⎪ ⎩ ∂φ ∂ν = 0

in Ω, (3.17) on ∂Ω.

Since uε is the least-energy solution of (1.4), the spectrum of the linearized operator Lε = p−1 ε2 − 1 + puε with homogeneous Neumann boundary conditions consists only of the eigenvalues {j,ε }∞ j =0 satisfying 0,ε > δ0 > 0 1,ε · · · → −∞ where the constant δ0 is independent of ε. Now, denote by {ϕj,ε }∞ j =0 the corresponding normalized eigenfunctions / {j,ε }∞ (with ϕ0,ε > 0) which form a complete orthonormal system in L2 (Ω). For λ ∈ j =0 , we can solve φ from the ﬁrst equation in (3.17) φ = qη(Lε − λ)−1 upε = qη

∞ p uε , ϕj,ε ! ϕj,ε , j,ε − λ

(3.18)

j =0

where ·, ·! denotes the inner product in L2 (Ω). Substituting (3.18) into the second equation in (3.17) we obtain, η = 0 if and only if λ satisﬁes qr − (s + 1) p−1 +

∞ ur−1 , ϕj,ε ! uε , ϕj,ε ! qrλ ε

− τ λ = 0, j,ε − λ (p − 1) Ω urε

(3.19)

j =0

p where denotes the summation over j satisfying j,ε = 0, since Lε uε = (p − 1)uε and therefore %

& j,ε uε , ϕj,ε !. upε , ϕj,ε = p−1

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Thus λ ∈ / {j,ε }∞ j =0 is an eigenvalue of Lε,τ if and only if λ satisﬁes the characteristic equation ∞ cj,ε − τ + α = 0, (3.20) χ(λ; ε, τ ) = λ j,ε − λ j =0

where cj,ε is given by (3.19) as follows cj,ε =

qr ur−1 ε , ϕ j,ε ! uε , ϕj,ε ! . r p−1 Ω uε

(3.21)

Now it is clear that c0,ε > 0 and, if r = p + 1, then cj,ε 0 for all j 1, which turns out to be crucial in analyzing the zeros of χ(λ; ε, τ ). We refer the interested readers to [NTY2] for details. Naturally, one would expect a stronger stability result than Theorem 3.10 when the dimension n = 1. Indeed, when n = 1, not only we obtain asymptotic stability (instead of weak stability), we are also able to enlarge the range of the parameter r in [NTY1]. T HEOREM 3.12 (Asymptotic stability). Suppose that n = 1. There exists a δ1 > 0 and an ε1 > 0 such that, for every |r − (p + 1)| < δ1 and ε ∈ (0, ε1 ), as long as α does not belong qr − 1), there exist positive to a certain ﬁnite ( possibly empty) set C of the interval (0, p−1 numbers τ1 > τ2 > · · · > τ2m−1 > 0, depending on p, q, r, s, ε, for which the following hold: (i) (Uε , ξε ) is asymptotically stable if τ ∈ (0, τ2m−1 ) ∪ (τ2m−2 , τ2m−3 ) ∪ · · · ∪ (τ2 , τ1 ); (ii) (Uε , ξε ) is unstable if τ ∈ (τ2m−1 , τ2m−2 ) ∪ · · · ∪ (τ3 , τ2 ) ∪ (τ1 , ∞). If, in addition C is empty, then m = 1. T HEOREM 3.13 (Asymptotic stability). Suppose that n = 1. There exists a δ1 > 0 and an ε1 > 0 such that for every |r − 2| < δ1 and ε ∈ (0, ε1), there is a τ1 0 for which the following hold: (i) if τ1 > 0 then (Uε , ξε ) is asymptotically stable for τ ∈ (0, τ1 ); (ii) if τ > τ1 then (Uε , ξε ) is unstable; (iii) if 1 < p < 2r + 1, then τ1 > 0 provided that α is sufﬁciently small. On the other hand, if p > 2r + 1 then τ1 = 0, provided that α is sufﬁciently small. Theorems 3.12 and 3.13 improve Theorem 3.10 in the one-dimensional case n = 1. Theorem 3.11, which guarantees the existence of periodic solutions, also gets improved in a similar fashion when n = 1. In fact, Theorems 3.12 and 3.13 hold for a more general activator–inhibitor system which allows self-production of the activator. The basic advantage for the case n = 1 is that now the linearized operator Lε ≡ ε2 − 1 + pup−1 ε

(3.22)

is invertible, where uε is the least-energy solution of (1.4), although the details become much more involved, for which we refer the reader to [NTY1].

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203

3.4. Diffusion systems Stability properties for diffusion systems have been studied for many important models. However, there seems to be no general results. For instance, the counterpart for the property that “stability implies triviality” for single equation, or that “stability implies monotonicity” for shadow systems, has not been established even for 2 × 2 diffusion systems. The situation here seems quite complicated. In this section, instead of surveying various stability and instability results for many diffusion systems, we shall restrict ourselves to mainly the activator–inhibitor system we have discussed in previous sections just to illustrate the methods involved. The ﬁrst stability results on spike solutions are due to [NTY1,NTY2]. (See [N2] for the announcement.) Based on the shadow system approach, it seems natural to expect that the stability and instability properties of the 2 × 2 diffusion system (3.9) be determined by its shadow system (3.10) when the diffusion coefﬁcient d2 is large. This is indeed the case. Again, to simplify our presentation, we deal with the case n = 1 in the 2 × 2 activator– inhibitor system (1.1) and its shadow system (1.3). Denoting d1 = ε2 , we ﬁrst obtain the single boundary-peak steady state solution for (1.1) from the least-energy pattern (Uε , ξε ) of the shadow system (1.3) given by (3.16). T HEOREM 3.14 [T]. Suppose that n = 1. There exist ε0 > 0 and D∗ > 0 such that for 0 < ε < ε0 and d2 > D∗ the system (1.1) has a steady state solution of the form U (x; ε, d2) = Uε (x) + Φ(x; ε, d2)

and (3.23)

V (x; ε, d2) = ξε + Ψ (x; ε, d2),

where (Uε , ξε ) is a steady state solution of the shadow system (1.3) given by (3.16) and, Φ and Ψ satisfy −q/(p−1) ξ Φ(·; ε, d2) ε

L∞

C d2

C and ξε−1 Ψ (·; ε, d2)L∞ d2

for some positive constant C independent of ε and d2 . Now, the stability and instability properties of the solution (U (·; ε, d2), V (·; ε, d2)) read as follows [NTY1]. T HEOREM 3.15 (Instability). There is a τ1 0 depending on (p, q, r, s), ε ∈ (0, ε0 ) and D > D∗ such that (U (x; ε, d2), V (x; ε, d2)) is unstable if τ > τ1 . T HEOREM 3.16 (Stability I). There exist δ > 0, ε1 > 0 and D2 > 0 such that for each r satisfying |r − (p + 1)| < δ and for ε ∈ (0, ε1) and d2 > D2 , unless s belongs to an exceptional set C consisting of at most ﬁnitely many points in the interval [0, r/(p − 1) − 1),

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there are positive numbers 0 < τ2m−1 < τ2m−2 < · · · < τ2 < τ1 for which the following hold: (i) if τ2j < τ < τ2j −1 for some j = 1, 2, . . . , m, then (U (x; ε, d2), V (x; ε, d2)) is asymptotically stable, where τ2m = 0; (ii) if τ2j −1 < τ < τ2j −1 for some j = 1, 2, . . . , m, then (U (x; ε, d2), V (x; ε, d2)) is unstable, where τ0 = +∞. If, in addition, C is empty, then m = 1. T HEOREM 3.17 (Stability II). There exist δ1 > 0, ε1 > 0 and D2 > 0 such that for each r satisfying |r − 2| < δ1 and for ε ∈ (0, ε1 ) and d2 > D2 , there is a nonnegative number τ1 for which the following hold: (i) if τ1 > 0, then (U (x; ε, d2), V (x; ε, d2)) is asymptotically stable for τ ∈ (0, τ1 ); (ii) if τ > τ1 then (U (x; ε, d2), V (x; ε, d2)) is unstable; (iii) suppose that 1 < p < 2r + 1, then τ1 > 0 provided that α is sufﬁciently small, qr i.e., s is sufﬁciently close to p−1 − 1. On the other hand, when p > 2r + 1, then τ1 = 0, provided that α is sufﬁciently small. T HEOREM 3.18 (Hopf bifurcation). Let δ1 , ε1 and D2 be the positive numbers given by Theorems 3.16 and 3.17. Assume that 0 < ε < ε1 , d2 > D2 , and that r satisﬁes |r − 2| < δ1 or |r − (p + 1)| < δ1 . Moreover, assume that s ∈ / C if |r − (p + 1)| < δ1 , on the other hand, assume that 1 < p < 2r + 1 and α is sufﬁciently small if |r − 2| < δ1 . Let τk be the positive number given by Theorems 3.16 and 3.17, where 1 k 2m − 1 if |r − (p + 1)| < δ1 and k = 1 if |r − 2| < δ1 . Then there is a one-parameter family of periodic solutions {(U (x; t; ε, d2; μ), V (x, t; ε, d2; μ))}|μ| 0 in the case when Ω is a ball of radius R in Rn , under either the homogeneous Dirichlet boundary condition (4.2) or the homogeneous Neumann boundary condition (4.3), where ν again denotes the unit outward normal to ∂Ω. Our goal here is to understand what kind of symmetries are being imposed to all positive solutions of (4.2) by different boundary conditions (4.2) and (4.3). Our ﬁrst result says that the Dirichlet boundary condition (4.2) is “rigid and coercive” – solutions of (4.1) and (4.2) basically inherit the symmetries of the domain Ω. (See Section 4.2 for exceptions and more discussions.) T HEOREM 4.1 [GNN1]. Let u be a solution of the Dirichlet problem (4.1) and (4.2) where f is locally Lipschitz continuous. Then u must be radially symmetric, i.e., u(x) = u(|x|), and u (r) < 0 for all 0 < r < R. Observe that there is essentially no condition imposed on f . Therefore, the radial symmetry of solutions to (4.1) and (4.2) seems to result from the symmetry of the domain Ω and the Dirichlet boundary condition. The proof makes use of the well-known “movingplane” method devised by A.D. Alexandroff in 1956. For simplicity we will only sketch the proof of Theorem 4.1 in the case 0 f ∈ C 1 . The general case can be proved in a similar manner with extra work. Deﬁne Σλ = x = (x1 , . . . , xn ) ∈ Ω | x1 > λ and let Tλ be the hyperplane which is perpendicular to x1 -axis at x1 = λ. Denote the following statement by (∗)λ : u(x) < u x λ for all x ∈ Σλ ,

and

∂u 0 on BR0 (0) by (4.6). On the other hand, from the choice of δ and (4.6) it follows that c(x) 0 in Σλ \ BR0 (0). Since w 0 on ∂(Σλ \ BR0 (0)) and limx→∞ w = 0 (for x ∈ Σλ ), we conclude from the maximum principle and the Hopf boundary point ∂w lemma that w > 0 in Σλ \ BR0 (0) and ∂x < 0 on Tλ . Thus λ ∈ Λ and our assertion is 1 established. The rest of the proof proceeds similarly as before, and is therefore omitted here. Although Theorem 4.3 is already quite general and covers a wide range of equations, the remaining borderline case f (0) = 0 and f > 0 in (0, δ) does include some important examples. For instance, the equation

u + up = 0 in Rn , u > 0 in Rn and u → 0 at ∞,

(4.7)

Qualitative properties of solutions to elliptic problems

211

where the exponent p n+2 n−2 , n 3, has attracted the attention of many mathematicians. All the radial solutions of (4.7) have been understood, and they possess remarkable, and perhaps unexpected, properties. (See [Wa,Li,GNW1,GNW2] and [PY].) However, the study of symmetry properties of (4.7) remains a major open problem. Only the critical case of (4.7), where p = n+2 n−2 , has been resolved. T HEOREM 4.5. All solutions of the problem n+2

u + u n−2 = 0

in Rn

and u > 0

in Rn ,

(4.8)

must take the form u(x) =

n(n − 2)λ2 λ2 + |x − x0 |2

n−2 2

(4.9)

where λ > 0 and x0 ∈ Rn . Note that no condition on the asymptotic behavior of the solution u is imposed in (4.8). We refer the reader to [CL] for a brief history and a short, ingenious proof of this remarkable theorem originally due to [CGS].

4.4. Related monotonicity properties, level sets and more general domains The publication of [GNN1] in 1979 has stimulated much research in this direction. In particular, there have been many variants of the “moving plane” method applied to various different domains and/or different types of solutions. (We have encountered one in Section 4.1 already.) Part of the conclusion resulting from the “moving plane” method is that the solution must be monotone (in addition to being radially symmetric). In 1991, a useful “sliding” method was devised by Beréstycki and Nirenberg [BN]. It was used, for instance, to establish the following result in [BCN1], which deals with more general unbounded domains than just Rn . Consider the following problem

u + f (u) = 0 in Ω, u > 0 in Ω and u = 0 on ∂Ω,

(4.10)

where Ω = {x = (x1 , . . . , xn ) ∈ Rn | xn > ϕ(x1 , . . . , xn−1 } is an unbounded domain in Rn , ϕ : Rn−1 → R is a locally Lipschitz continuous function, and f satisﬁes the following hypothesis: There exist 0 < s0 < s1 < μ such that f (s) δ0 s on [0, s0 ) for some δ0 > 0, nonincreasing on (s1 , μ), and f > 0 on (0, μ), f 0 on (μ, ∞).

(4.11)

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T HEOREM 4.6. Let u be a bounded solution of (4.10) with M = sup u < ∞. Suppose ∂u that (4.11) holds. Then u must be monotone in xn , i.e., ∂x > 0 in Ω. n In particular, the theorem above applies to domains including half-space. However, in this case, much stronger results for more general f (u) are available. For instance, the following theorem was proved in [BCN1]. T HEOREM 4.7. Let u be a bounded solution of

u + f (u) = 0 in H = x = (x1 , . . . , xn ) ∈ Rn | xn > 0 , u > 0 in H and u = 0 on ∂H,

(4.12)

where f is locally Lipschitz. If f (M) 0 where M = sup u, then u is a function of xn ∂u alone and ∂x > 0 in H . n Incidentally, in [BCN1] it was conjectured that if there is such a solution in Theorem 4.7, then necessarily f (M) = 0. This conjecture has been veriﬁed only in n = 2 by Jang [J] in 2002. In this connection we ought to mention a well-known conjecture of De Giorgi in 1978. C ONJECTURE (De Giorgi). Let u be a solution of u + u − u3 = 0 in Rn with |u| 1 and least for n 8.

∂u ∂xn

(4.13) > 0 in Rn . Then all level sets [u = λ] of u are hyperplanes, at

This conjecture was proved by Ghoussoub and Gui [GG1] for n = 2 in 1998, by Ambrosio and Cabré [AmC] for n = 3 in 2000, and, signiﬁcant progress was made by [GG2] for n = 4, 5 and the conjecture is established under an extra condition in [S] for n 8 recently. Here we will describe the basic ideas used in [GG1]. In fact, a much more general result was established in [GG1]. T HEOREM 4.8. Let f ∈ C 1 . Suppose that u is a bounded solution of u + f (u) = 0 on R2 with

∂u ∂x2

(4.14)

0 in R2 . Then u is of the form

u(x) = g(ax1 + bx2) for some appropriate constants a, b ∈ R. This is truly a beautiful theorem. Its proof makes use of the following result of [BCN2].

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P ROPOSITION 4.9. Let L = − − V be a Schrödinger operator on Rn with the potential V being bounded and continuous. If Lu = 0 has a bounded, sign-changing solution, then the ﬁrst eigenvalue

λ1 (V ) = inf

n − V ψ 2 ) ∞ ψ ∈ C 0 in R2 ; for otherwise, we will have ∂x ≡ 0 in R2 by the First, we may assume that ∂x 2 2 Maximum principle and we are done. ∂u Next, observe that ∂x satisﬁes the equation 2 ϕ + V (x)ϕ = 0

(4.15)

∂u in R2 , where V (x) = f (u(x)) is bounded and continuous. Since ∂x > 0 in R2 , it follows 2 that λ1 (V ) 0, and (4.15) has no bounded, sign-changing solution (by Proposition 4.9). On the other hand, given a point x0 ∈ R2 , we can choose a direction ν such that ν · ∂u 2 ∇u(x0 ) = 0. Since ∂u ∂ν also satisﬁes (4.15), we have ∂ν ≡ 0 in R , i.e., u is constant along the direction ν and our proof is complete.

Incidentally, Proposition 4.9 is false for n 3. (See [GG1] and [B].) Concerning properties of the level sets of solutions in bounded smooth domains without radial symmetry, some progress has been made as well. When Ω is convex, it seems natural to ask if the level sets of positive solutions, namely, {x ∈ Ω | u(x) μ}, to the Dirichlet problem (4.1) and (4.2) are convex. Even for the very special case f (u) = λ1 u, where λ1 is the ﬁrst eigenvalue of − on Ω under the zero Dirichlet boundary condition, it was a long-standing conjecture that the level sets of the ﬁrst eigenfunction for a convex domain are convex. This conjecture was proved by Brascamp and Lieb [BL] in 1976 by using the heat equation and log concave functions. Since then, techniques involving Maximum principles for elliptic equations have been developed by several authors, including Korevaar [K], Kennington [Ke], Caffarelli and Friedman [CF] and Korevaar and Lewis [KL]. The basic idea is to show that, instead of the solution u, v = g(u) is convex for some properly chosen transformation g, which implies that the level sets of v (and therefore that of u) are convex. The transformation g is suitably chosen so that v = g(u) satisﬁes the equation v = h(v, ∇v), where h satisﬁes 1 h > 0 and 0. h vv

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Then, it was established, in case n = 2 by [CF] and n 3 by [KL], that v = g(u) is convex in Ω. The effect of g is to “bend” the graph of u making it nearly vertical near ∂Ω. However, for given f , there seems to be no known algorithm for ﬁnding g.

4.5. Generalizations and other types of equations Many of the symmetry results in previous sections have been generalized to more general nonlinear equations – some may be established by essentially the same arguments, others require new ideas. Generally speaking, if we replace the term f (u) in (4.1) by f (r, u) = f (|x|, u), then symmetry results Theorems 4.1 and 4.3 still hold provided that f (r, u) is nonincreasing in r > 0. On the other hand, if f (r, u) is increasing in r, one cannot expect solutions to be radially symmetric anymore. For instance, the Dirichlet problem,

u − u + V |x| up = 0 in BR ⊆ Rn , u > 0 in BR and u = 0 on ∂BR ,

has nonradially symmetric solutions for R large, where p > 1, V (|x|) = 1 + |x| , and 0 < < (n − 1)(p − 1)/2. (See, e.g., [DN], Proposition 5.10.) Similarly, so does its counterpart for entire space. Signiﬁcant examples involving f (|x|, u) but not covered by the generalization of Theorem 4.3 include the Matukuma equation in astrophysics u +

1 up = 0 1 + |x|2

in Rn . The handling of symmetry properties of positive solutions to this kind of equations often requires a detailed knowledge of the asymptotic behaviors of the solutions at ∞ in order to get the “moving plane” process started. (See [NY] and [Y] for more details.) One can also replace the Laplace operator in (4.1) by more general operators; e.g., by fully nonlinear operators F (x, u(x), Du(x), D 2 u(x)) = 0, where F satisﬁes the following: (F1) F (x, s, pi , pij ), 1 i, j n, is continuous in all of its variables, C 1 in pij and 2u ∂u Lipschitz in s and pi , where pij ’s are position variables for ∂x∂i ∂x , pi for ∂x and j i s for u. ¯ x, pi , pij )|ξ |2 for all ξ ∈ Rn , where λ¯ > 0 in Rn × (F2) Fpij (x, s, pi , pij )ξi ξj λ(s, 2

R × Rn × Rn . (F3) F (x, s, pi , pij ) = F (|x|, s, pi , pij ) and F is nonincreasing in |x|, (F4) F (x, s, p1 , . . . , pi0 −1 , −pi0 , pi0 +1 , . . . , pn , p11 , . . . , −pi0 j0 , . . . , −pj0 i0 , . . . , pnn ) = F (x, s, pi , pij ) for 1 i0 , j0 n and i0 = j0 .

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T HEOREM 4.10 [LiN]. Suppose that F satisﬁes (F1)–(F4) and Fs 0 for |x| large, and for s small and positive. Let u be a positive C 2 solution of F x, u(x), Du(x), D 2 u(x) = 0 in Rn , n 2, (4.16) u(x) → 0 at ∞. Then u must be radially symmetric (up to a translation) and ur < 0 for r = |x| > 0. The proof uses essentially the same arguments as in that of Theorem 4.3, thus is omitted here. However, we wish to remark here that the elliptic operator F in Theorem 4.10 is not required to be uniformly elliptic, therefore is quite general. For instance, it includes the minimal surface operator, or, equations of mean-curvature type " div √ Du 2 + f (u) = 0 in Rn , 1+|Du| (4.17) u > 0 in Rn and u → 0 at ∞. Consequently, Theorem 4.10 also contains previous work [FL] on (4.17). In this direction we ought to discuss the p-Laplacian p u = div |Du|p−2 Du ,

(4.18)

where p > 1, which exhibits certain degeneracy or singularity depending on p > 2 or p < 2. (Note that the case p = 2 gives rise to the usual Laplace operator.) The case 1 < p < 2 is studied in [DPR]. It is proved there that essentially under the same hypothesis (4.5) a solution u of the problem p u + f (u) = 0 in Rn , (4.19) u → 0 at ∞, u > 0 in Rn , where 1 < p < 2, must be radially symmetric (up to a translation) and ur < 0 in r = |x| > 0. The method of proof in [DPR] still uses the “moving plane” technique, but with a weak comparison principle instead of the usual Maximum principle. The symmetry of solutions to (4.19) for the degenerate case p > 2 does not hold in general, however. See [SZ], Section 6, for a counterexample. 4.6. Symmetry of nonlinear elliptic systems Some of the symmetry results described in previous sections have been generalized to positive solutions of nonlinear elliptic systems. In this section, we will only mention two of them: One for balls, the other one for the entire space Rn . It turns out that Theorem 4.1 can be generalized to cooperative elliptic systems in a straightforward manner. (See [Ty].) The following elliptic system, ui + fi (u1 , . . . , um ) = 0 in Ω, i = 1, . . . , m, (4.20) ui > 0 in Ω and ui = 0 on ∂Ω,

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is said to be cooperative if fi is C 1 and ∂fi 0 ∂uj

for all i = j and 1 i, j m.

(4.21)

T HEOREM 4.11. Let Ω be a ball of radius R in Rn , f satisfy (4.21) and (u1 , u2 , . . . , um ) be a solution of (4.20). Then for each i, ui is radially symmetric and u (r) < 0 for 0 < r = |x| < R. Since the usual Maximum principle for single elliptic equations generalizes to cooperative elliptic systems [PW], the proof of Theorem 4.1 also generalizes naturally to establish Theorem 4.11. The second result here generalizes Theorem 4.3 for the entire space. This is more involved. Here we are dealing with solutions of the following problem

ui + fi (u1 , . . . , um ) = 0 in Rn , i = 1, . . . , m, ui > 0 in Rn and ui (x) → 0 as x → ∞.

(4.22)

In addition to (4.21), we will also assume fi , i = 1, . . . , m, satisfying the following hypothesis:

There exists ε > 0 such that the system (4.22) is fully coupled in 0 < u < ε; more precisely, for any I, J ⊆ {1, . . . , m} with I ∩ J = φ and I ∪ J = {1, . . . , m}, there exist i0 ∈ I and j0 ∈ J ∂fi such that ∂uj0 > 0 in 0 < u < ε.

(4.23)

All principal minors of −A(u1 , . . . , um ) have nonnegative determinants for 0 < u < ε, where ∂fi A(u1 , . . . , um ) = . ∂uj 1i,j m

(4.24)

0

Recall that the principal minors of a matrix (mij )1i,j m are the submatrices (mij )1i,j k ,

1 k m.

Observe that (4.23) is to guarantee that all ui , i = 1, . . . , m, are radially symmetric with respect to the same point, while (4.24) reduces to (4.5) in Theorem 4.3 in the single equations case. In [BS] the following result is proved. T HEOREM 4.12. Suppose that (4.21), (4.23) and (4.24) hold and u is a solution of (4.22). Then u must be radially symmetric (up to a translation), and u (r) < 0 for r = |x| > 0.

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The proof, still using the “moving plane” method, is more involved. We refer the interested readers to [BS].

4.7. Miscellaneous results In this section, we collect miscellaneous results concerning symmetry properties of solutions to various boundary value problems including singular boundary values, or overdetermined systems. In [Ta], the following problem was considered

u + f |x|, u = 0 u→∞

in Rn , n 3, at ∞.

(4.25)

Under the assumptions that f , roughly speaking, is monotone in u with superlinear growth in u when |x| and u both are large and positive, and, r 2n−2 f (r, u) is “asymptotically” monotone in r for u large, it is established in [Ta] that all solutions of (4.25) are radially symmetric. The proof consists of two parts: First, prove that the difference of any two solutions of (4.25) must tend to 0 as |x| → ∞; then apply the arguments of [LiN] described in Section 4.3. In case n = 2, the method in [Ta] yields a similar result with the “boundary value u → ∞ as |x| → ∞” in (4.25) replaced by u(x) →∞ log |x|

as |x| → ∞,

(4.26)

and with another technical condition imposed on the monotonicity of f with respect to r. It is curious to note that there is an earlier result, due to [CN], asserting that all solutions of (4.25) in R2 with f (r, u) = K(r)eu ,

(4.27)

where K 0 and K ∼ |x|− at ∞, for some > 2, are radially symmetric. In fact, in this case all solutions are completely understood and classiﬁed; in particular, there is no solution having the asymptotic behavior (4.26). Incidentally, the equation in (4.25) with the nonlinearity (4.27) is known as the conformal Gaussian curvature equation with K as the prescribed Gaussian curvature in R2 . To conclude this section, we mention the following over-determined system ﬁrst considered in [Se] in 1971. T HEOREM 4.13. Let Ω be a smooth domain in Rn . Suppose the over-determined system ⎧ in Ω, ⎨ u = −1 u=0 on ∂Ω, ⎩ ∂u = constant on ∂Ω, ∂ν

(4.28)

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has a solution. Then Ω must be a ball and u(x) =

a 2 − |x|2 , 2n

where a is the radius of Ω. Serrin used the “moving plane” method, described in previous subsections of this section, to establish this result. However, there is a much simpler proof for this particular theorem due to Weinberger [W1], also in 1971, using clever integral identities. Integral identity approach to symmetry properties of elliptic equations has also been used to handle p-Laplacian. (See [B].) 5. Graphics and visualization 1 With the advances of computing facilities in recent years, numerical simulations or scientiﬁc computations have become an integral part of modern mathematics, especially in the branches of differential equations, applied mathematics and related areas. One of the most, perhaps the most, direct ways to understand the behavior of solutions to elliptic equations is to visualize the shape of solutions by numerically graphing them. In this section we shall brieﬂy present the graphics numerically obtained for the solutions to some of the equations discussed in previous sections of this chapter. Again, we will focus only on positive solutions; moreover, graphics for positive solutions to nonlinear elliptic equations under homogeneous Dirichlet or Neumann boundary conditions are included here for comparison purposes. As far as numerical analysis of nonlinear elliptic equations is concerned, some early papers may be traced back to the 1950s and the 1960s [Be,P]. During that period, the mentality for studying nonlinear equations was mostly directed toward establishing the existence and uniqueness of the given system. A major goal was to establish convergence and error estimates of the (unique) numerical solution. Thus, the work was mostly analytical rather than computational in nature, because computers were few and unavailable, and of very limited number crunching power. The nonlinearities were of the kind satisfying the global Lipschitz condition and, thus, they were just perturbations of linear equations. Consequently, much of the work in the early era could not be directly applied to the problems considered here. Things began to change rapidly during the 1970s and the 1980s when the power of computing accelerated following Moore’s law. Supercomputers were made available to mathematicians at universities for computing numerical solutions of partial differential equations. Since the 1990s, powerful desktop workstations have appeared, which possessed the number crunching capability of the supercomputers of the earlier generations. Visualization software packages were also being perfected. Nowadays, medium to large tasks of numerical computation can be handled by a central processor in a mathematics department, with relative ease. Many problems can now be solved by computing on a home computer. 1 This section is written in collaboration with G. Chen at Texas A&M University, A. Perronnet at Université Pierre et Marie Curie and J. Zhou, also at Texas A&M University.

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Three basic types of numerical methods are commonly used to solve elliptic boundary value problems: (i) FDM (the ﬁnite difference method); (ii) FEM (the ﬁnite element method); (iii) BEM (the boundary element method). Each has its own advantages and disadvantages. Overall speaking, FEM is the leading and by far the most powerful numerical method among the three in that (i) it can handle the geometry of the domain very well; (ii) it has an inherent variational structure; (iii) many commercial packages are available for automatic mesh generation. Many nonlinear equations have multiple solutions. For semilinear elliptic boundary value problems, the Monotone Iteration Scheme (MIS), by Amman [A3] and Sattinger [Sa] (but originated much earlier, to Bierberbach [Bi]) and then generalized to various different forms [AC,P2], gives a systematic method for ﬁnding and determining multiple solutions of semilinear elliptic equations. The scheme itself is constructive in nature and its algorithmic realization is straightforward. Its numerical implementation can be done through FDM, FEM or BEM. Rigorous convergence and error estimates may be found in [CDNZ, DCNZ,HMW,I,P1–P3] along with many examples of proﬁles of numerical solutions given therein. Thus, the numerical analysis and computation of MIS is now a well-developed and understood subject. However, solutions obtainable through this scheme are all stable solutions. As far as unstable solutions are concerned, one needs to look beyond MIS in order to ﬁnd such solutions. The Mountain Pass Lemma (MPL) provides a powerful method for such a purpose. However, the proof of MPL contains ingredients which, we feel, either are not totally constructive in the algorithmic sense, or involve considerable complexity in order to be realized into algorithms. This is the major reason that has held up the numerical realization of MPL. To implement MPL, obviously some adaptation is required. Choi and McKenna’s paper [CM] in 1993 was the ﬁrst to succeed in the adaptation of MPL to numerical implementation by FEM, twenty years after the result of MPL [AR] was published. The algorithm in [CM] is a min–max iterative method. With the choices of different initial state satisfying the assumptions of MPL, multiple solutions can be computed. A reﬁned version of the min–max method of Choi–McKenna employed by us in [CNZ], called the Mountain Pass Algorithm (MPA), will be given in Section 5.1. A different idea for computing multiple solutions of semilinear elliptic boundary value problems, which is quite effective especially when the nonlinearity is a power law, utilizes scaling and is called the Scaling Iterative Algorithm (SIA) in [CNZ]. Numerical solutions of semilinear elliptic boundary value problems using MPA and FEM may be found in [CM,DCC,CNPZ], while those obtained by SIA and BEM may be found in [CNZ]. These two different algorithms and the different associated numerical treatments serve to corroborate the correctness and accuracy of the numerical solutions. Such numerical solutions all have Morse index one. To obtain numerical solutions of higher Morse indices, high-linking method [C], can be realized algorithmically and then implemented with FDM, FEM or BEM; see [DCC,CNZ,LZ1].

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5.1. Mountain-pass and scaling algorithms We describe MPA and SIA below. The Mountain Pass Algorithm (MPA). To solve u + f (x, u) = 0 in Ω

(5.1)

with prescribed homogeneous linear boundary conditions, we consider the problem of ﬁnding a critical point of the functional # J (u) = Ω

$ 2 1 ∇u(x) − F x, u(x) dx + γ u2 (x) dσ 2 ∂Ω

(5.2)

in the function space E = H01 (Ω) or E = H 1 (Ω), where γ 0 (γ = 0 if the function space is H01 (Ω)), and F (x, u(x)) deﬁned by ∂ F (x, u) = f (x, u) ∂u satisﬁes all the assumptions of the MPL. Note that if the underlying Banach space E is H01 (Ω), then the corresponding boundary condition is u = 0 on ∂Ω. Otherwise, the associated boundary condition is (γ u + ∂u/∂ν) = 0 on ∂Ω. We use (D), (N) and (R) to denote, respectively, the following boundary conditions: (D) u = 0 on ∂Ω; (N) (∂u/∂ν) = 0 on ∂Ω; (R) (γ u + ∂u/∂ν) = 0 on ∂Ω. Mountain Pass Algorithm (MPA). Step 1. Choose an initial state w0 ∈ E sufﬁciently smooth; set w1 = w0 . Step 2. If w1 satisﬁes the boundary condition (D), (N) or (R), and if w1 + f (x, w1 ) 2 ε L (Ω) ˆ for a prescribed small error limit ε > 0, stop and exit. Otherwise, from w1 , solve v:

vˆ = −f (x, w1 ) on Ω, subject to boundary condition (D), (N) or (R).

Step 3. For t: T > t > 0, let λ(t) be such that J λ(t) w1 + t vˆ = max J λ w1 + t vˆ . λ∈[0,1]

(5.3)

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221

Find tˆ: T tˆ 0 such that J λ tˆ w1 + tˆvˆ = min J λ(t) w1 + t vˆ . T ≥t 0

Step 4. Update: w1 := λ tˆ w1 + tˆvˆ ,

w1 := λ tˆ w1 + tˆvˆ .

(5.4)

Go to Step 2. Scaling Iterative Algorithm. Next, let us look at SIA. It deals with the BVP

u − au + bup = 0 on Ω, subject to boundary condition (D), (N) or (R),

(5.5)

where a, b > 0 and p > 1. Scaling Iterative Algorithm (SIA). Step 1. Choose any v0 (x) 0 on Ω, v0 ≡ 0; v0 sufﬁciently smooth. Step 2. Find αn+1 > 0 and vn+1 (·) such that ⎧ p ⎨ vn+1 (x) − avn+1 (x) = −αn+1 bvn (x) on Ω, v (x ) = 1, ⎩ n+1 0 subject to boundary condition (D), (N) or (R).

(5.6)

Step 3. If ε˜ n ≡ vn+1 − vn H 1 (Ω) < ε, 0

output and stop. Else go to Step 2. 1/(p−1)

Then u = α∞ v∞ is an approximate solution of (5.5), where v∞ and α∞ are the last iterate for (5.6). Rigorous proofs of convergence for MPA and SIA are truly challenging. There are good reasons to believe that without more restrictive assumptions convergence will not hold for the general nonlinearity f (x, u) in (5.1) and the general domain Ω. However, some progress has been made recently in establishing the convergence of MPA; see [LZ2,LZ3]. Additional assumptions that the problem be nondegenerate, i.e., J (u∗ ) is invertible at the critical point u∗ , and that adjustable stepsize (cf. λ(tˆ ) in (2.4)) be used are essential for the proof. For SIA, a modiﬁed version called OSA (Optimal Scaling Algorithm) has been studied in [CEZ]. Its convergence can be proved using the same ideas as in [LZ2,LZ3]. In spite of all the aforementioned progress in [CEZ,LZ2,LZ3], at present no error estimates are available when MPA, SIA or their variants are implemented by FDM, FEM or BEM. This is certainly an area worth more investigation.

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Note that numerical results obtained with MPA and SIA reconcile with total agreements [CNZ]. Thus, in the following section, we will only use SIA for our computational purpose, since the coding of its computer programs is somewhat simpler.

5.2. Visualization of solutions of singularly perturbed semilinear elliptic boundary value problems We compute and exhibit a few examples of multiple positive solutions of singularly perturbed semilinear elliptic boundary value problems in R2 . The equations treated here are of the form

ε2 u − u + up = 0 in Ω ⊆ R2 , p > 1, subject to boundary condition (D) or (N),

(5.7)

where ε2 > 0 is a small number. Here, we have chosen ε2 = 10−3 . For (5.7), its variational functional is # J (u) = Ω

$ 1 2 1 ε2 2 p+1 |∇u| + u − |u| dx, 2 2 p+1

u ∈ E.

(5.8)

The system in (5.7) constitutes a singularly perturbed nonlinear boundary value problem. Here we have achieved good success with the numerical computation of the (D) and (N) cases, which are actually the situations where the theoretical properties of the solutions of the singularly perturbed problem are known [NT2,NT3,NW]. However, the singularly perturbed Robin boundary value problem remains to be carefully investigated. We ﬁrst look at the domain shown in Figure 1. It consists of disk with radius 1/2 on the left, connected through a rectangular corridor to an elliptical domain with an elliptical cavity on the right. The two boundary ellipses are concentric with center at (2, 0) and have, respectively, major axes of lengths 1 and 1/2, and minor axes of lengths 1/2 and 1/10. A sample triangulation is also displayed in Figure 1, where noticeably on some parts of the domain, dense meshes are used while elsewhere the meshes are sparser. Our mesh generation (based on the commercial software FEMLAB) has the capability of manual adjustable local mesh reﬁnement. This is important because many solutions of the singularly perturbed problem here have spikes. Numerical data can be properly calculated only if dense mesh reﬁnements are made on the portion of the domain where the spike occurs. This, in our opinion, is the greatest challenge in obtaining high accuracy of numerical solutions of singularly perturbed boundary value problems (5.7). For the Dirichlet boundary condition, a total of six single-peak solutions have been captured. They are displayed in ascending order of the energy functional J in Figures 2–6. Note that the least energy solution (i.e., the ground state) as displayed in Figure 2 lives on the “largest open ball” contained in Ω, which is consistent with [NW], Theorem 2.2, p. 734. Also, the solution symmetric to the one in Figure 3 is not displayed. So the solution count for Figure 3 actually is 2.

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Fig. 1. The two-dimensional domain formed by a disk connected through a rectangular corridor to an elliptical region with an elliptical cavity. Note that this discretization is just a sample. In the actual computations of the solutions displayed in the following ﬁgures, dense grids are chosen in the neighborhoods of the domain where solution spikes occur in order to secure high accuracy of the singularly perturbed boundary value problem.

Fig. 2. The least-energy solution of the Dirichlet boundary value problem with p = 3 in (5.7), J = 5.8663 × 10−3 , max u = 2.2064, where (and in subsequent ﬁgures) max u denotes the maximum of u(x, y) on the domain.

R EMARK 5.1. It is not hard to see that the largest possible inscribed balls that would ﬁt into Ω near the peaks of the solutions in Figures 3 and 4 are of the same size – both have diameters of length 0.5. Presumably, this should force the numerical results of the two solutions extremely close. However, we do not understand the relatively large discrepancies appeared between the solutions in Figures 3 and 4.

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Fig. 3. A positive solution of the Dirichlet boundary value problem with p = 3 in (5.7), J = 5.8686 × 10−3 , max u = 2.2047. The maximum happens at the point (x, y) = (1.9991, 0.7504), whose distance to the boundary ∂Ω is computed to be 0.25. Note that there is another identical solution obtainable through reﬂection about the axis of symmetry of the domain.

Fig. 4. A positive solution of the Dirichlet boundary value problem with p = 3 in (5.7), J = 5.8735 × 10−3 , max u = 2.2035. The maximum happens at the point (x, y) = (1.6521, 0.0021), whose distance to the boundary ∂Ω is computed to be 0.23745. Here, we wish to point out that the errors in this case seem larger than those in the previous cases. We do not know if this is due to the presence of the corners. (See Remark 5.1.)

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Fig. 5. A positive solution of the Dirichlet boundary value problem with p = 3 in (5.7), J = 5.8923 × 10−3 , max u = 2.2048.

Fig. 6. A positive solution of the Dirichlet boundary value problem with p = 3 in (5.7), J = 5.9490 × 10−3 , max u = 2.1937. This is the only single-peak positive solution living on the corridor.

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Next, we display the graphics of three solutions of the Neumann boundary value problem (N) in Figures 7–9, again in ascending order of the energy functional value J . Note that the lowest energy solution is given in Figure 7, where the maximum happens at a boundary point with the largest curvature, consistent with the results in [NT2,NT3]. There is another solution obtained by reﬂection with respect to the axis of symmetry of the domain. Therefore, the solution count for Figure 7 is two. Similarly, the solution count for Figure 9 is also two. R EMARK 5.2. It seems that there should be a solution, to the Neumann boundary value problem with p = 3 in (5.7), which has its single-peak located near the far right point (2.5, 0) and has its energy J lower than that of the solution in Figure 9. However, this solution is more difﬁcult to capture by our numerical schemes. R EMARK 5.3. For a singularly perturbed problem considered in this section, the maximum of the solutions according to [CNZ, (98), p. 1601] is approximated to be 2.206205. All the positive solutions as displayed in Figures 2–9 take their max u values within 5% relative error of this value. Thus, these solutions may be said to lie quite well within the “asymptotic regime” as ε2 → 0.

Fig. 7. The least-energy solution of the Neumann boundary value problem with p = 3 in (5.7), J = 2.7493 × 10−3 , max u = 2.1507. Another positive solution is obtainable by reﬂection along the axis of symmetry of the domain.

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Fig. 8. A positive solution on the Neumann boundary value problem with p = 3 in (5.7), J = 2.8578 × 10−3 , max u = 2.1729. This is the only positive solution we have found that lives on the disk on the left of the domain.

Fig. 9. A positive solution of the Neumann boundary value problem with p = 3 in (5.7), J = 2.9421 × 10−3 , max u = 2.2066. This is the only positive solution we have found that lives on the corridor, up to symmetry. (Another positive solution is obtainable by reﬂection along the axis of symmetry of the domain.)

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5.3. Concluding remarks Numerical results and graphics obtained in this chapter can be extended to domains in R3 . (See [CNPZ].) Although we have only included graphics of solutions with single-peaks here, multipeak solutions can be treated as well. (See [CNZ].) However, numerical treatments for solutions with multidimensional concentration sets are very challenging and have not been studied. An obvious difﬁculty in this direction is that the Morse indices of those solutions are very large; in fact, they tend to inﬁnity as ε → 0. There is much to do in this direction numerically. In this section, we have only treated homogeneous Dirichlet or Neumann boundary value problems. For the Robin boundary value problem

ε2 u − u + up = 0 in Ω, γ u + ∂u on ∂Ω, ∂ν = 0

(5.9)

where γ 0, very little is known theoretically or numerically. However, given the opposite effects of Dirichlet and Neumann boundary conditions (cf. Section 1.3), it would seem extremely interesting if we could understand, as the parameter γ varies from 0 (which corresponds to the homogeneous Neumann boundary condition) to ∞ (which corresponds to the homogeneous Dirichlet boundary condition), how the solutions to (5.9) changes their qualitative properties.

Acknowledgment Research was supported in part by NSF.

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CHAPTER 4

On Some Basic Aspects of the Relationship between the Calculus of Variations and Differential Equations

Pablo Pedregal ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain E-mail: pablo.pedregal@uclm.es

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. A little bit of history . . . . . . . . . . . . . . . . . . . . . . . . 3. The Euler–Lagrange equation: From VP to EL . . . . . . . . . 4. Convexity: From EL to VP . . . . . . . . . . . . . . . . . . . . 5. Convexity: The direct method . . . . . . . . . . . . . . . . . . 6. Young measures . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Scalar problems under pointwise constraints . . . . . . . . . . 8. Vector problems and systems of PDE . . . . . . . . . . . . . . 9. Vector problems and quasiconvexity . . . . . . . . . . . . . . . 10. Second-order problems . . . . . . . . . . . . . . . . . . . . . . 11. Nonexistence: Lack of coercivity . . . . . . . . . . . . . . . . . 12. Nonexistence: Lack of convexity . . . . . . . . . . . . . . . . . 13. Generalized VP and generalized EL . . . . . . . . . . . . . . . 14. Dynamical problems: Lack of convexity and lack of coercivity 15. Numerical approximation . . . . . . . . . . . . . . . . . . . . . 16. Comments on other aspects of the CV . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract We describe the paradigmatic link between variational problems and differential equations through the classical Euler–Lagrange equations of optimality associated with a variational principle. Through the analysis of several standard and classical problems and situations, we try to convey the main ideas, methods and techniques. The exposition is somewhat informal, HANDBOOK OF DIFFERENTIAL EQUATIONS Stationary Partial Differential Equations, volume 1 Edited by M. Chipot and P. Quittner © 2004 Elsevier B.V. All rights reserved 235

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but statements have been written with care. Sharp results and further developments are left for specialists.

Keywords: Coercivity, Convexity, Direct method, Regularity, Weak lower semicontinuity MSC: 49J45, 49K, 35J20, 35J50

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1. Introduction The relationship between the calculus of variations (CV) and differential equations (DE) is best expressed through the connection between the paradigmatic problem of the CV Minimize I (u) = W x, u(x), ∇u(x) dx, u = u0 on ∂Ω, Ω

and the boundary value problem ∂W ∂W x, u(x), ∇u(x) = x, u(x), ∇u(x) div ∂A ∂u u = u0

in Ω, on ∂Ω.

We will identify these two fundamental problems as the variational problem (VP) and the associated Euler–Lagrange equation, or problem (EL). It is important for us to specify the following features of the different elements entering into those two problems. 1. Ω ⊂ RN is assumed to be a bounded, regular domain in RN with regular boundary ∂Ω. 2. The class of functions u is something important to clarify. Typically, they will belong to appropriate Sobolev spaces. It is also interesting to specify dimensions for the target space, u : Ω → Rm . 3. The integrand W (x, u, A) is a Carathéodory function (measurable in x and continuous in (u, A)) where W : Ω × Rm × Mm×N → R. Sometimes it is interesting to allow integrands W : Ω × Rm × Mm×N → R∗ ≡ R ∪ {+∞}

4. 5. 6.

7.

that may take on the value +∞ somewhere. This possibility is especially fruitful to enforce additional pointwise constraints on competing functions for VP. We will describe one such classical example. The function u0 is a prescribed one so that the condition u = u0 means that competing functions for the above variational problem must comply with such boundary values. For EL to have a precise meaning, we should enforce regularity assumptions on W at least with respect to the variables (u, A). The restrictions on the boundary ∂Ω could be of a different nature depending on the properties of W as we will see. In this way we could have Dirichlet, Neumann, mixed boundary conditions and even more involved situations. It is also important to make a distinction of these problems depending on the different values of N and m (the dimensions of the domain Ω and the number of components of u, respectively). Depending on the values of these two dimensions the properties of the two problems are amazingly different:

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(1) N = m = 1. We will refer to VP as a scalar (m = 1), one-dimensional (N = 1) variational problem. The differential equation EL is a single ordinary differential equation. (2) N = 1, m > 1. This case corresponds to a vector, one-dimensional variational problem. EL is a coupled system of differential equations. (3) N > 1, m = 1. This is a scalar, multidimensional variational problem while EL is a single partial differential equation (PDE). (4) N, m > 1. We simply refer to VP as a vector variational problem. EL is a coupled system of PDE. How does the connection between VP and EL arise? At this stage we will proceed formally. Suppose U is a minimizer for VP, i.e., I (U ) I (u), whenever u = u0 on ∂Ω. Take u admissible for VP in an arbitrary fashion, and let ϕ = u − U . Notice that ϕ = 0 on ∂Ω. Consider the function of a real parameter t, deﬁned by g(t) = I (U + tϕ). This test function ϕ is called an admissible “variation” of U and in this way the name Calculus of Variations was universally accepted for the ﬁeld. We notice that g has an absolute minimum at t = 0 because U is a minimizer for VP. If we further assume all necessary regularities on W so that g is differentiable, we must have g (0) = 0 and can formally compute $ # ∂W ∂W 0 = g (0) = (x, u, ∇u)ϕ + (x, u, ∇u)∇ϕ dx. ∂A Ω ∂u By using the divergence theorem (assuming even more regularity on W if necessary) on the second term, and keeping in mind that ϕ = 0 on ∂Ω, we arrive at $ # ∂W ∂W (x, u, ∇u) − div (x, u, ∇u) ϕ dx. 0= ∂A Ω ∂u The arbitrariness of u implies the arbitrariness of ϕ (except for the vanishing boundary values) and this in turn implies that ∂W ∂W (x, u, ∇u) − div (x, u, ∇u) = 0 ∂u ∂A in Ω. A rigorous derivation of EL from the minimizer property requires more rigor in caring about the technical points involved in the justiﬁcation of the different steps. We will do this later. A closer look at the previous ideas reveals a clear parallelism with the ﬁnite-dimensional situation. Suppose I : D ⊂ Rd → R

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is a scalar, real-valued mapping. We are interested in the minimization problem Minimize

I (x): x ∈ D.

If X is such a point of minimum, I is differentiable and x is an admissible direction (variation) in the sense that X + t (x − X) belongs to D for t small, then the derivative of I at X in the direction of x − X should vanish: ∇I (X) · (x − X) = 0. If we have a whole collection of admissible directions x for which this equation should be satisﬁed, then ∇I (X) ≡ 0 and X must be a critical point for I . Anyone having taken an elementary course in Vector Calculus knows that there may be a whole variety of situations and interesting issues regarding the relationship between the initial minimization problem and the set of critical points for I . Our intuition on the ﬁnite-dimensional case may lead us in asking interesting (and relevant) questions concerning the much more complex situation in inﬁnite dimension. There are four such basic issues: 1. When are there global minimizers for I ? When is there a unique global minimizer? 2. When are there critical points? When is there only one critical point? 3. When is a critical point a global minimizer? 4. What is the signiﬁcance of other type of critical points like local minima and saddle points when there are absolute minimizers and when there are no such points? Some of these same issues about VP and EL, and their relationship, can and must be addressed. Speciﬁcally, we would like to focus on the following points. Our efforts will lead to their (partial) answer throughout these pages. 1. When are there solutions to any of these two problems, VP and EL? 2. Under what circumstances can we go from solutions of one of them to solutions of the other? 3. What happens when one of them has solutions but the other one does not? 4. What is the role played by convexity and coercivity? 5. What can be done when there are no solutions? We will try to provide some insight on these points as well as illustrate our observations with standard examples. The answer will depend on the values of the dimensions N and m. It is worthwhile to write down some of the most important examples of VP and their associated EL. These are important because they have historically inspired, to a great deal, the effort to understand some of the basic topics we will explain. In writing some of these examples we have ignored irrelevant positive, multiplicative constants. 1. Brachistochrone: √ 1 + A2 u (x) = 0. , N = m = 1, W (x, u, A) = √ √ x x 1 + u (x)2 2. Laplacian: N > 1, m = 1,

1 W (x, u, A) = |A|2 , 2

u = 0.

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3. p-Laplacian: N > 1, m = 1,

W (x, u, A) =

div |∇u|p−2 ∇u = 0.

1 |A|p , p

4. Minimal surfaces: N > 1, m = 1,

W (x, u, A) =

1 + |A|2,

div

∇u 1 + |∇u|2

= 0.

5. Obstacle problem: N > 1, m = 1,

W (x, u, A) =

1

2 |A|

2,

+∞,

u ψ(x), else.

The graph of ψ is the obstacle. 6. Linear elasticity: N, m > 1,

1 W (x, u, A) = Eε(A) : ε(A) − P (x) · u, 2 div Eε(∇u) = P ,

where E is the (fourth-order) elasticity tensor of material constants, P is the density of bulk load, and ε(A) is the symmetrization operation ε(A) =

1 A + AT . 2

7. Nonlinear elasticity: N, m > 1,

W (x, u, A) = W0 (x, A) − f (x, u), ∂W0 (x, A) ∂f (x, u(x)) , div = ∂A ∂u

where W0 is the density of internal elastic energy and f is the density of ﬁeld forces. 8. Diffusion systems: N = m > 1,

1 W (x, u, A) = |A|2 + w(u), 2

u = ∇w(u).

9. Wave equation: N > 1, m = 1,

2

− A2N , W (x, u, A) = A

2

− ∂ u = 0, u 2 ∂xN

AN ), x = (x,

u is the Laplacian with respect to where we write A = (A, ˜ xN ) and the ﬁrst N − 1 variables.

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There are also some interesting examples of second order where VP incorporates explicit dependence on second derivatives. In this case we reserve the variable λ for ﬁrst derivatives and use A for second derivatives. EL is more complicated in this case. We will discuss this later. We just write down these three examples and defer the discussion until Section 10. There should also be a discussion about boundary conditions. 10. Bi-harmonic equation: N > 1, m = 1,

W (x, u, λ, A) =

2 1 trace(A) , 2

(u) = 0.

11. Plate equation: N = 2, m = 1,

1 W (x, u, λ, A) = EA : A − F (x)u, 2 ∂ 2 ∂ 2u Eij kl = F. ∂xi ∂xj ∂xk ∂xl i,j

k,l

Again, E is the tensor of elastic constants and F is the vertical load acting on the plate. 12. Monge–Ampère equation (under vanishing boundary data): N > 1, m = 1,

W (x, u, λ, A) = −u det A + (N + 1)f (x)u, det ∇ 2 u = f.

13. Another version of the Monge–Ampère equation (under vanishing boundary data): N > 1, m = 1,

W (x, u, λ, A) = (cof A)λ : λ + N(N + 1)f (x)u, det ∇ 2 u = f,

where cof(A) is the cofactor matrix A cof A = det A1, and 1 is the identity matrix. We have explicitly written down here the most basic form of the functionals and equations. Even so, they already incorporate the relevant ingredients from our perspective. By playing appropriately with the dependence of W on u and x, it is not hard to produce nonhomogeneous and more complicated versions of each of those examples. For instance, it is relatively easy to produce functionals corresponding to inhomogeneous diffusion equations, or certain nonlinear (semilinenar, quasilinear) versions of the Laplacian, the Monge–Ampère equation, the wave equation, mean curvature, etc. It is important to emphasize that the study of any variational problem or group of similar problems can be and has been the subject of whole treatises which we can hardly cover in these pages and which the author cannot claim to master. We will therefore restrict ourselves to the more general and broad aspects of the CV in connection with DE, as the title of this contribution pretends to convey. As such we have not tried to be exhaustive in any sense and the reader will discover many gaps. Our list of references also reﬂects this perspective of being as general as possible avoiding sources too specialized for a wide

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audience. Since most of the topics we will treat are well known, we have referred to textbooks and survey works instead of articles in specialized journals, when possible. Further bibliography can be found in the references in our ﬁnal section. The paper is organized according to some of the issues we have tried to raise in this introduction. In particular, we have tried to stress the importance of convexity, coercivity, the direct method, the nature of vector problems, second-order problems, as well as indicating how the lack of existence is typically due to lack of convexity or coercivity. Several discussions on vector variational problems and their application to nonlinear elasticity are also included. We have also tried to explain why Young measures is a convenient tool in dealing with variational principles of any kind. In particular, they let unify the treatment of convex and nonconvex variational problems. In the case of nonconvex problems, generalized VP and DE of optimality have timidly been indicated. Some remarks on computations and various examples and simulations are also described. The ﬁnal section includes some further but brief remarks on various other aspects of the CV, together with related bibliographical sources in case some readers would like to study some of these aspects. Proofs are rather sketches of proofs, so that emphasis is placed on main ideas and not on technicalities. Other results are however presented without any indication about their proofs. It may be appropriate to indicate here a few general references on the CV at various levels and covering different aspects: [2,21,23,28,32,37,49,56,62,66].

2. A little bit of history The interplay between the CV and DE is as old as the CV itself. Indeed, from the very beginning (18th century) the problems in the CV that attracted researchers were tackled and, in some cases, explicitly solved by looking at associated DE of optimality or EL equations. The issue of whether solutions found through EL were or were not the true minimizers for functionals was not really settled until the beginning of the 20th century with the works of Hilbert (see the famous speech [38]) and many others that culminated with the fundamental contribution of Tonelli [61] and the formalism of the direct method. It was tacitly assumed that typical problems in the CV would always admit optimal solutions, and hence it was legitimate to seek them by examining DE of optimality. The passage from minimizers to solutions of EL equations is valid under regularity and technical assumptions. However, the existence of minimizers (in general terms) and the passage from solutions of EL to minimizers requires as a main ingredient the convexity and ellipticity of the integrand and of the EL equation, respectively. At any rate, convexity seems to be a feature one cannot be dispensed with when dealing with variational problems from a general perspective. The initial problems in the CV were obviously scalar, one-dimensional problems like the brachistochrone, the hanging cable, the minimal surfaces of revolution, the least resistance problem, etc. (see some basic references with historical content like [2,9,16,21,32,47,62]). As we have pointed out, EL in this case is an ordinary differential equation which in some of those examples could be explicitly solved in some way. Thus optimal solutions were found. It was however assumed that those solutions were truly the sought minimizers and by that time (essentially all of 19th century) no one would apparently think about whether those solutions could really be the minimizers. Everybody thought that, after all, there

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must be minimizers for such regular, well-behaved functionals. Such minimizers ought to satisfy the corresponding EL equation. The path from minimizers to solutions of EL equations took much time and dedication until it was clearly understood and stated. It was achieved thanks to the clever minds of people like Euler, Gauss, Lagrange, Hamilton, Jacobi, Riemann, Weierstrass. It was a crucial and fundamental step. One of the ﬁrst examples where the possibility of a VP without optimal solutions was indicated, is due to Weierstrass. It is concerned with minimizing

1

xu (x)2 dx

0

under the end-point conditions u(0) = 1,

u(1) = 0.

It is elementary to show that the associated EL equation is incompatible with the conditions at end-points. In particular, there cannot be minimizers. This is again not too hard to show since the value of the inﬁmum vanishes but clearly it cannot be taken on by a single function u. This sort of examples produced a real upheaval on the foundations of the CV. It helped in pushing Hilbert to seek more solid foundations for the discipline, and all issues related to the CV motivated, to a good extent, his introduction of weak convergence and weak topologies in functional spaces. He set to himself the task of rigorously proving that there exist minimizers for the Dirichlet principle 1 ∇u(x)2 dx Minimize 2 Ω subject to u = u0

on ∂Ω.

He succeeded in doing so. In his famous speech at the turn of the 20th century, three of the twenty problems were related to the CV (see [38]). Another innocent-looking problem without minimizers is due to Bolza, also at the beginning of the 20th century. This time we try to

1

u (x)2 − 1

Minimize

2

+ u(x)2 dx

0

subject to u(0) = u(1) = 0. It is also elementary to obtain a minimizing sequence in the form of ﬁner and ﬁner saw-tooth functions taking the inﬁmum to zero. But this value cannot be achieved by a

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single function. The nature of this lack of minimizer is drastically distinct than the one in Weierstrass’ example. For scalar, multidimensional problems EL becomes a PDE. The old strategy of looking for minimizers of functionals by ﬁnding (proving the existence of ) solutions for these equations and then showing that they are the minimizers sought, was not viable any more as it required to have independent methods for showing the existence of solutions of PDE. Especially when these are nonlinear, that was a task beyond the reach of known techniques and methods. Little by little, ideas started to move towards the reverse direction: ﬁrst show that there are minimizers for functionals and then prove that those are solutions of the associated EL problem. When the direct method to show existence of minimizers independently was ready, this scheme became one of the more powerful methods of showing existence of solutions for difﬁcult, nonlinear PDE. The direct method was initiated essentially with the work of Hilbert and it culminated (always for scalar problems) in the fundamental contribution of Tonelli [61]. His famous theorem reads as follows. T HEOREM 2.1. Let f (x, u, A) be convex in A for each x ∈ [a, b] and u ∈ R, and lim

|A|→∞

f (x, u, A) = +∞ |A|

uniformly in (x, u). Then the corresponding variational problem with integrand f admits (global) minimizers. It is interesting to notice that there is no regularity assumptions on f . This is however a requirement one cannot be dispensed with when talking about EL. Vector variational problems were not systematically studied until the work of Morrey [48]. He immediately realized that these were not simply a generalization of the scalar problems but rather that intriguing and surprising facts might be hidden behind this type of problems, waiting to be appreciated and understood. Current research in this area is still struggling to reveal the wealth and complexity of vector variational problems. Morrey himself realized and proved that an apparently new convexity property was necessary and sufﬁcient for weak lower semicontinuity of vector integral functionals. He called this property quasiconvexity. A function W deﬁned on matrices Mm×N is called quasiconvex if 1 W A + ∇ϕ(x) dx W (A) |D| D for all matrices A and all test functions ϕ. D is any regular domain. This deﬁnition turns out to be domain-independent so that it is a good deﬁnition. From the very beginning this deﬁnition was hard to understand. Necessary and sufﬁcient conditions were sought. Morrey proved that all quasiconvex functions must be rank-one convex W tA + (1 − t)B tW (A) + (1 − t)W (B),

t ∈ [0, 1],

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provided the difference A − B is a matrix of rank one. After much experimentation he conjectured that rank-one convexity would not be equivalent to quasiconvexity. This conjecture has been settled by Sverak [60]. He produced a counterexample of a rank-one convex function not quasiconvex. This counterexample is only valid when m > 2. The case m = 2 is still open. Ball [6] made also a fundamental contribution introducing the class of polyconvex functions which are those integrands W (A) deﬁned on matrices that admit a representation of the form W (A) = w(A, cof A, det A), where w is a convex function (in the usual sense) of all its arguments. This class of integrands are fundamental in nonlinear elasticity. The study of systems of PDE through vector variational problems is however complex and, except for a few standard cases, has not been systematically explored. In the time going from the work of Tonelli until the contribution of Morrey, I would like to mention two important ﬁelds or schools: the Chicago school and the ﬁeld of minimal surfaces [47], and the introduction by Young [65] of Young measures or parametrized measures in the context of optimal control problems. This has turned out to be the main tool in analyzing nonconvex variational problems.

3. The Euler–Lagrange equation: From VP to EL The most favorable situation in which EL can be derived for a minimizer of VP involves all the needed regularity, both on the integrand W and on the minimizer itself, so that the formal computations written in the Introduction can in fact be justiﬁed. Under these regularity assumptions those formal calculations can be easily shown to be correct. T HEOREM 3.1. Suppose the domain Ω is bounded with regular (Lipschitz) boundary, and let W : Ω × Rm × Mm×N → R be twice differentiable in all its variables. Suppose, in addition, that a certain function u : Ω → Rm is also twice differentiable and it minimizes VP among all such twice differentiable functions respecting the appropriate boundary values. Then EL is identically veriﬁed over Ω. The proof has already been indicated in the Introduction. The reader may check that under the regularity hypotheses assumed in this theorem, all the different differentiations and integrations by parts can be justiﬁed. This a classical result in the sense that all the assumptions are speciﬁed in terms of smooth and regular functions. In fact, it is a result of a very limited applicability because it requires a priori much more regularity on the minimizer than one could anticipate. We

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know that in most cases, variational principles should be posed in much more general function spaces and minimizers are not typically expected to have such regularity. The key point in deriving EL in a more ﬂexible context is to ensure that in computing formally the derivative of g(t) = I (U + tϕ)

(3.1)

at t = 0, the resulting integral # Ω

$ ∂W ∂W x, u(x), ∇u(x) ϕ(x) + x, u(x), ∇u(x) ∇ϕ(x) dx ∂u ∂A

(3.2)

is well deﬁned. This demands, to begin with, that the integrand W be differentiable with respect to u and A. The well-posedness of these integrals will also depend on the class of variations ϕ we are willing to allow, and this in turn depends on the class of competing functions for VP. Usually, VP is set so that all functions in a certain Sobolev space complying with boundary conditions are admissible. The natural Sobolev space will also depend on the growth, or rather, on the coercivity of the integrand W with respect to A. If W (x, u, A) f (x) + c |A|p − 1 ,

p 1, c > 0, f ∈ L1 (Ω),

1,p

then I (u) is ﬁnite when u ∈ u0 + W0 (Ω). In this case u0 can also be taken in W 1,p (Ω). The exponent occurring in this lower bound will determine the Sobolev space in which we 1,p can work and allow variations. We may let ϕ ∈ W0 (Ω), and for such class of variations we would like to have that the integrals in (3.2) are well deﬁned. This task essentially involves Hölder inequality and the Sobolev embedding theorem to gain extra integrability. There are several sets of growth assumptions on the partial derivatives of W with respect to A and u to achieve that goal. We will simply describe one such general-purpose situation. A more ﬁne adjustment of exponents (in relation to space dimension) may lead to sharper results. T HEOREM 3.2. Suppose the integrand W is differentiable with respect to u and A, and ∂W f1 (x) + c 1 + |A|p−1 , (x, u, A) ∂A ∂W p ∂u (x, u, A) f2 (x) + c 1 + |A| , 1,p

where f1 ∈ Lp/(p−1) (Ω), f2 ∈ L1 (Ω), c 0, p 1. Let u ∈ u0 + W0 (Ω) be a minimizer 1,p for VP in this same class of competing functions. Then (3.2) holds for all ϕ ∈ W0 (Ω) ∩ L∞ (Ω).

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P ROOF. As pointed out before, all we need to show is that the claimed assumptions imply that the integrals appearing in (3.2) are well deﬁned. The expression in (3.2) is less than or equal to $ # ∂W ∂W |∇ϕ| dx. |ϕ| + (x, u, ∇u) (x, u, ∇u) ∂u ∂A Ω By the growth assumptions on the partial derivatives of W , we can write an upper bound p |ϕ| f2 + c 1 + |∇u| dx + |∇ϕ| f1 + c 1 + |∇u|p−1 dx. Ω

Ω

By Hölder inequality, we have that this expression is dominated by p ϕL∞ (Ω) f2 L1 (Ω) + cuW 1,p (Ω) p−1 + KϕW 1,p (Ω) 1 + f1 Lp/(p−1) (Ω) + uW 1,p (Ω) , where K is some ﬁxed constant depending on Ω. Since this last quantity is ﬁnite under our assumptions, we have proved our result. It is well known that a function u ∈ W 1,p (Ω), such that (3.2) holds true for all 1,p ϕ ∈ W0 (Ω), is called a weak solution of EL. Under the assumptions stated in our last theorem, all we can say is that a minimizer u ∈ W 1,p (Ω) for VP will be a weak solution of EL. Improving weak solutions to strong, or even classical, solutions of EL (Theorem 3.1) is a standard issue which is not directly related to variational problems [35]. It is also relevant to our discussion here the issue of the regularity of the minimizers of VP. Sometimes this regularity may be used to weaken the growth assumptions on the derivatives of the integrand W to show the differentiability of the auxiliary function g in (3.1) used in the derivation of EL. Once EL is established, typical techniques for DE to show further regularity of minimizers can be pursued. The regularity issue is also relevant to discard the occurrence of the Lavrentiev phenomenon (see [14,15]). Since regularity is a rather technical and involved ﬁeld, especially for vector problems, we simply refer the reader to [33]. We have already established with a bit of rigor the relationship between VP and EL. No structural assumptions on W are needed to show that minimizers for VP are (weak) solutions of EL. However, this connection is still not so helpful as we would need to have independent (direct) methods to ﬁnd (show the existence of ) solutions for VP. The fact that minimizers ought to be solutions of EL can be useful to discard the existence of minimizers in some cases when one can show that EL, together with boundary conditions, does not admit solutions. The following are two classical such examples. W EIERSTRASS ’ EXAMPLE . Let us go back to Weierstrass’ example where we would like to minimize 1 1 2 xu (x) dx 2 0

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among all functions respecting u(0) = 1,

u(1) = 0.

If we look at EL, it is elementary to arrive at d xu (x) = 0. dx This equation is easily integrated to ﬁnd u(x) = c log x + d, where c, d are arbitrary constants. Notice that there is no function among those which can possibly comply with u(0) = 1. Therefore there can be no global minimizer. Indeed, this can be shown in a complete elementary way by considering the sequence of functions 1, x ∈ (0, 1/j ), uj (x) = − log x/ log j, x ∈ (1/j, 1). It is easy to check that uj are admissible and that the value of the functional goes down to zero as j tends to ∞. The inﬁmum is thus zero. But it is also clear that this inﬁmum cannot be reached by a single function. E XAMPLE IN [16]. This time we try to minimize

1

u(x)2 + u (x)2 dx

0

under the end-point conditions u(0) = 0, u(1) = 1. When integrands do not depend explicitly on x, there is a more convenient form of EL which is $ # ∂W d W u(x), u (x) − u (x) u(x), u (x) = 0. dx ∂A It is a Calculus exercise to check this claim. This leads to ∂W W u(x), u (x) − u (x) u(x), u (x) = constant . ∂A In our situation, and after some algebra, we obtain u2 = c2 u2 + (u )2 . Further computations and typical decompositions to compute some primitives lead to u − c 1 1 − c 1 1 , log + = x + log 2 u + c u 2 1 +c

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where we have already taken into account u(1) = 1. This formula is incompatible with the boundary condition at 0. As before, we conclude that there can be no minimizer for this problem. Indeed, for any admissible function u we have

1

u(x)2

+ u (x)2 dx

0

1

u (x) dx = 1.

0

Equality is impossible for a single function, and yet the sequence " uj (x) =

0, 0 x 1 − j1 , j x − 1 + j1 , 1 − j1 x 1,

is minimizing since the values of the functional decrease to 1. At the beginning of the CV, ﬁnding minimizers for VP was the central issue and EL was used to ﬁnd such minimizers. Yet no attention was explicitly paid to the fact that main structural assumptions on the integrand W were required to make this move on ﬁrm ground.

4. Convexity: From EL to VP We would like to understand under what conditions solutions of EL are indeed minimizers for VP. Recalling the comparison with the ﬁnite-dimensional situation, we know that there might be other solutions to the optimality equations expressed in EL which are not true global minimizers for VP. In particular local minima, saddle points, or even maxima can be solutions of the equations of optimality. What is the assumption on W ensuring that the passage from solutions of EL to minimizers of VP is legitimate? There is one single word which, probably in different versions, will be by our side from now on: Convexity. T HEOREM 4.1. Suppose U ∈ W 1,p (Ω) is a weak solution of EL, i.e., (3.2) is true for all 1,p 1,p ϕ ∈ W0 (Ω), and U − u0 ∈ W0 (Ω) for some ﬁxed u0 ∈ W 1,p (Ω). If W (x, u, A) is a Carathéodory function, smooth and convex in (u, A) for ﬁxed x ∈ Ω, then U is a minimizer 1,p for VP over u0 + W0 (Ω). 1,p

P ROOF. The proof is in fact simple. Let ϕ ∈ W0 (Ω) be arbitrary and let us compare I (U ) with I (U + ϕ). Let us examine the difference I (U + ϕ) − I (U ) W x, U (x) + ϕ(x), ∇U (x) + ∇ϕ(x) − W x, U (x), ∇U (x) dx. = Ω

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By the convexity assumed on W and bearing in mind that EL requires the differentiability of W with respect to (u, A), we can directly write I (U + ϕ) − I (U ) $ # ∂W ∂W x, U (x), ∇U (x) ϕ(x) − x, U (x), ∇U (x) ∇ϕ(x) dx. ∂A Ω ∂u But this last integral vanishes precisely because u is a weak solution of EL. Indeed, this last integral is exactly (3.2). This implies that u is truly a global minimizer of VP. A relevant remark is the following. If we go back to the sections g(t) in (3.1), and suppose we have sufﬁcient regularity so that these functions are differentiable (as in Theorem 3.2), then a weak solution of EL will translate into the fact that t = 0 is a critical point 1,p for all these g(t) for arbitrary ϕ ∈ W0 (Ω). Assume in addition that all these sections g(t) are convex functions of t. Then it is clear that U will be a global minimizer for VP. From this perspective, it is the convexity of the functional itself which we need rather than the convexity of the integrand. It is easy to check that if W (x, u, A) is convex in (u, A) for a.e. x ∈ Ω then the functional I will be convex. However, the converse is not necessarily true when the dimension m (the number of components for competing functions for VP) is greater than unity, i.e., when EL is indeed a system of PDE instead of a single equation. In other words, there are nonconvex integrands deﬁned on matrices (m, n > 1) so that the corresponding functional I for VP is convex. This is our ﬁrst indication that scalar variational problems (m = 1) and vector variational problems (m > 1) are different. Equivalently, we may say that PDE are qualitatively different in some aspects than systems of PDE, and these are much more complex. We will devote a whole section (Section 7) to examine vector variational problems and to discover some of the surprises they reserve for us. On the other hand, it is true that for the scalar case only convex integrands give rise to convex functionals if we do not allow explicit dependence of W on u. The explicit dependence on u lets one build some interesting counterexamples. The proof of this fact is essentially technical and consists in appropriately localizing the convexity of I [23]. A typical example where a solution of EL can be shown to be a minimizer for VP is the brachistochrone. We are interested in determining the optimal proﬁle of a plane, vertical curve joining two given points at different heights, in such a way that a unit mass employs the least time possible in reaching the lowest point under the action of gravity without friction. After a convenient choice of axes, and putting the X-axis vertically in the direction of gravity, we seek to

a

Minimize 0

1 + u (x)2 dx √ x

subject to u(0) = 0, u(a) = A, where a, A > 0. In this formulation we have used several

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normalizations. Let us examine the associated EL equation. In this case we must solve u 1 = , √ x 1 + (u )2 c

(u )2 x = 2. 2 1 + (u ) c

This leads to x u (x) = 2 , c −x

2

x

u(x) =

0

c2

s ds, −s

where the constant c is to be determined in such a way that a

A=

c2

0

s ds. −s

In order to ﬁnd a more explicit form of the solution, we will use the change of variables in the integral for u given by s(r) =

c2 r (1 − cos r) = c2 sin2 . 2 2

Then u(t) = c

t

2 0

c2 r sin dr = (t − sin t), 2 2 2

where x(t) =

c2 t (1 − cos t) = c2 sin2 . 2 2

In parametric form,

x(t), u(t) = C(1 − cos t), C(t − sin t) ,

0 t t0 ,

is the solution. It already veriﬁes x(0) = u(0) = 0. The constants C and t0 must be found by imposing x(t0 ) = a, u(t0 ) = A. This curve is an arc of a cycloid. Because the integrand for the brachistochrone is a strictly convex function of A, Theorem 4.1 applies and we can conclude that this arc of cycloid is truly the optimal proﬁle. Sometimes, adjusting boundary conditions depends on their relative sizes, in the sense that when they run in a certain range it is possible to ﬁnd optimal proﬁles, but if they do not belong to this range, it is not possible to ﬁnd optimal curves complying with such boundary data. This is the situation for the classical problem of the minimal surfaces of revolution. There is a whole very interesting discussion about the relative sizes of the values at end-points so that boundary conditions can be met. In some cases, it is not possible

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to ﬁnd solutions complying with boundary conditions just as in the examples in the previous sections. In such cases, there are no optimal solutions [52]. It is interesting to stress that despite of having the appropriate convexity for all these one-dimensional examples, optimal solutions only exist when EL, together with boundary conditions, is solvable.

5. Convexity: The direct method We have emphasized that the path to show existence of solutions for VP through EL or for EL through VP requires starting out with a solution of one of the two. When the spatial dimension is N = 1 then EL is a set of ODE and it might be possible to ﬁnd solutions independently. Under convexity assumptions, these will be minimizers for VP. When N > 1, we need to solve a set of PDE. Even in the scalar case, when EL is a single equation, it may not be so easy to ﬁnd solutions of EL independently, and even if we succeeded in doing so, we would be forced to ask for convexity to ensure the existence of minimizers. At the end, it turns out that under this convexity (and without regularity assumptions) the direct method provides minimizers for VP in a simple, elegant and general way. It is for this reason that the variational approach to existence of nonlinear PDE has been one of the main tools in the last decades. The direct method can be treated in a rather abstract way. It is not related to the integral nature of VP. It can also be motivated by examining ﬁrst the ﬁnite-dimensional situation and try to translate it to inﬁnite dimensions. In order to appreciate the simplicity and elegance of the direct method, let us turn, as before, to the ﬁnite-dimensional situation. Let I : Rn → R∗ . We would like to ﬁnd x0 ∈ Rn such that I (x0 ) I (x) for all x ∈ Rn . The ﬁrst condition we need to ensure is that I be bounded from below, I (x) c > −∞, for all x ∈ Rn . Otherwise, there is nothing we can do about the analysis of the minimization problem: there can exist no minimizer. Put −∞ < m = inf I (x): x ∈ Rn , and let {xj } be a minimizing sequence: I (xj ) m. If {xj } is relatively compact in Rn (this is the case if lim infx→∞ I (x) > m) and I is continuous, for some appropriate subsequence, not relabeled, xj → x0 and I (xj ) → m. Therefore I (x0 ) = m and x0 is a minimizer. In fact, since we are interested in minimizers, it sufﬁces to demand the lower semicontinuity of I , I (x) lim inf I (xj ), j →∞

whenever xj → x. The direct method consists in imitating the ﬁnite-dimensional case in the inﬁnitedimensional situation. The different important ingredients are: (1) I is not identically +∞; (2) I is bounded from below; (3) compactness in the topology on the set of competing functions; (4) lower semicontinuity of I with respect to the chosen topology.

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The function spaces of competing functions usually are Banach spaces with integral norms Lp (Ω), W 1,p (Ω), and the appropriate topologies with good compactness properties are the weak topologies over these spaces. In particular, if X is one of these spaces and is reﬂexive, it is well known that uj X M < ∞

implies uj $ u,

u ∈ X,

possibly for a subsequence (Banach–Alaouglu–Bourbaki theorem). This property is extremely convenient and explains, from our perspective, why weak convergence is so important. The most difﬁcult step in applying the direct method is to enforce the sequential lower semicontinuity property with respect to weak topologies: uj $ u in X

implies I (u) lim inf I (uj ). j →∞

We can summarize the previous considerations in the following abstract theorem, the proof of which has been already indicated. T HEOREM 5.1. Let us consider the variational principle inf{I (u): u ∈ A}, where (i) A is a closed, convex subset of a reﬂexive Banach space X; (ii) I is coercive: I (u) CuX , C > 0, or limu→∞ I (u) = +∞; (iii) I is sequentially lower semicontinuous with respect to the weak topology in X; (iv) there exists u¯ ∈ A such that I (u) ¯ < ∞. Then there exists u0 ∈ A with I (u0 ) I (u) for all u ∈ A. In our context, the functional I is the one for VP. If we assume that c |A|p − 1 W (x, u, A),

c > 0, p > 1,

then minimizing sequences {uj } will converge weakly to some u in W 1,p (Ω) (remember p > 1 and uj = u0 on ∂Ω). According to the direct method, we must bother with the property of (sequential) weak lower semicontinuity of the functional I : uj $ u in W 1,p (Ω) implies I (u) lim inf I (uj ). j →∞

This property is inherited through the convexity of the functional I (see [23]). T HEOREM 5.2. Let Y be a linear submanifold of X, and I :Y ⊂ X → R

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be a convex, coercive functional, bounded over bounded sets of Y . Then I is weak lower semicontinuous. P ROOF. Let uj $ u in Y . By Mazur’s lemma [58], certain convex combinations of the uj ’s converge strongly to the same u: vj =

(j )

λk uk ,

vj → u.

k

Let tj = 1 − u − vj → 1,

wj = vj +

1 (u − vj ). 1 − tj

Notice how we need the restriction that Y be a linear submanifold so that wj ∈ Y . For j sufﬁciently large, wj u + 1 so that {I (wj )} is a bounded set of numbers because of the boundedness property of I . By the convexity of I , I (u) tj I (vj ) + (1 − tj )I (wj ) tj

(j )

λk I (uk ) + (1 − tj )I (wj ),

k

and taking limits in j , we obtain I (u) lim inf I (uj ).

j →∞

As a result of these two theorems, we have a well-established existence theorem for convex, coercive, bounded functionals deﬁned over linear submanifolds. T HEOREM 5.3. Let the functional I be coercive, convex, bounded and nontrivial over a weakly closed subset, linear submanifold Y = A of a reﬂexive Banach space. Then I admits global minimizers. The application of this result to our integral functionals in VP is immediate. T HEOREM 5.4. Let I be deﬁned as in VP. Suppose the integrand W is convex in (u, A) for ﬁxed x and c |A|p − 1 W (x, u, A) C |A|p + 1 ,

p > 1, C c > 0.

Then VP admits global minimizers. If in addition, W is sufﬁciently smooth and further technical assumptions (in the spirit of Theorem 3.2) hold, then there are (weak) solutions for the associated EL problem.

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There are two main improvements on this last result which can be proved by taking into account the integral nature of the functional I in VP and the relationship between the variables (u, A) as they are not unrelated since A = ∇u: (1) we only need the convexity of W with respect to A and not the joint convexity on the pairs (u, A); (2) the upper bound on W can be dropped altogether so that only coercivity is essential. One of the most direct ways of showing these results is by using Young measures, a very convenient tool when dealing with integral functionals in the CV.

6. Young measures Our main motivation to study here Young measures is to understand and relate the limiting behavior of the integrals

W x, uj (x), ∇uj (x) dx Ω

and the value of the integral

W x, u(x), ∇u(x) dx Ω

whenever uj $ u in some appropriate Sobolev space. Weak lower semicontinuity for the corresponding functional I amounts to showing

W x, uj (x), ∇uj (x) dx

lim inf j →∞

Ω

W x, u(x), ∇u(x) dx Ω

if uj $ u. Let us assume, for simplicity, that the integrand W depends only upon the gradient variable W = W (A) = W (∇u), so that we are interested in knowing when

W ∇u(x) dx

W ∇uj (x) dx

lim inf j →∞

Ω

(6.1)

Ω

if ∇uj $ ∇u. To see more clearly the issue, let us simplify the situation still further. Suppose that ∇u = A is constant throughout Ω. The weak convergence ∇uj $ A implies that 1 ∇uj (y) dy → A, |E| E for any measurable subset E ⊂ Ω. If W is continuous, we will have

1 W |E|

∇uj (y) dy → W (A). E

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In particular, when E = Ω,

1 W |Ω|

∇uj (y) dy → W (A). Ω

If we compare this with (6.1) we realize that we would conclude the weak lower semicontinuity if we could show, for instance, that

1 W ∇uj (x) dx W ∇uj (y) dy dx |Ω| Ω Ω Ω 1 ∇u(y) dy → |Ω|W |Ω| Ω

= |Ω|W (A) for all j , i.e., if for any appropriate arbitrary ﬁeld v we have

1 |Ω|

1 W ∇v(x) dx W |Ω| Ω

∇v(x) dx .

(6.2)

Ω

Thus the property of weak lower semicontinuity has been reduced to understanding the above inequality for the integrand W . The question is: what are the continuous integrands W deﬁned on matrices that respect the above inequality when commuting with integration? This deﬁnitely reminds us of the classical Jensen’s inequality [57]. T HEOREM 6.1. Let μ be a positive measure over a σ -algebra in a set Ω such that μ(Ω) = 1. Let f be a vector-valued function in L1 (μ) such that f (x) ∈ K for μ-a.e. x ∈ Ω where K ⊂ Rm is a convex set. If ϕ is a convex function deﬁned in K then

f dμ

ϕ Ω

ϕ(f ) dμ. Ω

We conclude that convexity will be a main ingredient in weak lower semicontinuity results. A very convenient way of making precise all of this informal discussion is by using Young measures. Some general textbooks on Young measures are [5,51,53,63]. We follow here the discussion in [55]. A Young measure is a family of probability measures ν = {νx }x∈Ω associated with a sequence of functions fj : Ω ⊂ RN → Rm such that supp(νx ) ⊂ Rm and they depend measurably on x ∈ Ω, which means that for any continuous ϕ : Rm → R the function of x, ϕ(x) ¯ =

Rm

ϕ(λ) dνx (λ) = ϕ, νx !,

(6.3)

is measurable. The fundamental property of this family of probability measures is that they can be used to represent weak limits of nonlinear quantities in the following sense:

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if {ϕ(fj )} converges weakly in L∞ (Ω) (or more generally, weakly in some Lp (Ω)), the weak limit can be identiﬁed with the function ϕ¯ in (6.3), lim ϕ(fj )h(x) dx = h(x) ϕ(λ) dνx (λ) dx, (6.4) j →∞ Ω

Rm

Ω

for all h ∈ L1 (Ω). Heuristically, the Young measure yields the limiting probability distribution of the values of {fj } when points are taken randomly around each ﬁxed x ∈ Ω. If BR (x) denotes the ball of radius R > 0 centered at x ∈ Ω, and E ⊂ Rm is any measurable set, then νx (E) = lim lim

R→0 j →∞

|{y ∈ BR (x): fj (y) ∈ E}| , |BR (x)|

where bars | · | denote the Lebesgue measure. This identiﬁcation clearly shows that the sequence {fj } is forced to oscillate near x among the different vectors in the support of νx with relative frequency given by the weights corresponding to such vectors. The formal result establishing that, under very mild growth conditions, we can always associate a Young measure with a given sequence of functions follows. T HEOREM 6.2 ([8]). Let Ω ⊂ RN be a measurable set and let zj : Ω → Rm be measurable functions such that sup g |zj | dx < ∞, j

Ω

where g : [0, ∞) → [0, ∞] is a continuous, nondecreasing function such that limt →∞ g(t) = ∞. There exists a subsequence, not relabeled, and a family of probability measures, ν = {νx }x∈Ω (the associated Young measure) depending measurably on x with the property that whenever the sequence {ψ(x, zj (x))} is weakly convergent in L1 (Ω) for any Carathéodory function ψ(x, λ) : Ω × Rm → R∗ , the weak limit is the function ¯ ψ(x, λ) dνx (λ). ψ(x) = Rm

If we go back to our discussion of variational principles, assume now that our sequence {W (∇uj )} is weakly convergent in L1 (Ω), where {uj } is minimizing for the functional I , so that lim W ∇uj (x) dx = W (A) dνx (A) dx. (6.5) j →∞ Ω

Ω

M

Here ν = {νx }x∈Ω is the Young measure associated with the sequence {zj = ∇uj } which is bounded in Lp (Ω) because of the coerciveness hypothesis assumed on W . Since ∇u(x) = A dνx (A) M

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must be the weak limit of {∇uj }, the weak lower semicontinuity property will hold true if Ω

M

W (A) dνx (A) dx

W Ω

M

A dνx (A) dx.

(6.6)

But this is Jensen’s inequality. In order to have a full result, we still have to deal with two issues: the weak convergence of {W (∇uj )} may not be valid in general and the full dependence of W on x and, more importantly, on u. There are two lemmas tailored to solve these difﬁculties. L EMMA 6.3 ([53]). If {zj } is a sequence of measurable functions with associated Young measure ν = {νx }x∈Ω , then ψ(x, λ) dνx (λ) dx lim inf ψ x, zj (x) dx j →∞

E

E Rm

for every Carathéodory function ψ, bounded from below, and every measurable subset E ⊂ Ω. This lemma asserts that even if the weak convergence of the compositions {W (∇uj )} does not hold, yet we always have an inequality which goes in the right direction for weak lower semicontinuity. L EMMA 6.4 ([53]). Let zj = (uj , vj ) : Ω → Rd × Rm be a bounded sequence in Lp (Ω) such that {uj } converges strongly to u in Lp (Ω). If ν = {νx }x∈Ω is the Young measure associated to {zj } then νx = δu(x) ⊗ μx for a.e. x ∈ Ω, where {μx }x∈Ω is the Young measure corresponding to {vj }. This lemma is applied to sequences {uj , ∇uj } for {uj } a weakly convergent sequence in W 1,p (Ω). In this situation we know that the functions themselves converge strongly to the weak limit by the compactness theorem of Sobolev spaces. If u ∈ W 1,p (Ω) is the weak limit and {μx }x∈Ω is the Young measure associated with the gradients {∇uj }, then νx = δu(x) ⊗ μx for a.e. x ∈ Ω. These two lemmas are somewhat technical though important. We do not include their proofs here. By using these two lemmas, and in the context of our previous discussion, we can summarize our main result as follows. T HEOREM 6.5. Let W : Ω × Rm × Mm×N → R∗ be a Carathéodory integrand (continuous in the variables (u, A) and measurable in x) which is convex in A for all ﬁxed pairs (x, u). Suppose further that W (x, u, A) c |A|p − 1 ,

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for some positive constant c and p > 1. Then the associated functional I is weak lower semicontinuous in W 1,p (Ω). In addition, if u0 ∈ W 1,p (Ω) is such that I (u0 ) < +∞ then the variational problem W x, u(x), ∇u(x) dx Minimize I (u) = Ω 1,p

subject to u − u0 ∈ W0 (Ω) admits at least one global minimizer in W 1,p (Ω). The existence part is a straightforward consequence of the direct method. A main corollary arises when we put together this existence result with Theorem 3.2. T HEOREM 6.6. Suppose W as above veriﬁes the following requirements: 1. Coercivity: W (x, u, A) c |A|p − 1 for some positive constant c and p > 1. 2. Convexity: W is convex in A for all ﬁxed pairs (x, u). 3. Regularity: W is differentiable with respect to u and A and ∂W p−1 , ∂A (x, u, A) f1 (x) + c 1 + |A| ∂W f2 (x) + c 1 + |A|p , (x, u, A) ∂u where f1 ∈ Lp/(p−1) (Ω), f2 ∈ L1 (Ω), c > 0, p 1. Then the associated EL admits at least one weak solution. For the scalar case, typical applications of this result include Laplace’s equation and the p-Laplacian case for p > 1, with all kinds of variants. Other nonlinear, intimidating examples can be written down for which this result directly applies. This is, for instance, the case for the integrands W=

1 + |A|p ,

W = |A|p + |A|p/2

for p > 2. The example of minimal surfaces is much more delicate precisely because the coercivity exponent degenerates to unity. The vector case will be treated later. The issue of uniqueness of solutions of EL or of minimizers of VP is very important, too. Often this is linked to some sort of strict convexity (or strict ellipticity). It is a standard exercise to prove the following. T HEOREM 6.7. Assume that, in addition to the hypotheses of Theorem 6.6, W is strictly convex in the pairs (u, A) for each ﬁxed x ∈ Ω. Then the global minimizer for VP and the weak solution for EL are unique.

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Notice how the strict convexity is assumed jointly in the pairs (u, A).

7. Scalar problems under pointwise constraints These problems are characterized by the fact that the corresponding integrand is only considered when an additional density is, for instance, less than zero. In general, we would have: W x, u(x), ∇u(x) dx Minimize I (u) = Ω

under u = u0

on ∂Ω

and, in addition, w x, u(x), ∇u(x) 0, for a new, known function w which typically has the same continuity properties as W . This type of variational problems incorporating these sorts of pointwise constraints can be recast into the typical variational form with a single integrand by simply allowing such integrands to take on the value +∞. Indeed, if we deﬁne W |w (x, u, A) =

W (x, u, A), if w(x, u, A) 0, +∞, else,

then the former problem is equivalent to W |w x, u(x), ∇u(x) dx Minimize I (u) = Ω

under the same boundary conditions over ∂Ω. Many examples and situations can be examined depending on the form and structure of both W and w. The best known example is, by far, the obstacle problem, and we will restrict our attention here to this case in which we take 1 W (x, u, A) = |A|2 , 2

w(x, u, A) = ψ(x) − u,

ψ(x) being a given function (the obstacle). The physical interpretation when we try to minimize the integral of W |w for this choice is clear: by minimizing the functional we seek the equilibrium conﬁguration of a membrane that is supposed to stay above the obstacle represented by the graph of ψ. The existence of a minimizer for the corresponding variational problem with integrand W |w is easy to establish provided that the set of competing functions is not empty. This

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only requires certain compatibility between the obstacle and the boundary datum in the sense that we must have ψ u0 over ∂Ω. When this condition holds, the direct method, just as has been indicated before, directly applies to this situation and there is no explicit modiﬁcation because of the fact that the integrand takes on, even abruptly, the value +∞. This issue is again much more delicate for vector problems. The reader is urged to check that convexity and coercivity hold and that there is no special difﬁculty in having integrands taking on the value +∞. Indeed, one can allow this possibility from the very beginning. Moreover, under strict convexity of W on (u, ∇u) the solution is unique. What is more interesting, and somehow requires a new framework, is the analysis of optimality conditions, i.e., the associated EL equation. It is clear that we will ﬁnd new ingredients here because integrands taking on inﬁnite values cannot be equally treated as their ﬁnite counterparts as far as optimality conditions are concerned. In fact, the study of optimality conditions for this sort of problems led to the birth of a new ﬁeld called Variational Inequalities, closely connected with Free Boundary Problems [31,43]. We can hardly explore all of this here. We will be contented with examining with a bit of care the typical obstacle problem. For a nontechnical discussion of this and many more topics, see [17]. There are, among others, two direct ways of examining optimality for the obstacle problem (and in general for all variational inequalities of this kind). One is focused on tailoring “variations” complying with the obstacle restriction. Indeed, if u is the sought minimizer and v is any other admissible function then, for t ∈ [0, 1], it is true that the convex combination (1 − t)u(x) + tv(x) as a function of x is admissible, too. Then we consider the function of t, 1 (1 − t)∇u(x) + t∇v(x)2 dx. g(t) = I (1 − t)u + tv = 2 Ω This time, because t is only allowed to move to the right of 0, t = 0 is a one-sided minimum point and, therefore, all we can say is g (0) 0. This information translates into 0

∇u(∇v − ∇u) dx Ω

whenever v ψ complies with the boundary datum. This inequality characterizes the minimizer u in a unique way. But to more clearly see the role played by the obstacle, let us assume that u is more regular so that we can apply the divergence theorem in the inequality above. Then u(v − u) dx. 0 Ω

If we let ϕ = v − u, we see that ϕ is permitted to be nonnegative, compactly-supported in Ω. This means that u 0 all over Ω. If in addition the obstacle ψ is continuous, and

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u > ψ in an open, compactly-supported subdomain then ϕ can also take on negative values in an arbitrary fashion as long as the parameter t is sufﬁciently small and the support of ϕ is contained in that subdomain. In this situation g (0) = 0 and we get u = 0. Altogether, we can summarize these observations by saying that the minimizer for the obstacle problem is determined by the conditions u(u − ψ) = 0

in Ω,

u 0,

in Ω.

u−ψ 0

The boundary of the set where u = ψ is the unknown, free boundary of the problem and it certainly is the crucial element to be determined since once this is known the solution is easily found. Another equivalent approach consists of introducing a multiplier p(x) associated with the obstacle restriction. In this way we have the augmented functional $ # 1 2 |∇u| + p(ψ − u) dx. I (u, p) = Ω 2 The optimal solution of the problem will be determined by the EL equation of this functional with respect to u together with the well-known conditions under pointwise restrictions in the form of inequalities u + p = 0 p(ψ − u) = 0,

in Ω, p 0,

ψ −u0

in Ω.

By eliminating the multiplier p we arrive again at the previous optimality conditions. 8. Vector problems and systems of PDE So far, we have not made the distinction between scalar and vector problems, i.e., between a PDE and a system of these although we have announced the striking differences between the two. In fact, most of the stated results can actually be applied to both situations so that under convexity and coerciveness of integrands with respect to the gradient variable we can achieve existence of minimizers for VP, and under further regularity properties such minimizers will be weak solutions of EL. Indeed, Theorem 6.6 can also be applied to a vector situation because under convexity assumptions of the integrand the same statement is valid in both situations. Hence a direct application of this result to integrands of the type 1 W = |A|2 + w(u) 2 for certain nonlinear functions w yields existence of weak solutions for diffusion systems of the type u = ∇w(u)

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under Dirichlet boundary conditions. More complex situations can also be tackled from this perspective. We have however remarked in Section 4 that, as a matter of fact, it is the convexity of the functional I itself rather than the convexity of the integrand what matters in going from solutions of EL to minimizers for VP, and have insisted that vector problems (m > 1) reserve some surprises for us. The analysis of systems of PDE has recently been undertaken by means of the typical tools from the CV [44]. Let us place ourselves in the simpler, genuine vector situation where competing functions for VP have two independent variables (N = 2) and two components (m = 2), so that u : Ω ⊂ R2 → R2 . EL will be a system of two (coupled) PDE in two variables (x1 , x2 ). The functional I (u) will look like W x, u(x), ∇u(x) dx, I (u) = Ω

where ∇u(x) ∈ M2×2 and W : Ω × R2 × M2×2 → R satisﬁes appropriate smoothness requirements. Imagine a situation where I (u), written as an integral over Ω, could be recast as an integral over ∂Ω by applying the divergence theorem. The structure of the integrand W must be such that the divergence theorem can be applied. Therefore, W x, u(x), ∇u(x) = div F x, u(x), ∇u(x) for some vector ﬁeld F : Ω × R2 × M2×2 → R2 . If we explicitly write, by the chain rule, that divergence in terms of partial derivatives of F , it is an elementary calculus exercise to get ∂Fi ∂Fi ∂Fi W x, u(x), ∇u(x) = + uj,i + uj,ki , ∂xi ∂uj ∂Aj k i

i,j

i,j,k

where all these terms are evaluated at (x, u(x), ∇u(x)). But since W (x, u(x), ∇u(x)) does not depend explicitly on second derivatives, the triple sum involving second derivatives should drop. Bearing in mind the equality of mixed partial derivatives, this condition leads

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to ∂F1 ∂F1 ∂F2 ∂F2 = = = = 0, ∂A11 ∂A21 ∂A12 ∂A22 ∂F1 ∂F2 ∂F1 ∂F2 + = + = 0. ∂A12 ∂A11 ∂A22 ∂A21 The ﬁrst set of equalities imply that F1 only depends on A12 and A22 , while F2 only on A11 and A21 . But the second set of equations links two functions with different independent variables. The only possibility is to have ∂F2 ∂F1 =− = constant (with respect to A) ∂A12 ∂A11 and the same for ∂F1 ∂F2 =− . ∂A22 ∂A21 Consequently, F1 = cA12 + dA22 + e, F2 = −cA11 − dA21 + f,

(8.1)

where c, d, e, f can depend on (x, u) but not on A. Once we have this information, we go back to ﬁnish the calculation of the divergence above to obtain, after a few careful computations, that W (x, u, A) = α(x, u) det A + β(x, u, A),

(8.2)

where β needs to be linear in A for ﬁxed (x, u). Our conclusion is that if W is of this form, essentially for arbitrary choices of α and β under the linearity restriction on β, then the functional can be rewritten as I (u) = F x, u(x), ∇u(x) · n(x) dS(x), ∂Ω

where n(x) is the unit, outer normal. If we recall the form of F in (8.1), we arrive at c x, u(x) ∇u1 (x) · T n(x) + d x, u(x) ∇u2 (x) · T n(x) I (u) = ∂Ω

+ e x, u(x) , f x, u(x) · n(x) dS(x) # du2 du1 + d x, u(x) c x, u(x) = dτ dτ ∂Ω $ + e x, u(x) , f x, u(x) · n(x) dS(x).

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Here, T is the counterclockwise (π/2)-rotation in the plane and τ is the corresponding unit tangent normal to ∂Ω. We immediately realize that this last integral is constant under a Dirichlet-type boundary condition for VP. In particular, when W is of the form given in (8.2) then the functional I for VP (under a Dirichlet-type boundary condition) is convex, in fact constant, despite the fact that the corresponding integrand is NOT convex! It is elementary to check that the determinant is not a convex function on matrices. The integrands of the form given in (8.2) are called null-Lagrangians because the functionals with such integrands only depend on the boundary values of competing functions. This important property of null-Lagrangians is reﬂected in the somewhat surprising fact that EL is identically satisﬁed for all functions in the appropriate space. For instance, it is easy to convince oneself that div adj(∇u) ≡ 0, where adj A =

∂ det A . ∂A

This is indeed a consequence of the constancy property of null-Lagrangians, because the functions g(t) in (3.1) are constant depending on the boundary values. Hence if M is such a null-Lagrangian, as long as u respects such boundary values and for arbitrary test functions ϕ, we will have

M ∇u(x) · ∇ϕ(x) dx = 0. Ω

Thus, ∂M div ∇u(x) = 0 ∂A in a weak sense for arbitrary u. In this way adding a null-Lagrangian to the integrand of a functional does not change the associated EL. We have checked above that in dimension two, (8.2) provides all null-Lagrangians. In dimension three, all null-Lagrangians are linear functions (with respect to A) of the different minors (of order one, two and three) of a 3 × 3 matrix. And so on and so forth in higher dimension. Null-Lagrangians are always linear functions of the different minors of a matrix. There is a main consequence of our previous discussion which leads to a new, important family of integrands. We will restrict our discussion here to the two-dimensional situation. Let us concentrate on the determinant so that we take α ≡ 1 and β ≡ 0 in (8.2.) The relationship between α and c and d in (8.1) is given by α=

∂c ∂d − . ∂u1 ∂u2

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Hence we can take d = u1 and c = 0, for instance. The corresponding vector ﬁeld F in (8.1) is F = (u1 u2,2 , −u1 u2,1 ), and det ∇u = div F u(x), ∇u(x) .

(8.3)

We shall use this fundamental identity in what follows. Suppose we have u(j ) $ u in W 1,p (Ω) for p > 2. Then det ∇u(j ) $ h in Lp/2 (Ω) for some h ∈ Lp/2 (Ω). Let ϕ be a test function in Ω. By applying (8.3), we can write det ∇u(j ) (x)ϕ(x) dx = div F u(j ) (x), ∇u(j ) (x) ϕ(x) dx Ω

Ω

= −

F u(j ) (x), ∇u(j ) (x) ∇ϕ(x) dx

Ω

→−

Ω

div F u(x), ∇u(x) ϕ(x) dx

=

F u(x), ∇u(x) ∇ϕ(x) dx

Ω

det ∇u(x)ϕ(x) dx.

= Ω

We have used the fact that u(j ) → u in Lp (Ω) and that F is linear on ∇u. The arbitrariness of ϕ implies that det ∇u(j ) $ det ∇u in the sense of distributions, and hence, we conclude that h = det ∇u. This is a most remarkable fact: u(j ) $ u implies that det ∇u(j ) $ det ∇u even if det is a nonlinear function. Again null-Lagrangians are the only nonlinear functions enjoying this weak continuity property. The weak continuity property of null-Lagrangians has important consequences. From the point of view of vector variational principles, the most important one is concerned with the class of polyconvex integrands. Recall that we insisted that convex functions enjoy the fundamental property that when commuting with integration the result is always bigger (Jensen’s inequality). This directly translates into the fact that convex integrands give rise to weak lower semicontinuous functionals. Based on the weak continuity of the minors, we can argue as follows. Suppose we have a convex function r : Rd → R

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where the dimension d is exactly the number of different minors for m × N matrices. Precisely, d=

min(m,N) i=1

m i

N i

.

If M(A) stands for the vector of all these minors in some preassigned order, we have just shown that uj $ u in some appropriate Sobolev space implies M(∇uj ) $ M(∇u). But then r M ∇u(x) dx lim inf r M ∇uj (x) dx, j →∞

Ω

Ω

a weak lower semicontinuity result. What is really remarkable is that the compositions r(M(A)) need not be convex. It is very easy to see, for example, that the square of the determinant for 2 × 2 matrices is not a convex function. The functions with this structure, a composition of a convex function with minors, are called polyconvex, and they provide the most important class of integrands for vector problems for which the direct method applies. Let us stress the fact that it is a much more broad class than the usual class of convex functions. T HEOREM 8.1. Let the integrand W : Ω × Rm × Mm×N → R be a Carathéodory function satisfying the following requirements: 1. Coercivity: there exists p > min{m, N} and c > 0 such that c |A|p − 1 W (x, u, A) for all (x, u) ∈ Ω × Rm . 2. Polyconvexity: for each ﬁxed pair (x, u) ∈ Ω × Rm the function of A, W (x, u, ·) is polyconvex. Then the corresponding variational problem VP admits at least one global minimizer in 1,p u0 + W0 (Ω) for arbitrary u0 ∈ W 1,p (Ω). This result is a typical example of existence theorem in nonlinear elasticity for hyperelastic materials with polyconvex energy densities. A main example where this theorem can be applied is the family of Ogden materials, the energy densities of which are of the form W (F ) =

r i=1

s α /2 β /2 ai tr F T F i + bj tr adj F T F j + r(det F ), j =1

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where r, s are positive integers, ai > 0, bj > 0, αi 1, βj 1 and r is a convex function. Showing the polyconvexity and the coercivity of these integrands is a very instructive exercise [19]. Nonlinear elasticity is also a good way to show the complexity of vector variational problems and their associated systems of PDE. Indeed, even in the case where the existence of global minimizers for energy functionals has been proved, yet the passage to weak solutions of the associated EL system still requires to overcome many difﬁculties. We can certainly apply a result like Theorem 6.6, but this requires regularity on the energy density and their partial derivatives, which is unrealistic in nonlinear elasticity. Indeed, a real energy density must comply with the condition W (A) = +∞ when det A 0, and this condition is incompatible with the typical growth assumptions on integrands and their derivatives. Notice however that it is compatible with the structure of energy densities of Ogden materials if r(t) is a convex function taking on the value +∞ for negative t. Another possibility to show that minimizers are weak solutions relies on the regularity of minimizers. But this again is a very delicate issue for vector problems (see comments and references in Section 9). Instead of dwelling here on arbitrary EL associated to polyconvex integrands, we will describe one of the most important situations where systems of PDE with variational structure are better understood: linear elasticity. In the next section, we will describe vector variational problems from a more general perspective. It is well known that the typical boundary value problem in linear elasticity is that of ﬁnding the equilibrium displacement u : Ω ⊂ R3 → R3 solution of the problem − div Eε(u) = P in Ω, Eε(u) · n = ψ

on Γ1 ⊂ ∂Ω,

u = u0

on Γ0 ⊂ ∂Ω,

where (1) Ω is the reference conﬁguration; (2) E is the elasticity tensor incorporating all elastic constants of the material; (3) ε(u) is the symmetrized gradient ε(u) = (∇u + ∇uT )/2; (4) P is the bulk density load over Ω; (5) n is the unit, outer normal to ∂Ω; (6) ψ is the density of surface load on Γ1 ; (7) u0 is a prescribed displacement on Γ0 . It is also well understood that this problem corresponds to EL for the functional 1 Eε(u) : ε(u) − P u dx. I (u) = Ω 2 It is interesting to point out that the functional is quadratic and hence EL is linear. Therefore one of the favorite ways in which existence of solutions for the equilibrium problem in

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elasticity is achieved is by showing the existence of a global minimum for I . Alternatively, we can rely directly on Theorem 6.6. The convexity and regularity hypotheses are not an issue since the functional is quadratic in the gradient variable. Technical restrictions must be imposed on the load P to comply with the appropriate requirements in that theorem. Because of the presence of the tensor ε(u) instead of ∇u, understanding coercivity is the real issue. This is essentially Korn’s inequality [19]. T HEOREM 8.2. For a given, regular domain in R3 , Ω, there exists a positive constant c > 0 such that ∇uL2 (Ω) c uL2 (Ω) + ε(u)L2 (Ω) for all u ∈ H 1 (Ω). Theorem 6.6 can be directly applied to obtain weak solutions of the system of linearized elasticity. When linear models are shown not to be a good approximation because nonlinear phenomena are involved, more complicated functionals than the quadratic one considered for linear elasticity are to be examined. In particular, it is explicitly assumed that systems of PDE for equilibrium are variational in nature, and indeed equilibrium conﬁgurations are postulated to be the result of minimizing energy. But as pointed out above, we fall back on all of the subleties of nonlinear elasticity.

9. Vector problems and quasiconvexity We have already introduced the concept of polyconvex integrand and how it gives rise to weak lower semicontinuous functionals in the vector case. A natural question is whether these are all integrands, in the vector case, for which we have weak lower semicontinuity. This is not so. A classical theorem [48] yields the exact class of integrands for which we have this property. T HEOREM 9.1. A continuous integrand W : Mm×N → R gives rise to a weak lower semicontinuous functional in W 1,∞ (Ω) if and only if W is quasiconvex in the sense that W (A)

1 |D|

W A + ∇ϕ(x) dx D

for all matrices, all bounded, regular domains (|∂D| = 0) and all test functions ϕ.

(9.1)

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In particular, polyconvex integrands are quasiconvex. Right after this concept was introduced, it was clear that it was not manageable. In particular, it was not an easy task to decide if a given integrand was or was not quasiconvex. Immediately, necessary conditions were sought. For this, one can try to build simple, nontrivial ﬁelds admissible in (9.1). A more convenient reformulation of the quasiconvexity condition follows. L EMMA 9.2. A function W : M → R is quasiconvex if and only if

W A + ∇ϕ(x) dx

W (A) Q

for all Q-periodic, Lipschitz deformations ϕ and every matrix A, where Q is any unit cube in RN . The converse of this result requires suitable upper bounds on W for ﬁnite p, or the ﬁniteness of W for p = +∞. With this formulation of the quasiconvexity condition, admissible vector ﬁelds can be constructed. The discussion that follows is the ﬁrst, nontrivial case. Let F ∈ M, a ∈ R3 and a unit vector n ∈ R3 be given. Let χ1/2 stand for the characteristic function of the interval (0, 1/2) in (0, 1), extended by periodicity to all of R, and put χ = 2χ1/2 − 1. Consider the vector function

x·n

u(x) =

χ(s) ds a,

x ∈ Q.

0

This function is Q-periodic if n is one of the three orthogonal axes of Q, and therefore is eligible in Lemma 9.2. Let us examine the gradient ∇u = ∇u(x) = χ(x · n)a ⊗ n a⊗n if 0 < x · n − x · n! < 1/2, = −a ⊗ n if 1/2 < x · n − x · n! < 1 (recall that the tensor product a ⊗ n is another way of writing the rank-one matrix anT , and r! designates the integer part of r). If W is quasiconvex, by Lemma 9.2, we must have 1 1 1 1 W (F ) W (F1 ) + W (F2 ) = W (F + a ⊗ n) + W (F − a ⊗ n), 2 2 2 2 for any matrix F and vectors a and n. A function verifying this inequality for every matrix F and vectors a and n, is called rank-one convex. We have shown that P ROPOSITION 9.3. Every quasiconvex function is rank-one convex.

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It is also interesting to write down the rank-one convexity condition in terms of second derivatives. If W is smooth then it is rank-one convex if and only if the sections g(t) = W (A + ta ⊗ n) are convex at t = 0 for every matrix A and vectors a and n. In terms of W , this is equivalent to saying i,j,k,l

∂ 2W (A)ai ak nj nl 0 ∂Aij ∂Akl

for all such A, a and n. This is called the Legendre–Hadamard condition. Morrey himself thought about the possibility that rank-one convexity could be equivalent to quasiconvexity. After a good deal of experimentation, he conjectured that in general it would not be so. This turned out to be correct. The counterexample is due to Sverak [60] but it is only valid when m 3. The case m = 2 is still open. On the other hand, examples of quasiconvex functions not polyconvex were known from the very beginning even among quadratic functions [25]. Indeed for quadratic integrands, quasiconvexity and rankone convexity are equivalent [23]. These different notions of convexity for the vector situations indicate that a general theory of systems of PDE should be much more involved than the theory for single equations. This is indeed so. As a matter of fact, one main application of vector variational problems is nonlinear elasticity, as indicated before. In this context, the variational problem corresponds to the nonlinear equations of equilibrium which are typical in Continuum Mechanics − div T x, ∇u(x) = F x, u(x) T x, ∇u(x) n(x) = G x, ∇u(x)

in Ω,

u(x) = u0 (x)

on Γ0 ,

on Γ1 ,

where T is the response function of the elastic material and it represents the Cauchy stress tensor, F and G are the body and surface forces, respectively, and they have been shown with explicit dependence on u and ∇u, respectively, to cover the more usual situation. When the material is hyperelastic and the body density force is conservative, i.e., T (x, A) =

∂W (x, A), ∂A

F (x, u) =

∂f (x, u), ∂u

then the system becomes ∂W ∂f div x, ∇u(x) + x, u(x) = 0, ∂A ∂u

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and solutions correspond to stationary functions of the variational problem with functional I (u) = W x, ∇u(x) dx − f x, u(x) dx. Ω

Ω

There are essentially two approaches to solving the above system of equilibrium. One is based on variational techniques when the integrand W is quasiconvex in the gradient variable. Another possibility is based on the implicit function theorem [19]. Regularity typically involves strict convexity. In the case that integrands are not even convex (although they may be quasiconvex) it is not clear how to proceed (see [1,30,33]). This is a very delicate and involved issue.

10. Second-order problems We would like to explicitly analyze scalar, second-order problems as some typical applications involve this kind of variational problems. This time we try to W x, u(x), ∇u(x), ∇ 2u(x) dx Minimize I (u) = Ω

subject to u = u0 ,

∂u = u1 ∂n

on ∂Ω.

As usual, Ω ⊂ RN is assumed to be a regular, bounded domain; competing functions u : Ω → R are scalar and W (x, u, λ, A) : Ω × R × RN × MN×N → R is assumed to be a Carathéodory integrand. u0 and u1 are given functions. To ﬁnd the form of the associated EL equation, we suppose U is a minimizer and, just as in the case of ﬁrst-order problems, consider the sections g(t) = I (U + tϕ), where admissible variations ϕ must comply with ϕ=

∂ϕ = 0 on ∂Ω. ∂n

t = 0 is a point of global minimum, and proceeding formally under suitable regularity assumptions so that g is differentiable, we arrive at $ # ∂W ∂W 2 ∂W + ∇ϕ +∇ ϕ dx = 0, ϕ ∂u ∂λ ∂A Ω

(10.1)

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where W and all its partial derivatives above are evaluated at

x, u(x), ∇u(x), ∇ 2u(x) .

By integrating by parts several times, and bearing in mind that boundary contributions drop due to the boundary values of ϕ, we obtain #

ϕ Ω

$ ∂W ∂W ∂W − div + div div dx = 0. ∂u ∂λ ∂A

Due to the arbitrariness of ϕ, we conclude that ∂W ∂W ∂W − div + div div =0 ∂u ∂λ ∂A in Ω. Explicitly, we can write

∂W x, u(x), ∇u(x), ∇ 2u(x) ∂Aij i,j ∂ ∂W x, u(x), ∇u(x), ∇ 2u(x) − ∂xk ∂λk

∂2 ∂xi ∂xj

k

+

∂W x, u(x), ∇u(x), ∇ 2u(x) = 0. ∂u

This fourth-order PDE is completed with the boundary conditions u = u0 ,

∂u = u1 ∂n

on ∂Ω.

This is the EL problem associated to the previous second-order variational problem. Notice that (10.1) is the weak form of EL. A parallel analysis to the one carried out in Sections 3, 4 and 5, can also be made for second-order problems. A typical result follows. T HEOREM 10.1. Suppose W (x, u, λ, A) : Ω × R × RN × MN×N → R is a Carathéodory integrand, convex in (u, λ, A) for a.e. ﬁxed x ∈ Ω and such that there are c > 0 and p > 1 with c |A|p − 1 W (x, u, λ, A). Further regularity and technical assumptions are needed on W so that EL is well posed. Then there is a one-to-one correspondence between minimizers of the associated VP and

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weak solutions of EL. In addition, if such convexity is strict then we have a unique minimizer and a unique weak solution. If the integrand explicitly depends on (u, λ) but the convexity does only hold with respect to the variable A for ﬁxed (x, u, λ), then we still have existence of minimizers and of weak solutions. T HEOREM 10.2. If W is as in Theorem 10.1 but it is convex in A, for ﬁxed (x, u, λ), then there are global minimizers for VP which are weak solutions of EL (under suitable regularity hypotheses). One of the ﬁrst examples is concerned with the integrand W=

2 1 tr(A) 2

under Dirichlet and Neuman boundary conditions. It is interesting to notice that coercivity for this integrand, as stated in the above theorems, is false. Yet, if we recall that coercivity is needed just to extract a weakly convergent sequence from a minimizing sequence, we realize that for an admissible sequence {uj } in H 2 (Ω) for which {uj } is bounded in L2 (Ω), due to the well-known regularity results for Laplace’s equation [35], the sequence is uniformly bounded in H 2 (Ω). It is easy to see that the corresponding EL equation is the bi-harmonic equation (u) = 0. Another usual example is concerned with the plate equation in Kirchhoff model where a vertical equilibrium conﬁguration of a clamped plate is assumed to be given by the vertical displacement function u(x), x ∈ Ω, solution of the variational problem: # Minimize Ω

$ 1 2 2 E∇ u(x) : ∇ u(x) − F (x)u(x) dx 2

subject to u=

∂u = 0 on ∂Ω. ∂n

Ω ⊂ R2 is the vertical projection of the plate, E is the fourth-order tensor of material constants, F is the vertical load and n is the outer, unit normal to Ω. The associated equilibrium equation (EL) is i,j

∂2 ∂ 2u = F. Eij kl ∂xi ∂xj ∂xk ∂xl k,l

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Material constants for linear elastic materials produce a tensor E so that the preceding existence results can be applied to the functional 1 W = EA : A − F (x)u, 2 and in this way existence of equilibrium conﬁgurations can be shown to exist. Another interesting and important problem is concerned with the Monge–Ampère equation det ∇ 2 u = f

in Ω,

u = u0

on ∂Ω,

where Ω is a regular domain in RN and the functions f , u0 and the unknown u are in appropriate function spaces. It turns out that this equation is the EL equation for the functional $ # 1 1 f u − uy uy uxx − ux ux uyy + ux uy uxy dx dy (10.2) 2 2 Ω when the dimension N = 2. This was known a few decades ago [21]. The generalization to higher dimensions is not straightforward. In fact, a different functional can be found which gives rise to the Monge–Ampère equation (under a vanishing Dirichlet boundary condition), and this time this functional is valid in any space dimension. We are talking about I (u) = u det ∇ 2 u − (N + 1)f u dx. (10.3) Ω

It is interesting to notice that despite the fact that these integrands depend on second derivatives and hence it is a second-order variational problem, yet the associated EL is also a second-order equation instead of fourth-order. This is a clear indication that some sort of degeneracy is taking place. Note that the boundary condition for the Monge–Ampère equation is a ﬁrst-order condition as it does not involve any derivative. Therefore the typical procedure to ﬁnd the associated EL equation for any of the functionals above should be examined with some care. As a matter of fact, only vanishing values can be treated in this way to compensate for the degeneracy. If we apply (10.1) to the functional in (10.3), we have that for an arbitrary test function ϕ vanishing on ∂Ω,

ϕ det ∇ 2 u − (N + 1)f + u adj ∇ 2 u∇ 2 ϕ dx = 0. Ω

Let us just focus on the term u adj ∇ 2 u∇ 2 ϕ dx Ω

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which we have to transform by means of the divergence theorem. We get u adj ∇ 2 u∇ϕ dS(x) − div u adj ∇ 2 u ∇ϕ dx. ∂Ω

Ω

This time, unless u vanishes on ∂Ω, the term on the boundary does not drop because we have to allow variations ϕ whose normal derivative on ∂Ω is arbitrary. But if we place ourselves in the situation where the boundary data u0 ≡ 0, then, regardless of ∇ϕ, the boundary term can be deleted. On the other hand, div u adj ∇ 2 u = u div adj ∇ 2 u + ∇u adj ∇ 2 u. But we have already pointed out in Section 8 that div adj ∇ 2 u = 0.

(10.4)

Hence the term we are analyzing equals (under a vanishing boundary condition) ∇u adj ∇ 2 u∇ϕ dx. − Ω

If we further integrate by parts, the term on the boundary vanishes again but this time because ϕ = 0 on ∂Ω, and we ﬁnally arrive, using again (10.4), at ∇ 2 u adj ∇ 2 u ϕ dx. Ω

The well-known relationship between det and adj leads to N det ∇ 2 uϕ dx. Ω

If we go back to EL, ϕ(N + 1) det ∇ 2 u − f dx = 0. Ω

EL reduces then to the Monge–Ampère equation when the boundary condition is u = 0 on ∂Ω. This discussion is valid for every space dimension. The functional in (10.3) can be rewritten, except for a multiplicative constant depending on dimension and always under vanishing boundary conditions, in a different way to recover (10.2) when N = 2, and to obtain a suitable generalization of it in higher dimensions (see [4]). It is in fact a matter of using the same representation with the determinant and the adjugate directly into the functional. Since div u∇u adj ∇ 2 u = u div ∇u adj ∇ 2 u + adj ∇ 2 u ∇u : ∇u = Nu det ∇ 2 u + adj ∇ 2 u∇u : ∇u,

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and, by the divergence theorem and the fact that u = 0 on ∂Ω,

div u∇u adj ∇ 2 u = 0, Ω

we have u det ∇ 2 u dx = − Ω

1 N

adj ∇ 2 u∇u : ∇u dx. Ω

Taking this identity into the functional I (u) we can also write 1 I (u) = − N

adj ∇ 2 u∇u : ∇u + N(N + 1)uf dx.

Ω

The EL equation for this functional is also the Monge–Ampère equation. In the case N = 2, we recover (10.2). It is easy to realize that the functional I is weak lower semicontinuous on W 2,p (Ω) with p > N due to the properties of the determinant indicated in Section 8. However, to apply the direct method to this case (which is not included in Theorem 10.2), we need coercivity. This ingredient fails to hold in this situation so that existence of solutions for the Monge–Ampère equation cannot be tackled in this way.

11. Nonexistence: Lack of coercivity We have emphasized throughout the preceding section that coercivity in appropriate function spaces is a main ingredient of the direct method to show existence of global minimizers and of weak solutions of the associated EL equation. Sometimes this requirement looks like a technical requisite, and indeed it is so in the sense that it is not a structural hypothesis in any sense. Yet when this requirement fails and no substitute can be found, the direct method cannot be applied directly, and typically, the original formulation of the problem should be reﬁned. By this we mean that the set of competing functions (or objects) ought to be either enlarged or restricted. In any case, a much more delicate analysis than the one carried out here must be performed to show existence of solutions in appropriate classes of functions. Two typical examples come to mind. The ﬁrst one is the (nonparametric) minimal surface problem. We try to 1 + |∇u|2 dx

Minimize I (u) = Ω

among all functions with ﬁxed boundary values over ∂Ω. For simplicity, we consider the case N = 2, so that Ω is a regular, bounded domain. For any smooth function u, I (u) measures the surface area of the part of the graph of u over Ω so that we are looking for

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the function of two variables u whose graph has the least area possible among all those complying with boundary conditions. The innocent-looking integrand W (x, u, A) =

1 + |A|2

has a property which keeps us from being able to apply the direct method: it has linear growth. The natural space where we would have to look for minimizers would be W 1,1 (Ω). But this space is not reﬂexive and hence uniformly bounded sequences need not have weakly convergent subsequences. Indeed, in some cases minimizers do not exist in this class of competing functions. This is easily shown in an elementary way in the case of minimal surfaces of revolution [52]. Observe how the above integrand is convex in A so that I is weak lower semicontinuous. In such a situation, the whole problem should be reformulated either enlarging the class of competing functions or objects (in this case treating surfaces in parametric form), or else restricting further the problem by adding some reasonable constraints. This last approach led to the fundamental contributions of Douglas and Radó (see an accessible account in [24]) on minimal surfaces. This fascinating subject is far beyond the scope of these notes. The other example has been indicated at the end of last section. We noticed that the Monge–Ampère equation, det ∇ 2 u = f

u=0

in Ω,

on ∂Ω,

is the EL problem associated to the functional

I (u) =

u det ∇ 2 u − (N + 1)uf dx

Ω

under vanishing boundary conditions. This functional is weak lower semicontinuous on appropriate Sobolev spaces, but there is no way one can bound from below the determinant det ∇ 2 u by its individual second derivative matrix ∇ 2 u. This is equivalent to the problem of a priori bounds of second derivatives for the Monge–Ampère equation. We know that these bounds are not correct without further requirements. If we restrict the set of competing functions to include, roughly speaking, the convexity of them, then this (interior) a priori bound is essentially correct [36] and, under these circumstances, the direct method could be applied. Other more geometrically-oriented approaches have also been explored (see [4,36]). This again exceeds the scope of this work.

12. Nonexistence: Lack of convexity The most basic example to understand nonconvexity is due to Bolza. Let us see what happens if we try to Minimize I (u) = 0

1

u (x)2 − 1

2

+ u(x)2 dx

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subject to u(0) = u(1) = 0. This is a scalar variational problem in dimension one. We claim that the inﬁmum value of the functional I over the set of competing functions (in appropriate Sobolev spaces) complying with boundary conditions vanishes. On the one hand, if m denotes such inﬁmum, we clearly have m 0, since the integrand can never be negative. On the other hand, by choosing a sequence of saw-tooth functions using exclusively slopes +1 and −1 over a partition of the unit interval in j equally-spaced subintervals, it is easily seen that the inﬁmum does indeed vanish. However, it is also clear that such inﬁmum cannot be attained as I (u) = 0 is impossible for a single function u. This is because of the incompatibility of the two contributions to I : the ﬁrst, depending on the derivative, favors slopes +1 and −1, but the second favors u = 0, and this two requirements are incompatible. Notice how the dependence of u is not convex. It is also interesting to look at the associated EL problem. It is elementary to ﬁnd u (x) 3u (x)2 − 1 = u(x),

u(0) = u(1) = 0.

Trivially u ≡ 0 is a solution. However, this solution is not a minimizer of our problem. The situation for higher-dimensional problems and even for vector variational problems is qualitatively the same when convexity, understood here in a broad sense to cover vector problems, is missing. The persistent oscillatory behavior between slopes +1 and −1 to approach the inﬁmum in Bolza’s example is typical of nonconvex problems lacking optimal solutions. Scalar, nonconvex (coercive) problems have not been extensively or systematically studied, probably because the need has not arisen yet. Some higher-dimensional versions of Bolza’s example have nonetheless been analyzed in the context of some models of micromagnetics [50,51]. Where nonconvexity has played a more prominent role for variational problems is the ﬁeld of nonlinear elasticity. Even though the perspective on this ﬁeld is essentially variational and no mention is made about the underlying equilibrium system, we believe saying a few words about it is worthwhile, at least to better grasp the signiﬁcance of nonconvexity as something natural in many situations in science and engineering. On the other hand, it has been the context where generalized variational problems deﬁned in terms of Young measures have been considered. The description that follows is taken from [55]. We will consider a crystal as a countable set of atoms arranged in a periodic fashion. Probably, the simplest way of describing this array is to place the origin at one of the atoms, and then refer the position of the remaining atoms to the chosen origin by using three independent lattice vectors {n1 , n2 , n3 }. We let N ∈ M be the matrix with columns ni . We postulate the existence of a nonnegative, free energy Φ that depends on the change of shape and on temperature, as well. It is a function of each particular periodic array of the atoms of the crystal given by a matrix N , as before. We assume that Φ is frame indifferent as usual (Φ(N) = Φ(QN) for any rotation Q) but it should also be invariant under any change of lattice basis: if N is an equivalent choice of lattice basis (equivalent in the sense that the positions of the atoms is the same for N and for N ) then Φ(N) = Φ(N ). Since N and N must be related by a matrix of the set GL Z3 = {M ∈ M: det M = ±1, mij ∈ Z},

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where Z is the set of integers, we can write Φ(N) = Φ(NM),

M ∈ GL Z3 .

Once a basis of lattice vectors has been chosen and the corresponding matrix N is ﬁxed, we deﬁne the energy density per unit reference volume by putting W (F ) = Φ(F N),

det F > 0.

Altogether we have the invariance W (F ) = W (QF H ), where Q is any rotation and H ∈ NGL(Z3 )N −1 , which is a conjugate group of GL(Z3 ). Furthermore, we also impose the conditions W (F ) 0,

W (1) = 0,

W (F ) → +∞

as det F → 0, +∞,

W (F ) = +∞ if det F 0; 1 is the identity matrix. In practice however, the above invariance is assumed only when H ∈ P where P is the point group of the reference crystal lattice consisting of all the matrices H ∈ NGL(Z3 ) N −1 that are rotations. This is a ﬁnite group. For example, if the atoms in the reference conﬁguration aligned themselves on cubic cells then P would include the 24 rotations that leave invariant a cube. As mentioned, W and Φ depend on temperature θ . Above certain critical temperature θ0 , there is a stable phase, taken as reference. By stable we simply mean that it minimizes W , so that Wθ (1) = 0 and θ > θ0 . At the transition temperature θ0 , there is a change of stability or of crystal structure, so that below θ0 , the stable phase is not represented by 1 any more but by some other nonsingular matrix U0 describing the change in crystal structure that has taken place. Thus Wθ (U0 ) = 0 but Wθ (1) > 0 for θ < θ0 . At the transition temperature θ = θ0 both phases may coexist Wθ0 (U0 ) = Wθ0 (1) = 0. Because of the invariance that the energy density W = Wθ0 must satisfy, we should have at the critical temperature W RH U0 H T = W (U0 ) = 0, for any H ∈ P and any rotation R. We have found many matrices for which the free energy density W vanishes: R1, RH1 U0 H1T , RH2 U0 H2T , . . . , RHn U0 HnT , where P = {1, H1 , H2 , . . . , Hn } and R is any rotation. We call each one of the sets RHi U0 HiT : R any rotation

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a potential well associated to Ui = Hi U0 HiT and make the further assumption that the free energy density W is positive outside the set of the wells: the zero set for W , {W = 0}, is exactly the set of the wells. Under these circumstances, we are looking for minimizers of the energy functional

W ∇u(x) dx,

I (u) = Ω

among all deformations u of the reference conﬁguration Ω ⊂ R3 satisfying appropriate boundary conditions. We are explicitly assuming that W is the same for all points in the reference domain (no dependence on x). The most striking consequence of the previous description is that the energy density for an elastic crystal cannot be quasiconvex. P ROPOSITION 12.1. Let W : M → R∗ be nonnegative and {W = 0} = RH U0 H T : R, a rotation, H ∈ P . If there exist matrices R, H and nonvanishing vectors a, n as before, such that U0 − RH U0 H T = a ⊗ n, the function W cannot be quasiconvex. P ROOF. The proof reduces to the observation that a nonnegative, convex function of one variable that vanishes at two points must vanish in the interval between them too. If we apply this argument to the function g(t) = W tU0 + (1 − t)RH U0 H T that vanishes for t = 0 and t = 1 and it is convex if we suppose W is quasiconvex, we conclude that

tU0 + (1 − t)RH U0 H T : t ∈ [0, 1] ⊂ {W = 0}.

However this is not possible. Due to the fact that each well is a compact set, and we have a ﬁnite number of them, the only possibility is that the whole segment be contained in the same well. In this case, RH U0 H T = QU0 for some rotation Q. This will clearly imply that 1 − Q is a rank-one matrix 1 − Q = b ⊗ v.

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This equation forces b to be an eigenvector of Q so that if b is not the zero vector must be the axis of rotation of Q. Then 0 = (1 − Q)b = b · vv, and if v is not the zero vector then b · v = 0. In this case we must also have b · Qv = 0, but 0 = b · Qv = b · v − |v|2 b = −|v|2 |b|2. Therefore, either b or v must vanish, but this contradicts our assumption.

This fact is a clear indication that there might be real difﬁculties in establishing the existence of equilibrium conﬁgurations for elastic crystals. Even though there are results on existence despite lack of convexity, this lack of quasiconvexity makes impossible to apply the direct method to show existence of equilibrium states in this case, and indeed leads us to think about nonexistence in many interesting cases. In fact, this lack of quasiconvexity is typically a precursor of the oscillatory behavior of minimizing sequences for nonconvex (nonquasiconvex) integrands. Fine phase mixtures provide minimizing sequences whose weak limit is not a minimizer. These highly oscillatory minimizing sequences represent the behavior of elastic crystals. The Young measures associated to the gradients of minimizing sequences may serve as a device to account for this behavior.

13. Generalized VP and generalized EL As indicated in the last paragraph, when the convexity properties on integrands are missing, one can set up a new generalized variational principle where we let Young measures compete in the minimization process. As an illustration of what we mean by this, we further examine the situation for models in elastic crystals. We still follow the discussion in [55]. Let A denote the set of admissible deformations for the old variational principle 1,p A = u ∈ W 1,p (Ω): u − u0 ∈ W0 (Ω) ,

stand for the set of Young measures where u0 ∈ W 1,p (Ω) is given with I (u0 ) < ∞. Let A generated by the gradients of bounded sequences in A. Deﬁne

I (ν) =

Ω

M

W (A) dνx (A) dx

Trivially, the choice νx = δ∇u(x) for u ∈ A takes us back to I (u) so that inf I for ν ∈ A. A infA I . The equality of the two inﬁma holds under upper bounds on W but as pointed out, this condition violates our basic hypothesis on W which takes the value +∞ when the determinant is nonpositive. Nevertheless, often times we seek stress-free microstructures.

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By this we refer to minimizing sequences such that I (uj ) 0. In this case, if ν = {νx }x∈Ω is the Young measure associated to {∇uj }, by Lemma 6.3 we have 0 I (ν) lim I (uj ) = 0. j →∞

Hence, I (ν) = 0 and because of the nonnegativity of W this can only happen if supp(νx ) ⊂ {W = 0} for a.e. x ∈ Ω. ν is called a stress-free microstructure and in real problems these are the ones that we are interested in. As we have argued they are the families of probability measures that, satisfying all the restrictions of the problem, have their support contained in the zero set of the energy density. For this reason the structure of that set is so important. We know that for an elastic crystal that set is a ﬁnite union of wells. In this way, we have reduced the problem of understanding the behavior of the material to ﬁnding all gradient Young measures supported in the set of the wells. Given a minimizing sequence such that I (uj ) 0, there exists an associated gradient Young measure that describes the behavior of {∇uj }. Conversely, if we ﬁnd a gradient Young measure supported in the set of wells, the sequences of functions whose gradients generate such a Young measure will be a stress-free minimizing sequence, and hence will describe a possible behavior of the material. What is crucial is the gradient requirement. We can ﬁnd many families of probability measures supported on the set of wells. But only those that are gradient Young measures are the ones that are associated to stress-free minimizing sequences and these are the ones relevant to our original variational problem. Said differently, only gradient Young measures are physically meaningful for our problem. The others do not have any physical signiﬁcance. We need to analyze the deﬁning properties of Young measures associated to the gradients of bounded sequences in W 1,p (Ω). These have been called W 1,p -Young measures. This is a deep issue, not completely understood except in an abstract way. Indeed an important result emphasizes that such characterization represents a duality between quasiconvex functions and gradient Young measures. In the statement that follows E p is essentially the space of functions with growth of order at most p ϕ(A) C 1 + |A|p . T HEOREM 13.1 [40,41]. Let ν = {νx }x∈Ω be a family of probability measures supported in the space of matrices M. ν is a W 1,p -Young measure

if and only if : 1,p (Ω) such that ∇u(x) = (i) there exists u ∈ W M A dνx (A) for a.e. x ∈ Ω;

(ii) M ϕ(A) dνx (A) ϕ(∇u(x)) for every ϕ ∈ E p quasiconvex and bounded from below, and a.e.

x ∈ Ω; (iii) Ω M |A|p dνx (A) dx < ∞. For the case p = ∞, the condition ϕ ∈ E p drops out (Jensen’s inequality ought to be true for all quasiconvex functions regardless of their growth) and the third requirement must be replaced by the condition of uniform compact support of νx . The class of probability measures for which Jensen’s inequality holds for all rank-one convex functions play a very important role. These are called laminates and follow a recursive, comprehensible (although at times very complex) construction pattern [53].

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In general and regardless of the convexity properties of integrands, variational principles can be generalized in terms of Young measures. To be more speciﬁc than in the preceding discussion, suppose we are to analyze the variational problem: W x, ∇u(x) dx Minimize I (u) = Ω

subject to a typical Dirichlet boundary condition on u. Admissible functions u belong to W 1,p (Ω) while we have upper and lower bounds on W c |A|p − 1 W (x, A) C |A|p + 1 , where p > 1, 0 < c < C. W is assumed to be a Carathéodory function. We will consider for simplicity the scalar case in which W : Ω × RN → R so that competing functions u are scalar (one single component). If, as before, we let 1,p A = u ∈ W 1,p (Ω): u − u0 ∈ W0 (Ω) ,

stands for the set of Young measures where u0 ∈ W 1,p (Ω) is given with I (u0 ) < ∞ and A generated by the gradients of bounded sequences in A, we can deﬁne

W (x, A) dνx (A) dx I (ν) = Ω

M

Since for any u ∈ A, the family of probability measures for ν ∈ A. ν = {νx }x∈Ω ,

νx = δ∇u(x) ,

it is obvious that belongs to A, inf I (ν) inf I (u). Under our growth assumptions on W , a typical relaxation theorem [22] establishes that these two inﬁma coincide, and that the inﬁmum for I is attained (no convexity is assumed). T HEOREM 13.2 [53]. Under our previous hypotheses, inf I (u) = min I (ν).

u∈A

ν∈A

P ROOF. We have already indicated that I (ν) inf I (u). inf

ν∈A

u∈A

The equality of these two inﬁma and the existence of a minimizer for I rely on a quite remarkable fact [53].

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L EMMA 13.3. Let {vj } be a bounded sequence in W 1,p (Ω). There always exists another sequence {uj } of Lipschitz functions such that {|∇uj |p } is equiintegrable and the two sequences of gradients, {∇uj } and {∇vj }, have the same underlying Young measure.

is generated by a sequence This result means that we can always assume that any ν ∈ A in A, {uj }, so that the pth powers of their gradients are equiintegrable in Ω. Due to the bounds we have on W , we conclude that {W (x, ∇uj (x))} is also equiintegrable or weak convergent in L1 (Ω), and, as a consequence, the representation of the weak limit in terms of the underlying Young measure is valid lim

j →∞ Ω

W x, ∇uj (x) dx =

Ω

RN

W (x, A) dνx (A) dx.

(13.1)

This implies that the two inﬁma are indeed equal. In addition, if we start out with a minimizing sequence for I , {vj }, Lemma 13.3 enables us to pass to a new minimizing

This ν is a sequence, {uj }, such that (13.1) holds for the associated Young measure ν ∈ A. minimizer for I. The proof of Lemma 13.3 is essentially technical. It utilizes some ideas on truncation operators for maximal functions and approximation. As usual, once we have minimizers for any variational problem, we can immediately talk about EL equations and optimality conditions. Further regularity conditions on W must be imposed just as in Theorem 3.2: W is differentiable with respect to A, and ∂W p−1 , ∂A (x, A) f1 (x) + c 1 + |A| where f1 ∈ Lp/(p−1)(Ω), c > 0, p 1. It is well known that the same hypotheses will hold for the convexiﬁcation of W , CW (x, A), with respect to the gradient variable.

be a minimizer. Then T HEOREM 13.4. Let μ ∈ A CW x, ∇v(x) = W (x, A) dμx (A), RN A dμx (A), v ∈ A, ∇v(x) = RN

and we have in a weak sense div CW x, ∇v(x) = 0.

are such that the above three conditions hold, then μ is a Conversely, if v ∈ A and μ ∈ A minimizer for the generalized variational principle.

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This theorem indicates the path to detect generalized minimizers for nonconvex variational principles. The Young measure minimizer encodes, in principle, all the information on how to construct oscillatory minimizing sequences for the original nonconvex problem in terms of mass points and weights, and in this way we can understand optimal behavior for such irregular problems. S KETCH OF PROOF. We can factor out the minimum over Young measures in two steps: min I (ν) = min

ν∈A

min

,∇v→ν v∈A ν∈A

I (ν),

with the ﬁrst moment ∇v(x), where ∇v → ν means that we are restricting only to ν ∈ A ∇v(x) =

W (x, λ) dνx (λ)

for a.e. x ∈ Ω.

Ω

We would like to examine the inner minimum min

,∇v→ν ν∈A

I (ν).

(13.2)

We claim that in fact min

,∇v→ν ν∈A

I (ν) =

CW x, ∇v(x) dx.

Ω

For ﬁxed x ∈ Ω, it is true that CW x, ∇v(x) =

min

∇v(x)→σ RN

W (x, λ) dσ (λ).

Let μx one such measure minimizer for such x. It can be proved that the family of proba and, by construction, bility measures μ = {μx } so chosen belongs to A

CW x, ∇v(x) dx Ω

equals the minimum for ﬁxed v ∈ A, in (13.2). This fact is true essentially because, for the scalar case, convexity is equivalent to weak lower semicontinuity and no further requirement is necessary. The vector case is however more complex. To end the proof of the theorem, notice that the variational principle,

CW x, ∇v(x) dx

Minimize Ω

for v ∈ A is now regular since we have all the convexity and regularity requirements.

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This result can, almost immediately, be generalized to allow explicit dependence on u, as well. A similar result holds for the vector case. The situation is however much more complicated because convexity should be replaced by quasiconvexity and the proof that the regularity assumptions on W are inherited by its quasiconvexiﬁcation is more delicate [13] as well as the fact that admissible families of probability measures for the inner minimum are characterized by Theorem 13.1.

14. Dynamical problems: Lack of convexity and lack of coercivity We have already pointed out the kind of difﬁculties one may encounter with VP and their associated EL equations when one of the two important ingredients does not hold. We have purposely avoided the situation where both elements are missing. We pretend in this section to focus on typical cases where this is exactly the situation: both coercivity and convexity are not present. Our examples are closely related to dynamical problems. We will restrict attention to some of the most basic cases. A more sophisticated survey would be needed to analyze more complex (and interesting) situations. See [46]. We start with the simplest possible situation where we assume competing functions to be scalar and depend on two variables called, intentionally, t and x. Our domain will now be Ω = (0, T ) × (0, L). The integrand for our ﬁrst example is W (t, x, u, ut , ux ) =

1 2 u − u2x , 2 t

so that we pretend to Minimize I (u) =

1 2

T 0

L

ut (t, x)2 − ux (t, x)2 dx dt

0

under boundary conditions to be unspeciﬁed for the time being. It is clear that the integrand is neither convex nor bounded. On the other hand, the EL equation is found to be the wellknown wave equation in its simplest form ut t − uxx = 0 in (0, T ) × (0, L). More complex versions of the wave equation, including higher-dimensional cases, can also be shown to correspond to noncoercive, nonconvex integrands. Does this mean that variational problems and techniques cannot be used to analyze and examine hyperbolic or parabolic problems? It is true that solutions to the wave equation are stationary points for the above functional and many properties could, in principle, be derived based on this stationarity nature. But the issue is whether minimization ideas and techniques can still be somehow used to analyze or even to approximate solutions of the wave equation. It is true that the dependence on the spatial variable has to be dealt with in a different fashion. In fact, we know that this distinguished variable is different from the rest of spatial variables.

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The key idea to be able to apply variational techniques to solve (show the existence and/or approximate) dynamical problems is discretization (or semidiscretization) in time. To explain better this idea, we will focus on the heat equation in dimension one. In this way the two terms of the equation ut − uxx = 0 are treated very differently. The time derivative is discretized and approximated by u(t + h, x) − u(t, x) h if h > 0 is the time step. On the other hand, the spatial second derivative is interpreted like the underlying EL equation of the typical quadratic energy. If we assume that u(t, x) is known at a given time t, we would like to determine u(t + h, x) as the solution of a certain well-posed variational problem. Some easy computations lead to see that u(t + h) should be determined as the minimizer of the problem 1 (v(x) − u(t, x))2 dx vx (x) + 2 2 h

L 1

Minimize I (v) = 0

2

subject to appropriate boundary conditions on 0 and L. t and h are considered parameters. Notice that the integrand for this variational problem is coercive and convex in both variables (v, v ). Therefore there is a unique solution v(x) = u(t + h, x) which is also a solution of the underlying EL vxx =

v(x) − u(t, x) h

which is the semidiscretized heat equation. By interpolating these approximations for times not belonging to the set of discrete times and studying the convergence as h → 0 one can show general existence theorems of the following type. T HEOREM 14.1. If ϕ is convex and of quadratic growth, there exists ∂u ∈ L2 Ω × R+ u ∈ L∞ R+ ; H01(Ω) with ∂t which satisﬁes − div ϕ(∇u) + and u = u0 for t = 0.

∂u =0 ∂t

in H −1 Ω × R+ ,

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An analogous result for the wave equation is also valid. In this case the second time derivative is approximated by a second-order ﬁnite difference u(t + 2h) − 2u(t + h) + u(t) , h2 and the semidiscretized variational principle is $ # 1 (v(x) − 2u(t + h, x) + u(t, x))2 ϕ(∇u) + dx. Minimize I (v) = 2 h2 Ω Appropriate boundary conditions on ∂Ω are to be enforced. What is also interesting is that under no convexity conditions for ϕ we can still show the existence of not a classical but a Young measure solution by using essentially the same underlying techniques. In fact, the study of Young measure solutions to PDE has lately received much attention. See for instance [26]. It is a way of generalizing the concept of solution for situations where typical structural hypotheses fail to hold. See [41] for a detailed analysis of one such typical, speciﬁc situation.

15. Numerical approximation Traditionally, the numerical approximation of many of the problems treated in these pages has been studied by discretizing the underlying EL equation. Indeed, some of these (elliptic) problems have been the favorite choices to show the validity of such numerical schemes. Typical diffusion equations, the minimal area problem, the p-Laplacian, the obstacle problem, the linear elasticity system, the plate problem, etc. [18], are among the well-known situations where ﬁnite element and/or ﬁnite difference analysis is carried out. In this section, we simply want to stress that it is possible to simulate many of these situations by using algorithms based directly on minimization so that the underlying variational principle is treated by discretizing it and using typical minimization algorithms instead of studying equilibrium equations. As far as we can tell this approach has not been systematically pursued. Our intention here is to show this procedure for the typical above examples (see [3,18]). In all of the examples, our domain is always the unit square in the plane. We always show the ﬁnal approximation for the minimizer.

Laplace equation We have two examples under Dirichlet boundary conditions. The difference between the two is the ﬁneness of the mesh and the boundary data.

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Fig. 1. Laplace equation.

(a)

(b)

Fig. 2. Poisson equation: (a) homogeneous; (b) nonhomogeneous.

Poisson equation The ﬁrst example corresponds to a homogeneous problem while in the second we have used a nonconstant equilibrium coefﬁcient.

Obstacle problem We show three different situations for three different types of obstacles.

p-Laplacian A simple situation for the p-Laplacian under Dirichlet boundary condition.

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Fig. 3. Obstacle problem.

Fig. 4. p-Laplacian equation.

Bi-harmonic equation Finally, two examples for the bi-harmonic equation for different vertical loads under a vanishing Dirichlet and Neumann boundary conditions.

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Fig. 5. Bi-harmonic equation.

16. Comments on other aspects of the CV The relationship between VP and EL has had a long and rich history throughout the 19th and 20th century to the point that variational methods to treat DE is a typical term to refer to the perspective of looking at DE from the ideas and techniques of the CV. On the other hand, a much better understanding of the relationship between the CV and DE has been explored by looking at the same issues from a different perspective. For instance, EL can be transformed into an equivalent Hamiltonian system by means of the Legendre transformation of the integrand. This point of view gives rise to the Hamilton–Jacobi theory and ﬁeld theories. Related to this, there is also a whole chapter of the Calculus of Variations on mechanics and geometry. All this material greatly exceeds the scope of these pages. Readers can ﬁnd a formal and rather complete description in [34]. The question of the sufﬁciently of EL to ﬁnd minimizers for VP led to investigate what would later be called the second variation. This amounts going back to (3.1) g(t) = I (U + tϕ) and, assuming these are twice differentiable, we must have in a point of minimum g (0) 0. Then we can write this local condition again in terms of U and ϕ and their derivatives up to order two, and arrive at the formal expression of the second variation of the functional. The study of this is relevant when one is concerned about local minimizers of the functional I . We have avoided the whole issue as we have been interested in global minimizers so that we have replaced the analysis of the second variation by concentrating on global convexity properties of the integrand and/or the functional itself. Yet the analysis of the second variation is important precisely for local minimizers which may not be global. The Legendre–Jacobi theory of the second variation as well as the Weierstrass ﬁeld theory are relevant in this regard. See [34,37]. One can also look at a different way of “making variations” for I with respect to changes in the independent variables rather than with respect to the dependent variables as we have

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done here. The resulting equilibrium equations are called Noether equations and they lead to Noether theorem on conservation laws. All this is again much more specialized and can be found in [34]. With regard to the analysis of critical points of I which may not be minimizers (neither local nor global), we can say that it is a whole ﬁeld of research especially for nonlinear, scalar problems. Palais–Smale theory, Morse theory and many other ﬁelds are relevant here. A recent good account on new developments in this direction is [29]. [59] is a more classic book. See also [27]. We also pointed out that in dealing with existence of (global) minimizers we do not need any smoothness hypothesis on the integrand. But the analysis of EL obviously requires this differentiability. Nonsmooth analysis is concerned with the study of equilibrium laws when such smoothness is relaxed and, in particular, with the relationship between VP and this generalized (nonsmooth) versions of EL. See [39,64].

Acknowledgments This work is supported by research projects BFM2001-0738 of the MCyT and GC-02-001 of Castilla-La Mancha (Spain).

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[47] E.J. McShane, The calculus of variations from the beginning through optimal control theory, Optimal Control and Differential Equations, A.B. Schwarzkopf, W.G. Kelley and S.B. Eliason, eds, Academic Press, New York (1978), 3–49. [48] Ch.B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals, Paciﬁc J. Math. 2 (1952), 25–53. [49] Ch.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin (1966). [50] S. Müller, Minimizing sequences for non-convex functionals, phase transitions and singular perturbations, Lecture Notes in Phys., Springer-Verlag (1990), 31–44. [51] S. Müller, Variational models for microstructure and phase transitions, Lecture Notes in Math., Vol. 1713, Springer-Verlag, Berlin (1999), 85–210. [52] J. Oprea, The Mathematics of Soap Films, Amer. Math. Soc., Providence, RI (2000). [53] P. Pedregal, Parametrized Measures and Variational Principles, Birkhäuser, Basel (1997). [54] P. Pedregal, Equilibrium conditions for Young measures, SIAM J. Control. Optim. 36(3) (1998), 797–813. [55] P. Pedregal, Variational Methods in Nonlinear Elasticity, SIAM, Philadelphia, PA (2000). [56] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ (1972). [57] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York (1974). [58] H.H. Schaefer, Topological Vector Spaces, Springer-Verlag, New York (1966). [59] M. Struwe, Variational Methods, Springer-Verlag, Berlin (1990). [60] V. Sverak, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 185–189. [61] L. Tonelli, Fondamenti di calcolo delle variazioni, Zanichelli, Bologna (1921). [62] J.L. Troutman, Variational Calculus and Optimal Control, Springer-Verlag, New York (1998). [63] M. Valadier, Young measures, Methods of Nonconvex Analysis, Lecture Notes in Math., Vol. 1446, SpringerVerlag, Berlin (1990), 152–188. [64] R. Vinter and H. Zheng, The extended Euler–Lagrange condition for nonconvex variational problems, SIAM J. Control. Optim. 35(1) (1997), 56–77. [65] L.C. Young, Generalized surfaces in the calculus of variations, I and II, Ann. Math. 43 (1942), 84–103 and 530–544. [66] L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia, PA (1969).

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CHAPTER 5

On a Class of Singular Perturbation Problems Itai Shafrir Department of Mathematics, Technion, Israel Institute of Technology, 32000 Haifa, Israel

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Problems involving a double-well potential . . . . . . . . . . . . . . 2.1. BV spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. %-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Vector valued problems with a double-well potential . . . . . . 2.4. The Dirichlet boundary value problem . . . . . . . . . . . . . . 3. The work of Bethuel–Brezis–Hélein . . . . . . . . . . . . . . . . . . 3.1. The case of zero degree boundary condition . . . . . . . . . . 3.2. The case of nonzero degree boundary condition . . . . . . . . 4. Minimization of Ginzburg–Landau energy when g is not S 1 -valued 4.1. The case of boundary condition without zeros . . . . . . . . . 4.2. The case of boundary condition with zeros . . . . . . . . . . . 5. The case of a general “circular-well” potential . . . . . . . . . . . . 5.1. A study of a degenerate metric . . . . . . . . . . . . . . . . . . 5.2. The singular perturbation problem . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF DIFFERENTIAL EQUATIONS Stationary Partial Differential Equations, volume 1 Edited by M. Chipot and P. Quittner © 2004 Elsevier B.V. All rights reserved 297

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On a class of singular perturbation problems

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1. Introduction The objective of these notes is to discuss several singular perturbation variational problems which have a common feature. In all these problems we are interested in the minimizer for a problem of the form min W (u): u ∈ C ,

(1.1)

G

where C is a certain class of functions deﬁned on a domain G ⊂ RN . Typically C consists of functions in a Sobolev space W 1,p (G, Rk ), k 1, or in the space BV (G, Rk ), which satisfy some boundary condition or a mass constraint. The potential W is a nonnegative function on Rk whose zero set Γ = {W −1 (0)} consists of, either a ﬁnite number of points, or a smooth closed curve. The common feature of the problems is that in each of them there are many minimizers to problem (1.1), and the question of selection of the “right” minimizer arises. As a ﬁrst example, we consider a problem motivated by the Cahn–Hilliard theory of phase transitions (see [34] and the references therein). A ﬂuid is conﬁned to a bounded container G ⊂ RN , whose Gibbs free energy, per unit volume, is given by a function W0 of the density distribution u. In order to determine the stable conﬁgurations of the ﬂuid we look for minimizers of the total energy

W0 u(x) dx

E0 (u) =

u(x) dx = m.

under the mass constraint

G

(1.2)

G

On the potential W0 we assume that there exist constants c0 , c1 such that the function W (u) = W0 (u) − (c0 u + c1 ) is a double-well potential, with two minima a and b (i.e., W 0 and W −1 (0) = {a, b}, see Figure 1). Replacing W0 by W in (1.2) leads to an equivalent minimization problem: W u(x) dx: u(x) dx = m . min E(u) := G

(1.3)

G

Clearly, for any m ∈ (a|G|, b|G|), there are inﬁnitely many solutions to problem (1.3), which are piecewise constant functions that take the values a and b only. This multiplicity is due to the lack of reference in the energy to the shape of the interface between the sets {u = a} and {u = b}. The Cahn–Hilliard model proposes to take this into account by adding to the energy a term of interfacial energy, multiplied by a small coefﬁcient ε2 . In mathematical terms, we denote by uε a minimizer for the problem: 2 2 W u(x) + ε |∇u| dx: u(x) dx = m , min Eε (u) := G

G

(1.4)

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I. Shafrir

Fig. 1. A double-well potential.

and study the limit of {uε } as ε goes to 0 (in order to be consistent with other problems, we look at the energy ε12 Eε instead). This approach was justiﬁed mathematically by the results of Modica [34] and Sternberg [45] who proved that uε converges to a function u0 which takes only the values a and b with minimal interface between the sets {u = a} and {u = b}. A rigorous statement and proof of the last assertion is given in Sections 2.1 and 2.2. We remark only that in this case the class C is given by C = u ∈ BV G, {a, b} : a {u = a} + b {u = b} = m . A study of an analogous problem with a Dirichlet boundary condition is the subject of Section 2.4 and a generalization for vector-valued problems is described in Section 2.3. Next we turn to a problem of seemingly different nature. Let G be a smooth, bounded and simply connected domain in R2 , and g : ∂G → S 1 a smooth boundary condition. We → S 1 of g. To put the problem in our general framelook for a “natural” extension u0 : G work we choose any smooth function W : R2 → [0, ∞) with W −1 (0) = S 1 (i.e., a “S 1 -well potential”) and consider the minimization problem W u(x) dx: u = g on ∂G . min E(u) :=

(1.5)

G

For the problem to be well deﬁned we must specify the class C of admissible functions. One possibility is to take the set Hg1 (G, S 1 ) = u ∈ H 1 (G, C), |u| = 1 a.e. in G, u = g on ∂G . But in the case where the degree D of g is nonzero, say D > 0, a case on which we concentrate now, this set is empty (see Proposition 3.1). On the other hand, we may take 1,p 1,p C = Wg (G, S 1 ) for any p ∈ [1, 2), but then E ≡ 0 on Wg (G, S 1 ). Note that every map 1,p u ∈ Wg (G, S 1 ) must be singular (since the degree of g is nonzero). Therefore, we should look for a singular perturbation that in the limit will favor the simplest possible structure of singularities. In the approach of Bethuel, Brezis and Hélein [8,9], that we shall describe next, one adds to the energy a Dirichlet energy term, multiplied by a small coefﬁcient that

On a class of singular perturbation problems

301

will go to 0. Choosing for simplicity the Ginzburg–Landau potential W (u) = (1 − |u|2 )2 gives an energy of the form 2 Fε (u) = 1 − |u|2 + ε2 |∇u|2 , G

but we prefer to study equivalently the energy divided by ε2 as before. Therefore, we are led to study the limit of {uε } as ε → 0, where for each ε > 0 we denote by uε a minimizer for 2 1 |∇u|2 + 2 1 − |u|2 over Hg1 (G, C). (1.6) Eε (u) := ε G Note that as in the previous case, we have enlarged the class of admissible functions for the problem involving Eε . In the current case this was done by allowing complex-valued functions. This ensures that Hg1 (G, C) = ∅, and the penalization term “forces” |u| to be close to 1 as ε goes to 0. It was proved in [9] in the case of star-shaped G (an assumption that was later shown to be unnecessary by Struwe [48]) that a subsequence {uεn } converges 3 z−aj to a map of the form u∗ = eiφ D j =1 |z−aj | , where a1 , . . . , aD are distinct points in G, and φ a smooth harmonic function which is determined by the requirement u∗ = g on ∂G. 1,α The convergence takes place in W 1,p (G) ∀p < 2 and in Cloc (G \ {a1 , . . . , aD }) ∀α < 1. Moreover, the asymptotic behavior of the energies is given by Eε (uε ) = 2πD| log ε| + O(1),

as ε → 0.

(1.7)

This result is proved in Section 3.2 while the easier case D = 0 is the subject of Section 3.1. Another important motivation for the study of this problem is a physical one. Indeed, the functional (3.1) is a simpliﬁed version of the Ginzburg–Landau energy in superconductivity, see the survey [38] and the references therein. In Section 4 we look at the minimization problem (1.6) in a more general setting where g is not assumed to be S 1 -valued. First, in Section 4.1 we treat the case of g which does not take the value 0 so that D = deg(g/|g|) is well deﬁned, and we assume again that D 0. The result of [2] for this case is that a subsequence uεn converges to a map of the 3 z−aj k form u∗ = eiφ D j =1 |z−aj | as above, but only away form the boundary, i.e., in Cloc (G \ {a1 , . . . , aD }) ∀k. Here, in contrast with the case of S 1 -valued g, the boundary values of u∗ are different from those of uε , since we have u∗ = g/|g| on ∂G. This is due to a boundary layer effect. In fact, a boundary layer of width of the order O(ε) is used by uε to pass from the boundary condition g to values very close to those of g/|g|. This phenomenon contributes a term of order O( 1ε ) to the energy, so that 2 Eε (uε ) = ε

∂G

|g|3 2 − |g| + 3 3

+ 2πD log

1 + O(1). ε

(1.8)

We see here a sort of combination of the results of Section 3 with those of Section 2. Indeed, on the one hand we ﬁnd point singularities in the interior, but also a transition layer near

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the boundary as was encountered already in Section 2.4. It should be noted that |g| has an effect only on the energy, but the limit u∗ is determined completely by the projection of g on S 1 , namely g/|g|. In fact, u∗ is the solution of the problem treated in [9] for the boundary condition g/|g|. In Section 4.2 the more difﬁcult case of g with zeros is discussed. More precisely, it is assumed that g has a ﬁnite number of zeros with a certain behavior around them. The analysis of this case is much more involved than before and only some ideas of the proofs are presented (the full details are given in [4]). The new feature here is that the limit u∗ has boundary singularities. In contrast to Section 4.1, here u∗ is determined not only by the map g/|g| (which has singularities at the zeros) but also by |g|, and more speciﬁcally, by the orders of the zeros of |g|. The Ginzburg–Landau potential (1 − |u|2)2 , studied in Sections 3 and 4, is a special case of a circular-well potential, by which we mean a smooth nonnegative function W on R2 whose zeros set is a closed smooth curve Γ , with length l(Γ ). We are led to consider more generally the following minimization problem, 1 min Eε (u) := |∇u|2 + 2 W (u): u ∈ Hg1 G, R2 (1.9) ε G for every ε > 0, where g : ∂G → R2 is a given smooth boundary condition. We denote by uε a minimizer in (1.9) and we are interested as usual in the asymptotic behavior of the minimizers {uε } and their energies as ε goes to 0. This is the subject of Section 5 which describes the results of [5]. We must add some assumptions on W , regarding its behavior near Γ and at inﬁnity (see (5.2) and (5.3)) and on g. In fact, the image of g should be close enough to Γ (this is in analogy with the assumption g(x) = 0 ∀x ∈ ∂G in Section 4.1). In order to be more precise about this assumption and the result we deﬁne the following function on R2 by Ψ (ζ ) =

inf

γ ∈Lip([0,1],R2 ) 0 γ (0)∈Γ,γ (1)=ζ

1

1/2 γ (t) dt. W γ (t)

The function Ψ can be viewed as a distance to Γ w.r.t. a degenerate Riemannian metric. It is shown in Section 5.1 that there exists a neighborhood of Γ of the form Ωλ0 = x ∈ R2 : Ψ (x) < λ0 in which Ψ is a C 2 -function. For each y ∈ Ωλ0 we denote by s˜ (y) the intersection with Γ of the gradient line of Ψ which passes through y. In that way we have deﬁned a new projection map s˜ : Ωλ0 → Γ . Our assumption on g is that image(g) ⊂ Ωλ0 . Denoting by D the degree of the map s˜ (g) : ∂G → Γ , and assuming w.l.o.g. that D 0, the main result of Section 5.2, Theorem 5.2, states that a subsequence {uεn } converges in Cloc (G \ {a1, . . . , aD }) to a limit of the form D 4 z − aj iφ0 u∗ = τ e , |z − aj | j =1

On a class of singular perturbation problems

303

) 1 where τ : S 1 → Γ satisﬁes |τ (s)| = l(Γ 2π ∀s ∈ S , for some D points a1 , . . . , aD ∈ G. Here the (smooth) harmonic function φ0 is determined by the requirement u∗ = s˜(g) on ∂G.

2. Problems involving a double-well potential In this section we shall consider minimization problems involving a double-well potential. The simplest example is the Ginzburg–Landau potential W (t) = (1 − t 2 )2 , for which the energy takes the form 2 1 Eε (u) = |∇u|2 + 2 1 − u2 , (2.1) ε G where u ∈ H 1 (G) = H 1 (G, R) and G is a bounded smooth domain in RN (N 2). The ﬁrst problem that we describe, following Modica [34] and Sternberg [45], is a Neumann boundary value problem with a mass constraint. Given a number m ∈ (−|G|, |G|) we denote for every ε > 0 by uε a minimizer for the problem: u=m . (2.2) min Eε (u): u ∈ H 1 (G) s.t. G

We are interested in the asymptotic behavior of uε as ε goes to zero. We shall follow quite closely the description of [45], using at some points [24]. It turns out that a natural space for the limit is the space of functions of bounded variation (BV). So we begin by recalling some of the basic properties of this space. 2.1. BV spaces Below we shall describe brieﬂy the basic properties of BV spaces. Our main sources are the books [21,25,49] where much more information on these spaces can be found. Let G be a domain in RN . The space BV(G) consists of the functions u ∈ L1 (G) whose weak derivatives ux1 , . . . , uxN are signed measures with ﬁnite variation. Equivalently, for u ∈ L1 (G) we deﬁne (2.3) |∇u| = sup u div g: g ∈ Cc1 G, RN with g(x) 1 ∀x ∈ G G

G

and set |∇u| < ∞ . BV(G) = u ∈ L1 (G): G

for u ∈ W 1,1 (G) the deﬁnition (2.3) coincides with the Lebesgue integral

Notethat N 2 1/2 . It is not difﬁcult to see that BV(G) is a Banach space with the norm G ( i=1 uxi ) |u| + |∇u|. uBV = G

G

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In the special case where u = χA with A a subset of G we have, by (2.3), PerG A := |∇χA | G

1 N div g: g ∈ Cc G, R with g(x) 1 ∀x ∈ G . = sup A

A subset A for which PerG A < ∞ is called a set of ﬁnite perimeter in G. If ∂A ∩ G is a Lipschitz-continuous hypersurface, then PerG A = H N−1 (∂A ∩ G) (with H N−1 denoting the (N − 1)-dimensional Hausdorff measure). Next we state two of the main properties of BV space: lower semicontinuity and compactness of BV in L1 . T HEOREM 2.1. If vn → v in L1 (G) then lim infn→∞

G |∇vn | G |∇v|.

P ROOF. For any g ∈ Cc1 (G, RN ), we have v div g = lim vn div g lim inf |∇vn | G

n→∞ G

n→∞

G

and the result follows by taking the supremum on g.

T HEOREM 2.2. If vn BV C ∀n then there exists a subsequence {vnk } such that vnk → v in L1 (G). S KETCH OF PROOF. The main tool of the proof is the following result about approximation of BV-functions by smooth functions (see [25,49]): for every u ∈ BV(G) there exists a sequence {ui } ⊂ C ∞ (G) satisfying |ui − u| = 0 and lim |∇ui | = |∇u|. lim i→∞ G

i→∞ G

G

It follows that for each n there exists un ∈ C ∞ (G) satisfying 1 and |un − vn | |∇un | 2C. n G G

(2.4)

Hence the sequence {un } is bounded in W 1,1 (G), so by the Rellich–Kondrachov theorem there is a subsequence {unk } such that unk → v in L1 (G). By (2.4) it follows that also vnk → v in L1 (G). 2.2. %-convergence A basic tool in the analysis of the minimizers {uε } of (2.2) is the notion of %-convergence of a family of functionals, which is due to Di Giorgi. We restrict ourselves to the special case needed here, namely convergence with respect to the L1 topology.

On a class of singular perturbation problems

305

D EFINITION 2.1. Consider a family of functionals Fε : L1 (G) → (−∞, ∞] ∀ε > 0 and another functional F0 : L1 (G) → (−∞, ∞]. We shall say that F0 is the %-limit of {Fε } as ε → 0, with respect to the L1 -topology, if the following two conditions hold: (i) ∀v0 ∈ L1 (G), ∀{vεn } ⊂ L1 (G) with εn → 0 such that limn→∞ vεn = v0 we have lim inf Fεn (vεn ) F0 (v0 ). n→∞

(ii) ∀v ∈ L1 (G), ∀{εn } such that εn → 0, there exists a sequence {ρεn } ∈ L1 (G) such that ρεn → v

in L1 (G)

and F0 (v) = lim Fεn (ρεn ). n→∞

The next lemma motivates the choice of F0 in our problem. L EMMA 2.1. Let I = inf

L −L

g

2

2 + 1 − g2 :

L > 0, g is piecewise C on [−L, L] with g(±L) = ±1 . 1

Then I = 83 . P ROOF. For any admissible g we have, by Cauchy–Schwarz inequality, L L L 2 8 2 2 2 2 1 − g g 2 g + 1−g 1−g g = . 2 3 −L −L −L For each L > 1 deﬁne ⎧ t ∈ [1 − L, L − 1], ⎨ tanh(t), gL (t) = (t + 1 − L) + (L − t) tanh(L − 1), t ∈ (L − 1, L], ⎩ (L + t) tanh(1 − L) + (t − 1 + L), t ∈ [−L, 1 − L). Since gL (t) = 1 − gL2 (t) on [1 − L, L − 1], we have

L−1

2 gL

1−L

+

2 1 − gL2

=2

L−1

1−L

1 = 4 tanh(L − 1) − tanh3 (L − 1) . 3

From this it is easy to conclude that lim

1 − gL2 gL

L

L→∞ −L

gL

2

2 8 + 1 − gL2 = 3

306

I. Shafrir

and the result follows. Next we deﬁne F0 : L1 (G) → [0, ∞] by 8

PerG x ∈ G: u(x) = −1 , u ∈ BV G, {−1, 1} , G u = m, F0 (u) = ∞, otherwise 3

(2.5)

and, for every ε > 0, Fε (u) =

G ε|∇u|

2

+

∞,

1 ε

2

1 − u2 , u ∈ H 1 (G), G u = m, otherwise.

(2.6)

Next we claim the following: T HEOREM 2.3. F0 is the %-limit of {Fε }, as ε → 0, with respect to the L1 -topology. Before proving Theorem 2.3, we show how it implies the following characterization of the possible limits of sequences of minimizers of (2.2). T HEOREM 2.4. Suppose uεn → u0 in L1 (G) for a sequence εn → 0, where, for each εn , uεn is a minimizer for (2.2) with ε = εn . Then, u0 = χG\A − χA where A is a minimizer for the problem inf PerG A: A ⊂ G, |G| − 2|A| = m .

(2.7)

P ROOF. Take any w0 ∈ L1 (G). By property (ii) of the %-convergence there exists a sequence ρεn satisfying ρεn → w0

in L1 (G)

and

lim Fεn (ρεn ) = F0 (w0 ).

n→∞

Therefore, by property (i) of the %-convergence we get that F0 (u0 ) lim inf Fεn (uεn ) lim Fεn (ρεn ) = F0 (w0 ), n→∞

n→∞

and it follows that u0 is a minimizer for F0 .

Next we prove the %-convergence of {Fε } to F0 . P ROOF OF T HEOREM 2.3. We need to verify both properties (i) and (ii) in Deﬁnition 2.1. (i) Consider a sequence vεn → v0 in L1 (G), and suppose that lim infn→∞ Fεn (vεn ) < ∞ (the result is clear otherwise). Note that the truncated function v˜εn = min 1, max(−1, vεn ) ,

On a class of singular perturbation problems

307

satisﬁes Fεn (v˜εn ) Fεn (vεn ). Therefore, we may assume a priori that vε (x) 1 n

in G, ∀n.

(2.8)

1 − s 2 ds,

(2.9)

Put φ(t) = 2

t

−1

so that t s 3 φ(t) = 2 s − 3

−1

t3 2 =2 t − + ∀t ∈ [−1, 1], 3 3

and note that by Cauchy–Schwarz inequality we have

∇φ(vεn ).

Fεn (vεn )

(2.10)

G

From the convergence vεn → v0 in L1 and (2.8) we deduce that φ(vεn ) → φ(v0 )

in L1 (G).

(2.11)

By (2.11), (2.10) and Theorem 2.1 it follows that

∇φ(v0 ) lim inf n→∞

G

∇φ(vε ) lim inf Fε (vε ). n n n n→∞

G

(2.12)

By Fatou’s lemma,

G

1 − v02

2

lim inf n→∞

G

2 1 − vε2n = 0,

and we obtain that v0 (x) ∈ {−1, 1} a.e. in G. Hence, from (2.12) we deduce that v0 ∈ BV(G, {−1, 1}) and consequently, 0 φ v0 (x) = 8 3

on {v0 = −1}, on {v0 = 1}.

Therefore

∇φ(v0 ) = 8 PerG {v0 = −1} = F0 (v0 ), 3 G

and the result follows from (2.12).

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I. Shafrir

(ii) It sufﬁces to consider v ∈ L1 (G) for which F0 (v) < ∞ (otherwise we may take ρεn = v ∀n) and we assume then that v ∈ BV G, {−1, 1} with

v = m, G

i.e., denoting by A = {x ∈ G: v(x) = −1} we have v = (−1)χA + χG\A

with PerG A < ∞ and |G \ A| − |A| = m.

Since by part (i), for any sequence {vεn } converging to v in L1 (G), we have lim inf Fεn (vεn ) F0 (v), n→∞

it is enough to construct a sequence {ρεnk } corresponding to a subsequence {εn k }. This will be achieved by several applications of a diagonalization argument, and for simplicity we will keep each time the notation {εn } for the subsequence. First, by [45], Lemma 1, there exists a sequence of open sets {Ak } satisfying the following properties: (i) ∂Ak ∩ G ∈ C 2 , (ii) |(Ak ∩ G)A| → 0 as k → ∞, (iii) PerG Ak → PerG A as k → ∞, (iv) H N−1 (∂Ak ∩ ∂G) = 0, (v) |Ak ∩ G| = |A|. Hence, by a diagonalization argument we can assume that ∂A ∩ G ∈ C 2

and H N−1 (∂A ∩ ∂G) = 0.

Consider L > 0 and a piecewise C 1 -function g on [−L, L] such that g(±L) = ±1. For each εn put ⎧ d(x) < εn L, ⎨ −1, ρεn (x) = gεn d(x) = g d(x) , −εn L d(x) εn L, εn ⎩ 1, d(x) > εn L, where d denotes the signed distance to Γ := ∂A ∩ ∂(G \ A), i.e., d(x) =

−dist(x, Γ ), x ∈ A, dist(x, Γ ), x ∈ G \ A.

We have |∇d(x)| = 1 a.e. on G and for some s0 > 0, d is a C 2 -function in {|d(x)| < s0 } (see [26], Section 14.6). Further, lim H N−1 d(x) = s0 = H N−1 (∂A ∩ G)

s→0

(see [34]).

(2.13)

On a class of singular perturbation problems

309

Recall the co-area formula [23]:

f h(x) |∇h| dx =

R

G

f (s)H N−1 x: h(x) = s ds,

(2.14)

which holds for any measurable f and Lipschitz-continuous h. Applying (2.14) and (2.13) yields

v(x) − ρε (x) dx = n G

0 −εn L

= εn

1 − gε (s)H N−1 d(x) = s n

εn L

+

−1 − gε (s)H N−1 d(x) = s n

0 0 −L

−1 − g(t)H N−1 d(x) = εn t

1 − g(t)H N−1 d(x) = εn t

L

+ εn 0

εn L −1 − g∞ + 1 − g∞ (PerG A + 1). Therefore ρεn → v in L1 (G). Similarly, Fεn (ρεn ) = =

εn L −εn L

2 2 1 1 − gε2n (s) εn gε n (s) + H N−1 d(x) = s εn

g (s)2 + 1 − g 2 (s) 2 H N−1 d(x) = εn s

L −L

→ H N−1 (∂A ∩ G)

L

−L

g (s)2 + 1 − g 2 (s) 2

as n → ∞.

By a diagonalization argument and Lemma 2.1 we get a sequence {ρεn } satisfying property (ii) of Deﬁnition 2.1, except for the fact that we only have lim

n→∞ G

ρεn = m

and not the actual equality sequence

G ρεn

= m ∀n. In fact, it is not difﬁcult to see that the modiﬁed

5 ρ˜εn = ρεn + m − |G| ρεn G

satisﬁes all the requirements.

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I. Shafrir

A natural question is whether there exists a sequence of minimizers {uεn } for which Theorem 2.4 can be applied. This is the question of compactness of minimizers of (2.2) which is treated in the following theorem. T HEOREM 2.5. For any sequence of minimizers {uεn } to (2.2), there exists a converging subsequence in L1 (G) (whose limit u0 is necessarily a minimizer of F0 , by Theorem 2.4). P ROOF. Let A be a subset of G with PerG A < ∞ and |G \ A| − |A| = m, and consider the corresponding {ρεn } as in the proof of Theorem 2.3. Then

∇φ(uε ) Fε (uε ) Fε (ρε ) C n n n n n

(2.15)

G

and

uεn (x) 2 1 − s 2 ds dx −1 G 3 2 2 C 1+ C. 1 − |uεn | |uεn | C 1 +

φ(uε ) = n G

G

G

Therefore {φ(uεn )} is bounded in BV(G) and by Theorem 2.2 we deduce that for a subsequence, still denoted by {εn }, we have vεn := φ(uεn ) → v0

in L1 (G),

(2.16)

for some v0 ∈ BV (G) (by Theorem 2.1). For |u| large we have φ(u) ∼ u3 so for |v| large, φ −1 (v) ∼ v 1/3 and (φ −1 ) (v) ∼ v −2/3 . Hence, φ −1 is uniformly continuous on R, and we deduce from (2.16) that uεn = φ −1 (vεn ) → u0 := φ −1 (v0 ) in measure. But since (2.15) implies that {uεn } is bounded in L4 (G) we obtain that uεn → u0 also in L1 (G).

2.3. Vector valued problems with a double-well potential The results of the previous subsection remain true, with essentially the same proof, when we replace the potential W (t) = (1 − t 2 )2 by a more general double-well potential W : R → R, i.e., a nonnegative continuous function, with exactly two zeros (some assumptions on the behavior of W at inﬁnity are needed to ensure a compactness result like Theorem 2.5, see [34,45] for details). More generally, for k 1 one may consider a potential satisfying: 1,∞ k W ∈ Wloc R , [0, ∞) with

W (u) = 0

iff u ∈ {a, b}, where a = b, (2.17)

On a class of singular perturbation problems

311

and for each ε > 0 deﬁne the energy 1 Eε (u) = |∇u|2 + 2 W (u), ε G for G a smooth bounded domain in RN and u ∈ H 1 (G, Rk ). Analogously to the problem (2.2) one may then consider the following problem: given m ∈ Rk of the form m = |G| θ a + (1 − θ )b for some θ ∈ (0, 1), study the asymptotic behavior, as ε goes to zero, of the minimizers {uε } of 1 k min Eε (u): u ∈ H G, R s.t. u=m .

(2.18)

G

We shall sketch the solution as given by Fonseca and Tartar [24], see also Sternberg [46]. We shall make two additional assumptions on W . The ﬁrst is concerned with the behavior of W near a and b: There exist α, δ > 0 such that " α|u − a|2 W (u) α1 |u − a|2 if |u − a| < δ, (2.19) α|u − b|2 W (u) α1 |u − b|2 if |u − b| < δ. The second assumption is on the behavior at inﬁnity: There exist C, R > 0 such that W (u) C|u|

if |u| > R.

(2.20)

We start by presenting a generalization of Lemma 2.1. We denote by I0 the distance between a and b with respect to a certain Riemannian metric, namely: I0 = 2 inf

1

−1

W 1/2 γ (t) γ (t) dt:

γ : [−1, 1] → Rk is piecewise C 1 , with γ (−1) = a, γ (1) = b .

(2.21)

Note that the integral in (2.21) is scaling invariant, and thus we can replace the interval [−1, 1] by any other interval [α, β]. L EMMA 2.2. Let I = inf

g (t)2 + W g(t) dt:

L −L

L > 0, g : [−L, L] → Rk is piecewise C 1 , with g(−L) = a, g(L) = b .

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I. Shafrir

Then I = I0 . P ROOF. For each admissible g : [−L, L] → Rk we have by Cauchy–Schwarz inequality

g (t)2 + W g(t) dt 2

L −L

L

−L

W 1/2 g(t) g (t) dt I0

and therefore I I0 . On the other hand, for any δ > 0 there exists a C 1 curve γ : [−1, 1] → Rk with γ (−1) = a, γ (1) = b and γ (t) = 0 ∀t, such that

1 −1

W 1/2 γ (t) γ (t) dt I0 + δ.

s Re-parameterizing γ with respect to the arc-length parameter τ (s) = −1 |γ (t)| dt yields

1 a curve g : [0, L] → Rk , L = −1 |γ (t)| dt, with |g (t)| = 1 ∀t, satisfying

L

W

1/2

g(t) dt =

0

1 −1

W 1/2 γ (t) γ (t) dt I0 + δ.

(2.22)

Next, consider the initial value problem "

h (s) = F h(s) ,

(2.23)

h(0) = L2 ,

where F (t) := W 1/2 (g(t)) is locally Lipschitz by (2.17). Therefore, there exists an interval (T0 , T1 ) (possibly unbounded) on which there is a solution to (2.23) with h(T0 ) = 0 and h(T1 ) = L. Extending h on (−∞, T0 ) by 0 and on (T1 , ∞) by L and setting g(s) ˜ = g h(s)

on (−∞, ∞),

yields a Lipschitz function satisfying g(−∞) ˜ = a, g(∞) ˜ = b and g˜ (s) = g h(s) h (s) = F h(s) = W 1/2 g(s) ˜

∀s ∈ (−∞, ∞),

which implies by (2.22) that

g˜ (s)2 + W g(s) ˜ =2

∞ −∞

∞

−∞

=2

0

L

g˜ (s) = 2 ˜ W 1/2 g(s)

T1

g˜ (s) ˜ W 1/2 g(s)

T0

W 1/2 g(s) g (s) I0 + δ.

(2.24)

On a class of singular perturbation problems

313

L]

→ Rk such that g˜

Finally, deﬁning a truncated function g˜L : [−L, L (−L) = a and

g˜ ( L) = b (as in the proof of Lemma 2.1) we get for L large enough, L

L − L

g˜ (s)2 + W g˜ (s)

L L

g˜ (s)2 + W g(s) ˜ + δ.

∞

−∞

(2.25)

Combining (2.24) with (2.25) gives

L −L˜

g˜ (s)2 + W g˜ (s) I0 + 2δ,

L L

which implies the desired result since δ can be chosen arbitrary small.

Next we deﬁne a “geodesic distance” function from a by φ(x) = 2 inf

1 −1

T g(s) g (s) ds:

g : [−1, 1] → Rk is piecewise C 1 , with g(−1) = a, g(1) = x ,

(2.26)

where T (u) := min(W 1/2 (u), M) and the constant M is determined as follows. Put f (r) = inf W 1/2 (u):

u − a + b = r 2

a − b . and r0 = 2

By (2.20) there exists r1 > r0 such that

r1

f (r) dr > r0

I0 . 2

(2.27)

Finally we set M = max W 1/2 (u):

u − (a + b) r1 . 2

It is easy to see that φ is a Lipschitz function on Rk satisfying ∇φ(u) 2 min M, W 1/2 (u) ∀u ∈ Rk .

(2.28)

Moreover, φ(b) = I0 .

(2.29)

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I. Shafrir

Indeed, clearly φ(b) I0 . Consider any admissible g : [−1, 1] → Rk with g(−1) = a, g(1) = b. If |g(s) − (a + b)/2| r1 ∀s ∈ [−1, 1], then clearly 2

1

−1

T g(s) g (s) ds I0 .

If for some s0 ∈ (−1, 1) and ε > 0, |g(s0 ) − (a + b)/2| = r1 + ε then by (2.27), 2

1

−1

T g(s) g (s) ds 2

r1 +ε

f (r) dr > I0 . r0

Now we are ready to prove analogous results to Theorems 2.3 and 2.4. In our case the relevant functionals are J0 , Jε : L1 (G, Rk ) → [0, ∞] given by

I0 PerG {x ∈ G: u = a}, u ∈ BV G, {a, b} , G u = m, J0 (u) = ∞, otherwise

(2.30)

and, for every ε > 0,

Jε (u) =

G ε|∇u|

+ 1ε W (u),

2

∞,

u ∈ H 1 G, Rk , G u = m, otherwise.

(2.31)

T HEOREM 2.6. J0 is the %-limit of {Jε }, as ε → 0, with respect to the L1 -topology. If uεn → u0 in L1 (G, Rk ) for a sequence εn → 0, where for each εn , uεn is a minimizer for (2.18), then u0 = χA a + χG\A b where A is a minimizer for the problem inf PerG A: A ⊂ G, |A|a + |G \ A|b = m . P ROOF. It is enough to prove the assertion on the %-convergence, since the characterization of the limits of {uεn } then follows exactly as in the proof of Theorem 2.4. We start with property (i) in Deﬁnition 2.1. Let vεn → v0 in L1 (G, Rk ) such that lim infn→∞ Jεn (vεn ) < ∞ (if the limit is ∞, the result is clear). By Fatou’s lemma we have

W (v0 ) lim inf G

n→∞

W (vεn ) = 0, G

so that v0 ∈ {a, b} a.e. By (2.28), φ is Lipschitz (with Lipschitz constant M). Hence, φ(vεn ) → φ(v0 ) in L1 , and since C Jεn (vεn ) 2

W G

1/2

vεn (x) ∇vεn (x) dx

∇φ vε (x) , n

(2.32)

G

we get from Theorem 2.2 that (up to a subsequence) φ(vεn ) → φ(v0 ) in L1 . By Theo-

On a class of singular perturbation problems

315

rem 2.1

∇φ(v0 ) lim inf

n→∞

G

∇φ(vεn ),

(2.33)

G

and it follows, using (2.29), that φ(v0 ) = I0 χ{v0 =b} ∈ BV(G). Therefore, also v0 ∈ BV(G) and combining (2.32) with (2.33) we are led to

∇φ(v0 ) = I0 PerG {v0 = b} lim inf Jεn (vεn ).

J0 (v0 ) =

n→∞

G

The proof of property (ii) follows exactly as the corresponding proof for Theorem 2.3, where we use Lemma 2.2 instead of Lemma 2.1. R EMARK 2.1. Fonseca and Tartar also proved a compactness result for the minimizers, i.e., an analogue to Theorem 2.5. Baldo [6] proved a generalization of the results of this subsection for a potential W with an arbitrary ﬁnite number of zeros.

2.4. The Dirichlet boundary value problem In this section we shall describe brieﬂy the results of Owen, Rubinstein and Sternberg [35] which deal with a similar problem to the one described in the previous subsections, but with Dirichlet boundary condition instead of Neumann. Again for simplicity we consider the Ginzburg–Landau energy (2.1), although more general double-well potentials are allowed in [35] (Ishige [27] generalized the results of [35] for vector-valued problems involving double-well potentials as considered in Section 2.3). As before G will denote a smooth (i.e., C 2 ) bounded domain in RN and g : ∂G → R a boundary condition. Since we want to study also boundary conditions g which are not the trace of a function in H 1 (G) (i.e., g ∈ / H 1/2 (∂G)), in order to allow, for example, jump discontinuities, we shall only require g ∈ L∞ (∂G)

(2.34)

and consider a family of maps {gε }ε>0 satisfying: lim gε = g

ε→0

in L1 (∂G),

gε L∞ (∂G) C, ∂gε C 1/4 , ∂σ ∞ ε L (∂G) ∂gε C. G ∂σ

(2.35) (2.36) (2.37) (2.38)

316

I. Shafrir

For each ε > 0 we denote by uε a minimizer for Eε (deﬁned in (2.1)) over Hg1ε = u ∈ H 1 (G): u = gε on ∂G , and we are interested in the limiting behavior of {uε } as ε goes to 0. This will be done by the technique of %-convergence, and for that matter we deﬁne a family of functionals Hε : L1 (G) → R by Hε (u) =

G ε|∇u|

2

+

∞,

1 ε

2 1 − u2 , u ∈ Hg1ε (G), otherwise.

(2.39)

For small ε there will be two contributions to the energy of uε . The ﬁrst corresponds to the energy in an interior layer, as in problem (2.2), the second corresponds to the energy in a boundary layer. The two are reﬂected in the candidate for the %-limit: ⎧ dH N−1 (x), ˜ ⎨ G ∇φ(u) + ∂G φ g(x) − φ u(x) (2.40) H0 (u) = u ∈ BV G, {−1, 1} , ⎩ ∞, otherwise, where φ is deﬁned in (2.9), and u˜ denotes the trace of u on ∂G. Note that the ﬁrst integral in (2.40) equals 83 PerG {x ∈ G: u = −1} (compare with (2.5)). The main result of [35] is summarized below. T HEOREM 2.7. H0 is the %-limit of {Hε }, as ε goes to 0, with respect to the L1 -topology. Moreover, for any sequence of minimizers {uεn } with εn → 0 there exists a subsequence which converges in L1 to a minimizer u0 of H0 . S KETCH OF PROOF. As in the previous cases treated in Sections 2.2 and 2.3 the main point is to prove %-convergence, as the other assertions then follow easily. The proof of property (ii) in Deﬁnition 2.1 is technically involved (see [35]) so we omit it. We shall only sketch the proof of property (i) in Deﬁnition 2.1 (i.e., the lower semicontinuity), and this too, for simplicity, only in the special case gε = g ∀ε. We are given a sequence {vεn } such that vεn → v0 in L1 (G). First, thanks to assumption (2.34) it is easy to see by truncation that we may assume that vεn L∞ (G) C

∀n.

(2.41)

It follows that φ(vεn ) → φ(v0 ) in L1 . For a small δ > 0 let G(δ) denote the union of G with a tubular neighborhood of ∂G of width δ. Fix a function gˆ ∈ BV(G(δ) \ G) whose trace on ∂G is g (see [25]) and extend the deﬁnition of each vεn to a function vˆεn deﬁned on G(δ) by setting vˆεn = gˆ on G(δ) \G. Similarly, use gˆ to extend v0 to a function vˆ0 deﬁned on G(δ) . We have then (see [25], Chapter 2) that vˆεn , vˆ0 ∈ BV(G(δ) ) and, for example, for vˆ0 : ∇ vˆ0 = ∇ gˆ + v˜0 − g . |∇v0 | + (2.42) G(δ)

G

G\G(δ)

∂G

On a class of singular perturbation problems

317

By (2.41) and Theorem 2.1 we have lim inf Hεn (vεn ) lim inf ∇φ(vεn ) n→∞

n→∞

= lim inf n→∞

G(δ)

G

G(δ)

∇φ vˆεn −

∇φ vˆ0 −

∇φ(v0 ) +

= G

G\G(δ)

G\G(δ)

∇φ gˆ

∇φ gˆ

φ u˜ 0 − φ(g), ∂G

where in the last step we applied (2.42) for φ(v0 ).

E XAMPLE 2.1. Let G = B(0, 1), the unit disc in R2 , and g(z) = Re z on ∂G. In this case it can shown that the minimizer for H0 must be of the form 1, Re z > 0, u0 (z) = sgn(Re z) = −1, Re z < 0. Therefore, by Theorem 2.7, for a subsequence of minimizers we have uεn → u0 in L1 (G) (see Figure 2), and further, Eε (uε ) ∼

2 2 8 · +2· ε 3 ε

3 cos θ − cos θ − 2 dθ. 3 3 −π/2 π/2

Fig. 2. Minimization for g(z) = Re z over R-valued maps.

(2.43)

318

I. Shafrir

We shall come back to this example later when we study the minimization problem with the same boundary condition, but for complex valued maps.

3. The work of Bethuel–Brezis–Hélein Let G be a smooth, bounded, simply connected domain in R2 and g : ∂G → S 1 a smooth boundary condition. For ε > 0 and u ∈ H 1 (G, C) we deﬁne the Ginzburg–Landau type energy by Eε (u) =

|∇u|2 + G

2 1 1 − |u|2 . 2 ε

(3.1)

For each ε > 0 we denote by uε a minimizer for the problem min Eε (u): u ∈ Hg1 (G, C) ,

(3.2)

where Hg1 (G, C) = u ∈ H 1 (G, C): u = g on ∂G (we shall often identify C with R2 ). We are interested in the asymptotic behavior, as ε → 0, of the minimizers {uε } as ε goes to 0. It turns out that this behavior depends in a crucial manner on the degree D = deg(g, ∂G), i.e., the winding number or Brouwer degree of g, as a map from ∂G to S 1 . From the properties of the Brouwer degree it follows that g has a continuous S 1 -valued extension if and only if D = deg(g, ∂G) = 0. The fact that this is also true for H 1 -maps is not to G obvious, and can deduced, for example, from the degree theory for maps in H 1/2(G, S 1 ) (see [14,17]). We present below a simple proof taken from [15]. P ROPOSITION 3.1. For any smooth g : ∂G → S 1 , the set Hg1 (G, S 1 ) is nonempty if and only if D = 0. P ROOF. First we claim that D=

1 π

G

ux1 ∧ ux2 dx1 dx2

R2 s.t. u = g on ∂G. ∀u ∈ C 2 G,

(3.3)

Indeed, denoting by τ the tangent unit vector to ∂G (in the positive sense) and by n the

On a class of singular perturbation problems

319

external unit normal vector, we compute 1 ux ∧ ux 2 π G 1 1 1 (u ∧ ux2 )x1 + (ux1 ∧ u)x2 = = div(u ∧ ux2 , ux1 ∧ u) 2π G 2π G 1 1 (u ∧ ux2 , ux1 ∧ u) · n ds = (u ∧ ux2 )nx1 + (ux1 ∧ u)nx2 ds = 2π ∂G 2π ∂G 1 1 (u ∧ ux2 )τ x2 − (ux1 ∧ u)τ x1 ds = u ∧ (ux1 τ x1 + ux2 τ x2 ) ds = 2π ∂G 2π ∂G 1 g ∧ gτ ds = D. = 2π ∂G Next we prove that (3.3) remains valid for any u ∈ Hg1 (G, R2 ). For that matter it sufﬁces to show that (3.4) ux 1 ∧ ux 2 = vx1 ∧ vx2 ∀u, v ∈ Hg1 G, R2 . G

G

Put w = v − u ∈ H01 (G, R2 ). Then (vx1 ∧ vx2 − ux1 ∧ ux2 ) G

= G

wx1 ∧ ux2 +

G

ux1 ∧ wx2 +

G

wx1 ∧ wx2 .

(3.5)

In view of (3.4) and (3.5) it sufﬁces to show that wx1 ∧ fx2 = wx2 ∧ fx1 ∀w ∈ H01 D, R2 , ∀f ∈ H 1 D, R2 . D

(3.6)

D

In fact, applying (3.6) ﬁrst with f = u, and then with f = w, and plugging the results in (3.5) yields (3.4). Consider ﬁrst w ∈ Cc∞ (G, R2 ). Then wx1 ∧ fx2 = (wx1 ∧ f )x2 − wx1 x2 ∧ f = − wx1 x2 ∧ f G

and

G

G

G

wx2 ∧ fx1 =

G

G

(wx2 ∧ f )x1 −

G

wx1 x2 ∧ f = −

G

wx1 x2 ∧ f,

so that (3.6) holds. The general case follows by approximation. Next assume that g has an S 1 -valued H 1 -extension, u˜ ∈ Hg1 (G, S 1 ). Since, d 2 d 2 u˜ = u˜ = 0 dx1 dx2

a.e. in G

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I. Shafrir

we conclude that u˜ x1 ∧ u˜ x2 = 0 a.e. in G. Hence, applying (3.3) to u˜ yields D = 0 which proves the necessity of this condition for the existence of an extension. On the other hand, if D = 0 we may write g = eiφ0 for some smooth function φ0 : ∂G → R, and then u = eiφ is a smooth S 1 -valued extension of g, for → R of φ0 . any smooth extension φ : G Any minimizer uε is a solution of the Euler–Lagrange equation

−uε = uε = g

2 ε2

1 − |uε |2 uε

in G, on ∂G.

(3.7)

The next lemma provides two basic estimates for uε . L EMMA 3.1. Any solution uε of (3.7) satisﬁes |uε | 1 in G and ∇uε L∞ (G)

(3.8) C ε

(3.9)

for some constant C > 0 independent of ε. P ROOF. We have 1 |uε |2 = uε · uε + |∇uε |2 2 2 2 = 2 |uε |2 |uε |2 − 1 + |∇uε |2 2 |uε |2 |uε |2 − 1 . ε ε

(3.10)

Therefore the function v = |uε |2 − 1 satisﬁes

−v + v=0

4 |u |2 v ε2 ε

0 in G, on ∂G,

so by the maximum principle v 0 in G, and (3.8) follows. Next, the function u˜ ε (x) = uε (εx) satisﬁes " 2 −u˜ ε = 2 1 − u˜ ε u˜ ε in Gε , u˜ ε = g˜ε on ∂Gε , where Gε = G/ε and gε (x) = g(εx) on ∂Gε . By standard elliptic estimates we get that ∇ u˜ ε L∞ (Gε ) C, and rescaling back we obtain (3.9).

On a class of singular perturbation problems

321

In the sequel we shall distinguish between the cases D = 0 and D = 0, starting with the easier case D = 0, following [8]. 3.1. The case of zero degree boundary condition In the case D = 0 one smooth S 1 -valued extension of g is of special interest. It is given ˜ by u0 = eiφ0 , where φ˜ 0 denotes the harmonic extension of φ0 in G (see the last part of the proof of Proposition 3.1). The map u0 satisﬁes |∇u0 |2 = min |∇u|2 , (3.11) u∈Hg1 (G,S 1 ) G

G

and it is the unique minimizer in (3.11). It is a natural candidate for the limit of {uε }. Indeed, we have the following: P ROPOSITION 3.2. uε → u0 in H 1 as ε → 0. P ROOF. By deﬁnition, Eε (uε ) Eε (u0 ) ∀ε, i.e., 2 1 |∇uε |2 + 2 1 − |uε |2 |∇u0 |2 . ε G G

(3.12)

Therefore {uε } is bounded in H 1 , and for a subsequence we have weakly in H 1 for some u ∈ Hg1 (G, C).

uε n $ u By (3.12),

2 1 − |uε |2 Cε2

∀ε,

(3.13)

G

and

2

|∇u0 |2 .

|∇uε | G

(3.14)

G

From (3.13) it follows that u ∈ Hg1 (G, S 1 ), and then by (3.14) and lower semicontinuity we deduce that 2 |∇u| |∇u0 |2 . G

G

This clearly implies that u = u0 . Further, the strong convergence uεn → u0 in H 1 follows from (3.14). Finally, the full convergence uε → u0 follows from the uniqueness of u0 as a minimizer in (3.11).

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I. Shafrir

L EMMA 3.2. |uε | → 1 uniformly on G. P ROOF. By (3.12) and the strong convergence uε → u0 in H 1 it follows that 1 ε2

2 1 − |uε |2 → 0.

(3.15)

G

with |uε (x0 )| < 1. By (3.9) we have Fix any x0 ∈ G uε (x) uε (x0 ) + γ C ρ ε

∀x ∈ B(x0 , ρ) ∩ G, ∀ρ ρ0 ,

(3.16)

with γ > 0 depending only on G, for some ρ0 > 0. Put ρ=

ε(1 − |uε (x0 )|) . 2γ C

(3.17)

For small ε we have ρ ρ0 and we get that 1 1 − uε (x) 1 − uε (x0 ) 2

∀x ∈ B(x0 , ρ) ∩ G,

(3.18)

and therefore,

2 1 − |uε |2 G

1 − |uε |2

2

B(x0 ,ρ)∩G

2 1 meas B(x0 , ρ) ∩ G 1 − uε (x0 ) . 4

(3.19)

But by the smoothness of ∂G there exists a constant α > 0 such that meas B(x0 , ρ) ∩ G αρ 2

∀ρ > 0, ∀x0 ∈ G.

(3.20)

Combining (3.19) and (3.20) with (3.15) we get that 1 − |uε (x0 )| → 0 uniformly in x0 ∈ G. Next we prove: L EMMA 3.3. {uε } is bounded in H 2 (G). P ROOF. We shall only prove the local estimate: 2 {uε } is bounded in Hloc (G).

(3.21)

On a class of singular perturbation problems

323

The argument for a global H 2 -bound is technically more involved and can be found in [8]. Setting Aε = 12 |∇uε |2 , we claim that 2 1 4 −Aε + D 2 uε A2 2 |uε |2 ε

on G,

(3.22)

where 2 2 2 ∂ 2 uε 2 D u ε = . ∂xi ∂xj i,j =1

Indeed, dropping the ε for simplicity we get from the Euler equation (3.7) that uxi = uxi

4 2(|u|2 − 1) + 2 u(u · uxi ). 2 ε ε

Then we compute 2 2 A = D 2 u + uxi (uxi ) i=1

2 4 (|u|2 − 1) = D 2 u + 2|∇u|2 + 2 (u · ∇u)2 2 ε ε 2 2 |u| . D u − |∇u|2 |u| √ Since |u| 2|D 2 u|, we have √ |D 2 u| 1 2 2 2 A2 D u + 4 2 , −A + D 2 u 2 2A |u| 2 |u| and (3.22) follows. By the strong convergence uε → u in H 1 (G) (see Proposition 3.2) it follows that for every δ > 0 there exists R > 0 such that |∇uε |2 < δ ∀x0 ∈ G, ∀ε. (3.23) B(x0 ,R)∩G

Let us ﬁx for the moment any δ > 0 (to be determined later) with the associated R. Fix any point x0 > 0 and set d = dist(x0 , ∂G) and r = min(d/2, R). Let ζ be a smooth function with support in B(x0 , r) such that ζ ≡ 1 on B(x0 , r/2). Multiplying (3.22) by ζ 2 and integrating yields 1 2

2 ζ D uε 4 2

G

2

G

ζ2 2 A + |uε |2 ε

2 ζ Aε . G

(3.24)

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I. Shafrir

By Lemma 3.2 we have |uε | 1/2 in G for ε ε0 , and we deduce from (3.24) and Proposition 3.2 that 2 2 2 ζ D uε C1 ζ 2 |∇uε |4 + C. (3.25) G

G

Applying the Sobolev inequality, 1/2

φ

|∇φ| + |φ| ∀φ ∈ W 1,1 (G),

C

2

G

G

for φ = ζ |∇uε |2 and then using Cauchy–Schwarz inequality and (3.23) we get

ζ |∇uε |D 2 uε

ζ |∇uε | C2 2

4

G

G

2 +C

C2

|∇uε |2 G∩B(x0 ,R)

2 ζ 2 D 2 u ε + C

G

2 ζ 2 D 2 uε + C.

C2 δ

(3.26)

G

Choosing δ =

1 2C1 C2

we deduce from (3.25) and (3.26) that

2 2 D uε C, B(x0 ,r/2)

and (3.21) follows. Next we are ready to state the main convergence result from [8]. ∀α < 1. T HEOREM 3.1. As ε → 0 we have uε → u0 in C 1,α (G)

P ROOF. Using the computation in (3.10) we obtain that the function ψ = (1 − |uε |2 )/ε2 satisﬁes −ε2 ψ + 4|uε |2 ψ = 2|∇uε |2 . By Lemma 3.2 we may assume that |uε | 1/2 in G and we deduce that −ε2 ψ + ψ 2|∇uε |2 . Multiplying (3.27) by ψ q−1 for any q > 1 and integrating yields

ψ 2

|∇uε |2 ψ q−1 .

q

G

G

(3.27)

On a class of singular perturbation problems

325

Applying Hölder inequality gives ψq 2∇uε 2L2q Cq ,

(3.28)

since by Lemma 3.3 and Sobolev embedding, {∇uε } is bounded in Lr (G) for every r < ∞. Plugging (3.28) in (3.7) we obtain that uε Lq Cq , and in particular, choosing q > 2, we deduce by Sobolev embedding that ∇uε L∞ C.

(3.29)

Using (3.29) in (3.27), and applying the maximum principle we get that ψL∞ 2∇uε 2L∞ C, which combined with (3.7) leads to uε L∞ C.

(3.30)

Finally, the convergence in C 1,α follows from (3.30) and Sobolev embedding.

since by (3.7), R EMARK 3.1. One cannot expect the convergence uε → u0 in C 2 (G) 2 uε = 0 on ∂G, while u0 = −u0 |∇u0 | . However, it is proved in [8] that, for every compact K ⊂⊂ G and every k, we have uε → u0 in C k (K). A more general version of Theorem 3.1, allowing boundary data which depends on ε, was also proved in [8]. It is useful in the study of the case of nonzero degree, see Section 3.2. We state the result without proof. T HEOREM 3.2. Consider for every ε > 0 a smooth boundary condition gε : ∂G → C satisfying gε L∞ (∂G) 1, gε H 1 (∂G) C and

1 − |gε |

2

C

∂G

for some constant C independent of ε. Up to a subsequence we may assume that gε → g

in C(∂G),

for some g : ∂G → S 1 , and we suppose that deg(g, ∂G) = 0. We may write then g = eiφ0 ˜ and set u0 = eiφ0 where φ˜ 0 is the harmonic extension of φ0 . Then uε → u0 strongly in k (G) for all k. 1 H (G), in C(G) and in Cloc

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I. Shafrir

3.2. The case of nonzero degree boundary condition Let us assume without loss of generality that D > 0. In view of Proposition 3.1 we must have limε→0 Eε (uε ) = ∞ and it would be useful to know the rate in which Eε (uε ) goes to inﬁnity. As a motivation, we start with a simple lemma. L EMMA 3.4. Let uε denote a minimizer for Eε over Hg1 (B(0, R), C) with g(z) = z/|z| on ∂B(0, R). Then Eε (uε ) 2π log

R + C, ε

(3.31)

with C independent of ε. P ROOF. Clearly Eε (uε ) Eε (v) for any v ∈ Hg1 (B(0, R), C). Next we choose v of a special form: v(reiθ ) = f (r)eiθ (using polar coordinates) with f (r) =

r

for r t, for t < r R,

t

1

where t ∈ (0, R) is a parameter to be determined in an optimal way. By a direct computation, Eε (v) = 2π

R

f

2

+

0

= 2π + 2π log

2 f2 2π R 1 − f 2 r dr r dr + 2 2 r ε 0

R t2 + 2πc0 2 := h(t), t ε

1 with c0 = 0 (1 − s 2 )2 s ds. It is easy to see that the minimum of h is achieved for t0 = √ ε/ 2c0 which gives, Eε (uε ) Eε (v) = h(t0 ) = 2π log

R + C, ε

and (3.31) follows. Using Lemma 3.4 we can prove an upper bound for the energy in the general case.

P ROPOSITION 3.3. Let g : G → S 1 be a smooth boundary condition of degree D > 0. Then Eε (uε ) 2πD log with C = C(G, g).

1 +C ε

∀ε,

(3.32)

On a class of singular perturbation problems

327

P ROOF. It sufﬁces to construct for each ε a map vε ∈ Hg1 (G, C) satisfying, Eε (vε ) 2πD| log ε| + C.

(3.33)

Fix D distinct points a1 , . . . , aD ∈ G and then R > 0 satisfying R < min |ai − aj |, min dist(ai , ∂G). i=j

i

1 Let w : G \ D i=1 B(ai , R) → S be any smooth map satisfying w = g on ∂G and w(z) = z−ai |z−ai | on ∂B(ai , R), i = 1, . . . , D. Such a map exists since D = deg(g). Finally deﬁne vε by

vε (z) =

⎧ ⎪ ⎨ w(z) z−ai

|z−ai | ⎪ ⎩ z−ai ε

in G \ D i=1 B(ai , R), in B(ai , R) \ B(ai , ε), i = 1, . . . , D, in B(ai , ε), i = 1, . . . , D.

By the computation of Lemma 3.4 we get that vε satisﬁes (3.33).

The proof of the lower bound, which shows that Eε (uε ) is really of the order 2πD log 1ε + O(1) is much more involved. We begin with a basic estimate, that will be proved at ﬁrst under the assumption that G is star-shaped about the origin, i.e., for some constant α > 0, x ·nα>0

∀x ∈ ∂G.

(3.34)

We shall later show how to remove this technical assumption. P ROPOSITION 3.4. Assume G is star-shaped about the origin. Then there exists a constant C0 = C0 (G, g) such that any solution of (3.7) satisﬁes

∂uε 2 2 + 1 1 − |uε |2 C0 . ∂n 2 ε G ∂G

(3.35)

P ROOF. The proof is based on Pohozaev identity (see [9,47]), i.e., all we need to do is to ∂uε ε multiply both sides of (3.7) by x · ∇uε = x1 ∂u ∂x1 + x2 ∂x2 and integrate by parts. Dropping the subscript ε for simplicity we have |∇u|2 2 . u(x · ∇u) = div ∇u(x · ∇u) − |∇u| − x · ∇ 2 Therefore, setting F (u) = −

2 1 1 − |u|2 2 2ε

and f (u) =

2 1 − |u|2 u, 2 ε

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I. Shafrir

yields 0 = u + f (u) (x · ∇u)

|∇u|2 = div ∇u(x · ∇u) − |∇u|2 − x · ∇ + x · ∇F (u) 2 |∇u|2 + xF (u) − 2F (u). = div ∇u(x · ∇u) − x 2

(3.36)

Integrating (3.36) over G yields 1 ε2

2 ∂u 2 1 1 − |u|2 + (x · n) 2 ∂G ∂n G 2 ∂g 1 ∂u ∂g . (x · n) − (x · τ ) = ∂τ ∂n ∂τ ∂G 2

(3.37)

Using (3.34) and Cauchy–Schwarz in (3.37) leads to (3.35). The next lemma provides a basic tool for locating the zeros of uε .

P ROPOSITION 3.5. There exist positive constants λ0 , μ0 and ε0 (depending only on G and g) such that if uε is a solution of (3.7) with ε ε0 satisfying 2 1 (3.38) 1 − |uε |2 μ0 2 ε G∩B(z0 ,2l) for some z0 ∈ G and l such that λ0 ε l,

(3.39)

then uε (x) 1 2

∀x ∈ G ∩ B(z0 , l).

(3.40)

P ROOF. Assume by contradiction that |uε (x0 )| < 1/2 for some x0 ∈ B(z0 , l) ∩ G. Arguing ε (x0 )|) (hence, ρ ρ0 for ε ε0 , as in the proof of Lemma 3.2, we choose ρ = ε(1−|u 2γ C see (3.16)) and we deduce as in (3.18) that 1 1 1 − uε (x) 1 − uε (x0 ) > 2 4

∀x ∈ B(x0 , ρ) ∩ G,

which implies, as in (3.19) and (3.20) that 1 ε2

B(x0 ,ρ)∩G

1 − |uε |2

2

>

αρ 2 1 α meas B(x0 , ρ) ∩ G . 2 16ε 16ε2 256C 2 γ 2

On a class of singular perturbation problems

329

Now, if ρ l then B(x0 , ρ) ⊂ B(z0 , 2l). Therefore, we may take 1 2γ C

λ0 =

and μ0 =

α . 256C 2 γ 2

Next we deﬁne the set of “bad points” by 1 Sε = x ∈ G: uε (x) < . 2

(3.41)

The next proposition shows that Sε can be covered by a ﬁnite number of discs with radii of the order O(ε). P ROPOSITION 3.6. Let G be a star-shaped domain. There exist an integer N and a posε itive constant λ such that, for each ε there is a collection of discs {B(xiε , λε)}N i=1 such that: Sε ⊂

Nε ! B xiε , λε i=1

and

ε x − x ε 8λε i

j

∀i, j, i = j,

with Nε N for all ε. P ROOF. By a simple recursive argument we can ﬁnd for each ε a collection of mutually ε ε disjoint discs {B(yiε , 2λ0 ε)}K i=1 , with yi ∈ Sε ∀i, such that Sε ⊂

Kε ! B yiε , 10λ0 ε . i=1

By Proposition 3.5, 1 ε2

G∩B(yiε ,2λ0 ε)

2 1 − |uε |2 > μ0

∀i,

hence, applying Proposition 3.4 yields Kε

C0 μ0

∀ε.

(3.42)

Set λ1 = 10λ0 . If for some pair i = j we have |yiε − yjε | < 8λ1 ε then we let λ2 = 9λ1 , multiply all radii by 9 and remove j from our collection. After a ﬁnite number of such

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I. Shafrir

iterations (whose number is bounded uniformly in ε by the desired collection of discs.

C0 μ0 ,

thanks to (3.42)) we arrive at

By Proposition 3.6, given any sequence εn → 0, we may extract a subsequence (still denoted by εn ) such that Nεn = N1

∀n,

(3.43)

and xiεn → li ∈ G,

i = 1, 2, . . . , N1 .

(3.44)

Some of the limit points may coincide, so we denote by (3.45)

a1 , a2 , . . . , aN2 1 the distinct N2 ( N1 ) points in {li }N i=1 and then

Λj = i: xiεn → aj ,

j = 1, . . . , N2 .

(3.46)

We may further assume that diεn = di

∀i, ∀n,

(3.47)

diεn

(3.48)

and we denote Kj =

∀j = 1, . . . , N2 .

i∈Λj

The next lemma (taken from [16]) provides a basic estimate for the energy of maps deﬁned on an annulus, which will be needed for establishing the lower bound for Eε (uε ). → C, with A := B(0, R1 ) \ B(0, R0 ), be a C 1 -map satisfying, L EMMA 3.5. Let u : A |u| a > 0 in A, d = deg u, ∂B(0, R0 ) = deg u, ∂B(0, R1 ) and

1 R02

1 − |u|2

2

K.

(3.49) (3.50) (3.51)

A

Then there exists a constant C = C(a, K, d) such that R1 |∇u|2 2πd 2 log − C. R 0 A

(3.52)

On a class of singular perturbation problems

331

P ROOF. Thanks to (3.49) we may write in A: u = ρei(dθ+ψ)

R. with ρ = |u| and ψ ∈ C 1 A,

Since |∇u| = |∇ρ| + ρ 2

2

2

d2 + 2d∇θ · ∇ψ + |∇ψ|2 r2

with r = |x|, we have

|∇u|2 A

ρ2 A

d2 2 := I1 + I2 + I3 . + 2d∇θ · ∇ψ + |∇ψ| r2

(3.53)

Next we write d2 R1 I1 = 2πd 2 log 1 − ρ2 2 . − R0 r A

(3.54)

Using Cauchy–Schwarz and (3.51), we obtain 2 dx 1/2 2 d 2 1/2 1 − ρ R K C. d 0 4 r2 A A |x| Again by Cauchy–Schwarz, we have 2d 2 ∂ψ |I2 | = ρ −1 ∂τ A r 1/2 2|d| 1/2 4d 2 K a 2 2 K R0 |∇ψ| + |∇ψ|2 . R0 4 A a2 A Finally, by (3.49), 2 2 2 I3 = ρ |∇ψ| a |∇ψ|2 . A

(3.55)

(3.56)

(3.57)

A

Combining (3.53) and (3.57) we are led to (3.52).

C) R EMARK 3.2. The conclusion of Lemma 3.5 remains valid for u ∈ H 1 (A, C) ∩ C(A, (by the same proof ). Next we generalize Lemma 3.5 for a ﬁnite union of annuli. P ROPOSITION 3.7. For σ δ > 0 and m distinct points x1 , . . . , xm such that |xi − xj | > 2δ

∀i = j,

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I. Shafrir

we denote Aj = B(xj , σ ) \ B(xj , δ) ∀j and let u ∈ H 1 ( satisfy |u| a > 0 on

m !

Aj

m

j =1 Aj , C)

deg u, ∂B(xj , σ ) = dj

and

∩ C(

m

j =1 Aj , C)

∀j,

j =1

and

1 δ2

2 1 − |u|2 K.

m

j=1 Aj

m

Then denoting d =

j =1 dj

we have

m

|∇u|2 2π|d| log

j=1 Aj

with C = C(a, K, m,

σ − C, δ

(3.58)

m

j =1 |dj |).

P ROOF. We use an argument from [48]. Denote, (1)

(1)

dj = dj , m(1) = m

xj = xj ,

∀j,

and R (1) = δ,

and J (1) = {1, . . . , m}. (1)

(1)

(1)

(1)

(1)

Set r (1) = 12 mini=j |xi −xj | and Aj = B(xj , r (1) )\B(xj , R (1) ) ∀j . By Lemma 3.5,

(1) m (1)2 r 2 dj |∇u| 2π log −C m (1) (1) R j=1 Aj j =1

(1) r 2π|d| log (1) − C. R

(3.59)

Next deﬁne R (2) as the minimal number R > r (1) for which there exists a subset J (2) ⊂ J (1) such that m ! (1) ! (1) B xj , R (1) ⊂ B xi , R j =1

and |xi1 − xi2 | 2R,

i1 = i2 in J (2) .

i∈J (2)

Clearly, R (2) Cr (1)

for some constant C = C(m). (1)

(2)

(2)

(2) = |J (2) |. If R (2) σ then we We denote the points {xi }i∈J (2) by {xj }m j =1 with m conclude by (3.59). Otherwise, we continue with the above construction which yields

R (1) < r (1) < R (2) < r (2) < · · · < R (k−1) < r (k−1) < R (k)

On a class of singular perturbation problems

333

with R (l+1) Cr (l)

∀l,

(3.60)

where k is the ﬁrst index satisfying R (k) σ , and the corresponding sets of points and degrees:

xj(l)

m(l) j =1

and

m(l) dj(l) = degr u, B xj(l), R (l) j =1 ,

l = 1, . . . , k − 1.

We also denote, 6 (l) (l) (l) Aj = B xj , r (l) B xj , R (l) , Note that

m(l)

(l) j =1 dj

= d for all l. Applying Lemma 3.5 and using (3.60) gives,

m

j=1 Aj

j = 1, . . . , m(l) , l = 1, . . . , k − 1.

|∇u| 2

k−1

(l)

m(l) l=1

2π|d|

j=1

k−1

(l)

Aj

log

l=1

= 2π|d| log

|∇u| 2π 2

m k−1

dj(l)

2

l=1 j =1

log

r (l) −C R (l)

R (l+1) −C R (l)

σ R (l) − C 2π|d| log − C. δ R (1)

Next we derive an optimal lower bound for the energy. T HEOREM 3.3. Let G be a smooth simply connected bounded domain in R2 (not necessarily star-shaped) and let g : ∂G → S 1 be a smooth boundary condition of degree D 0. Then Eε (uε ) 2πD log

1 −C ε

∀ε,

(3.61)

with C = C(G, g). P ROOF. Let R > 0 be large enough so that G ⊂ B(0, R). Fix any smooth map U : B(0, R)\ G → S 1 such that U |∂G = g and let g˜ = U |∂B(0,R) (which has necessarily degree D). Denote for each ε by u˜ ε a minimizer for Eε over Hg˜1 (B(0, R), C). Clearly,

∇ u˜ ε 2 + 1 1 − u˜ ε 2 2 ε2 B(0,R) 2 1 |∇uε |2 + 2 1 − |uε |2 + |∇U |2 = Eε (uε ) + C. ε G B(0,R)\G

(3.62)

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I. Shafrir

Since B(0, R) is star-shaped, we know by Proposition 3.6 that ! Nε 1

B xiε , λε , Sε := x ∈ B(0, R): u˜ ε (x) < ⊂ 2 j =1

with Nε N, ∀ε. It will be convenient to ﬁx R1 > R and to consider a smooth S 1 -valued extension U1 of U to B(0, R1 ) \ B(0, R). This induces an extension of each u˜ ε to B(0, R1 ) (only a constant is added to its energy). Fix any σ < R1 − R. Applying Proposition 3.7 we get for ε < σ/λ:

∇ u˜ ε 2 + 1 1 − u˜ ε 2 2 2 ε B(0,R1 )

∇ u˜ ε 2 + 1 1 − u˜ ε 2 2 2 ε ε j=1 B(xi ,σ )

Nε

2πD log

σ − C. λε

Combining it with (3.62) we are led to (3.61).

C OROLLARY 3.1. Let G be a simply connected bounded domain in R2 (not necessary star-shaped) and g a smooth S 1 -valued boundary condition of degree D 0. Then there exists a constant C such that 2 1 1 − |uε |2 C ∀ε. (3.63) ε2 G P ROOF. We use an elegant observation of del Pino and Felmer [20]. Applying Theorem 3.3 for 2ε instead of ε yields |∇uε |2 + G

2 1 1 − |uε |2 2 4ε

|∇u2ε |2 + G

2πD log

2 1 1 − |u2ε |2 2 4ε

1 − C. 2ε

(3.64)

On the other hand, by the upper bound (3.32), |∇uε |2 + G

1 1 2 2 1 − |u | 2πD log + C. ε ε2 ε

Subtracting (3.64) from (3.65) yields the result.

(3.65)

Thanks to Corollary 3.1 we get that the conclusion of Proposition 3.6 remains valid without assuming that G is star-shaped. So for a subsequence εn → 0 we have (3.43)–(3.48). In the next lemmas we obtain some further properties of {uεn }. L EMMA 3.6. We have Kj = 1 ∀j = 1, . . . , N2 .

On a class of singular perturbation problems

335

P ROOF. As in the proof of Theorem 3.3 we may use a smooth extension and assume that uεn is deﬁned on a larger domain than G. Fix R > 0 such that |ai − aj | > 2R ∀i = j . Applying Proposition 3.7 with σ = R/2 and δ = λεn gives (for εn small enough): N1

εn i=1 B(xi ,R/2)

|∇uεn |2 2π

N2

|Kj | log

j =1

R − C. εn

(3.66)

Combining it with the upper bound (3.32) we get by letting εn → 0 that |Kj | D. j ∈N2

Since

Kj = D, it follows that

j ∈N2

Kj 0,

j = 1, . . . , N2 .

(3.67)

Next we claim that Kj ∈ {0, 1},

j = 1, . . . , N2 .

(3.68)

Fix any δ ∈ (0, R). For εn small enough we have B xiεn , λεn ⊂ B(aj , δ) ∀i ∈ Λj , j = 1, . . . , N2 . Applying Lemma 3.5 gives R |∇uεn |2 2πKj2 log − C δ B(aj ,R)\B(aj ,δ)

∀j = 1, . . . , N2 .

(3.69)

Combining (3.69) with (3.66), applied with δ instead of R, yields

2 2 R R Kj − Kj log − C. + 2π εn δ

N

|∇uεn |2 2πD log G

(3.70)

j =1

Choosing δ small enough in (3.70) would yield a contradiction with (3.32), unless N2 2 Kj − Kj = 0, j =1

which clearly implies (3.68). Finally we have to rule out the possibility Kj = 0. Assume ﬁrst that Kj = 0 for some aj ∈ G. Fix R > 0 such that 2R < min |ak − aj | k=j

and 2R < dist(aj , ∂G).

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By Proposition 3.7 and our assumption Kj = 0 it follows that

|∇uεn |2 2π

k=j

B(ak ,R)

k=j

R R Kk log − C = 2πD log − C. εn εn

This together with (3.32) gives |∇uεn |2 + B(aj ,2R)

2 1 1 − |uεn |2 C. εn 2

By Fubini theorem we can ﬁnd R0 ∈ (R, 2R) such that (up to passing to a subsequence): |∇uεn |2 + ∂B(aj ,R0 )

2 2C 1 1 − |uεn |2 2 εn R

∀n.

We can now apply Theorem 3.2 to {uεn } on the domain B(aj , R0 ) to conclude in particular that |uεn | → 1 uniformly on B(aj , R0 ). But since there exists at least one i ∈ Λj , so that |uεn (xiεn )| 1/2 ∀n, we get a contradiction. The case aj ∈ ∂G is treated similarly, see [9]. Note that from Lemma 3.6 it follows that N2 = D. Next we rule out singularities on the boundary. L EMMA 3.7. We have aj ∈ G, j = 1, . . . , D. P ROOF. Assume by contradiction that aj ∈ ∂G for some j . A generalization of Lemma 3.5 to half-annulus (similar to [9], Lemma VI.1) gives |∇uεn |2 4π log B(aj ,R)∩G\B(aj ,δ)

R − C, δ

(3.71)

for all small δ, R with 0 < δ < R. But as in the proof of (3.68), (3.71) contradicts the upper bound for δ small enough. We are now in position to state and prove the main convergence result for {uεn }. T HEOREM 3.4. Up to a subsequence we have

uε n → u∗ for some u∗ ∈

7

\ {a1 , . . . , aD } ∀α < 1 and in W 1,p (G) ∀p ∈ [1, 2), in C 1,α G (3.72)

1p 0 we have by Proposition 3.7 and (3.32): G\

|∇uεn |2 2πD log

D

j=1 B(aj ,η)

1 + C. εn

\ {a1 , . . . , aD }) by Theorem 3.2 and Fubini, as This leads to the convergence in C 1,α (G in the proof that Kj = 0 above, see [9] for details. We now describe the argument of [48] for the proof of the global W 1,p -convergence, for any p ∈ (1, 2). Fix a small σ > 0 so that B(aj , σ ) ⊂ G ∀j . For any integer k = 0, 1, . . . we have by Proposition 3.7 that D

j=1 B(aj ,σ/2

k+1 )

|∇uεn |2 2πD log

σ 2k+1 ε

n

− C.

(3.73)

By (3.73) and (3.32) we obtain D

j=1 B(aj ,σ/2

k )\B(a

j ,σ/2

k+1 )

|∇uεn |2 2πD log

2k+1 + C. σ

(3.74)

By Hölder inequality and (3.74), we get D

j=1 B(aj ,σ )

|∇uεn | = p

∞ D

j=1 B(aj ,σ/2

k=0

C

k )\B(a

j ,σ/2

k+1 )

|∇uεn |p

∞ p/2 σ 2(1−p/2) log 2k /σ + 1 2k k=0

C

∞

(1 + k)p/2 2(p−2)k < ∞.

(3.75)

k=0

By (3.75) it follows that {uεn } is bound in W 1,p and therefore a subsequence converges weakly to u∗ . But given any δ > 0 we can ﬁnd by (3.75) a k0 such that

δ |∇uεn |p , k0 2 j=1 B(aj ,σ/2 )

D

k0 1 while on G \ D j =1 B(aj , σ/2 ) we have even convergence in C norm, hence certainly in W 1,p . These two facts clearly imply the strong convergence in W 1,p (G). From what we proved so far it follows that the limit u∗ is a smooth S 1 -valued harmonic \ {a1 , . . . , aD } (i.e., it satisﬁes −u∗ = |∇u∗ |2 u∗ , or equivalently, it can be map in G written locally as u∗ = eiφ with φ a harmonic function) with degree 1 around each aj . We conclude by citing without proof two more precise results from [9] on the limit u∗ .

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I. Shafrir

T HEOREM 3.5. The limit u∗ is the canonical harmonic map associated with a1 , . . . , aD and the degrees 1, . . . , 1, so that in particular it can be written as u∗ (z) = e

iφ(z)

D 4 z − aj , |z − aj |

j =1

which is determined (up to an additive conwhere φ is a smooth harmonic function in G stant, a multiple of 2π ) by the requirement u∗ = g on ∂G. As for the location of the points a1 , . . . , aD we have the following: T HEOREM 3.6. The conﬁguration (a1 , . . . , aD ) minimizes the renormalized energy W over (b1 , . . . , bD ) ∈ GD which is deﬁned by W (b1 , . . . , bD ) = −2π

i=j

log |bi − bj | − 2π

R(bi , bj ),

i,j

where R(x, y) = Ψ (x, y) − log |x − y| and Ψ (x, y) is the solution of ⎧ ⎪ ⎨ x Ψ (x, y) = 2πδy in G, g×gτ ∂Ψ on ∂G, ∂νx = D ⎪ ⎩

∂G Ψ (g × gτ ) dσ (x) = 0. R EMARK 3.3. A different and more general method to derive the lower bound (3.61) was found independently by Jerrard [28] and Sandier [39]. One of its advantages is that it applies to maps which are not necessarily minimizers, but merely satisfy the upper bound (3.32). In particular, this method is very useful in the study of the full Ginzburg– Landau functional, i.e., including the magnetic ﬁeld. On this subject see the works of Serfaty [43,44], and Sandier and Serfaty [40–42]. More precise results on the convergence of {uε } can be found in the work of Comte and Mironescu [19], and Pacard and Rivière [36]. Questions of symmetry of minimizers on a disc and of global solutions for the Euler–Lagrange equation are addressed in papers by Mironescu [32,33] and in [36]. A study of the asymptotic behavior of nonminimizing solutions of (3.7) is carried out in [9], Chapter X. Analogous problems in higher dimension, i.e., of complex valued maps deﬁned on domains G in RN , N 3, were subject of intensive recent research, starting from the works of Rivière [37] and Lin and Rivière [31], see also Bethuel, Brezis and Orlandi [11,12], Bourgain, Brezis and Mironescu [10], and Jerrard and Soner [29]. 4. Minimization of Ginzburg–Landau energy when g is not S 1 -valued In the previous section we described the results of [8,9] for the problem (3.2) when the boundary condition g is S 1 -valued. In this section we shall investigate what happens when we remove this requirement, i.e., when we allow a general (smooth) g : ∂G → C. It turns

On a class of singular perturbation problems

339

out that the situation becomes extremely delicate in case we allow g to vanish. In this case we shall be able to give a solution only under some precise assumptions on the zeros of g and the behavior of g near these zeros (see Section 4.2). The easier case, of nonvanishing g, is the subject of the next section. 4.1. The case of boundary condition without zeros In this section we describe the results of [2] for a boundary condition g satisfying g ∈ C ∞ ∂G, C \ {0} .

(4.1)

As before, G is supposed to be a smooth, simply connected bounded domain in R2 . Since the boundary condition g is no more “compatible” with the potential, we expect a contribution of boundary interaction to the energy. This energy is expected to concentrate in a thin boundary layer near ∂G, of the same type as the one we encountered in the scalar problems in Section 2. In the interior of the domain we expect the minimizer to behave like in the case of S 1 -valued boundary condition, as described in Section 3. In order to separate between these two different behaviors, near the boundary, and in the interior, an energy decomposition formula, which is based on an argument of Mironescu and Lassoued [30], plays an important role. It involves the minimizer ρε for the scalar minimization problem, 2 1 1 (G) , where Eε (ρ) = |∇ρ|2 + 2 1 − ρ 2 . (4.2) min Eε (ρ): ρ ∈ H|g| ε G This is a special case of the problem described in Section 2.4 which is much simpler than the general case since the positivity of the boundary condition implies positivity of the minimizer. Therefore, instead of a two-phase problem (with the wells ±1) we have a onephase problem (with the only well +1)! We remark that the minimizer ρε is unique (this follows, for example, from [18]). In fact, it is the unique solution to ⎧ ⎨ −ρ = ρ 0 ⎩ ρ = |g|

2 ε2

1 − ρ2 ρ

in G, in G, on ∂G.

(4.3)

Further, as in (3.9) we have the gradient estimate ∇ρε L∞ (G)

C . ε

(4.4)

The decomposition formula is given in the next lemma. L EMMA 4.1. For any u ∈ Hg1 (G, C), we have, with v = u/ρε , Eε (u) =

|∇ρε |2 + G

1 2 2 1 − ρ + ε ε2

G

ρε2 |∇v|2 +

1 4 2 2 1 − |v| ρ . ε ε2

(4.5)

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I. Shafrir

P ROOF. We ﬁrst calculate

2 2 1 ∇(ρε v) + 2 Eε (u) = Eε (ρε v) = 1 − |ρε v|2 ε G G 2 1 = Eε (ρε ) + ρε2 |∇v|2 + 2 ρε4 1 − |v|2 ε G G 2 1 2 2 2 2 |v| − 1 |∇ρε | + 2 ρε ρε − 1 + + ∇(ρε2 )∇ |v|2 − 1 . ε 2 G G Denoting 8ε (v) := E

G

ρε2 |∇v|2

1 + 2 ε

G

2 ρε4 1 − |v|2 ,

(4.6)

we get by Green’s theorem: 8ε (v) Eε (u) = Eε (ρε ) + E 2 1 2 + |v| − 1 − ρε2 + |∇ρε |2 + 2 ρε2 ρε2 − 1 . 2 ε G But the last integral is zero since the Euler–Lagrange equation (4.3) implies that 1 2 ρ2 ρε = 2 2ε ρε2 − 1 + |∇ρε |2 . 2 ε

A direct consequence of the decomposition (4.5) is that uε is a minimizer for Eε over 8ε over H 1 (G, C). This minimization Hg1 (G, C) iff vε := uε /ρε is a minimizer for E g/|g| 8ε resembles the minimization problem of Section 3 since the boundary conproblem for E 8ε depend dition g/|g| is again S 1 -valued. On the other hand, the coefﬁcients of the energy E on the function ρε and it is not clear a priori what is the effect on the behavior of the minimizers {vε }. However, the next proposition shows that the values of ρε are very close to 1, except for a boundary layer of width O(ε). This fact will imply indeed that the behavior 1 (G, C). In the sequel we of {vε } is very similar to that of the minimizers of Eε over Hg/|g| shall denote by δ = δ(x) the distance of the point x from ∂G. P ROPOSITION 4.1. There exists a constant C > 0 such that ρε (x) − tanh tanh−1 g(x) + δ(x) Cε ∀x ∈ G, ∀ε. ε

(4.7)

We shall not give the proof of Proposition 4.1 since it follows from the same technique used in Lemma 4.4 (this estimate can also be deduced from a result of Berger and Fraenkel [7] and from a more general result, [3], Proposition 2.1). Proposition 4.1 is a global result. Away from the boundary, we can even show that ρε tends to 1 in an exponential rate as a function of εδ :

On a class of singular perturbation problems

341

L EMMA 4.2. There exists a constant C > 0 such that for all x ∈ G and all ε we have 1 − ρε (x) Ce −δ 2ε

(4.8)

and ∇ρε (x) C δ

# 2 $ −δ δ + 1 e 2ε . ε

(4.9)

P ROOF. Put a = min g(x): x ∈ ∂G and b = max g(x): x ∈ ∂G . Since w ≡ min(a, 1) (w ≡ max(b, 1)) is a subsolution (respectively, super solution) in (4.3), we obtain that min(a, 1) ρε (x) max(b, 1) ∀x ∈ G. Fix any x ∈ G and denote δ = δ(x). For y ∈ B(x, δ) deﬁne

−1 w1 (r) = tanh tanh a + 1

δ 2 −r 2 3δε

if a < 1, if a 1,

with r = r(y) = |y − x|. When a < 1 a direct computation gives w1 8r 2 4 = 2 2 1 − w12 w1 + 1 − w12 r 9δ ε 3δε 8 4 1 − w12 . 2 1 − w12 w1 + 9ε 3δε

−w1 = −w1 −

We may consider only δ 12ε a (otherwise (4.8) is clear while (4.9) follows from (4.4)). 4 a Then 3δε 9ε2 and −w1 ε12 (1 − w12 )w1 . Since w1 = a ρε on ∂B(x, δ) it follows that w1 is a subsolution for ⎧ ⎨ −ρ = ρ 0 ⎩ ρ = ρε

1 ε2

1 − ρ2 ρ

in B(x, δ), in B(x, δ), on ∂B(x, δ).

Clearly w1 ≡ 1 is a subsolution in case a 1. Similarly we deﬁne −1 w2 (r) = coth tanh b + 1

δ 2 −r 2 3δε

if b > 1, if b 1.

(4.10)

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I. Shafrir

The same calculation as above gives (when b > 1) 8r 2 4 1 − w22 w2 + 1 − w22 2 2 3δε 9δ ε 4 8 2 1 − w22 w2 + 1 − w22 , 3δε 9ε

−w2 =

1 2 and for δ 12ε a we have −w2 ε 2 (1 − w2 )w2 . It follows that w2 is a super solution for (4.10) (this is trivial for w2 ≡ 1 in the case b 1). As remarked above, the solution for (4.10) is unique, and we conclude that w1 ρε w2 , hence

|1 − ρε | Ce−

2(δ 2 −r 2 ) 3δε

on B(x, δ).

In particular, δ

|1 − ρε | Ce− 2ε

on B(x, δ/2),

which implies (4.8). In order to prove (4.9) we deﬁne the function ρ˜ε (y) = ρε (x + 2δ y) on B(0, 1). It satisﬁes − ρ˜ε − 1 = 2

δ 2ε

2

1 − ρ˜ε2 ρ˜ε .

By standard elliptic estimates we have ∇ ρ˜ε (0) C ρ˜ε − 1

L∞ (B(0,1))

+ ρ˜ε − 1L∞ (B(0,1))

# 2 $ δ δ C + 1 e− 2ε . ε The result follows since ∇ ρ˜ε (0) = 2δ ∇ρε (x).

The next lemma provides an estimate for the energy of ρε . L EMMA 4.3. Eε (ρε ) =

2 ε

∂G

|g|3 2 − |g| + 3 3

+ O(1).

(4.11)

P ROOF. The upper-bound in (4.11) is proved by an explicit construction, in the spirit of the results of Section 2.2; see [2], Appendix, for details. For the lower-bound we ﬁx a smooth which satisﬁes |V (x)| 1 on G and V (x) = n(x) on ∂G, where vector ﬁeld V (x) on G n(x) denotes the unit normal to ∂G at x. Then we have by Cauchy–Schwarz inequality and

On a class of singular perturbation problems

343

Green’s theorem: 2 Eε (ρε ) ε

1 − ρ 2 ∇ρε G

ε

2 ρε3 − ρε + ∇ ·V 3 3 G |g|3 ρε3 2 2 2 2 − |g| + − ρε + − div V . = ε ∂G 3 3 ε G 3 3

2 ε

Since

2 3

− ρε +

ρε3 3

(4.12)

= 13 (1 − ρε )2 (ρε + 2), we get from (4.12) that

2 Eε (ρε ) ε

∂G

|g|3 2 − |g| + 3 3

− CεEε (ρε ),

and (4.11) follows.

Using Lemma 4.2 and Proposition 4.1 the asymptotic analysis of the minimizers {vε } 8ε over H 1 (G, R2 ) can be carried out using similar techniques to those of [9,48]. for E g/|g| This leads to the following theorem which is proved in [2]. We shall not give the details of the proof here since we shall prove later in Section 5 an analogous result for a more general potential. We only remark that Lemmas 4.1 and 4.3 are used in (4.13). T HEOREM 4.1. Let g : ∂G → R2 \ {0} be a smooth boundary condition with D = deg(g/|g|) 0. Then there is a subsequence εn → 0 and exactly D points a1 , . . . , aD in G such that uεn → u∗ = eiφ0

D 4 z − aj |z − aj |

k in Cloc G \ {a1 , . . . , aD } ∀k,

j =1

where φ0 is a smooth harmonic function which is determined by the condition u∗ = g/|g| on ∂G. Moreover, Eε (uε ) =

2 ε

∂G

|g|3 2 − |g| + 3 3

+ 2πD log

1 + O(1). ε

(4.13)

It should be noted that the energy Eε (uε ) is composed of two singular parts. The ﬁrst, of the order 1ε , depends only on |g|, while the second and smaller one, of the order 2πD log 1ε , is determined completely by the “projected” boundary condition, g˜ := g/|g|. The location of a1 , . . . , aD is determined by g˜ alone by the requirement to be a minimizing conﬁguration for the renormalized energy associated with g˜ (see Theorem 3.6).

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4.2. The case of boundary condition with zeros A challenging open problem is to describe the limiting behavior of the minimizers {uε } of (3.2) for an arbitrary smooth boundary condition g : ∂G → R2 , i.e., when g is allowed to have zeros. In [4] a special class of these boundary conditions was studied, namely those with a ﬁnite number of zeros. As an example, consider the boundary condition g(z) = z on G = B(1, 1) (= the unit disc centered at the point 1 = (1, 0)) which has a single zero at 0 ∈ ∂G. In contrast with Section 4.1, the projection of g, g˜ = g/|g|, which is expected to determine the form of the limit u∗ , is now singular. It has a “phase jump” of order π at 0, see Figure 3. This is typical for the kind of boundary data that we shall consider. The assumptions we make on g are as follows. The zero set of g is ﬁnite and is denoted by Z = σ ∈ ∂G; g(σ ) = 0 = {b1 , . . . , bk }.

(4.14)

Near each zero bj the following behavior is assumed, g(σ ) = |σ − bj |αj hj (σ )

for some αj > 0

(4.15)

with hj a positive C 2 -function. We also assume that g˜ := g/|g| = eiΘ ∈ C 2 ∂G \ {b1 , . . . , bk } (hence also Θ ∈ C 2 (∂G \ {b1 , . . . , bk })), and that for some {Φj }kj =1 , {Θj }kj =1 we have lim Θ(σ ) = Θj

σ →bj−

and

lim Θ(σ ) = Θj + Φj .

σ →bj+

(4.16)

Here the one-sided limits at bj± are taken in accordance with the positive sense on ∂G. Note that in general (i.e., for k > 1, see (4.17)) the value of each Φj is determined only modulo 2π , but the following relation must hold: k

Φj +

j =1

∂G\{b1 ,...,bk }

g˜ ∧ g˜σ = 0.

(4.17)

Therefore, the quantity kj =1 Φj is completely determined by g. It will be convenient to

j }k of admissible jumps (i.e, satisfying (4.16) and (4.17)). As in Section 4.1 ﬁx a set {Φ j =1

we expect the energy Eε (uε ) to decompose into two terms: the ﬁrst of the order 1ε , as in (4.11), and the second, of order log 1ε , depending on the singular limit u∗ . Let us try to “guess” then what are the possible candidates for limits of {uε }. We expect such a limit u∗ to be a smooth S 1 -valued map in G, except for a ﬁnite number of singularities. A certain number of them, D (possibly zero), are interior singularities a1 , . . . , aD ∈ G, z−a all of the same degree s = ±1, such that near each aj , u∗ (z) ∼ eicj ( |z−ajj | )s (as in Section 3). Singularities at the points {bj }kj =1 are expected as well. At each bj we expect

On a class of singular perturbation problems

345

Fig. 3. The boundary condition g(z) = z on B(1, 1).

j + 2πdj , for some integer dj , so that in a neighbora “jump of phase” of the order Φ z−b

j ic −( hood of bj , u∗ (z) ∼ e j ( |z−bj | ) Φj /π+2dj ) , for some integer dj . A priori, u∗ may have singularities at points in ∂G \ {b1 , . . . , bk }, but the proof in [4] shows that this possibility is excluded. The choice of D, s and {dj }kj =1 is not arbitrary: Using (4.17) it is not difﬁcult to see that we must have: k

dj = sD.

(4.18)

j =1

Let us denote the class of “admissible limits” by "

D k 4 z − aj s 4 z − bj −(Φj /π+2dj ) · : A = u∗ = e · |z − aj | |z − bj | iφ

j =1

j =1

φ is a smooth harmonic function in G,

'

u∗ = g˜ on ∂G \ Z, s = ±1 s.t. (4.18) holds .

(4.19)

In order to determine which u∗ ∈ A is actually “chosen” by {uε } we should compute its “energy cost”. First we note that as in Section 4.1, we can write for each u ∈ Hg1 (G, C), u = ρε v, where ρε is the minimizer in (4.2). Then, a variant of the proof of Lemma 4.1 yields: 8ε (v), Eε (u) = Eε (ρε ) + E

(4.20)

8ε as deﬁned in (4.6). Since the quantity Eε (ρε ) does not depend on u (an estimate for with E 8ε (vε ), it is given by (4.11)), the asymptotic behavior of Eε (uε ) is determined by that of E

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I. Shafrir

with vε = uε /ρε . For that matter we should have on our hands good pointwise estimates for ρε , especially near the zeros of g. The next lemma from [4] is used in the proof of a 8ε (vε ). lower-bound for E We shall assume in the sequel for simplicity that g has only one zero, i.e., k = 1 and Z = {b1 }.

(4.21)

We denote for any η > 0, Gη = {x ∈ G: δ(x) < η}. By [26], Section 14.6, there exists b0 > 0 such that δ ∈ C 2 (Gb0 ) and any x ∈ C 2 (Gb0 ) has a unique nearest point projection σ (x) ∈ ∂G. L EMMA 4.4. There exists a positive constant C1 such that ρε (x) tanh tanh−1 g σ (x) + δ(x)/ε − C1 εα0 ∀x ∈ G, ∀ε, with α0 = min(α1 , 1).

(4.22)

P ROOF. For each ε ∈ (0, 1) let fε denote a function in C 2 (∂G) satisfying: (i) fε (σ ) = |g(σ )| if |σ − b1 | > ε, (ii) 0 fε (σ ) |g(σ )| if |σ − b1 | ε, (iii) |fε (σ )| Cεα1 −1 and |fε (σ )| Cεα1 −2 if |σ − b1 | ε. The existence of such fε is clear from our assumptions on g. Let ρ¯ε denote the minimizer of Eε with the boundary condition fε (again, it is unique by [18]). Put δ(x) −1 ρ˜ε (x) = tanh tanh fε σ (x) + . ε Since ρε ρ¯ε , it is enough to show that ρ¯ε ρ˜ε − C1 εα0

on G.

(4.23)

On Gη0 we may use the coordinates (σ, δ) and write for every w ∈ C 2 (Gη0 ), w = wδδ + wδ div n + wσ σ + wσ div s, where n = ∇δ and s is a unit vector ﬁeld, orthogonal to n. By a direct computation we have on Gη0 , −ρ˜ε =

2 1 − ρ˜ε2 ρ˜ε + eε , ε2

with eε (x) Cεα0 −2 .

(4.24)

On a class of singular perturbation problems

We claim that there exists a positive constant C > 0 such that ρ˜ε (x) − ρ¯ε (x) Cεα0 ∀x ∈ G, ∀ε ∈ (0, 1).

347

(4.25)

This will certainly imply (4.23). Our proof of (4.25) follows a similar argument to the one used in [3], Proposition 2.1. Suppose by negation that (4.25) does not hold. Then, for a sequence εn → 0, we have (4.26) lim εn−α0 ρ˜εn − ρ¯εn L∞ (Ω) = +∞. n→∞

Let xn denote a maximum point of |ρ˜εn − ρ¯εn | over G. Passing to a subsequence if necessary, we may assume without lost of generality that ρ˜εn (xn ) > ρ¯εn (xn ) for all n. By (4.8) we have δ(x) 1 − ρ¯ε (x) Ce− 2ε ∀x ∈ G, (4.27) and a similar estimate clearly holds for ρ˜ε . Thus, if xn ∈ G \ Gη0 for an inﬁnite number η0

of n’s, then |ρ˜εn (xn ) − ρ¯εn (xn )| Ce− 2εn , which clearly contradicts (4.26) for n large. Therefore we may assume that xn ∈ Gη0 for all n. Next we note that by (4.27) and the corresponding estimate for ρ˜ε we have for some K > 0, ρ˜εn (x), ρ¯εn (x)

2/3 ∀x ∈ G \ GKεn .

(4.28)

Assume ﬁrst that xn ∈ G \ GKεn for an inﬁnite number of indices n. Then, 2 3rn − 1 0 − ρ˜εn − ρ¯εn (xn ) = eεn (xn ) − 2 ρ˜εn − ρ¯εn (xn ), 2 εn where rn is a number lying between ρ¯εn (xn ) and ρ˜εn (xn ). This yields ρ˜εn (xn ) − ρ¯εn (xn ) 1 2 2 eεn (xn )εn , contradicting (4.26) and (4.24). In the remaining case we can assume that xn ∈ GKεn for all n. Hence, passing to a subsequence we may suppose that xn → σ¯ ∈ ∂G and that the following limit exists, t˜ = lim

tn

n→∞ εn

(4.29)

.

Using the coordinates (σ, δ) we denote xn = (σn , δn ) and then deﬁne two sequences of rescaled functions on the domain Dεn = {(s, t): (σn + εn s, εn t) ∈ G} by w˜ εn (s, t) = ρ˜εn (σn + εn s, εn t)

and w¯ εn (s, t) = ρ¯εn (σn + εn s, εn t).

1 (R2 ) (with From standard elliptic estimates it follows that w˜ εn → w˜ and w¯ εn → w¯ in Cloc + 2 R+ = {(s, t); t > 0}) where w(s, ˜ t) and w(s, ¯ t) are both solutions of

"

−w = 2 1 − w2 w w = g(σ¯ )

in R2+ , on ∂R2+ .

(4.30)

348

I. Shafrir

But by a result of Angenant [1], the nonnegative solution of (4.30) is unique, and it is a function of the variable t only. In fact, in our case we have the explicit formula, w(s, ˜ t) = w(s, ¯ t) = tanh(a + t) with a = tanh−1 g(σ¯ ) . Next we deﬁne, Vεn (x) =

ρ˜εn (x) − ρ¯εn (x) . ρ˜εn (xn ) − ρ¯εn (xn )

By assumption, |Vεn (x)| 1 ∀x ∈ G and Vεn (xn ) = 1. The equation satisﬁed by Vεn is −Vεn =

2 eε n , 1 − 3Rε2n Vεn + 2 ρ˜εn (xn ) − ρ¯εn (xn ) εn

(4.31)

where Rεn (x) is a point lying between ρ˜εn (x) and ρ¯εn (x). Deﬁning a rescaled sequence by

εn (s, t) = Vεn (σn + εn s, εn t) as above, we may pass to the limit in (4.31), using (4.24), V

where V

satisﬁes

εn → V (4.25) and (4.26) and the fact that w¯ = w˜ to infer that V ⎧ 2 2

⎪ ⎨ −V = 2 1 − 3w˜ V in R+ ,

=0 V on ∂R2+ , ⎪ ⎩ ˜ V 0, t = 1 (see (4.29)).

(4.32)

But by [1] there is no solution to (4.32). This contradiction completes the proof of the proposition. Using Lemma 4.4 we shall deduce the following estimate (4.35) which is essential for 8ε (vε ). Recall that we make the simplifying assumpestablishing the lower-bound for E tion (4.21). We shall need some notation. First, we ﬁx β ∈ (0, 1) satisfying max

1 α0 9 ,1 − , < β < 1. 1 + α0 2 10

The reasoning for this choice is given in [4]. We then denote by b˜1 the unique point on ∂(G \ Gεβ ) satisfying σ (b˜1 ) = b1 and let 1

ε = σ˜ ∈ ∂(G \ Gεβ ): σ˜ − b˜1 > Kεmin(1, α1 ) , Σ

(4.33)

for some K > 0 whose value will be ﬁxed in the course of the proof of the next proposition.

ε → R+ by Finally, we deﬁne a function pε : Σ ⎧ 1−β min(1, α1 ) ⎪ ε ⎨ ˜1 ε α1 , 1 σ if Kε ˜ − b α1 ˜ (4.34) pε (σ˜ ) = |σ˜ −b1 | 1−β ⎪ ⎩ εβ α1 ˜ if σ˜ − b1 ε ,

On a class of singular perturbation problems

349

P ROPOSITION 4.2. There exists a constant c0 > 0 such that for every small ε > 0 we have 8ε (vε ) c0 E

ε Σ

|g(σ ˜ (σ˜ )) − vε (σ˜ )|2 8ε (vε , G \ Gεβ ). dσ˜ + E pε (σ˜ )

(4.35)

P ROOF. We identify each point x ∈ Gεβ with the pair (σ˜ , δ) = (σ˜ (x), δ(x)), where σ˜ (x) denotes the nearest point projection of x on ∂(G \ Gεβ ). By the Cauchy–Schwarz inequality we get Gε β

ρε2 |∇vε |2

1 2 1 2

ε Σ

εβ 0

2

∂vε ρε2 ∂δ

ε β ∂v ε

dδ dσ˜ 2

εβ

dδ dδ ∂δ ρε2

ε 0 0 Σ 2 1 β g˜ σ (σ˜ ) − vε σ˜ , ε 2

−1

εβ

0

Σε

dσ˜

dδ ρε2

−1

dσ˜ .

(4.36)

ε we have, by Lemma 4.4, denoting r = r(x) = |σ˜ (x) − b˜1 |, For x ∈ Gεβ with σ˜ (x) ∈ Σ ρε (x) tanh tanh−1 g σ (x) + δ/ε − Cεmin(α1 ,1) tanh c r α1 + δ/ε − Cεmin(α1 ,1) tanh cr α1 + δ/ε .

(4.37)

Indeed, the last inequality in (4.37) holds, provided we choose K large enough in (4.34), as can be veriﬁed by considering separately the cases α1 1 and α1 > 1, and using the

ε . A direct calculation gives deﬁnition (4.33) of Σ

εβ 0

dδ tanh2 (cr α1 + εδ )

= ε εβ−1 − coth εβ−1 + cr α1 + coth cr α1 ,

which together with (4.37) leads to,

ε , ∀σ˜ ∈ Σ

εβ 0

⎧ ⎨ cε dδ r α1 ρε2 (σ˜ , δ) ⎩ β cε

Plugging (4.38) in (4.36) yields the result.

for r ε for r > ε

1−β α1

,

1−β α1

.

(4.38)

8ε (vε ). We shall see later how to continue from Proposition 4.2 to get a lower bound for E Now we show how it motivates an upper-bound construction, which is more involved than the corresponding ones in Sections 3.2 and 4.1. We describe in Proposition 4.3 the basic construction in a special case which demonstrates the basic idea of the general case.

350

I. Shafrir

P ROPOSITION 4.3. Let ∂G be ﬂat near the point 0 ∈ ∂G, with g(0) = 0 being the unique zero of g, so that for some R > 0, ∂G ∩ B(0, R) coincides with the half-disc B + (0, R) := B(0, R) ∩ {x1 > 0}. Let g be given in {(x2 , 0): x2 ∈ [−R, R]} by g(0, x2 ) =

|x2 |α1 eiΦ/2 |x2 |α1 e−iΦ/2

for 0 < x2 R, for − R x2 < 0.

(4.39)

Let φ such that φ ≡ Φ(mod 2π) be given. Then for every ε > 0 there exists a map V = Vε on B + (0, R) satisfying V (x) =

x |x|

φ/π

on ∂B + (0, R) \ {0}

(4.40)

1 φ2 log + O(1). π(α1 + 1) ε

(4.41)

and 8ε V , B + (0, R) = E

S KETCH OF THE PROOF. We shall assume the following pointwise upper bound for ρε on B + (0, R), for small enough R: x1 . ρε (x1 , x2 ) min 1, C |x2 |α1 + ε

(4.42)

This estimate can be justiﬁed, at least for α1 large enough, by the argument of Lemma 4.4. We denote by 0˜ the point (ε, 0) and let (r, θ ) denote polar coordinates centered at the ˜ It will be convenient to describe the construction on a domain slightly larger then point 0. B + (0, R), namely +

ε ∪ D

ε ∪ G− DR,ε = A ε ∪ Gε ∪ Bε ,

where G− ε = [0, ε] × [−R, −ε], ˜ ε ∩ {x1 > ε},

ε = B 0, D

G+ Bε = [0, ε] × [−ε, ε], ε = [0, ε] × [ε, R], ˜ R \ B 0, ˜ ε ∩ {x1 > ε};

ε = B 0, A

see Figure 4. In DR,ε \ Bε , we shall deﬁne V as V = eif , with a properly chosen scalar function f . In

Aε we set f (r, θ ) = f˜(r)

θ π/2

(4.43)

On a class of singular perturbation problems

351

Fig. 4. An upper-bound construction near a zero of g.

with a function f˜ : [ε, R] → R that will be prescribed below. And now comes the point where the lower-bound estimate (4.35) motivates the upper-bound construction. The function f˜ is chosen as to minimize ∂V

ε ∂τ A

2 +c

[−R,−ε]∪[ε,R]

|V (ε, x2 ) − g(0, x2 )|2 dx2 ε/|x2 |α1

or rather 2 −ε ∂f |f (ε, x2 ) + φ/2|2 +c dx2 ε/|x2 |α1

ε ∂τ −R A R |f (ε, x2 ) − φ/2|2 +c dx2. ε/|x2 |α1 ε

(4.44)

Note that if we take f of the form (4.43) then it is an even function on the x2 -axis, and we can rewrite the expression (4.44) as 2 2$ ˜ ∂f + 2c |f (r) − φ/2| dr ε/r α1 ε Sr+ ∂τ $ R R# 4 ˜2 |f˜(r) − φ/2|2 dr := hr f˜ dr. f (r) + 2c = α1 πr ε/r ε ε

J f˜ :=

R #

(4.45)

352

I. Shafrir

In view of (4.45), a minimizing f˜ will be obtained by choosing f˜(r), for each ﬁxed r, as a minimum point of the function hr (f˜). A simple computation gives 0=

8 ˜ 4c(f˜ − φ/2) dhr ⇒ , f+ πr ε/r α1 df˜

which gives, with c˜ =

πc 2 ,

f˜(r) = 1 −

φ ε φ 8/(πr) = 1 − . 8/(πr) + 4cr α1 /ε 2 ε + cr ˜ α1 +1 2 1

R EMARK 4.1. Note that for r ε α1 +1 we have f˜(r) ∼ = 1 α1 +1

, f˜(r) differs signiﬁcantly from to the order of ε ˜ characteristic length of the problem near 0.

φ 2.

φ 2

(4.46)

and only when r is decreased 1

Therefore we see that ε α1 +1 is the

ε in such a way that E 8ε (V , DR,ε \ A

ε ) C. C LAIM . V can be extended to DR,ε \ A − First, in G+ ε (and analogously in Gε ) we deﬁne f in such a way that equality will hold in the following Cauchy–Schwarz inequality (for all x2 ∈ [ε, R]):

|f (ε, x2 ) − φ/2|2

ε α1 x1 −2 dx 1 0 (x2 + ε ) In fact, since

ε 0

2 ε x1 2 ∂f α1 x2 + (x1 , x2 ) dx1 . ε ∂x1 0

(x2α1 + x1 /ε)−2 dx1 =

ε , α α x2 1 (x2 1 +1)

we simply take f which satisﬁes,

1 ∂f (x1 , x2 ) is proportional to α1 . ∂x1 (x2 + x1 /ε)2 So using the constraints f (0, x2 ) =

φ 2

and f (ε, x2 ) = f˜(x2 ) we are led to

x1 α dt/(x2 1 + t/ε)2 φ φ 0 ˜ f (x1 , x2 ) = − ε − f (x2 ) α1 2 2 2 0 dt/(x2 + t/ε) α α φ x1 /(x2 1 (x2 1 + x1 /ε)) φ ˜ − f (x ) = − 2 2 2 ε/(x2α1 (x2α1 + 1)) x1 ε(x2α1 + 1) φ = 1− . 2 (εx2α1 + x1 )(ε + cx ˜ 2α1 +1 )

By a direct computation we then ﬁnd (using (4.42)) that G+ε ρε2 |∇f |2 C and similarly,

ε ∪ Bε is possible with cost of ρε2 |∇f |2 C. Further, a special extension of V to D G− ε energy of the order O(1) only, see [4] for details.

On a class of singular perturbation problems

353

In view of the claim, the proof would be completed once we show that,

ε A

|∇f |2 =

1 φ2 log + O(1). π(α1 + 1) ε

(4.47)

First, it is not difﬁcult to verify that 2 ∂f = O(1).

ε ∂r A

(4.48)

Next, by (4.43) and (4.46) we have 2 ˜ ∂f = f (r) ∂τ πr ε φ , 1− = πr ε + cr ˜ α1 +1 which yields 2 cR 2 R ˜ α1 +1 ∂f φ2 (cr ˜ α1 +1 )2 dr t dt =φ = ∂τ α1 +1 )2 r α +1 π π(α + 1) (ε + cr ˜ (ε + t)2

1 1 ε Aε cε ˜ cR cR ˜ α1 +1 dt ˜ α1 +1 εφ 2 φ2 dt − = α +1 α +1 π(α1 + 1) cε ε+t π(α1 + 1) cε (ε + t)2 ˜ 1 ˜ 1 R α1 +1 φ2 + O(1). log = π(α1 + 1) ε Combining (4.48) and (4.49) we are led to (4.47).

(4.49)

Next, we return to the general setting, allowing g with several zeros. For each admissible u∗ ∈ A, i.e., u∗ = eiφ

k D 4 z − aj s 4 z − bj −(Φj /π+2dj ) , |z − aj | |z − bj |

j =1

(4.50)

j =1

we can construct a family of maps {Uε = ρε Vε } ⊂ Hg1 (G, C), converging to u∗ , with the asymptotic behavior of the energies given by Eε (Uε ) = Eε (ρε ) + L(u∗ ) log

1 + O(1), ε

(4.51)

where L(u∗ ) = 2πD +

k

j + 2πdj )2 (Φ . π(1 + αj ) j =1

(4.52)

354

I. Shafrir

Note that L(u∗ ) depends only on {dj }kj =1 , since the value of D is then determined by (4.18). In order to construct such a family {Uε }, we ﬁrst use a construction of the type

j + 2πdj . given in the proof of Proposition 4.3 around each bj for a jump of phase φ = Φ This gives a contribution of

j + 2πdj )2 (Φ 1 log + O(1) for each bj . π(1 + αj ) ε Then we use the construction of Proposition 3.3 for D vortices of degrees ±1 around the points a1 , . . . , aD which yields a contribution of 2π| log ε| + O(1) to Eε (Uε ). Note that L(u∗ ) takes into account only the number D of interior singularities {aj }D j =1 , but not their location which is expected to affect only the O(1) term, as in [9]. Motivated by the above, we deﬁne Λ = min L(u∗ ).

(4.53)

u∗ ∈ A

Although the minimization is taken over an inﬁnite set, it is clear that the minimum in (4.53) is attained. From the above discussion we deduce the following upper bound for the energy: 2 Eε (uε ) ε

∂G

2 |g|3 1 − |g| + + Λ log + O(1). 3 3 ε

(4.54)

We can now state our main result. T HEOREM 4.2. Let G and g be as above. Then, there exists a subsequence εn → 0 and u∗ ∈ A of the form (4.50) which realizes the minimum in (4.53), such that uεn → u∗ m (G \ {a , . . . , a }) ∀m. Moreover, strongly in Cloc 1 D 2 Eε (uε ) = ε

∂G

2 |g|3 1 − |g| + + Λ log + O(1). 3 3 ε

(4.55)

The proof of the upper bound in (4.55) (i.e., (4.54)) was sketched above. The proof of the lower bound is much more complicated. The starting point is the estimate (4.35) which motivates the deﬁnition of a new energy Eε (w) := c0

ε Σ

|g(σ ˜ (σ˜ )) − w(σ˜ )|2 8ε (w, G \ Gεβ ). dσ˜ + E pε (σ˜ )

(4.56)

Clearly it is enough to prove that 1 min Eε (w): w ∈ H 1 (G \ Gεβ , C) Λ log + O(1). ε

(4.57)

On a class of singular perturbation problems

355

Note that there is no boundary condition for the problem on the left-hand side of (4.57). Yet, the boundary condition g˜ = g/|g| is “forced”, in the limit, by the penalization (dividing by pε ) in the boundary integral on the right-hand side of (4.56). It seems difﬁcult to prove (4.57) directly, the main difﬁculty being the presence of two 8ε and a pε -scale in the boundary integral. different scalings in the energy Eε : an ε-scale in E It is more convenient to consider instead yet another energy, namely ε (w) = c0 E

ε Σ

|g(σ ˜ (σ˜ )) − w(σ˜ )|2 dσ˜ pε (σ˜ )

+

|∇w|2 + G\Gεβ

2 1 1 − |w|2 , p˜ε

(4.58)

where p˜ε is a certain extension of pε to all of Gεβ , which satisﬁes p˜ε (x) ε ∀x ∈ Gεβ , ε (w) ∀w and p˜ε (x) = ε for δ(x) εβ (see [4] for the precise deﬁnition). Since Eε (w) E it sufﬁces to prove then that ε (w): w ∈ H 1 (G \ Gεβ , C) Λ log 1 + O(1). min E ε

(4.59)

ε is that the scale varies continuously over The advantage in working with the energy E G \ Gεβ . For each ε, we denote by wε a minimizer for the problem on the left-hand side of (4.59). The proof of (4.59) relies on a careful analysis of the minimizers {wε }. Some of the techniques of [9,48,13], as described in Section 3.2, are useful here as well but there are some additional difﬁculties. Here, there are two kinds of “bad points”. The ﬁrst, are “useful bad points”, where the modulus of wε is smaller then (say) 1/2, i.e.,

Sε(i)

1 = x ∈ G \ Gεβ : wε (x) < 2

(cf. (3.41)).

(4.60)

But here we should also take into account the boundary bad points where wε differs signiﬁcantly from g˜ = g/|g|, or more precisely, 1 Sε(b) = σ˜ ∈ ∂(G \ Gεβ ): g˜ σ σ˜ − wε σ˜ > . 2

(4.61)

First, we associate with each b˜j (deﬁned analogously to b˜1 above, i.e., as the unique point on ∂(G \ Gεβ ) satisfying σ (b˜j ) = bj ) a bad half-disc 1 B b˜j , ε 1+αj ∩ (G \ Gεβ ).

(4.62)

356

I. Shafrir 1

The signiﬁcance of the value ε 1+αj , the characteristic length near bj , was already indicated in Remark 4.1. We set Γ εβ := (G \ Gεβ )

9! k

1 B b˜j , ε 1+αj .

(4.63)

j =1

The ﬁrst main step of the proof consists of covering the bad points (Sε ∪ Sε ) ∩ Γ εβ by a ﬁnite number of “bad discs” or “bad half-discs” of the form (i)

B xiε , λp˜ ε xiε ∩ (G \ Gεβ ),

(b)

i = 1, . . . , Nε , with Nε N.

In the second step we use this covering to prove the lower bound ε (wε ) Λ log 1 + O(1), E ε which implies of course (4.59) and (4.57) and then the lower bound in (4.55) follows (see [4] for details of the two steps). Once the energy estimate (4.55) is proved, we can turn to the proof of the convergence assertion in Theorem 4.2. Here we start by applying a variant of the del Pino–Felmer trick [20] (see Corollary 3.1) to get the bound 1 ε2

2 1 − |vε |2 + G\Gεβ

∂(G\Gεβ )

|g(σ ˜ (σ˜ )) − vε (σ˜ )|2 dσ˜ C. p˜ε (σ˜ )

(4.64)

Using (4.64) we can now deﬁne the “bad points” of vε in analogy with (4.60) and (4.61) and show that they can be covered by a ﬁnite number of bad discs/half-discs. By arguments similar to the ones of Section 3.2, but technically more involved (see [4]) we can prove an energy bound away from the singularities and deduce convergence, again away from the singularities. We conclude this section with two examples. E XAMPLE 4.1. This example demonstrates that here, in contrast with Section 4.1, the singular limit u∗ depends not only on g˜ = g/|g| but also on the order of each zero of g. For example, consider the domain G = B(1, 1) with the boundary conditions g (1) (z) = z4 and g (2) (z) = |z|5 z4 so that g (2) (z) g (1) (z) = = |g (1) (z)| |g (2) (z)|

z |z|

4 .

Applying Theorem 4.2 gives different limits for the two problems: for g (1) (z) the limit is z−a1 iφ1 z 2 u(1) ∗ (z) = e ( |z| ) · ( |z−a1 | ), for some a1 ∈ G (and some smooth harmonic function φ1 ) iφ2 z 4 while for g (2) (z) it is of the form u(2) ∗ (z) = e ( |z| ) .

An example with a different ﬂavor is the following.

On a class of singular perturbation problems

357

E XAMPLE 4.2. Consider G = B(0, 1) and the boundary condition g(z) = Re z on ∂G which has two zeros at z = ±i. Note that this g takes only real values, so it makes sense to consider also the scalar minimization problem of the same Eε (u), but over the class Hg1 (G) = Hg1 (G, R) = u ∈ H 1 (G); u = g on ∂G . Indeed, this was the object of Example 2.1 where we saw that the minimizers {uε } of the scalar problem satisfy: uε → u0 in L1 (G), where u0 (z) = sgn(Re z), and further, the asymptotic behavior of the energies is given by (2.43). Next consider the same problem, but now for complex-valued maps, as we do throughout this section. It follows from Theorem 4.2 that uεn → u0 or u¯ 0 , where u0 (z) =

F0 (z) |F0 (z)|

and F0 (z) = i

z−i . z+i

One can see in Figure 5 the level curves of u0 which are circles passing through the two points (0, ±1) = ±i. Moreover, the energy estimate of Theorem 4.2 gives in this case 2 Eε (uε ) = 2 · ε

π/2

3 cos θ − cos θ 3 −π/2

π 1 2 − dθ + 2 · log + O(1). 3 2 ε

(4.65)

Comparing (4.65) to (2.43) we see that a term of the order 1ε , coming from a boundary layer where |uε | varies from the boundary condition |g| to 1, is common for both problems.

Fig. 5. Minimization for g(z) = Re z over C-valued maps.

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I. Shafrir

The difference is that instead of another term of the order 1ε (the ﬁrst term on the righthand side of (2.43)), we have in (4.65) only a term of the order | log ε|. Indeed, this is the cost of having as a limit an S 1 -valued map, with two point singularities, compared with a {−1, +1}-valued map for the scalar problem, which must have line singularities (see Figure 2).

5. The case of a general “circular-well” potential A smooth function W : R2 → [0, ∞) will be called a “circular-well” potential if it satisﬁes W > 0 on R2 \ Γ,

W = 0 on Γ,

(5.1)

for some closed smooth curve (at least of class C 3 ) Γ in R2 . We shall also assume that we are in the generic case, i.e., Wnn > 0

on Γ

(5.2)

(Wnn denotes the second derivative in the normal direction to Γ ), and a technical assumption on the behavior at inﬁnity: ∂W 0 for |z| > R0 . ∂|z|

(5.3)

As in Sections 3 and 4 we consider a bounded, smooth, simply connected domain G in R2 and a smooth boundary condition g : ∂G → R2 (later we shall impose more conditions on g). For each ε > 0 we deﬁne the energy W (u) |∇u|2 + 2 , Eε (u) = ε G and denote by uε a minimizer for the problem min Eε (u): u ∈ Hg1 G, R2 .

(5.4)

As before, we are interested in the asymptotic behavior of the minimizers {uε } and their energies {Eε (uε )}, as ε → 0. Note that the Ginzburg–Landau energy (3.1) is a special case corresponding to the potential W (u) = (1 − |u|2 )2 (and then Γ = S 1 ). If we assume in addition that g is Γ -valued, then the methods of Section 3 can be adapted without too many difﬁculties to prove analogous results to Theorems 3.1 and 3.5. Therefore, we shall concentrate on the more difﬁcult case where g is not Γ -valued, looking for analogous results to those of Section 4.1. There the basic tool was the decomposition formula (4.5). One cannot expect a result of this type to hold for general W , so that another idea is required. An important role in the problem is played by a certain degenerate metric associated with W (of the same type as φ in (2.26)). The next section is devoted to the study of this metric.

On a class of singular perturbation problems

359

5.1. A study of a degenerate metric Let W satisfy conditions (5.1)–(5.3). We deﬁne a function Ψ on R2 by Ψ (ζ ) =

inf

γ ∈Lip([0,1],R2 ) 0 γ (0)∈Γ, γ (1)=ζ

1/2 γ (t) dt. W γ (t)

1

(5.5)

Since the integral in (5.5) is invariant w.r.t. rescaling, we may replace the interval [0, 1] by any other closed interval. The function Ψ can be viewed as a new distance function to Γ , with respect to a degenerate Riemannian metric. It is not difﬁcult to see that Ψ ∈ Lip(R2 ) and that it is a solution of the eikonal-type equation, ∇Ψ (ζ )2 = W (ζ )

a.e. on R2 .

(5.6)

Functions of this type appeared in works on related problems by many authors (c.f. [24,45]) as we saw in Section 2.3. One cannot expect Ψ to be smooth everywhere. For example, when W is a function of the distance to Γ , i.e., W (u) = F (dist(u, Γ )) as in the case of the Ginzburg–Landau energy, Ψ is not differentiable on the skeleton of Γ , see [3]. However, we shall see below that Ψ is smooth in a small neighborhood of Γ . ˜ First we introduce some notations. We denote by δ(x) the (signed) distance function to Γ (with the convention that δ˜ is negative inside Γ and positive outside) and then, for any η > 0, ˜ 0 such that δ˜ ∈ C 2 (Γη0 ) and any x ∈ C 2 (Γη0 ) has a unique nearest point projection σ˜ (x) ∈ Γ . In Γη0 it will be convenient to work with the ˜ coordinates. In particular, from our assumptions (5.1) and (5.2) it follows that we (σ˜ , δ) may write in Γη0 , using these coordinates, W σ˜ , δ˜ = a σ˜ , δ˜ δ˜2 ,

(5.8)

with a a positive C 2 -function. The derivatives of W are then given by ∂W = aσ˜ σ˜ , δ˜ δ˜2 ∂ σ˜

and

∂W ˜ = aδ˜ σ˜ , δ˜ δ˜2 + 2a σ˜ , δ˜ δ. ˜ ∂δ

(5.9)

We may write ∇W =

∂W ∂W τ+ n, ∂ σ˜ ∂ δ˜

with n = ∇ δ˜ and τ = ∇ σ˜ . Since |n| = |τ | = 1, we have ∂n = c σ˜ τ ∂ σ˜

and

∂τ = −c σ˜ n, ∂ σ˜

(5.10)

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I. Shafrir

where c(σ˜ ) denotes the curvature of Γ at σ˜ . From (5.9) and (5.10) we obtain ∂ ∇W σ˜ , 0 = 0 ∂ σ˜

and

∂ ∇W (σ˜ , 0) = 2a σ˜ , 0 n. ∂ δ˜

(5.11)

The main result of this section is the following. P ROPOSITION 5.1. There exists η1 > 0 such that the equation

|∇U |2 = W, U = 0 on Γ,

(5.12)

has a unique C 2 -solution in Γη1 , which moreover coincides there with Ψ . P ROOF. In the nondegenerate case, the proof of such result is classical via the characteristic method, see [22], Section 3.2. We shall use a variant of this method in order to overcome the difﬁculty caused by the degeneracy. We deﬁne an Hamiltonian, H (X, P ) = |P |2 − W (X),

(5.13)

so that the ﬁrst equation in (5.12) can be written as H (x, ∇U ) = 0. We are looking for a solution (X, P ) : Γ × (−∞, c) → Γη1 × R2 of the characteristics system ⎧ ⎪ ⎨ X(x0 , −∞) = x0 ∀x0 ∈ Γ, X˙ = ∂H ∂P = 2P , ⎪ ⎩ P˙ = − ∂H = ∇W (X),

(5.14)

∂X

where dot represents derivative w.r.t. t. The construction of a solution U from (X, P ) is then standard, see (5.30). In order to deﬁne a problem √ on a bounded domain we make the change of variables r = eαt , where α = α(x0 ) = 2 a(x0) (see (5.8)). Using this new variable (5.14) becomes: ⎧ X(x , 0) = x ⎪ ⎨ ∂X 0 2P 0 ∂r = α(x0 )r , ⎪ ⎩ ∂P = ∇W (X) . ∂r

and P (x0 , 0) = 0 ∀x0 ∈ Γ, (5.15)

α(x0 )r

We shall construct a solution X of (5.15) with image in a one-sided neighborhood of Γ , ˜ of the form {x ∈ R2 : δ(x) ∈ [0, η)}, but an analogous argument will give a solution on the ˜ ∈ (−η, 0]}. other side of Γ , namely in {x ∈ R2 : δ(x) Integrating the equations in (5.15) yields an equivalent form:

r

X(x0 , r) − x0 = 0

2P (x0 , s) ds, α(x0 )s

r

P (x0 , r) = 0

∇W (X(x0 , s)) ds. α(x0 )s

On a class of singular perturbation problems

361

(5.16) Let Y and Q be deﬁned by Y (x0 , r) =

X(x0 , r) − x0 r

and Q(x0 , r) =

P (x0 , r) . r

(5.17)

If Q and Y are associated with a solution (X, P ) to (5.16), then by the regularity of W and (5.11), we get

∇W (x0 + sY (x0 , s)) ds α(x0 )s 0 1 r 1 2 = D W (x0 ) sY (x0 , s) + so Y (x0 , s) ds r 0 α(x0 )s r n 2a(x0)Y (x0 , s) · n ds + o(1) = α(x0 )r 0

Q(x0 , r) =

1 r

r

(5.18)

and Y (x0 , r) =

1 r

r 0

2Q(x0 , s) ds. α(x0 )

(5.19)

From (5.18) and (5.19) we deduce that "

Q(x0 , 0) = Y (x0 , 0) =

2a(x0 ) α(x0 ) Y (x0 , 0) · n n, 2Q(x0 ,0) α(x0 ) .

(5.20)

Thus (5.20) implies a compatibility condition on the initial values Q(x0 , 0), Y (x0 , 0). We deduce in particular that Y (x0 , 0)√ must be parallel to n, and we may choose Y (x0 , 0) = n (and then necessarily Q(x0 , 0) = a(x0 )n).

, Q,

by Next we introduce yet another pair of unknowns, Y

(x0 , r), Y (x0 , r) = n(x0 ) + r Y √

0 , r) . Q(x0 , r) = a(x0) n(x0 ) + r Q(x

(5.21)

From (5.18) and (5.19) we obtain the following system of equations that must be satisﬁed

and Q:

by Y

(x0 , r) = 1 Y r2 1 = 2 r

r

0 r 0

1 √ Q(x0 , s) − n(x0 ) ds a(x0)

0 , s) ds s Q(x

(5.22)

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I. Shafrir

and

0 , r) = √ 1 Q(x0 , r) − a(x0 )n(x0 ) Q(x r a(x0) r 1 1

(x0 , s) ∇W x0 + sn(x0 ) + s 2 Y = 2 2a(x0)r 0 s − 2a(x0)n(x0 ) ds.

(5.23)

, Q)

the right-hand side of (5.22) and by T2 (Y

, Q)

the right-hand side Denoting by T1 (Y 2 of (5.23), we deﬁne a map T = (T1 , T2 ) from (C(Γ × [0, R], R ), · ∞ ))2 to itself by

, Q)

= (T1 (Y

, Q),

T 2 (Y

, Q)).

Clearly (Y

, Q)

is a solution to (5.22) and (5.23) if and T (Y only if it is a ﬁxed point of T . Next we claim: C LAIM . T is a strict contraction (and therefore has a unique ﬁxed point) provided that R is chosen small enough.

) ∈ (C(Γ × [0, R], R2 ))2 . Using (5.22) we get, for R small

, Q),

(Y

, Q Consider any (Y enough, 1 r s Q − Q , s) ds (x 0 2 0 0 0 satisfy λ0 . Ψ ∈ C2 Ω

(5.39)

Note that thanks to Proposition 5.1 (5.39) does hold for λ0 small enough. Throughout this subsection we shall make the following assumption on the smooth boundary condition g : ∂G → R2 : Image(g) ⊂ Ωλ0 .

(5.40)

We denote λ1 := max Ψ g(x) : x ∈ ∂G ,

(5.41)

so that 0 λ1 < λ0 . The neighborhood Ωλ0 of Γ can be covered by a system of nonintersecting gradient lines of Ψ . In particular for each x ∈ Ωλ0 there exists a unique gradient line which passes through it and we shall denote by s˜ (x) its intersection point with Γ . Equivalently, we look at the solution X(x0 , r) of ∂X

(X) = 2∇Ψ α(x0 )r , X(x0 , 0) = x0 ∀x0 ∈ Γ ∂r

(5.42)

On a class of singular perturbation problems

367

(compare with (5.15)). There exist unique x0 = x0 (x) ∈ Γ and r = r(x) > 0 such that λ0 x = X(x0 , r) and we then deﬁne s˜ (x) = x0 . We shall denote by γx0 : (−∞, t (x0)] → Ω the path deﬁned by γx0 (t) = X x0 , eαt /2 (with X given by (5.42)),

(5.43)

with Ψ (γx0 (t (x0 ))) = λ0 . Hence γx0 satisﬁes

γ˙x0 = ∇Ψ (γx0 ), γx0 (−∞) = s˜(x)

and γx0

2 log r(x) α

= x.

(5.44)

The map s˜ can be viewed as a projection from Ωλ0 onto Γ . Note that in general, unless W is a function of the Euclidean distance to Γ , s˜ differs from the Euclidean nearest point projection. Using s˜ we now deﬁne the degree D of a boundary condition g satisfying (5.40) by D = deg s˜ (g), ∂G .

(5.45)

In other words, D is the Brouwer degree of the map s˜ (g) : ∂G → Γ . We shall assume in the sequel, without loss of generality, that D 0. The next lemma provides two basic estimates on uε and its gradient. The proof is similar to the one of Lemma 3.1, and is therefore omitted. L EMMA 5.1. Any minimizer uε of problem (5.4) satisﬁes |uε | C1 in G

(5.46)

and ∇uε L∞ (G)

C2 ε

(5.47)

for some constants C1 , C2 > 0 independent of ε. In order to demonstrate the relevance of Ψ to our problem we present a simple (but nonoptimal) lower-bound for the energy, which uses an argument similar to the one of Lemma 4.3. P ROPOSITION 5.2. We have 2 Ψ (g) − C. Eε (uε ) ε ∂G P ROOF. Fix a vector ﬁeld V ∈ C 1 (G, R2 ) such that: (i) V = n ( = the unit exterior normal to ∂G) on ∂G,

(5.48)

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I. Shafrir

(ii) |V (x)| 1 ∀x ∈ G. Then, for any u ∈ Hg1 (G, R2 ) we have, by the Cauchy–Schwarz inequality and Green formula, |∇u|2 + G

W (u) 2 2 ε ε

2 ε

2 = ε

W (u)|∇u|

G

G

2 ε

∇ Ψ (u) G

∇ Ψ (u) · V Ψ (g) − ∂G

2 ε

Ψ (u) · div V . G

Now, 2 c c Ψ (u) · div V Ψ (u) W (u) c εEε (u), ε ε G ε G G i.e, 2 Eε (u) ε

Ψ (g) − c εEε (u), ∂G

and (5.48) follows.

Although (5.48) provides the right O( 1ε )-order term of Eε (uε ), it does not give any information on the term of order O(log 1ε ) (which is related to the degree). Nevertheless, a reﬁnement of the above argument plays an important role in the proof of the optimal lower bound that we shall give below. As usual, the proof of the upper bound is much easier. This is the object of the next proposition. We shall denote by l(Γ ) the length of the curve Γ . P ROPOSITION 5.3. We have Eε (uε )

2 ε

1 l 2 (Γ ) log + C. Ψ g(σ ) dσ + D 2π ε ∂G

P ROOF. Clearly it is enough to construct {vε } ⊂ Hg1 (G, R2 ) such that 2 Eε (vε ) ε

1 l 2 (Γ ) log + C. Ψ g(s) ds + D 2π ε ∂G

Recall that the map x → (σ (x), δ(x)) is a C 1 -diffeomorphism of Gb0 = x ∈ G: δ(x) b0

(5.49)

On a class of singular perturbation problems

369

on ∂G × [0, b0 ] (see [26], Section 14.6). The map Ht (σ ) : ∂G → {x ∈ G: δ(x) = t} given by Ht (σ ) = σ + tn is also a C 1 -diffeomorphism and its Jacobian satisﬁes JacHt (σ ) − 1 ct ˜ ∀(t, σ ) ∈ (0, b0) × ∂G. (5.50) In Gb0 we shall identify a point x with its (σ, δ)-coordinates: (σ (x), δ(x)). For ε small we use the notations of (5.44) and deﬁne vε by: ⎧ 2 log r(g(σ )) ⎪ − δε γs˜(g(σ )), δ ε1/2 , ⎪ α ⎨ 1/2 1/2 vε (σ, δ) = 2ε 1/2−δ vε σ, ε1/2 + δ−ε1/2 s˜ g(σ ) , ε1/2 < δ 2ε1/2 , ε ε ⎪ ⎪ ⎩ s˜ g(σ ) , 2ε1/2 < δ b0 . It remains to deﬁne vε on G \ Gb0 . We use a similar construction to the one of Proposition 3.3. We choose D points a1 , . . . , aD ∈ G \ Gb0 and r0 such that 0 < r0 On (G \ Gb0 ) \

1 0 1 min min |ai − aj |, min δ(ai ) − b0 . i=j i 2

D

j =1 B(aj , r0 )

we set vε = f0 where f0 is a Γ -valued C 1 -map such that x−a

f0 (x) = vε (x) = s˜ (g(σ (x))) on ∂(G \ Gb0 ) and f0 (x) = τ ( |x−ajj | ) on ∂B(aj , r0 ), j = 1, . . . , D, where τ : S 1 → Γ satisﬁes |τ (s)| =

l(Γ ) 2π

∀s ∈ S 1 .

(5.51) x−a

Finally, on each B(aj , r0 ) we deﬁne vε (x) = f (z) · τ ( |x−ajj | ) where the scalar function f is deﬁned by: 1 for ε < r r0 , f (r) = r for 0 r ε. ε As in the proof of Proposition 3.3, it is easy to verify that l 2 (Γ ) 1 log + C, Eε vε , B(aj , r0 ) = 2π ε

j = 1, . . . , D,

which implies that Eε (vε , G \ Gb0 ) D

1 l 2 (Γ ) log + C. 2π ε

(5.52)

It remains to estimate Eε (vε , Gb0 ). From the proof of Proposition 5.1 we deduce the estimates γx (t) − x0 , γ (t) Cect and Ψ γx (t) Cect ∀x0 ∈ Γ, (5.53) x0 0 0

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I. Shafrir

for some positive constants c, C. Using these estimates we conclude easily that Eε (vε , Gb0 \ Gε1/2 ) C.

(5.54)

Finally, on Gε1/2 we have 1 ∂vε 2 log r(g(σ )) δ 1 (σ, δ) = − γs˜(g(σ )) − = − ∇Ψ vε (σ, δ) . ∂δ ε α ε ε Since by the construction of vε , − ∂Ψ∂δ(vε ) 0, we get, using (5.50), that ∂vε 2 W (vε ) 2 ∂vε I1 := ∇Ψ (vε ) ∂δ + ε2 = − ε ∂δ Gε1/2 Gε1/2

2 = ε

∂(Ψ (vε )) 2 − ∂δ ε G 1/2

ε

∂G 0

ε 1/2

−

∂(Ψ (vε )) 1 + cδ ˜ dδ dσ. ∂δ

(5.55)

Next, for each σ ∈ ∂G we have

ε 1/2

∂(Ψ (vε (σ, δ))) 1 + cδ ˜ dδ ∂δ 0 = Ψ vε (σ, 0) − Ψ vε σ, ε1/2 1 + cε ˜ 1/2 ε1/2 cΨ ˜ vε (σ, δ) dδ C, + −

(5.56)

0

where C is independent of σ and ε. An immediate consequence of (5.56) is that I1

In particular, G 1/2 W (vε ) Cε and using Remark 5.1 we obtain that

C ε.

ε

Ψ (vε ) Cε.

(5.57)

Gε1/2

Integrating (5.56) on ∂G and using (5.57) in (5.55) yields 2 I1 ε

Ψ g(σ ) dσ + O(1).

(5.58)

∂G

ε It is easy to verify that | ∂v ∂σ | C and therefore

∂vε 2 C. I2 := G 1/2 ∂σ

(5.59)

ε

The result follows by combining (5.58) and (5.59) with (5.54) and (5.52).

On a class of singular perturbation problems

371

Next we turn to the proof of the lower-bound for Eε (uε ), which is essential also for the convergence result. T HEOREM 5.1. We have Eε (uε )

2 ε

1 l 2 (Γ ) log − C. Ψ g(σ ) dσ + D 2π ε ∂G

(5.60)

The proof relies on several lemmas. The ﬁrst is a reﬁned version of Proposition 5.2. L EMMA 5.2. For each α ∈ (1/2, 1), there exist constants c0 (α), C0 (α), C1 (α) and a(α) ∈ (0, 1) such that

W (uε ) 2 |∇uε | + ε ε2

Ψ g(σ ) dσ − C0

2

Gc0 εα

and

(5.61)

∂G

Ψ uε σ, c0 εα dσ C1 ε1+a .

(5.62)

∂G

Moreover, for α in a compact subinterval of (1/2, 1), the constants c0 , C0 , C1 can be chosen uniformly bounded. P ROOF. As in the proof of Proposition 5.2, for any c > 0, we have

W (uε ) 2 |∇uε | + ε2 ε

1/2 |∇uε | W (uε )

2

Gcεα

2 ε 2 ε

Gcεα

∇ Ψ (uε ) Gcεα

∇ Ψ (uε ) · V Gcεα

for any C 1 vector ﬁeld V such that |V | 1 on Gcεα . Choosing V = −∇δ yields, |∇uε |2 + Gcεα

2 ε

W (uε ) ε2

2 Ψ g(σ ) dσ − ε ∂G

:= I1 + I2 + I3 .

2 Ψ uε σ, cεα dσ − ε ∂G

Ψ (uε ) div V Gcεα

(5.63)

By the upper bound (5.49) we know that G W (uε ) Cε, so that by (5.46) and (5.38) we deduce that W (uε ) ∼ Ψ (uε ) and therefore I3 is bounded (uniformly in ε).

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I. Shafrir

Using G W (uε ) Cε again, we have G Ψ (uε ) Cε and there exists then some c1 ∈ (0, 1) such that

Ψ uε σ, c1 εα dσ Cε1−α . ∂G

For c = c1 we get I2 Cε−α and (5.63) becomes |∇uε |2 + Gc1 εα

W (uε ) 2 ε2 ε

Ψ g(σ ) dσ − Cε−α . ∂G

Using the upper bound again, we obtain |∇uε |2 + G\Gc1 εα

W (uε ) Cε−α . ε2

In particular, G\G α W (uε ) Cε2−α and then there exists c2 ∈ (1, 2) such that c1 ε

α 2−2α , and therefore also ∂G W (uε (σ, c2 ε )) dσ Cε

Ψ uε σ, c2 εα dσ Cε2−2α . ∂G

This last estimate is then plugged back in (5.63) and the argument is repeated. Let n be such that n n−1 α< . n n+1 Applying the above argument n times we obtain the existence of some cn ∈ (n − 1, n) such that, |∇uε |2 + Gcn εα

W (uε ) 2 ε2 ε

Ψ uε (σ, 0) dσ − Cεn−1−nα . ∂G

Using the upper bound once more, we get |∇uε |2 + G\Gcn εα

which leads to G\Gcn εα

W (uε ) C εn−1−nα + | log ε| , 2 ε

Ψ (uε ) Cε2 εn−1−nα + | log ε| ,

On a class of singular perturbation problems

373

and to the existence of a c0 such that

Ψ uε σ, c0 εα dσ Cε(n+1)(1−α) + Cε2−α | log ε| Cε1+a

(5.64)

∂G

for any a satisfying 0 < a < min(1 − α, (n + 1)(1 − α) − 1). We therefore proved (5.62) and using (5.64) in (5.63) with c = c0 yields that I2 C, and (5.61) follows as well. The next lemma provides a simple pointwise lower bound for |∇uε |. We denote Gε0 := x ∈ G: uε (x) ∈ Ωλ0 .

(5.65)

L EMMA 5.3. We have |∇uε |2 and

|∇(Ψ (uε ))|2 W (uε )

in G

2 |∇(Ψ (uε ))|2 |∇uε |2 β ∇ s˜ (uε ) + W (uε )

(5.66)

in Gε0

(5.67)

for some β > 0. S KETCH OF THE PROOF. First it is clear that, for any x ∈ G, ∇ Ψ (uε ) 2 = ∇Ψ (uε ) · ∇uε 2 ∇Ψ (uε )2 |∇uε |2 = W (uε )|∇uε |2 , (5.68) and (5.66) follows. Next, on Ωλ0 there exists a continuous orthogonal frame (ν, τ ), where at each point y ∈ Ωλ0 , ν = ν(y) is a unit vector in the direction of ∇Ψ (y) and τ = τ (y) is an orthogonal unit vector. For x ∈ Gε0 we may write |∇uε |2 = |∇ν uε |2 + |∇τ uε |2 , and an exact form of (5.68) is simply |∇ν uε |2 =

|∇(Ψ (uε ))|2 . W (uε )

(5.69)

Finally, it is not difﬁcult to prove that 2 |∇τ uε |2 β ∇ s˜ (uε )

for some β > 0 (see [5]).

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I. Shafrir

An important role in the proof of Theorem 5.1 is played by the scalar function d0ε which is deﬁned as the minimizer for the problem min Gc0 εα

|∇d|2 1 α ), d = Ψ (uε ) on ∂Gc ε α . : d ∈ H (G c ε 0 0 F (uε ) + ε2

(5.70)

The existence and uniqueness of d0ε is standard. The Euler–Lagrange equation for d0ε is given by "

∇d0ε = 0 in Gc0 εα , div W (u )+ε 2

(5.71)

ε

d0ε = Ψ (uε )

on ∂Gc0 εα .

Next we prove: L EMMA 5.4. For ε small enough we have 0 d0ε λ1

in Gc0 εα .

(5.72)

P ROOF. It is enough to show that d0ε = Ψ (uε ) λ1

on ∂Gc0 εα (for ε small),

(5.73)

and then apply the maximum principle to (5.71). The inequality on ∂G is clear from (5.41). For the bound on ∂Gc0 εα \ ∂G = {x ∈ G: δ(x) = c0 εα }, note ﬁrst that by (5.62) and (5.38) we have W (uε ) Cε1+a . ∂Gc0 εα \∂G

Let x0 ∈ ∂Gc0 εα \ ∂G satisfy δ˜ uε (x0 ) = m :=

max

∂Gc0 εα \∂G

δ˜ uε (x) .

˜ ε (x))| m/2 for every x ∈ ∂Gc0 εα \ ∂G satisfying |x0 − x| By (5.47) we have |δ(u for some c > 0. For such x we obtain, for some a0 > 0 (see (5.2) and (5.46)): a 0 m2 , W uε (x) a0 δ˜2 uε (x) 4 so that, for ε small, m3 ε mε a0 m2 = a0 8c 2c 4

∂Gc0 εα \∂G

W (uε ) Cε1+a ,

mε 2c

On a class of singular perturbation problems

375

which leads to m εb for some b > 0. Therefore also Ψ (uε ) CW (uε ) Cε2b < λ1 on ∂Gc0 εα \ ∂G, for ε small enough, and (5.73) follows. By the deﬁnition of d0ε , Lemma 5.3 and the upper bound (5.49), we deduce that Gc0 εα

|∇d0ε |2 W (uε ) + ε2

Gc0 εα

|∇(Ψ (uε ))|2 C . ε W (uε ) + ε2

(5.74)

Let d1ε ∈ H01 (Gc0 εα ) be the scalar function deﬁned by d1ε = Ψ (uε ) − d0ε . Then |∇(Ψ (uε ))|2 1 |∇d0ε |2 + 2∇d0ε · ∇d1ε + |∇d1ε |2 . = 2 2 W (uε ) + ε W (uε ) + ε

(5.75)

The motivation for introducing d1ε is the following simple consequence of (5.75), (5.71) and Green’s formula: |∇(Ψ (uε ))|2 |∇d0ε |2 |∇d1ε |2 = + . (5.76) 2 2 W (uε ) + ε2 Gc εα W (uε ) + ε Gc εα W (uε ) + ε 0

0

In fact, from (5.69) and (5.76) we conclude that

|∇ν uε |2 Gc0 εα

Gc0 εα

|∇d0ε |2 |∇d1ε |2 + . W (uε ) + ε2 W (uε ) + ε2

(5.77)

The next lemma provides a crucial lower bound for the right-hand side of (5.77). L EMMA 5.5. For every α ∈ (1/2, 1) there exists a constant C2 = C2 (α) such that Gc0 εα

|∇d0ε |2 W (uε ) + ε2 2 + 2 ε W (uε ) + ε ε2

Ψ g(s) ds − C2 ,

∂G

with C2 (α) uniformly bounded for α in a compact subinterval of (1/2, 1). P ROOF. As in the proof of Lemma 5.2 we have for V = −∇δ:

|∇d0ε |2 W (uε ) + ε2 + 2 ε2 Gc0 εα W (uε ) + ε 2 2 |∇d0ε | ∇d0ε · V ε Gc ε α ε Gc ε α 0 0 2 2 Ψ uε (σ ) dσ − d0ε div V = ε ∂Gc εα ε Gc ε α 0 0 2 2 Ψ g(σ ) dσ − d0ε div V − C, ε ∂G ε Gc ε α 0

(5.78)

376

I. Shafrir

where in the last inequality we applied (5.62). Therefore we only need to prove that d0ε Cε.

(5.79)

Gc0 εα

Fix any δ ∈ (0, c0 εα ) and let δ0 = δ/5. By the upper bound (5.49) there exists δ1 ∈ (δ0 , 2δ0) such that ∂Gδ1 \∂G

W (uε )

Cε . δ

By the same argument as in the proof of Lemma 5.2, we get that W (uε ) G\Gδ1

Cε2 + Cε2 | log ε|. δ

Repeating the argument we ﬁnd δ2 ∈ (2δ0 , 3δ0) such that W (uε ) G\Gδ2

Cε3 + Cε2 | log ε|. δ2

Repeating the argument one last time we deduce that for any δ, there exists δ3 ∈ (3δ0 , 4δ0) such that W (uε ) G\Gδ3

Cε4 + Cε2 | log ε|. δ3

Hence, for any δ ∈ (ε, c0 εα ), we have W (uε ) G\Gδ

Cε4 + Cε2 | log ε|. δ3

(5.80)

Next, using (5.50) and (5.62), we obtain c0 ε α

Gc0 εα \Gε

d0ε C ε

C

∂G c0

εα

c0

εα

ε

C ε

d0ε (σ, δ) dσ dδ #

d0ε σ, c0 εα +

∂G

$ ∇d0ε (σ, t) dt dσ dδ

c0 ε α

δ

|∇d0ε | dδ + Cε1+a+α . G\Gδ

(5.81)

On a class of singular perturbation problems

377

By the Cauchy–Schwarz inequality, (5.74) and (5.80), we get

c0 ε α

ε

|∇d0ε | dδ G\Gδ

c0 ε α

ε

C 1/2 ε

ε

G\Gδ

|∇d0ε |2 W (uε ) + ε2

c0 ε α c0 ε α

1/2

1/2 W (uε ) + ε2

dδ

G\Gδ

1/2 W (uε ) + ε

2

dδ

G\Gδ

ε2 1/2 + ε| log ε| + ε dδ δ 3/2 ε c ε α −Cε3/2 · s −1/2 ε0 + cεα+1/2| log ε| + cε1/2+α Cε.

C ε1/2

(5.82)

Combining (5.82) with (5.81) we obtain that d0ε Cε. Gc0 εα \Gε

On the other hand, the inequality d0ε Cε Gε

is obvious since |Gε | = O(ε) and d0ε is bounded by (5.72). This completes the proof of (5.79) and the result of the lemma follows. The next proposition establishes a lower bound for the energy on Gc0 εα which is the basis for the proof of Theorem 5.1. P ROPOSITION 5.4. There exists a constant K > 0 such that for all α ∈ (1/2, 1) there holds: W (uε ) |∇uε |2 + ε2 Gc0 εα |uε (σ, c0 εα ) − s˜ (g(σ ))|2 2 Ψ g(σ ) dσ + K dσ − C3 (5.83) ε ∂G εα ∂G with C3 = C3 (α) that is bounded uniformly for α in a compact subinterval of (1/2, 1). P ROOF. By Lemma 5.5 and (5.76) we have Gc0 εα

|∇Ψ (uε )|2 W (uε ) 2 + W (uε ) ε ε2

∂G

Ψ g(s) ds +

Gc0 εα

|∇d1ε |2 − C. W (uε ) + ε2

378

I. Shafrir

Combining it with (5.66) and (5.67) yields |∇uε |2 + Gc0 εα

2 ε

W (uε ) ε2

Ψ g(σ ) dσ +

∂G

Gc0 εα

+β

Gc0 εα ∩Gε0

|∇d1ε |2 W (uε ) + ε2

∇ s˜ (uε )2 − C.

(5.84)

Fix any σ0 ∈ ∂G. We distinguish two cases. Case 1. For all δ ∈ (0, c0 εα ) we have d1ε (σ0 , δ) λ0 − λ1 (see (5.41)). In this case, since d0ε (σ0 , δ) λ1 by (5.72), we have (σ0 , δ) ∈ Gε0 for every δ ∈ (0, c0 εα ) (see (5.65)). Using the Cauchy–Schwarz inequality we get

c0 ε α

β

α 2 ∇ s˜ uε (σ0 , δ) 2 dδ C |˜s (g(σ0 )) − s˜ (uε (σ0 , c0 ε ))| . α ε

0

By (5.38), uε σ0 , c0 εα − s˜ uε σ0 , c0 εα 2 = O Ψ uε σ0 , c0 εα = O W uε σ0 , c0 εα . So in this case we obtain, for some constants K0 , K1 > 0,

c0 ε α

∇ s˜ uε (σ0 , δ) 2 dδ

β 0

K0

|uε (σ0 , c0 εα ) − s˜ (g(σ0 ))|2 W (uε (σ0 , c0 εα )) − K1 . α ε εα

(5.85)

Case 2. There exists δ ∈ (0, c0 εα ) such that d1ε (σ0 , δ ) > λ0 − λ1 . In this case, since uε is bounded thanks to (5.46), we obtain again by Cauchy–Schwarz inequality

c0 ε α 0

|∇d1ε (σ0 , δ)|2 dδ c W (uε (σ0 , δ)) + ε2 c

δ 0

2 ∇d1ε (σ0 , δ)2 dδ c (λ0 − λ1 ) δ

(λ0 − λ1 )2 c0 ε α

K2

|uε (σ0 , c0 εα ) − s˜(g(σ0 ))|2 εα

(5.86)

On a class of singular perturbation problems

379

for some K2 > 0. Integration over σ0 ∈ ∂G of either (5.85) or (5.86) yields for some con 1 > 0: stants K, K

|∇d1ε |2 ∇ s˜ (uε )2 + β 2 Gc0 εα W (uε ) + ε Gc0 εα ∩Gε0 |uε (σ, c0 εα ) − s˜ (g(σ ))|2 W (uε (σ, c0 εα ))

1 K dσ − K dσ α ε εα ∂G ∂G := I1 − I2 .

Since I2 is bounded thanks to (5.62), the result of the lemma follows from (5.84).

We can now describe the main idea of the proof of Theorem 5.1 which is similar to that of Theorem 4.2 as described in Section 4.2. Motivated by Proposition 5.4 we deﬁne a new energy W (w) |∇w|2 + Eε (w) = ε2 G\Gc0 εα |w(σ, c0 εα ) − s˜ (g(σ ))|2 +K dσ, ∀w ∈ H 1 G \ Gc0 εα , R2 . α ε ∂G (5.87) By Proposition 5.4, W (uε ) 2 2 |∇uε | + − Ψ g(σ ) dσ Eε (uε ) − C. 2 ε ∂G ε G Therefore, it sufﬁces to show that 1 l 2 (Γ ) min Eε (w): v ∈ H 1 G \ Gc0 εα , R2 D log − C. 2π ε

(5.88)

The disadvantage of working with (5.87) (as was the case with the energy (4.56)) is the presence of two different scales in the energy. To overcome this difﬁculty we deﬁne another energy by ε (w) = E

|∇w|2 + G\Gc0 εα

W (w) + pε2 (x)

∂G

|w(σ, c0 εα ) − s˜ (g(σ ))|2 dσ, pε (σ )

where pε is deﬁned by ⎧ 1/2 α α ⎪ ε + 1 − δ(x) − c0 εα /ε1/2 εK ⎨ δ(x) − c0 ε /ε pε (x) = if c0 εα δ(x) ε1/2 + c0 εα , ⎪ ⎩ ε if δ(x) > ε1/2 + c0 εα .

(5.89)

380

I. Shafrir

ε (w) ∀w, it sufﬁces to prove then that Since Eε (w) E 2 ε (wε ) D l (Γ ) log 1 − C, E 2π ε

(5.90)

where wε is a minimizer for ε (w): w ∈ H 1 G \ Gc0 εα , R2 . min E As in the proof of Theorem 4.2, the main step in the proof of (5.90) consists of showing that the set of “bed points” of wε can be covered by a ﬁnite number of “bad discs/halfdiscs” with radii of the order pε . Similarly to Section 4.2, we have again two kinds of bad points: (1) Points x where wε (x) is far from Γ , i.e., for some r1 > 0. Sε(i) = x ∈ G \ Gc0 εα : dist wε (x), Γ > r1

(5.91)

(2) Boundary points where wε differs signiﬁcantly from s˜(g) = g, i.e., − wε σ˜ > r2 for some r2 > 0. Sε(b) = σ˜ ∈ ∂(G \ Gc0 εα ): s˜ g σ σ˜ (5.92) The proof is quite technical, but easier than the proof of the analogous assertion in Theorem 4.2, since here pε (x) is a bounded function which, in addition, depends only on δ(x). Therefore, we do not expect bad points on the boundary, and these can be indeed ruled out by the analysis whose details can be found in [3,5]. The combination of the upper-bound (5.49) with the lower-bound (5.60) leads also to the following convergence result (again we refer the reader to [3,5] for the proof). T HEOREM 5.2. Let W be a smooth function on R2 satisfying (5.1)–(5.3) and let g be a smooth boundary condition on ∂G satisfying (5.40) with D = deg(˜s (g)) 0. Then there is a subsequence εn → 0 and exactly D points a1 , . . . , aD in G such that uε n → u∗ = τ e

iφ0

D 4 z − aj |z − aj |

in Cloc G \ {a1 , . . . , aD } ,

j =1

where φ0 is a smooth harmonic function which is determined by the constraint u∗ = s˜ (g) on ∂G, and τ : S 1 → Γ satisﬁes (5.51). We close with an example which demonstrates the importance of using the projection s˜ in Theorem 5.2 (instead of the usual Euclidean projection).

On a class of singular perturbation problems

381

Fig. 6. Example 5.1.

E XAMPLE 5.1. Take Γ = S 1 and ﬁx 0 = a ∈ B(0, 1). Set 2 z − a Ψ (z) = − + 3 1 − az ¯

1 z − a 3 3 1 − az ¯

and deﬁne W on B(0, 1) by 2 (1 − |a|2 )2 z − a 2 2 W (z) = ∇Ψ (z) = , 1− |1 − az| ¯ 4 1 − az ¯

(5.93)

and complete the deﬁnition of W outside B(0, 1) in such a way that W will be a C 3 -function on R2 satisfying (5.1)–(5.3). Since Ψ ∈ C ∞ (B(0, 1) \ {a}), we know, thanks to Proposition 5.1, that Ψ coincides in B(0, 1) \ {a} with the function deﬁned in (5.5). The level curves of Ψ inside B(0, 1) are the circles which are the images, by the Möbius z+a transformation ma (z) = 1+ az ¯ , of the circles centered at 0, see Figure 6. Now consider G = B(0, 1) and the boundary condition g(eiθ ) = beiθ for some b ∈ (0, |a|). Then, applying Theorem 5.2 to this example we see that the degree D = deg(g) ˜ is zero since image(˜s (g)) covers only part of Γ = S 1 . Hence, the limit map u∗ = eiφ0 is smooth. Note that deg(g/|g|) = 1, so using the Euclidean nearest point projection instead of s˜ would lead to a wrong result! Acknowledgments I thank Michel Chipot and Pavol Quittner for inviting me to write this survey, and especially for their patience. Sections 4 and 5 describe joint results with Nelly André which were obtained during the last eight years. I am grateful to her for this long and fruitful collaboration.

382

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CHAPTER 6

Nonlinear Spectral Problems for Degenerate Elliptic Operators Peter Takáˇc Fachbereich Mathematik, Universität Rostock, D-18055 Rostock, Germany E-mail: takac@hades.math.uni-rostock.de

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. A priori regularity results . . . . . . . . . . . . . . . 2.3. Maximum and comparison principles . . . . . . . . . 3. The ﬁrst eigenvalue λ1 . . . . . . . . . . . . . . . . . . . . 3.1. Convexity on the cone of positive functions . . . . . 3.2. The inequality of Díaz and Saa . . . . . . . . . . . . 3.3. The ﬁrst eigenfunction ϕ1 . . . . . . . . . . . . . . . 4. Subcritical spectral problems (λ < λ1 ) . . . . . . . . . . . . 4.1. Existence and uniqueness for λ < λ1 . . . . . . . . . 4.2. Nonexistence for λ = λ1 . . . . . . . . . . . . . . . . 4.3. Anti-maximum principle for λ > λ1 . . . . . . . . . 5. Linearization about the ﬁrst eigenfunction . . . . . . . . . . 5.1. Linearization and quadratization . . . . . . . . . . . 5.2. The weighted Sobolev space Dϕ1 . . . . . . . . . . . 5.3. A compact embedding with a weight for p > 2 . . . 5.4. Simplicity of the ﬁrst eigenvalue for the linearization 5.5. Another compact embedding for 1 < p < 2 . . . . . 5.6. A few geometric inequalities . . . . . . . . . . . . . 6. An improved Poincaré inequality for p > 2 . . . . . . . . . 6.1. Statement and proof of Poincaré’s inequality . . . . . 6.2. Fredholm alternative at λ1 . . . . . . . . . . . . . . . 1,p 6.3. Application to the embedding W0 !→ Lp . . . . . 7. A saddle point method for p < 2 . . . . . . . . . . . . . . . 7.1. Simple saddle point geometry . . . . . . . . . . . . . 7.2. A Palais–Smale condition . . . . . . . . . . . . . . . 7.3. Fredholm alternative at λ1 . . . . . . . . . . . . . . . HANDBOOK OF DIFFERENTIAL EQUATIONS Stationary Partial Differential Equations, volume 1 Edited by M. Chipot and P. Quittner © 2004 Elsevier B.V. All rights reserved 385

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389 392 392 393 396 397 398 400 401 402 403 406 407 409 410 413 414 417 422 423 427 427 433 435 436 437 438 440

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8. Asymptotic behavior of large solutions . . . . . . . . 8.1. An approximation scheme . . . . . . . . . . . . 8.2. Convergence of approximate solutions . . . . . 8.3. First-order estimates . . . . . . . . . . . . . . . 8.4. Second-order estimates . . . . . . . . . . . . . . 8.5. A priori bounds . . . . . . . . . . . . . . . . . . 8.6. Nonexistence for λ = λ1 . . . . . . . . . . . . . 9. A variational approach . . . . . . . . . . . . . . . . . 9.1. A minimax method . . . . . . . . . . . . . . . . 9.2. Asymptotic behavior of the constrained minima 9.3. Asymptotic behavior of jλ near ±∞ . . . . . . 9.4. Existence of a solution for λ near λ1 . . . . . . 9.5. Existence of two or three solutions . . . . . . . 10. (Un)ordered pairs of sub-/supersolutions . . . . . . . 10.1. Existence results using ordered pairs . . . . . . 10.2. Existence results using unordered pairs . . . . . 10.3. (Un)ordered sets of solutions for λ = λ1 . . . . 11. Bifurcations and the Fredholm alternative . . . . . . . 11.1. An abstract global bifurcation result . . . . . . 11.2. Bifurcations from inﬁnity . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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441 442 445 446 451 457 460 461 462 463 465 468 470 475 476 478 479 482 483 485 487 487

Abstract This work surveys analytical methods and results for nonlinear spectral problems for degenerate elliptic operators of the following type: Jλ (u) = 0, which is the Euler equation for the energy functional λ def 1 A(x, ∇u) dx − B(x)|u|p dx − F (x, u) dx Jλ (u) = p Ω p Ω Ω 1,p

deﬁned on the Sobolev space W0 (Ω) or W 1,p (Ω). Here, Ω ⊂ RN is a bounded domain, 1 < p < ∞, and λ ∈ R is the spectral parameter (e.g., a control parameter). The energy density in the ﬁrst (and second) integral in Jλ (u) is assumed to be positively p-homogeneous in the variable u ∈ R, whereas the reaction function F (x, ·) is assumed to be asymptotically p-subhomogeneous, F (x, u) →0 |u|p

as |u| → ∞, uniformly for x ∈ Ω.

The work begins with the properties of the ﬁrst (smallest) eigenvalue λ1 of the corresponding nonlinear eigenvalue problem, Jλ (u) = 0 with F ≡ 0. Then the Euler equation Jλ (u) = 0 is studied for λ < λ1 . Finally, the solvability of this equation (existence, nonexistence and multiplicity of weak solutions) is investigated for any λ near λ1 . Employed are variational methods (also with constraint), monotonicity methods (pairs of sub- and supersolutions) and asymptotic bifurcation methods (from inﬁnity). A number of very recent results on the Fredholm 1,p alternative for the (quasilinear) p-Laplacian p on W0 (Ω) is surveyed, most of them with complete proofs.

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Keywords: Anti-maximum principle, Bifurcation from inﬁnity, Degenerate or singular quasilinear Dirichlet problem, First eigenvalue, Fredholm alternative, Global minimizer, Improved Poincaré inequality, Minimax principle, Nonlinear eigenvalue problem, p-Laplacian, Saddle point, Sub- and supersolutions MSC: Primary 35P30, 47J10; secondary 35J20, 49J35

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1. Introduction The main purpose of this survey is to review some of the most recent developments in the spectral theory of quasilinear elliptic operators of second order and their immediate consequences on solvability of quasilinear elliptic partial differential equations with a spectral parameter. An important class of such equations is represented by the Euler equations for the critical points of the energy functional λ def 1 p A(x, ∇u) dx − B(x)|u| dx − F (x, u) dx (1.1) Jλ (u) = p Ω p Ω Ω 1,p

deﬁned for every function u : Ω → R from the Sobolev space W0 (Ω) or W 1,p (Ω), where Ω is a bounded domain in RN (N 1), 1 < p < ∞, and λ ∈ R is the spectral parameter. In applications to engineering problems, λ can be viewed as a control parameter. The most typical restriction we impose on Jλ (u) will be the positive p-homogeneity of the integrands (energy densities) in the ﬁrst two integrals in (1.1) with respect to the variable u = u(x) ∈ R. This means that also in the ﬁrst integral we require A(x, tξ ) = |t|p A(x, ξ ) for all t ∈ R

(1.2)

∂A (x, 0) = 0. and for all (x, ξ ) ∈ Ω × RN . As a consequence, we obtain A(x, 0) = 0 and ∂ξ i N The function A(x, ·) : R → R is assumed to be strictly convex and coercive for every x ∈ Ω. Furthermore, we assume that the weight function B : Ω → R is in L∞ (Ω), such that B 0 and B ≡ 0 in Ω. Finally, the methods presented in this work apply only to asymptotically p-subhomogeneous integrands in the last integral of Jλ (u), that is, to

F (x, u)/|u|p → 0 as |u| → ∞, uniformly for x ∈ Ω.

(1.3)

A canonical example of the energy functional (1.1) that we use in a good part of this work is given by 1 λ Jλ (u) = |∇u|p dx − |u|p dx − F (x, u) dx (1.4) p Ω p Ω Ω 1,p

on W0 (Ω), where def

u

F (x, u) =

f (x, t) dt

for x ∈ Ω and u ∈ R.

0

The function f : Ω × R → R may take, for example, one of the following four forms: ⎧ g(x); ⎪ ⎪ ⎨ c(x) arctan u + g(x); f (x, u) = ⎪ c(x)|u|q−2u + g(x); ⎪ ⎩ c(x)|u|q−1 + g(x)

(1.5)

390

P. Takáˇc

for (x, u) ∈ Ω × R, where 1 < q < p, and c, g ∈ L∞ (Ω) are given functions which are not both identically vanishing. The corresponding Euler equation for the critical points of the functional Jλ deﬁned in (1.4) reads as follows: −p u = λ|u|p−2 u + f x, u(x) in Ω;

u=0

on ∂Ω,

(1.6)

where f (x, u) = (∂F /∂u)(x, u). Here, p stands for the Dirichlet p-Laplacian deﬁned by def

p u = div(|∇u|p−2 ∇u). The ﬁrst alternative in (1.5), in which f (x, u) ≡ f (x) = g(x) is independent from the state variable u ∈ R for each x ∈ Ω, appears to be a typical example suitable for presenting all our basic ideas: Here we develop appropriate analytic tools that can be applied to treat also the three remaining alternatives for f (x, u) without much change. Thus, we will focus our attention mostly on the solvability of the Dirichlet boundary value problem −p u = λ|u|p−2 u + f (x)

in Ω;

u = 0 on ∂Ω.

(1.7)

Since λ ∈ R is a spectral parameter taking values near the ﬁrst (smallest) eigenvalue λ1 of −p , which is given by (see Section 3) 1,p |∇u|p dx: u ∈ W0 (Ω) with |u|p dx = 1 , λ1 = inf Ω

(1.8)

Ω

one may regard (1.7) as a problem whose solvability (i.e., existence, nonexistence and mul1,p tiplicity of weak solutions in W0 (Ω)) should be described by some kind of a nonlinear version of the Fredholm alternative; cf. [37], Chapter II. To investigate the critical points of the functional Jλ , ﬁrst we need to realize that Jλ is 1,p coercive on the Sobolev space V whenever λ < λ1 ; if V = W0 (Ω) then λ1 > 0, whereas if V = W 1,p (Ω) then λ1 = 0. Hence, the existence of a critical point, that is a global minimizer for Jλ , follows by a standard minimization argument ([53], Theorem 1.2, p. 4). Furthermore, our strict convexity hypothesis on A(x, ·) guarantees that the ﬁrst integral in (1.1) is strictly convex on any linear subspace of W 1,p (Ω) not containing the constant functions. The second integral in (1.1) is obviously convex on Lp (Ω) and strictly convex on the linear subspace of all constant functions. Finally, if the function F (x, ·) : R → R in the third integral in (1.1) happens to be convex for each x ∈ Ω, then Jλ turns out to be not only coercive (for λ < λ1 ) but also strictly convex on V whenever λ 0. Another wellknown result ([53], pp. 58–60) then guarantees that Jλ possesses precisely one critical point, namely, the global minimizer. This is as far as one can get by applying the “general theory” to the functional Jλ . 1,p If V = W0 (Ω), p = 2, and 0 < λ < λ1 , the critical points of Jλ are not unique, in general: Besides a global minimizer there might also be a saddle point; see [29], Example 2, p. 148, for 1 < p < 2, and [49], Eq. (5.26), p. 12, for 2 < p < ∞, where such examples with the function F (x, u) = f (x)u are constructed in an open interval Ω ⊂ R1 . However, if f 0 and f ≡ 0 in Ω, uniqueness still holds, by a result due to Díaz and Saa [17] and generalized later by Takáˇc, Tello and Ulm [60]. The case λ < λ1 is treated in Section 4.

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For λ = λ1 , the coercivity of Jλ1 and consequently the existence of a global minimizer for Jλ1 are lost, in general; see [48], Theorem 1.2, p. 390. Thus, one of the aims of our presentation will be to provide reasonable necessary and/or sufﬁcient conditions on the function F (x, u) such that the functional Jλ1 have a critical point. In fact, we will obtain additional information on the “geometry” of the functional Jλ for any λ near λ1 [21,59]. Finally, we apply topological bifurcation methods to obtain continua of pairs (λ, uλ ) in R × V consisting of a parameter value λ (near λ1 ) and a critical point uλ for Jλ . A standard tool in a number of variational methods is the Palais–Smale condition (at 1,p some critical level). Let us now consider only the case V = W0 (Ω). In [48], Theorem 1.2(ii), p. 390, it is shown that for the functional (1.4) with p > 2 and F (x, u) = f (x)u, the Palais–Smale condition fails to hold at the zero level. Therefore, in order to obtain a priori bounds for the critical points of Jλ for λ near λ1 , we simply admit possible, a priori large critical and “almost critical” points of Jλ and then determine their precise asymptotic behavior as λ approaches λ1 . Our method is based on the following well-known fact: λ1 is a simple eigenvalue of the positive Dirichlet p-Laplacian −p with the associated eigenfunction ϕ1 normalized by ϕ1 > 0 in Ω and ϕ1 Lp (Ω) = 1, by a result due to [2], Théorème 1, p. 727, and later generalized in [46], Theorem 1.3, p. 157. The corresponding result remains valid also for the ﬁrst two terms of the more general functional (1.1) on 1,p W0 (Ω), as shown in [60], Theorem 2.6, p. 80. Moreover, the eigenvalue λ1 is positive and isolated. As a consequence, it is not difﬁcult to show ([27], Proof of Théorème 2, p. 732, or [28], Section 6, p. 69) that a possible large critical point uλ of Jλ for λ near λ1 must take the form uλ = t −1 (ϕ1 + vt) ), where t ∈ R is a number with |t| > 0 small enough, and 1,p vt) ∈ W0 (Ω) is a function orthogonal to ϕ1 in L2 (Ω) with the norm vt) W 1,p (Ω) → 0 0

as |t| → 0. This forces λ = λ(t) → λ1 as well. But we need much stronger results on the rate of decay of both, vt) → 0 (in a suitable norm) and λ − λ1 → 0 as |t| → 0, which have been established recently in [23], Theorem 4.1, and [57], Propositions 5.2 and 8.3, and [58], Proposition 6.1. These results describe asymptotic bifurcations from inﬁnity of the form uλ = t −1 (ϕ1 + vt) ) as |t| → 0, which are easily transformed to bifurcations from zero where the unknown function vt) in tuλ = ϕ1 + vt) has to be investigated as |t| → 0. Recalling the positive p-homogeneity in u of the ﬁrst two terms in the functional Jλ , we notice immediately that the linearization of (1.6) about the eigenfunction ϕ1 together with the “quadratization” of functional (1.4) about ϕ1 play the key role in determining the asymptotic behavior of vt) as |t| → 0. In contrast to related methods for the semilinear case p = 2 ( [35], Chapter 18, or [36]), our linearization and quadratization are exact: They use the (precise) integral versions of the ﬁrst- and second-order Taylor formulas, respectively, rather than Taylor approximations by linear or quadratic expressions. This method was introduced recently by the author [57] and is presented in Section 5. For the special choice f (x, u) ≡ f (x) the method yields vt) /|t|p−2 t → V ) as |t| → 0, in W01,2 (Ω) if 1 < p < 2 1,p and in a suitable weighted Sobolev space Dϕ1 (W0 (Ω) !→ Dϕ1 ) if 2 < p < ∞. The limit function V ) is the unique solution of the corresponding limit equation under the condition that V ) is orthogonal to ϕ1 in L2 (Ω). The limit equation is linear with the nonhomogeneous term equal to f (x), so that the classical Fredholm alternative for a selfadjoint linear operator in a Hilbert space applies. In a number of important applications we will often be able to show that the asymptotic behavior of large solutions uλ = t −1 (ϕ1 + vt) ) as |t| → 0

392

P. Takáˇc

leads, in fact, to a contradiction; for instance, if Ω f ϕ1 dx = 0. Hence, if this happens, there can be no large solutions to problem (1.6) or, in other words, we get an a priori bound on the set of all critical points of functional (1.4). This is the connection between the non) linear

problem (1.6) and the linear problem for V obtained in the limit |t| → 0. Even if Ω f ϕ1 dx = 0, our method is precise enough to exclude large solutions, for instance, if λ = λ1 . This is the main difference between the linear case p = 2 and the nonlinear case p = 2; see Section 8. The solvability itself of the spectral problem (1.7) is treated by “easier” methods in Sections 6 and 7 (for λ = λ1 ), whereas more difﬁcult and complicated tools are developed in Sections 9, 10 and 11 (for λ near λ1 ). To summarize the state-of-the-art work on problem (1.7) up to now, many interesting new results have been obtained for λ near λ1 . The work of Anane and Tsouli [4] is one of the very few dealing with the second eigenvalue λ2 of −p . A variational characterization of all eigenvalues of −p is a challenging open question in space dimension N 2 (cf. Drábek and Robinson [26]). In space dimension N = 1, when Ω ⊂ R1 is an open interval, signiﬁcant progress for λ = λk (any eigenvalue, k = 1, 2, . . . ) has been achieved in the recent work of Manásevich and Takáˇc [47]. 2. Preliminaries 2.1. Notation The closure, interior and boundary of a set S ⊂ RN are denoted by S, int(S) and ∂S, def

respectively, and the characteristic function of S by χS : RN → {0, 1}. We write |S|N =

χ (x) dx if S is also Lebesgue measurable. We set R+ = [0, ∞) and N = {1, 2, 3, . . .}. RN S We denote by Ω a bounded domain in RN (N 1). Given an integer k 0 and 0 α 1, the Hölder space of all k-times continuously differentiable funcwe denote by C k,α (Ω) tions u : Ω → R whose all (classical) partial derivatives of order k possess a continuous The norm uC k,α (Ω) extension up to the boundary and are α-Hölder continuous on Ω. k,α is deﬁned in a natural way. As usual, we abbreviate C k (Ω) ≡ C k,0 (Ω). The in C (Ω) consisting of all C k functions u : Ω → R with compact support linear subspace of C k (Ω) 7 k k is denoted by C0 (Ω); we set C0∞ (Ω) = ∞ k=0 C0 (Ω). Given 1 p ∞, we denote by p L (Ω) the Lebesgue space of all (equivalence classes of ) Lebesgue measurable functions u : Ω → R with the norm " u(x)p dx 1/p < ∞ if 1 p < ∞, def Ω p up ≡ uL (Ω) = if p = ∞. ess supx∈Ω u(x) < ∞ Finally, for an integer k 1, we denote by W k,p (Ω) the Sobolev space of all functions u ∈ Lp (Ω) whose all (distributional) partial derivatives of order k also belong to Lp (Ω). Again, the norm uk,p ≡ uW k,p (Ω) in W k,p (Ω) is deﬁned in a natural way. The closure in W k,p (Ω) of the set of all C k functions u : Ω → R with compact support is denoted by k,p W0 (Ω). We refer to Kufner, John and Fuˇcík [44] for details about these and other similar function spaces. All Banach and Hilbert spaces used in this article are real.

Nonlinear spectral problems

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The positive and negative parts of a real-valued function u are denoted by u+ and u− , 1,p respectively, where u+ = max{u, 0} and u− = max{−u, 0}. If u ∈ W0 (Ω) then also u± ∈ 1,p W0 (Ω); see [39], Theorem 7.8, p. 153. More precisely, we have ∇u+ = ∇u almost everywhere in Ω+ = {x ∈ Ω: u(x) > 0} and ∇u+ = 0 almost everywhere in Ω \ Ω+ . The corresponding result holds for u− as well. def

We work with the standard inner product in L2 (Ω) deﬁned by u, v! = Ω uv dx for u, v ∈ L2 (Ω). The orthogonal complement in L2 (Ω) of a set M ⊂ L2 (Ω) is denoted 2 by M⊥,L , 2 def M⊥,L = u ∈ L2 (Ω): u, v! = 0 for all v ∈ M . The inner product ·, ·! in L2 (Ω) induces a duality between the Lebesgue spaces Lp (Ω) and Lp (Ω), where 1 p, p ∞ with p1 + p1 = 1, and between the Sobolev space 1,p W0 (Ω) and its dual W −1,p (Ω), as well. Finally, this inner product induces also the canonical duality between the space of test functions D(Ω) ≡ C0∞ (Ω) and the space of distributions D (Ω). We keep the same notation also for the duality between the Cartesian products [Lp (Ω)]N and [Lp (Ω)]N .

2.2. A priori regularity results We always assume that the function A of (x, ξ ) ∈ Ω × RN and its partial gradient ∂ξ A ≡ ∂A N ( ∂ξ ) with respect to ξ ∈ RN satisfy the following hypotheses, upon the substitution i i=1 def 1 p ∂ξ A(x, ξ )

a(x, ξ ) =

def 1 ∂A p ∂ξi .

where ai =

H YPOTHESIS (A). A : Ω × RN → R+ veriﬁes the positive p-homogeneity hypothe∂A sis (1.2), A ∈ C 1 (Ω × RN ), and its partial gradient ∂ξ A : Ω × RN → RN satisﬁes p1 ∂ξ = i 1 N ai ∈ C (Ω × (R \ {0})) for all i = 1, 2, . . . , N , together with the following ellipticity and growth conditions: There exist some constants γ , Γ ∈ (0, ∞) such that N ∂ai (x, ξ ) · ηi ηj γ · |ξ |p−2 · |η|2 , ∂ξj

(2.1)

i,j =1

N ∂ai Γ · |ξ |p−2 , (x, ξ ) ∂ξ

i,j =1

N ∂ai Γ · |ξ |p−1 , (x, ξ ) ∂x

i,j =1

(2.2)

j

j

for all x ∈ Ω, all ξ ∈ RN \ {0} and all η ∈ RN .

(2.3)

394

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(It is evident that it sufﬁces to require inequalities (2.1)–(2.3) for |ξ | = 1 only; the general case ξ ∈ RN \ {0} follows from the positive p-homogeneity hypothesis (1.2).) It follows that A(x, · ) is strictly convex and satisﬁes γ Γ |ξ |p A(x, ξ ) |ξ |p p−1 p−1

for all ξ ∈ RN .

(2.4)

1,p

Hence, the functional Jλ on W0 (Ω) is coercive and bounded on bounded sets. Indeed, the inequalities in (2.4) follow from γ Γ |ξ |p A(x, ξ ) − A(x, 0) − ξ · ∂ξ A(x, 0) |ξ |p p−1 p−1 for all (x, ξ ) ∈ Ω × RN . This is, in turn, a direct consequence of Taylor’s formula combined with (2.1) and (2.2). Recall that the positive p-homogeneity hypothesis (1.2) forces ∂A (x, 0) = 0 for all x ∈ Ω and i = 1, 2, . . . , N . A(x, 0) = 0 and ∂ξ i Finally, we assume that F satisﬁes: H YPOTHESIS (F). F : Ω × R → R is given by the integral u f (x, t) dt for x ∈ Ω and u ∈ R, F (x, u) = 0

where f : Ω × R → R is a Carathéodory function, i.e., f (·, u) : Ω → R is Lebesgue measurable for each u ∈ R and f (x, ·) : R → R is continuous for almost every x ∈ Ω, and there exists a constant C ∈ (0, ∞) such that f (x, u) C 1 + |u|p−1 for all x ∈ Ω and all u ∈ R. (2.5) 1,p

Now we are ready to state the main regularity result for a weak solution u ∈ W0 (Ω) of the Dirichlet boundary value problem − div a(x, ∇u) = f x, u(x) in Ω;

u = 0 on ∂Ω.

(2.6)

We will use this a priori regularity throughout the entire article. P ROPOSITION 2.1. Let 1 < p < ∞ and let hypotheses (A) and (F) be satisﬁed. Assume 1,p that u ∈ W0 (Ω) is a weak solution of problem (2.6). Then u ∈ C 1,β (Ω) where β ∈ (0, 1) is a constant independent from u. If, in addition, ∂Ω is a compact manifold of class C 1,α Moreover, β is for some α ∈ (0, 1), then β ∈ (0, α) can be chosen such that u ∈ C 1,β (Ω). C where C > 0 is some constant depending again independent from u, and uC 1,β (Ω) solely upon Ω, A, f , N , p, and the norm uLp0 (Ω) with p0 =

p∗ = 2p

Np N−p

if p < N, if p N.

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1,p

Notice that, owing to the Sobolev embedding W0 (Ω) !→ Lp0 (Ω), we have also uC 1,β (Ω) C , where the constant C depends solely upon Ω, A, f , N , p, and the norm uW 1,p (Ω) . Similarly, one obtains uC 1,β (Ω) C as well, where the constant C 0 depends solely upon Ω, A, f , N , p, and the norm uL∞ (Ω) . These two consequences of Proposition 2.1 will be used quite often in the sequel. Proposition 2.1 is, in fact, a combination of the following two lemmas, in which we keep our hypotheses and notation from the proposition. L EMMA 2.2. Let g : Ω × R → R be a Carathéodory function such that g(·, s) ∈ L1loc (Ω) for every s ∈ R, and the following inequality holds with some constants a > 0 and b 0: s · g(x, s) a|s|p + b|s| for all s ∈ R and a.e. x ∈ Ω. 1,p

Assume that u ∈ W0 (Ω) satisﬁes

% & a(x, ∇u), ∇φ dx = Ω

g x, u(x) φ dx Ω

for all φ ∈ C0∞ (Ω).

Then u ∈ L∞ (Ω) and there exists a constant c > 0 such that uL∞ (Ω) c, where c depends solely upon a, b, N , p, and uLp0 (Ω) . This is a special case of a more general result shown in Anane’s thesis [3], Théorème A.1, p. 96. Although his proof is carried out only for a(x, ξ ) ≡

1 ∂ξ A(x, ξ ) = |ξ |p−2 ξ, p

(x, ξ ) ∈ Ω × RN ,

(2.7)

one can rewrite it directly for our more general case. 1,p

L EMMA 2.3. Assume that u ∈ W0 (Ω) is a weak solution of problem (2.6) such that u ∈ L∞ (Ω). Then u ∈ C 1,β (Ω) where β ∈ (0, 1) is a constant independent from u. If, in addition, ∂Ω is a compact manifold of class C 1,α for some α ∈ (0, 1), then β ∈ (0, α) Moreover, β is, again, independent from u, and can be chosen such that u ∈ C 1,β (Ω). uC 1,β (Ω) C where C > 0 is some constant depending solely upon Ω, A, f , N , p, and the norm uL∞ (Ω) . The ﬁrst statement of this lemma, interior regularity in C 1,β (Ω), was established independently by DiBenedetto [18], Theorem 2, p. 829, and Tolksdorf [62], Theorem 1, p. 127. The second statement, regularity near the boundary, is due to Lieberman [45], Theorem 1, p. 1203. The constant β depends solely upon α, N and p. We keep the meaning of the constants α and β throughout the entire article and denote by β an arbitrary but ﬁxed number such that 0 < β < β < α < 1. Last but not least, Lieberman’s regularity results have been shown for the Neumann boundary conditions as well.

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While Anane’s proof of Lemma 2.2 is based on the special form of a(x, ξ ) ≡ p1 ∂ξ A(x, ξ ) with the positively p-homogeneous potential A(x, ·) satisfying also hypothesis (1.2), N N Lemma 2.3 is valid with any vector ﬁeld a ≡ (ai )N i=1 : Ω × R → R satisfying ai ∈ C 0 Ω × RN ∩ C 1 Ω × RN \ {0} (i = 1, 2, . . . , N) together with the ellipticity and growth conditions (2.1)–(2.3).

2.3. Maximum and comparison principles The strong maximum principle for the critical points of the functional J0 (i.e., λ = 0) will turn out to be essential in proving the simplicity of the ﬁrst eigenvalue λ1 . 1,p We begin with the weak comparison principle for the weak solutions u, v ∈ W0 (Ω) of the following Dirichlet boundary value problems, respectively, − div a(x, ∇u) + b(x, u) = f (x) in Ω; − div a(x, ∇v) + b(x, v) = g(x) in Ω;

u=0

on ∂Ω,

(2.8)

v=0

on ∂Ω.

(2.9)

As a direct consequence, the uniqueness of these solutions follows. We assume that A satisﬁes Hypotheses (A), and b : Ω × R → R is a Carathéodory function that is increasing in the second variable, i.e., u v in R implies b(x, u) b(x, v) for almost every x ∈ Ω, and it satisﬁes the growth condition (2.5) with b in place of f . L EMMA 2.4. Let f, g ∈ L∞ (Ω) satisfy f g in Ω. Then any weak solutions u, v ∈ 1,p W0 (Ω) of problems (2.8) and (2.9), respectively, satisfy u v almost everywhere in Ω. This result is shown in [61], Lemma 3.1, p. 800. Its proof is standard: Consider the def 1,p function w = (u − v)+ = max{u − v, 0}; hence, w ∈ W0 (Ω). Subtract the second equation (2.9) from the ﬁrst one (2.8), multiply the difference by w, and subsequently integrate over Ω. The integrals over Ω+ = {x ∈ Ω: w(x) > 0} force |Ω+ |N = 0. 1,p To obtain the strong maximum principle for a weak solution u ∈ W0 (Ω) of problem (2.8), we strengthen our hypotheses on b as follows: H YPOTHESIS (b). b : Ω × R → R is a Carathéodory function that is increasing in the second variable and satisﬁes the growth condition b(x, u) C|u|p−1

for a.e. x ∈ Ω and all u ∈ R,

(2.10)

with a constant C ∈ (0, ∞). Recall that Ω is said to satisfy an interior sphere condition at a point x0 ∈ ∂Ω if there exists an open ball Br (y) = {x ∈ RN : |x − y| < r}, centered at some point y ∈ Ω and with radius 0 < r < ∞, such that Br (y) ⊂ Ω and x0 ∈ ∂Br (y).

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L EMMA 2.5. Let f ∈ L∞ (Ω) satisfy f 0 and f ≡ 0 in Ω. Then any weak solution 1,p u ∈ W0 (Ω) of problem (2.8) veriﬁes u > 0 almost everywhere in Ω. If, in addition, Ω satisﬁes an interior sphere condition at a point x0 ∈ ∂Ω and, for some ε > 0, ∂Ω ∩ ∩ Bε (x0 )), then the outer normal derivative Bε (x0 ) is a manifold of class C 1 and u ∈ C 1 (Ω on ∂Ω of u at x0 veriﬁes the Hopf maximum principle (∂u/∂ν)(x0 ) < 0. This result is due to Tolksdorf [61], Propositions 3.2.1 and 3.2.2, p. 801, for a(x, ξ ) ≡ a(ξ ) and to Vázquez [65], Theorem 5, p. 200, for a(x, ξ ) ≡

1 ∂ξ A(x, ξ ) = |ξ |p−2 ξ, p

(x, ξ ) ∈ Ω × RN .

The proof given in [61], p. 802, extends directly to our more general case. R EMARK 2.6. One may ask if the strong comparison principle, u ∂ν ∂ν

on ∂Ω,

(2.11)

is valid in the setting of Lemma 2.4, provided f ≡ g, ∂Ω is of class C 1,α for some α ∈ (0, 1), and Ω satisﬁes the interior sphere condition at every point of ∂Ω. We refer the reader to [12,13] for a positive answer to this question in a number of special cases; for example, if b satisﬁes hypothesis (b) and, in addition, b(x, ·) is locally Lipschitz continuous on R \ {0} for almost every x ∈ Ω, and ∂b Γ · |u|p−2 if 1 < p < 2, (x, u) (2.12) 0 Γ if 2 p < ∞, ∂u holds for almost all (x, u) ∈ Ω × (0, ε0 ], with some constants Γ ∈ (0, ∞) and ε0 > 0, these hypotheses guarantee only u < v near the boundary of Ω together with ∂u/∂ν > ∂v/∂ν on ∂Ω for any 1 < p < ∞ and ∂Ω connected ([13], Proposition 2.4, p. 728). For 1 < p 2 they guarantee also (2.11) provided either N = 1 ([13], Theorem 3.1, p. 733) or else N 2 and u(x) ≡ u(|x|) and v(x) ≡ v(|x|) are radially symmetric in a ball Ω = BR (0) ⊂ RN ([13], Theorem 3.3, p. 737). However, for p > 2, Hypotheses (b) and (2.12) may not be sufﬁcient for (2.11) to be valid throughout the domain Ω: A counterexample to (2.11) with u(0) = v(0) is given in [13], Example 4.1, pp. 740–741, where Ω = B1 (0) ⊂ RN is the unit ball and b(x, u) ≡ λ|u|p−2 u with a constant λ > 0 large enough. Unlike in [61], Proposition 3.3.2, p. 803, and in a number of other articles on this topic, in [12,13] it is not assumed that |∇u| > 0 1,p must attain its or |∇v| > 0 throughout Ω. In fact, any function u ∈ W0 (Ω) ∩ C 1 (Ω) maximum or minimum in Ω at some point xˆ ∈ Ω; hence, ∇u(x) ˆ = 0. 3. The ﬁrst eigenvalue λ1 Let us consider the energy functional Jλ deﬁned by (1.1). We assume that A satisﬁes Hypotheses (A), and the weight function B satisﬁes:

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H YPOTHESIS (B). B : Ω → R+ belongs to L∞ (Ω) and does not vanish identically (almost everywhere) in Ω. 1,p

We treat the more difﬁcult case of Jλ deﬁned on W0 (Ω) in detail, leaving the trivial case Jλ : W 1,p (Ω) → R to the reader. The ﬁrst (smallest) eigenvalue λ1 for the Euler 1,p equation corresponding to the energy functional Jλ on W0 (Ω) is given by 1,p p λ1 = inf A(x, ∇u) dx: u ∈ W0 (Ω) with B(x) |u| dx = 1 . Ω

(3.1)

Ω

1,p

Since the Sobolev embedding W0 (Ω) !→ Lp (Ω) is compact by Rellich’s theorem, the inﬁmum above is attained and satisﬁes 0 < λ1 < ∞. Furthermore, it is easy to see that 1,p if u ∈ W0 (Ω) is a minimizer for λ1 , then so is αu+ provided u+ ≡ 0 in Ω and α =

±( Ω B(x)(u+ )p dx)−1/p . The corresponding claim holds also for u− , with the constant α

replaced by β = ±( Ω B(x)(u− )p dx)−1/p . Indeed, if both u± ≡ 0 in Ω, then we have

+ p + − p − Ω B(u ) Ω A(x, ∇u ) dx Ω B(u ) Ω A(x, ∇u ) dx

λ1 = + p + p p − p Ω B|u| Ω B(x)(u ) dx Ω B|u| Ω B(x) (u ) dx

B(x)(u+ )p B(x)(u− )p

Ω + Ω λ1 p p Ω B(x)|u| Ω B(x)|u| = λ1 . Consequently, both αu+ and βu− are (nontrivial) minimizers for λ1 and therefore satisfy the Euler equation − div a(x, ∇u) = λ1 B(x)|u|p−2 u

in Ω;

u=0

on ∂Ω.

(3.2)

We apply the strong maximum principle (Lemma 2.5) to conclude that u+ ≡ 0 in Ω forces 1,p u > 0 almost everywhere in Ω, and analogously for u− . Thus, a minimizer u ∈ W0 (Ω) for λ1 is either almost everywhere positive or else almost everywhere negative in Ω. 1,p Our next goal is to show that a minimizer u ∈ W0 (Ω) for λ1 is unique up to the sign ±. In other words, we wish to show that the eigenvalue λ1 in problem (3.2) is simple. Notice that the case Jλ : W 1,p (Ω) → R is trivial due to the fact that λ 1 = 0. Hence, the only minimizers for λ1 over W 1,p (Ω) are the constant functions u = ±( Ω B(x) dx)−1/p . 3.1. Convexity on the cone of positive functions

1,p Notice that the functional u → Ω A(x, ∇u) dx is strictly convex on W0 (Ω), by the 1,p ellipticity condition (2.1) (hypothesis (A)). Knowing that any eigenfunction u ∈ W0 (Ω) 1,p associated with the ﬁrst eigenvalue λ1 , that is to say, any weak solution u ∈ W0 (Ω) to problem (3.2), must be either positive or else negative throughout Ω, we may replace u

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399

by −u if necessary and thus assume u > 0 in Ω. Hence, the minimization constraint in

def p p formula (3.1) reads Ω B(x)

u(x) dx = 1. Upon the substitution v = u , the constraint becomes linear in v, i.e., Ω B(x) v(x) dx = 1. It follows that λ1 = inf A x, ∇ v 1/p dx: v ∈ V˙+ with B(x)v(x) dx = 1 , Ω

(3.3)

Ω

where def 1,p V˙+ = v : Ω → (0, ∞): v 1/p ∈ W0 (Ω) . The following property of the functional v → step in proving that the eigenvalue λ1 is simple.

Ω

A(x, ∇(v 1/p )) dx on V˙+ is the crucial

D EFINITION 3.1. A functional K : C → R ∪ {+∞} on a convex cone C ⊂ X \ {0} in a vector space X (over the ﬁeld R) is called ray-strictly convex if, for all v0 , v1 ∈ C and θ ∈ (0, 1), we have K (1 − θ )v0 + θ v1 (1 − θ )K(v0 ) + θ K(v1 ), where the equality may hold only if v0 and v1 are co-linear, i.e., v1 = αv0 for some α ∈ (0, ∞). Recall that C is called a convex cone in X \ {0} if C ⊂ X \ {0} is convex and satisﬁes v ∈ C ⇒ αv ∈ C for all α ∈ (0, ∞). def L EMMA 3.2. The functional K : V˙+ → R, deﬁned by K(v) = v ∈ V˙+ , is ray-strictly convex on V˙+ .

Ω

A(x, ∇(v 1/p )) dx for

Notice that the statement of the lemma includes also the convexity of the set V˙+ . This lemma is shown in [60], Lemma 2.4, p. 79. For the special case A(x, ξ ) = |ξ |p , (x, ξ ) ∈ Ω × RN , it is due to Díaz and Saa [17]. P ROOF OF L EMMA 3.2. Using the positive p-homogeneity hypothesis (1.2), we observe that A x, ∇ v 1/p = p−p v A x, v −1 ∇v for all v ∈ V˙+ and a.e. x ∈ Ω. Take v = (1 − θ )v0 + θ v1 and replace ∇v by ξ = (1 − θ )ξ0 + θ ξ1 , where v0 , v1 ∈ (0, ∞) and ξ0 , ξ1 ∈ RN are arbitrary positive numbers and arbitrary vectors in RN , respectively, and θ ∈ (0, 1). Next we rewrite (1 − θ )ξ0 + θ ξ1 ξ ξ0 ξ1 (1 − θ )v0 θ v1 = = + v (1 − θ )v0 + θ v1 (1 − θ )v0 + θ v1 v0 (1 − θ )v0 + θ v1 v1 =

(1 − θ )v0 ξ0 θ v1 ξ1 + . v v0 v v1

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P. Takáˇc

Now ﬁx any x ∈ Ω. The function A(x, ·) being strictly convex on RN , by the ellipticity condition (2.1) (Hypothesis (A)), we compute (1 − θ )v0 ξ0 ξ1 θ v1 ξ A x, A x, + A x, v v v0 v v1 v0 ξ0 ξ1 v1 = (1 − θ ) A x, + θ A x, . (3.4) v v0 v v1 The equality holds if and only if ξ0 /v0 = ξ1 /v1 . Furthermore, the function (v, ξ ) → vA(x, v −1 ξ ) is convex on (0, ∞) × RN . In particular, taking v0 , v1 ∈ V˙+ and ξi = ∇vi (i = 0, 1), multiplying (3.4) by v and integrating over Ω, we arrive at (1 − θ )v0 + θ v1 ∈ V˙+ , proving the convexity of the set V˙+ , and K (1 − θ )v0 + θ v1 (1 − θ )K(v0 ) + θ K(v1 ) where the equality holds if and only if v0−1 ∇v0 = v1−1 ∇v1 in Ω. The latter equality is equivalent to v1 /v0 ≡ const in Ω. The lemma is proved. 3.2. The inequality of Díaz and Saa An important consequence of the ray-strict convexity of the functional K : V˙+ → R established in Lemma 3.2 is the following ray-strict monotonicity of its subdifferential ∂K : V˙+ → D (Ω) deﬁned formally for each “suitable” v0 ∈ V˙+ by 1/p

def

∂K(v0 ) = −

div(a(x, ∇(v0 ))) (p−1)/p

.

(3.5)

v0

For v0 ∈ V˙+ we write v0 ∈ dom(∂K) if and only if (i) ess infK v0 > 0 on every compact set 1/p K ⊂ Ω, and (ii) the expression in (3.5) belongs to D (Ω). Substituting u0 = v0 above we get p

∂K(u0 ) = −

div(a(x, ∇u0)) p−1

.

u0

R EMARK 3.3. We claim V˙+ ∩ C 0 (Ω) ⊂ dom(∂K). Indeed, if v0 ∈ V˙+ ∩ C 0 (Ω) then also its “neighborhood” N(v0 ) = v0 + BK,δ = {v0 + φ: φ ∈ BK,δ }, where BK,δ = φ ∈ C01 (Ω): φ = 0 in Ω \ K and φC 1 (Ω) 0. It is now easy to see that the functional K has the directional derivative at v0 in every direction φ ∈ C01 (Ω) \ {0}. According to (3.5), this derivative is given by %

& ∂K(v0 ), φ =

Ω

1/p −(p−1)/p · ∇ v0 a x, ∇ v0 φ dx.

Hence, ∂K(v0 ) is in the dual of the Fréchet space C01 (Ω) and, in particular, in D (Ω). The following ray-strict monotonicity is a generalized version of the well-known inequality of Díaz and Saa established in [17] for the special case A(x, ξ ) = |ξ |p , (x, ξ ) ∈ Ω × RN . Their hypotheses have been weakened by Lindqvist [46]. Here we state this inequality under the hypotheses convenient to us. 1,p

L EMMA 3.4. Let u0 , u1 ∈ W0 (Ω) be such that u0 > 0 and u1 > 0 in Ω and both u0 /u1 and u1 /u0 are in L∞ (Ω). Then we have div(a(x, ∇u0 )) div(a(x, ∇u1)) p p − u0 − u1 dx 0 (3.6) + p−1 p−1 Ω u0 u1 where the equality holds if and only if v1 /v0 ≡ const in Ω. Of course, the integral in (3.6) has to be understood as a(x, ∇u0) · ∇ u0 − (u1 /u0 )p−1 u1 Ω

+ a(x, ∇u1) · ∇ u1 − (u0 /u1 )p−1 u0 dx 0.

(3.7)

The last integral converges absolutely as a Lebesgue integral. Moreover, its integrand is always nonnegative owing to inequality (3.4); it vanishes if and only if (∇u0 )/u0 = (∇u1 )/u1 . However, if we know that both expressions div(a(x, ∇u0 )) and div(a(x, ∇u1 )) are, say, in L∞ (Ω), then also the integral in (3.6) converges absolutely. The proof of our generalized version of the Díaz–Saa inequality is analogous to those in [17] and [46]; we leave the details of proving (3.4) ⇒ (3.7) to the reader. 3.3. The ﬁrst eigenfunction ϕ1 The inequality of Díaz and Saa is often used to show that the eigenvalue λ1 in problem (3.2) is simple, cf. [17] and [46]. In order to avoid any smoothness hypothesis on the boundary ∂Ω, we prefer to apply Lemma 3.2 directly. 1,p

C OROLLARY 3.5. Let u0 , u1 ∈ W0 (Ω) be two eigenfunctions associated with the eigenvalue λ1 , i.e., let u0 and u1 be two nontrivial weak solutions to problem (3.2). Then either u0 > 0 or else u0 < 0 throughout Ω, and u1 /u0 ≡ const in Ω.

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P. Takáˇc

This corollary is a nonlinear version of the well-known Kre˘ın–Rutman theorem for linear operators. If ∂Ω is of class C 1,α for some α ∈ (0, 1), and Ω satisﬁes the interior sphere condition at every point of ∂Ω, then the Hopf maximum principle (Lemma 2.5) applies to (3.2), and so the abstract nonlinear Kre˘ın–Rutman theorem from [55], Theorem 3.5, p. 1763, can be used to derive our corollary. An alternative proof, using Picone’s identity for the p-Laplacian, can be found in [1], Theorem 2.1, p. 821. Another abstract version of the nonlinear Kre˘ın–Rutman theorem is given in [41]. P ROOF OF C OROLLARY 3.5. Owing to formula (3.1) we observe that both u0 and u1 are minimizers for the functional def

Jλ(0) (u) = 1

1 p

A(x, ∇u) dx − Ω

λ1 p

B(x)|u|p dx Ω

1,p

deﬁned for u ∈ W0 (Ω). By our reasoning at the beginning of this section, we know that the function ui (i = 0, 1) has deﬁnite sign = ±1 throughout Ω; we may assume ui > 0 p (0) 1/p in Ω. Hence, the function vi = ui satisﬁes vi ∈ V˙+ and Jλ1 (vi ) = 0. Now we apply Lemma 3.2 to conclude that the functional def (0) A x, ∇ v 1/p dx − λ1 B(x)v dx, v ∈ V˙+ , Kλ1 (v) = pJλ1 v 1/p = Ω

Ω

is ray-strictly convex on V˙+ . Consequently, if v1 /v0 ≡ const in Ω, then for any convex combination v = (1 − θ )v0 + θ v1 with θ ∈ (0, 1) we must have Kλ1 (v) < (1 − θ )Kλ1 (v0 ) + θ Kλ1 (v1 ) = 0, a contradiction to formula (3.1). We have proved u1 /u0 ≡ const in Ω as desired.

R EMARK 3.6. According to Corollary 3.5, from now on we denote by ϕ1 the positive solution to problem (3.2) normalized by the condition ϕ1 Lp (Ω) = 1. In this way ϕ1 is determined uniquely. Recall that the strong maximum principle (Lemma 2.5) guarantees ϕ1 > 0 almost everywhere in Ω. Moreover, if the boundary ∂Ω is of class C 1,α for some for some β ∈ (0, α), by Proposition 2.1. Finally, if Ω satα ∈ (0, 1), then ϕ1 ∈ C 1,β (Ω) isﬁes also an interior sphere condition at a point x0 ∈ ∂Ω, then (∂ϕ1 /∂ν)(x0 ) < 0, by the Hopf maximum principle (Lemma 2.5). We will need these facts throughout the rest of this work.

4. Subcritical spectral problems (λ < λ1 ) 1,p

We are concerned with the critical points of the energy functional Jλ on W0 (Ω) deﬁned by (1.1) for the “subcritical” values of the spectral parameter λ, −∞ < λ < λ1 . We assume

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403

that A and B satisfy Hypotheses (A) and (B), respectively, and the function F satisﬁes Hypothesis (F) together with the decay condition f (x, u) → 0 as |u| → ∞, uniformly for x ∈ Ω. |u|p−1

(4.1)

This means that f = ∂F /∂u is assumed to be asymptotically (p − 1)-subhomogeneous; cf. (1.3). 1,p 1,p The existence of a critical point u0 ∈ W0 (Ω) for Jλ on W0 (Ω), that is a global minimizer, is a textbook result obtained by a standard minimization argument ([53], Theorem 1.2, p. 4). However, the uniqueness of this critical point depends on the geometry of the functional Jλ . If, for instance, λ 0 and the function u → f (x, u) is decreasing on R for a.e. x ∈ Ω, then both functions u → −λ|u|p and u → −F (x, u) are convex, and there1,p fore the functional Jλ is strictly convex on W0 (Ω). This shows that the global minimizer is the only critical point for Jλ ([53], pp. 58–60). The case 0 < λ < λ1 is more delicate and so one needs to be more speciﬁc in addressing the question of uniqueness of a critical point. Let us consider only the case f (x, u) ≡ f (x) independent from u ∈ R where f ∈ L∞ (Ω). Examples constructed in [29], Example 2, p. 148, for 1 < p < 2 and [49], Eq. (5.26), p. 12, for 2 < p < ∞, both in an open interval Ω ⊂ R1 , show that besides a global minimizer there also might be a saddle point for Jλ . In these counterexamples to uniqueness, the function f changes sign in the interval Ω. The next theorem shows that this is essential.

4.1. Existence and uniqueness for λ < λ1 T HEOREM 4.1. Let −∞ < λ < λ1 . Assume that A and B satisfy Hypotheses (A) and (B), respectively, and the function F satisﬁes Hypothesis (F) together with the decay condition (4.1). In addition, let f 0 in Ω × R and assume that the function u → f (x, u)/up−1 is decreasing on (0, ∞) for a.e. x ∈ Ω and strictly decreasing for all x ∈ Ω from a set Ω ⊂ Ω of positive Lebesgue measure. Then the Dirichlet boundary value problem

− div a(x, ∇u) = λ B(x)|u|p−2 u + f x, u(x) in Ω, u=0 on ∂Ω,

(4.2)

1,p

possesses a weak solution u ∈ W0 (Ω). Moreover, if f (·, 0) ≡ 0 in Ω then u > 0 in Ω, and this solution is unique. On the other hand, if f (·, 0) ≡ 0 in Ω then, besides the trivial solution ≡ 0 in Ω, problem (4.2) possesses at most one nontrivial solution u; it satisﬁes u > 0 in Ω again. Finally, the nontrivial solution (if it exists) is the global minimizer for Jλ 1,p over W0 (Ω). For the special case A(x, ξ ) = |ξ |p , (x, ξ ) ∈ Ω × RN , this theorem was ﬁrst obtained by Díaz and Saa [17], Théorème 1 et 2, p. 521. The method of proof we present below is taken from [29], Appendix and [60], Proof of Theorem 2.5.

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P ROOF OF T HEOREM 4.1. We begin with a trivial observation: The positive p-homogeneity hypothesis (1.2) implies A(x, ξ ) = ξ · a(x, ξ ) for (x, ξ ) ∈ Ω × RN , where def 1 p ∂ξ A(x, ξ ).

a(x, ξ ) =

This is an easy computation,

A(x, ξ ) − A(x, 0) = p

1

ξ · a(x, tξ ) dt

0

1

=p

t

p−1

dt ξ · a(x, ξ ) = ξ · a(x, ξ ).

0 1,p

1,p

Let u ∈ W0 (Ω) be any critical point for Jλ on W0 (Ω), i.e., let u be a weak solution of problem (4.2). We claim that u 0 in Ω. Indeed, we can multiply (4.2) by 1,p u− ∈ W0 (Ω) and then integrate by parts over Ω, thus arriving at

a(x, ∇u) · ∇u− dx = λ Ω

B(x)|u|p−2 uu− dx + Ω

f x, u(x) u− dx. Ω

(Recall u− = max{−u, 0}.) Employing A(x, ξ ) = ξ · a(x, ξ ) and f 0 in Ω × R, we get

A x, ∇u− dx λ Ω

p B(x) u− dx. Ω

Formula (3.1) and λ < λ1 then force u− ≡ 0 in Ω, i.e., u 0 in Ω. The a priori regularity result in Proposition 2.1 applied to problem (4.2) guarantees u ∈ C 1,β (Ω) for some β ∈ (0, 1). The strong maximum principle (Lemma 2.5) leaves us with the following two alternatives: Either u > 0 throughout Ω, or else both u ≡ 0 and f ( · , 0) ≡ 0 in Ω. It remains to treat only the former alternative; we have to show that Jλ 1,p has at most one critical point u ∈ W0 (Ω) with u > 0 in Ω. To this end, let u0 denote 1,p a global minimizer for Jλ on W0 (Ω). Hence, u0 0 in Ω and Jλ (u0 ) Jλ (0) = 0. 1,p Moreover, we have either u0 > 0 in Ω, or else u0 ≡ 0 in Ω. Suppose that u1 ∈ W0 (Ω) 1,p is another critical point for Jλ on W0 (Ω) such that u1 ≡ 0 in Ω; hence u1 > 0 in Ω. We p will arrive at a contradiction, i.e., we show that u1 ≡ u0 in Ω. Set vi = ui (i = 0, 1); so v0 ∈ V˙+ ∪ {0} and v1 ∈ V˙+ . Next we show that the functional v → Jλ (v 1/p ) is strictly convex on V˙+ ∪ {0}. Indeed, deﬁne λ def 1 1,p (0) Jλ (u) = A(x, ∇u) dx − B(x)|u|p dx, u ∈ W0 (Ω). p Ω p Ω (0) Lemma 3.2 shows that the functional v → Jλ (v 1/p ) is ray-strictly convex on V˙+ . Furthermore, employing our hypothesis that the function u → f (x, u)/up−1 is decreasing on (0, ∞) for a.e. x ∈ Ω and strictly decreasing for all x ∈ Ω from a set Ω ⊂ Ω of positive Lebesgue measure, we obtain that the function v → −F (x, v 1/p ) is convex on

Nonlinear spectral problems

405

R+ = [0, ∞) for a.e. x ∈ Ω and strictly convex for all x ∈ Ω . We conclude that the functional def Kλ (v) = pJλ v 1/p = pJλ(0) v 1/p − p

F x, v 1/p dx, Ω

v ∈ V˙+ ∪ {0},

(4.3)

is strictly convex. Now we are ready to show u1 ≡ u0 in Ω. Suppose that u1 ≡ u0 in Ω and consider the function κ(θ ) = Kλ (1 − θ )v0 + θ v1 for 0 θ 1. p Recall vi = ui (i = 0, 1) with v0 ∈ V˙+ ∪ {0} and v1 ∈ V˙+ . The function u0 being a global 1,p minimizer for Jλ on W0 (Ω), we must have κ(θ ) κ(0) for all κ ∈ [0, 1]. Moreover, κ is strictly convex on [0, 1], by the strict convexity of Kλ . Elementary analysis results then imply

0 lim

θ→0+

κ(θ ) − κ(0) κ(1) − κ(1 − t) < lim , t →0+ θ t

(4.4)

for the one-sided derivatives of a convex function exist and are increasing. Since u1 ∈ C 1,β (Ω) and u1 > 0 in Ω, the subdifferential ∂Kλ (v1 ) of Kλ at v1 exists in D (Ω) and is given by (see Remark 3.3) ∂Kλ (v1 ) = − =−

1 (p−1)/p v1

1 (p−1)/p v1

1/p 1/p div a x, ∇ v1 + f x, v1 (x) − λB(x)

1/p Jλ v1

in Ω,

(4.5)

where Jλ : W0 (Ω) → W −1,p (Ω) stands for the Fréchet derivative of Jλ on W0 (Ω). 1,p In the norm of W0 (Ω) we approximate the difference v0 − v1 by a test function φ from D(Ω) (hence, with compact support) in order to guarantee v1 + tφ 12 v1 > 0 for all 0 t t1 , where t1 ∈ (0, 1) is a sufﬁciently small number. If φ is taken close enough to v0 − v1 , then the second inequality in (4.4) remains valid also when v0 − v1 is replaced by φ. That inequality shows that we cannot have ∂Kλ (v1 ) = 0 in D (Ω). Consequently, 1/p 1/p neither can the equation Jλ (v1 ) = 0 hold in W −1,p (Ω). We conclude that u1 = v1 is not a critical point for Jλ , a contradiction to our assumption. 1,p We have veriﬁed that u0 , a global minimizer for Jλ over W0 (Ω), is the only nontrivial critical point for Jλ if u0 ≡ 0 in Ω, and the only critical point if u0 ≡ 0. This ﬁnishes our proof of the theorem. 1,p

1,p

Even though the two remaining paragraphs of this section deal with two special problems which are critical (λ = λ1 ) and even supercritical (λ − λ1 > 0 small enough), we have

406

P. Takáˇc

decided to include them here because these problems can be treated by the same methods that we have used in the proof of Theorem 4.1. 4.2. Nonexistence for λ = λ1 Much of the present article is devoted to the question of solvability of the critical spectral problem

− div a(x, ∇u) = λ1 B(x)|u|p−2 u + f (x) in Ω, u = 0 on ∂Ω.

(4.6)

The following nonexistence result complements Theorem 4.1. It will turn out to be extremely useful later in a detailed analysis of certain asymptotic properties of large solutions to problem (4.2) for λ near λ1 (λ λ1 ). T HEOREM 4.2. Assume that A and B satisfy hypotheses (A) and (B), respectively, and f ∈ L∞ (Ω) satisﬁes 0 f ≡ 0 in Ω. Then problem (4.6) has no weak solution 1,p u ∈ W0 (Ω). For the special case A(x, ξ ) = |ξ |p , (x, ξ ) ∈ Ω × RN , this theorem is due to Fleckinger et al. [27], Théorème 1, p. 731, or [28], Theorem 2.3, p. 54. A different proof thereof, based on Picone’s identity, is given in [1], Theorem 2.4, p. 824. 1,p

P ROOF OF T HEOREM 4.2. On the contrary, suppose that u0 ∈ W0 (Ω) is a weak solution of problem (4.6). One shows u0 ∈ C 1,β (Ω) for some β ∈ (0, 1) and u0 > 0 throughout Ω p exactly as in the proof of Theorem 4.1. Set v0 = u0 ; so v0 ∈ V˙+ . def Again, the functional v → Kλ1 (v) = pJλ1 (v 1/p ) is strictly convex on V˙+ ∪ {0}. Recall that the subdifferential ∂Kλ1 (v0 ) of Kλ1 at v0 exists in D (Ω) and is given by formula (4.5) with v0 in place of v1 , λ = λ1 and f (x, u0 ) ≡ f (x). Since u0 is a critical point for Jλ1 , so is v0 for Kλ1 . But this means that v0 is the global minimizer for Kλ1 over V˙+ ∪ {0} and the unique critical point of Kλ1 as well. As a consequence, Kλ1 is bounded from below on V˙+ ∪ {0}. On the other hand, from (4.3) we compute p 1/p τ ϕ1 − p Kλ1 τ ϕ1 = pJλ(0) 1

F x, τ 1/p ϕ1 dx = −pτ 1/p Ω

f ϕ1 dx Ω

p

for τ ∈ R+ , which shows Kλ1 (τ ϕ1 ) → −∞ as τ → +∞, a contradiction to the boundedness of Kλ1 from below. The theorem is proved. R EMARK 4.3. In Theorem 4.2, the hypothesis 0 f ≡ 0 in Ω is not necessary for the nonexistence in problem (4.6); it can be “slightly” perturbed, see [57], Corollaries 2.4 and 2.9, for the special case A(x, ξ ) = |ξ |p , (x, ξ ) ∈ Ω × RN . We will treat this generalization later in Section 8.6 (Theorem 8.14).

Nonlinear spectral problems

407

4.3. Anti-maximum principle for λ > λ1 We combine the Hopf maximum principle (Lemma 2.5) with the nonexistence result for λ = λ1 (Theorem 4.2) to derive the so-called anti-maximum principle for the supercritical spectral problem

− div a(x, ∇u) = λB(x)|u|p−2 u + f (x) in Ω, u = 0 on ∂Ω,

(4.7)

with λ − λ1 > 0 small enough, which is due to Fleckinger et al. [27], Théorème 2, p. 732, or [28], Theorem 2.4, p. 55 (see also [56], Theorem 7.2, p. 154), again, for the special case A(x, ξ ) = |ξ |p , (x, ξ ) ∈ Ω × RN . The anti-maximum principle was ﬁrst obtained by Clément and Peletier [9], Theorem 1, p. 222, for the linear Dirichlet problem (p = 2) using spectral analysis for λ near λ1 . T HEOREM 4.4. Let Ω ⊂ RN be a bounded domain with C 1,α boundary for some α ∈ (0, 1), and let Ω satisfy the interior sphere condition at every point of ∂Ω. Assume that A and B satisfy Hypotheses (A) and (B), respectively, and f ∈ L∞ (Ω) satisﬁes 0 f ≡ 0 in Ω. Then there exists a constant δ ≡ δ(f ) > 0 depending on f such that every weak solution u to problem (4.2) satisﬁes the anti-maximum principle u0 ∂ν

on ∂Ω

(4.8)

whenever λ1 < λ < λ1 + δ. 1,p

Recall that any weak solution u ∈ W0 (Ω) to problem (4.2) with any λ ∈ R satisﬁes for some β ∈ (0, α), by Proposition 2.1. u ∈ C 1,β (Ω) P ROOF OF T HEOREM 4.4. We proceed by contradiction. Suppose there is no such constant δ ≡ δ(f ) > 0. Then there exists a sequence {αk }∞ k=1 ⊂ (λ1 , ∞) with αk → λ1 as k → ∞, such that for every k = 1, 2, . . . , problem (4.7) with λ = αk has a weak solution 1,p uk ∈ W0 (Ω) which does not satisfy inequalities (4.8). This means

− div a(x, ∇uk ) = αk B(x)|uk |p−2 uk + f (x) uk = 0 on ∂Ω.

in Ω,

(4.9)

We claim uk ∞ → ∞

as k → ∞.

(4.10)

∞ Suppose not; then there is a subsequence of {uk }∞ k=1 , denoted by {uk }k=1 again, that ∞ is bounded in L (Ω). The regularity result in Proposition 2.1 implies that {uk }∞ k=1 is for some β ∈ (0, 1). Moreover, by Arzelà–Ascoli’s theorem, {uk }∞ bounded in C 1,β (Ω) k=1 ∗ for any β ∗ ∈ (0, β). Thus, we may extract a convergent is relatively compact in C 1,β (Ω)

408

P. Takáˇc

∗ as k → ∞. Letting k → ∞ in the weak formulation subsequence unk → u∗ in C 1,β (Ω) of (4.9), we arrive at p−2 ∗ a x, ∇u∗ · ∇w dx = λ1 B(x)u∗ u w dx + f (x)w dx

Ω

Ω

Ω

∗

is a weak solution of problem (4.7) with λ = λ1 , for all w ∈ W0 (Ω). So u∗ ∈ C 1,β (Ω) a contradiction to Theorem 4.2. We have proved our claim (4.10). Now set vk = uk /uk ∞ for k = 1, 2, . . . , obviously vk ∞ = 1. Thus, problem (4.9) becomes ⎧ ⎨ − diva(x, ∇v ) = α B(x)|v |p−2 v + f (x) in Ω, k k k k p−1 (4.11) uk ∞ ⎩ ∂Ω. vk = 1,p

1,β ∗ (Ω), let us extract a convergent subsequence Since {vk }∞ k=1 is relatively compact in C ∗ ∗ 1,β as k → ∞. Again, letting k → ∞ in the weak formulation of (4.11), (Ω) vnk → v in C we arrive at p−2 ∗ ∗ a x, ∇v · ∇w dx = λ1 B(x)v ∗ v w dx Ω

Ω

∗ 1,p is an eigenfunction for the nonlinear for all w ∈ W0 (Ω). We conclude that v ∗ ∈ C 1,β (Ω) ∗ eigenvalue problem (3.2), with v ∞ = 1. Hence, v ∗ = cϕ1 with some constant c ∈ R, c = 0. We distinguish between the following two cases: Case c > 0. There is an integer k0 1 such that each vnk , for k k0 , satisﬁes the Hopf maximum principle (Lemma 2.5), that is,

vnk > 0

in Ω

and

∂vnk 0

in Ω,

we may apply our nonexistence result, Theorem 4.2, to problem (4.13) with fnk in place of f , thus arriving at a contradiction.

Nonlinear spectral problems

409

Case c < 0. Again, there is an integer k0 1 such that each −vnk , for k k0 , satisﬁes inequalities (4.12). But this contradicts our assumption that unk = unk ∞ vnk does not satisfy inequalities (4.8). Theorem 4.4 is proved. R EMARK 4.5. The hypothesis 0 f ≡ 0 in Ω in Theorem 4.4 is not necessary for the (weaker) anti-maximum principle u < 0 in Ω. It can be proved under the (weaker) hypoth

esis Ω f ϕ1 dx > 0 combined with the nonexistence for problem (4.6) (i.e., the conclusion of Theorem 4.2). For the special case A(x, ξ ) = |ξ |p , (x, ξ ) ∈ Ω × RN , this improvement is due to Arcoya and Gámez [5], Theorem 27, p. 1908. We will give a different proof thereof later in Section 8.5 (Theorem 8.13). We have seen in the proof of Theorem 4.4 that the Hopf maximum principle (Lemma 2.5), called also Hopf ’s lemma, applied to the eigenfunction ϕ1 (cf. Remark 3.6) played a crucial role in obtaining the anti-maximum principle (4.8). This fact has been explored further in greater detail for domains Ω with nonsmooth boundary (e.g., with corners in R2 ) and for p = 2 in the works of Birindelli [6] and Sweers [54]. They studied the questions of the validity of Hopf’s lemma and the anti-maximum principle separately. R EMARK 4.6. The anti-maximum principle has played an important role in the recent studies on the Fuˇcík spectrum of the p-Laplacian p with various boundary conditions; the reader is referred to [11,27,28] and numerous references therein for analytical results, and to [8] for numerical results. R EMARK 4.7. Last but not least, let us mention that many of the results presented in this section have been generalized to systems of equations involving the p-Laplacian p . Most of them remain valid for strictly cooperative systems; see [10,28,30–33,56].

5. Linearization about the ﬁrst eigenfunction We would like to answer the question of existence and uniqueness or multiplicity of weak 1,p solutions u ∈ W0 (Ω) to problem (4.2) also in the “resonant” case λ = λ1 . Recall that for p = 2 the problem in (4.2) is semilinear and has been extensively studied. In particular, if f (x, u) ≡ f (x) is independent from u ∈ R where f ∈ L2 (Ω), then one can apply the standard Fredholm alternative for a selfadjoint linear operator on the Hilbert space L2 (Ω) in order to conclude that either (i) f, ϕ1 ! = 0 in which case the set of all weak solutions to problem (4.2) is a straight 1,p line {u0 + τ ϕ1 : τ ∈ R} in W0 (Ω) with u0 , ϕ1 ! = 0, or else (ii) f, ϕ1 ! = 0 in which case problem (4.2) has no solution. Arguing by intuition from bifurcation theory, one should expect that the straight line of solutions from case (i) becomes “deformed” for p = 2. To describe this phenomenon, another parameter (besides τ ∈ R) has to be introduced into problem (4.2). In [22,24,58],

410

P. Takáˇc

the orthogonality condition f, ϕ1 ! = 0 has been replaced by f = f · ϕ1 + f ) ,

& % def where f = ϕ1 −2 f, ϕ1 ! and f ) , ϕ1 = 0, L2 (Ω)

(5.1)

with ζ = f ∈ R being the parameter and f ) ∈ L∞ (Ω) ﬁxed, f ) ≡ 0 in Ω. Let u ∈ 1,p W0 (Ω), u = τ ϕ1 + u) with τ ∈ R, be a solution of (4.2). The following a priori relation between τ and ζ was established in [58], Proposition 6.1, p. 331, for p = 2: · Q0 (u0 , u0 ) = 0, lim |τ |p−2 τ ζ = (p − 2)ϕ1 −2 L2 (Ω)

|τ |→∞

(5.2)

where Q0 (u0 , u0 ) is a positive number depending on f ) but not on ζ ∈ R. This number corresponds to the quadratic form associated with the linearization of problem (4.2) about the ﬁrst eigenfunction ϕ1 , with λ = λ1 , described in Remark 3.6. In order to present this linearization and its important consequences in a tractable manner, from now on we restrict ourselves to the special case A(x, ξ ) = |ξ |p and B(x) = 1 for (x, ξ ) ∈ Ω × RN treated in [21,23,24,57–59] and many other articles. In particular, λ1 and ϕ1 satisfy −p ϕ1 = λ1 |ϕ1 |p−2 ϕ1

in Ω;

ϕ1 = 0

on ∂Ω.

(5.3)

The eigenvalue λ1 is given by the variational formula (1.8). The eigenfunction ϕ1 associated with λ1 is normalized by ϕ1 > 0 in Ω and ϕ1 Lp (Ω) = 1. We would like to stress that practically all our results presented below apply to the general case as well, with the obvious necessary adjustments, provided A and B satisfy Hypotheses (A) and (B), respectively. We leave details to the reader.

5.1. Linearization and quadratization In order to determine the asymptotic behavior of “large” solutions u to problem (1.6), 1,p u = τ ϕ1 + u) with τ ∈ R and u) ∈ W0 (Ω) as |τ | → ∞, we will estimate the functional u) → Jλ1 (τ ϕ1 + u) ) by suitable quadratic forms. Recall that Jλ1 has been deﬁned in (1.4). To this end, we need to compute the ﬁrst two Fréchet derivatives of the functional Jλ1 . Our computations are rigorous for p > 2; we leave a few formal corrections for 1 < p < 2 to the reader. We deﬁne the functional def 1 1,p F (u) = |∇u|p dx, u ∈ W0 (Ω). p Ω The ﬁrst Fréchet derivative F (u) of F at u ∈ W0 (Ω) is given by F (u) = −p u in W −1,p (Ω), where p1 + p1 = 1. This follows from 1,p

% & F (u), φ =

|∇u|p−2 ∇u · ∇φ dx, Ω

1,p

φ ∈ W0 (Ω).

(5.4)

Nonlinear spectral problems

411

The second Fréchet derivative F (u) is a bit more complicated; if 1 < p < 2, it might have to be considered only as a Gâteaux derivative which is not even densely deﬁned: % & F (u)ψ, φ = |∇u|p−2 (∇φ · ∇ψ) + (p − 2)|∇u|−2 (∇u · ∇φ)(∇u · ∇ψ) dx

Ω

=

|∇u|

p−2

Ω

; : ∇u ⊗ ∇u , ∇φ ⊗ ∇ψ dx, I + (p − 2) |∇u|2 RN×N

1,p

φ, ψ ∈ W0 (Ω).

(5.5)

Of course, I is the identity matrix in RN×N , the tensor product a ⊗ b stands for the N N (N × N)-matrix T = (ai bj )N i,j =1 whenever a = (ai )i=1 and b = (bi )i=1 are vectors N N×N . from R , and · , · !RN×N denotes the Euclidean inner product in R For a ∈ RN (a = ∇u in our case), a = 0 ∈ RN , we introduce the abbreviation def

A(a) = |a|

p−2

a⊗a I + (p − 2) . |a|2

(5.6)

def

If p > 2, we set also A(0) = 0 ∈ RN×N . For a = 0, A(a) is a positive deﬁnite symmetric matrix. Its positive deﬁnite square root is equal to def

A(a) = |a|

(p−2)/2

I + −1 +

p−1

a ⊗ a |a|2

.

The spectrum of the matrix |a|2−p A(a) consists of the eigenvalues 1 and p − 1; moreover, we have for v ∈ RN , with v · a = 0,

A(a)v = |a|p−2 v

A(a)a = (p − 1)|a|p−2a. For all a, v ∈ RN \ {0} we thus obtain min{1, p − 1}

A(a)v, v!RN max{1, p − 1}. |a|p−2 |v|2

(5.7)

Notice that for the general version of the energy functional (1.1) the matrix A(a) takes the form def

A(x, a) = |a|p−2

2 N N ∂ai ∂ A 1 (x, ξ ) = |a|p−2 (x, ξ ) , ∂ξj p ∂ξi ∂ξj i,j =1 i,j =1

where x ∈ Ω and ξ = |a|−1 a for a ∈ RN \ {0}.

412

P. Takáˇc

From this point on, until the end of this paragraph, we restrict ourselves to p > 2. The case 1 < p < 2 is somewhat different and will be taken care of in detail in the second half of the next subsection (Section 5.2). We rewrite the p-homogeneous part of the energy functional (1.4) with λ = λ1 using the integral forms of the ﬁrst- and second-order Taylor 1,p formulas. Let φ ∈ W0 (Ω) be arbitrary. We combine (5.3) and (5.4) to obtain 1 p

∇(ϕ1 + φ)p dx − λ1 |ϕ1 + φ|p dx p Ω Ω 1 ∇(ϕ1 + sφ)p−2 ∇(ϕ1 + sφ) · ∇φ dx ds = 0

Ω

1

− λ1 0

|ϕ1 + sφ|p−2 (ϕ1 + sφ)φ dx ds.

(5.8)

Ω

Similarly, using (5.3) and (5.5), we get 1 ∇(ϕ1 + φ)p dx − λ1 |ϕ1 + φ|p dx = Qφ (φ, φ), p Ω p Ω

(5.9) 1,p

where Qφ is the symmetric bilinear form on the Cartesian product [W0 (Ω)]2 deﬁned as follows, using the matrix abbreviation (5.6): Qφ (v, w) :# def = Ω

1

$ ; A ∇(ϕ1 + sφ) (1 − s) ds ∇v, ∇w

0

#

1

− λ1 (p − 1) Ω

|ϕ1 + sφ|

p−2

dx RN

$ (1 − s) ds vw dx

(5.10)

0

1,p

for v, w ∈ W0 (Ω). In particular, one has 2 · Q0 (v, v) % & p−2 A(∇ϕ1 )∇v, ∇v RN dx − λ1 (p − 1) ϕ1 v 2 dx = Ω

Ω

2 ∇ϕ1 · ∇v dx = |∇ϕ1 |p−2 |∇v|2 + (p − 2) |∇ϕ1 | Ω p−2 1,p − λ1 (p − 1) ϕ1 v 2 dx, v ∈ W0 (Ω).

(5.11)

Ω

Furthermore, our deﬁnition (1.8) of λ1 and (5.9) guarantee Qt φ (φ, φ) 0 for all t ∈ R\{0}. Letting t → 0, we arrive at 1,p

Q0 (φ, φ) 0 for all φ ∈ W0 (Ω).

(5.12)

Nonlinear spectral problems

413

Next we wish to show that the symmetric bilinear form Q0 is closable in L2 (Ω) and to characterize the domain Dϕ1 of its closure; see, e.g., Kato [42], Chapter VI, §1.3, p. 313. 5.2. The weighted Sobolev space Dϕ1 In the sequel, we always assume the following hypothesis on the domain Ω: H YPOTHESIS (H1). If N 2 then Ω is a bounded domain in RN whose boundary ∂Ω is a compact manifold of class C 1,α for some α ∈ (0, 1), and Ω satisﬁes also the interior sphere condition at every point of ∂Ω. If N = 1 then Ω is a bounded open interval in R1 . It is clear that for N 2, Hypothesis (H1) is satisﬁed if Ω ⊂ RN is a bounded domain with C 2 boundary. The Hopf maximum principle (Lemma 2.5) guarantees (see Remark 3.6) ϕ1 > 0 in Ω

and

∂ϕ1 2. We start with the degenerate case 1,p 2 < p < ∞. Let us introduce a new norm on W0 (Ω) by def

1/2 |∇ϕ1 |p−2 |∇v|2 dx

vDϕ1 =

1,p

for v ∈ W0 (Ω),

Ω

(5.14)

1,p

and denote by Dϕ1 the completion of W0 (Ω) with respect to this norm. That the 1,p seminorm (5.14) is in fact a norm on W0 (Ω) follows from inequality (5.18) below. The Hilbert space Dϕ1 coincides with the domain of the closure of the quadratic form 1,p 1,p Q0 : W0 (Ω) → R given by Q0 (φ) = Q0 (φ, φ) for φ ∈ W0 (Ω), cf. formula (5.11). The singular case 1 < p < 2 is more complicated. The Hilbert space Dϕ1 , endowed with the norm (5.14) for p > 2, needs to be redeﬁned for 1 < p < 2 as follows. We deﬁne v ∈ Dϕ1 if and only if v ∈ W01,2 (Ω), ∇v(x) = 0 for almost every x ∈ Ω \ U = {x ∈ Ω: ∇ϕ1 (x) = 0}, and def

vDϕ1 =

1/2 |∇ϕ1 |

p−2

2

|∇v| dx

< ∞.

(5.15)

U

Consequently, Dϕ1 endowed with the norm · Dϕ1 is continuously embedded into

W01,2 (Ω). We conjecture that Dϕ1 is dense in L2 (Ω). This conjecture would immediately follow from |Ω \ U |N = 0. The latter holds true if Ω is convex; then also Ω \ U

414

P. Takáˇc

is a convex set in RN with empty interior, and hence of zero Lebesgue measure, see [28], Lemma 2.6, p. 55. If the conjecture is false, we need to consider also the orthogonal complement 2 Dϕ⊥,L = v ∈ L2 (Ω): v, φ! = 0 for all φ ∈ Dϕ1 . 1 Notice that v ∈ Dϕ⊥,L implies v = 0 almost everywhere in U . This means that Dϕ⊥,L 1 1 is isometrically isomorphic to a closed linear subspace of L2 (Ω \ U ). Consequently, if 2 and v is continuous in an open set G ⊃ Ω \ U , then v ≡ 0 in Ω. Indeed, this v ∈ Dϕ⊥,L 1 claim follows from the fact that Ω \ U has empty interior, by (5.3) combined with (5.13). In 2 contrast, if v ∈ L2 (Ω) satisﬁes v, ϕ1 ! = 0 then v ∈ / Dϕ⊥,L . Furthermore, we have χΩ\U ∈ 1 2 2 L 2 ⊥,L Dϕ1 , the closure of Dϕ1 in L (Ω), which implies that Dϕ1 is isometrically isomorphic to a proper subspace of L2 (Ω \ U ). This can be seen as follows. Fix any ε > 0. Since U = Ω \ U is a compact subset of Ω, and the Lebesgue measure ⊂ Ω, and |G \ U |N ε. In is regular, there is an open set G ⊂ RN such that U ⊂ G, G 2

2

def

particular, δ = dist(U , Ω \ G) > 0. Now deﬁne dist(x, Ω \ G) δ/3 , K0 = x ∈ Ω: dist(x, U ) δ/3 ; K1 = x ∈ Ω: hence, dist(K0 , K1 ) δ/3. By Tietze’s extension theorem, there exists a continuous func → [0, 1] such that v = 0 on K0 and v = 1 on K1 . This function can be molliﬁed tion v : Ω in a standard way (using a convolution of v with a smooth nonnegative function with compact support in a ball of radius < δ/3 centered at the origin) to obtain another C 1 function → [0, 1] such that w = 0 in an open neighborhood of Ω \ G and w = 1 in an open w:Ω neighborhood of U . Clearly, w ∈ Dϕ1 and 2 |w − χU | dx dx ε. G\U

Ω

ϕL2 as desired. It follows that χU ∈ D 1 Several important properties of Dϕ1 established in [57] are presented below. The following result is obvious [57], Lemma 4.1. L EMMA 5.1. Let 1 < p < ∞, p = 2, and let Hypothesis (H1) be satisﬁed. Then we have Q0 (ϕ1 , ϕ1 ) = 0 and 0 Q0 (v, v) < ∞ for all v ∈ Dϕ1 . 5.3. A compact embedding with a weight for p > 2 We assume 2 < p < ∞ throughout this paragraph. Notice that (5.7) entails & % 2 vDϕ A(∇ϕ1 )∇v, ∇v RN dx (p − 1)v2Dϕ for v ∈ Dϕ1 . 1

Ω

1

(5.16)

Nonlinear spectral problems

415

For 0 < δ < ∞, we denote by def Ωδ = x ∈ Ω: dist(x, ∂Ω) < δ

(5.17)

the δ-neighborhood of ∂Ω. Its complement in Ω is denoted by Ωδ = Ω \ Ωδ . The following compact embedding result was ﬁrst proved in [57], Lemma 4.2, p. 199. L EMMA 5.2. Let 2 < p < ∞ and assume that Hypothesis (H1) is satisﬁed. Then we have: (a) For every δ > 0 small enough, · Dϕ1 is an equivalent norm on W01,2 (Ωδ ). (b) The embedding Dϕ1 !→ L2 (Ω) is compact. P ROOF. Part (a) follows immediately from (5.13). To prove (b), we start with the proof of continuity of Dϕ1 !→ L2 (Ω). We take advantage of the Dirichlet boundary value problem (5.3) to compute, for every v ∈ C01 (Ω),

p−2 2

λ1 Ω

v dx = λ1

ϕ1

Ω

p−1 2 −1 v ϕ1 dx

ϕ1

|∇ϕ1 |p−2 ∇ϕ1 · ∇ v 2 ϕ1−1 dx

= Ω

=2

|∇ϕ1 |

p−2

Ω

(∇ϕ1 · ∇v)vϕ1−1 dx

− Ω

|∇ϕ1 |p v 2 ϕ1−2 dx.

Adding the last integral and estimating the second to the last by the Cauchy–Schwarz inequality, we arrive at

p−2 2

λ1 Ω

v dx +

ϕ1

Ω

|∇ϕ1 |p v 2 ϕ1−2 dx 1/2

|∇ϕ1 |p−2 |∇v|2 dx

2

Ω

|∇ϕ1 |p−2 |∇v|2 dx +

2 Ω

1 2

Ω

Ω

|∇ϕ1 |p v 2 ϕ1−2 dx

1/2

|∇ϕ1 |p v 2 ϕ1−2 dx,

and therefore, λ1 Ω

p−2 ϕ1 v 2 dx

1 + 2

Ω

|∇ϕ1 |p v 2 ϕ1−2 dx 2 v2Dϕ . 1

(5.18)

Since C01 (Ω) is dense in Dϕ1 , the last inequality holds also for every v ∈ Dϕ1 . Using (5.13) we conclude that the embedding Dϕ1 !→ L2 (Ω) is continuous.

416

P. Takáˇc

To prove the compactness of Dϕ1 !→ L2 (Ω), we take advantage of the Dirichlet boundary value problem (5.3) again to compute, for every v ∈ Dϕ1 ,

p−1 2

λ1 Ω

|∇ϕ1 |p−2 ∇ϕ1 · ∇ v 2 dx

v dx =

ϕ1

Ω

2

|∇ϕ1 |p−1 |∇v| · |v| dx Ω

1/2

2vDϕ1

|∇ϕ1 | v dx p 2

,

(5.19)

Ω

by the Cauchy–Schwarz inequality. Let {vn }∞ n=1 be any weakly convergent sequence in Dϕ1 ; we may assume vn $ 0. Hence, vn $ 0

weakly in L2 (Ω) and

(5.20)

weakly in L2 (Ω)

|∇ϕ1 |(p−2)/2∇vn $ 0

N

(5.21)

as n → ∞, where ∇vn ∈ [W −1,2 (Ω)]N . We will show that, indeed, vn → 0 strongly in L2 (Ω). Given any 0 < η < ∞ small enough, let us decompose Ω = Uη ∪ Uη where def Uη = x ∈ Ω: ∇ϕ1 (x) > η

def and Uη = x ∈ Ω: ∇ϕ1 (x) η .

(5.22)

We deduce from (5.20) and (5.21) that the restrictions vn |Uη of vn to Uη form a weakly convergent sequence in W 1,2 (Uη ). It follows that vn L2 (Uη ) → 0 as n → ∞, by Rellich’s theorem. Next, in (5.19) we replace v by vn . Owing to (5.21), there is a constant C > 0 independent from n such that vn Dϕ1 Cλ1 /2, and consequently, (5.19) yields

Ω

p−1 2 vn dx

ϕ1

1/2 |∇ϕ1 |p vn2 dx

C Ω

.

(5.23)

We split the integral on the right-hand side using Ω = Uη ∪ Uη . The two integrals are estimated by p p 2 |∇ϕ1 | vn dx ∇ϕ1 ∞ vn2 dx, (5.24)

Uη

Uη

Uη

|∇ϕ1 |p vn2 dx

η

p Uη

vn2 dx

η

p Ω

vn2 dx.

(5.25)

Now choose any 0 < ε < ∞. First, ﬁx η0 > 0 small enough so that p/2

η0

· sup vn L2 (Ω) n1

ε √ . C 2

(5.26)

Nonlinear spectral problems

417

Second, ﬁx η > 0 and δ > 0 sufﬁciently small such that 0 < η η0 and Ωδ ⊂ Uη , where the set Ωδ has been deﬁned in (5.17). This choice is possible by the Hopf maximum principle (5.13) for ϕ1 . Third, recalling vn L2 (Uη ) → 0 as n → ∞, ﬁx an integer n0 1 large enough such that p/2

∇ϕ1 ∞ · vn L2 (Uη )

ε √ C 2

for all n n0 .

(5.27)

The numbers η, δ and n0 being ﬁxed, we ﬁrst apply (5.26) and (5.27) to (5.24) and (5.25), respectively, and then combine the last two with the inequality (5.23), thus arriving at p−1 ϕ1 vn2 dx ε for all n n0 . (5.28) Ω

In particular, setting Ωδ = Ω \ Ωδ , we infer from (5.13) and (5.28) that vn L2 (Ω ) → 0 δ as n → ∞. Finally, we make use of Uη ∪ Ωδ = Ω to conclude that vn L2 (Ω) → 0 as n → ∞. The proof of the lemma is ﬁnished. R EMARK 5.3. For N = 1, Lemma 5.2 follows from [37], Lemma 1.3, p. 238. Also the idea of using the bilinear form Q0 was introduced there. The case N = 1 is much simpler to handle because one can compute the asymptotic behavior of the derivative ϕ1 (x) near its zeros very precisely, see (5.34) below. Now we are able to construct the closure of the symmetric bilinear form Q0 given by (5.11); see [42], Chapter VI, §1.3, p. 313. We extend the domain of Q0 to Dϕ1 × Dϕ1 . This extension of Q0 is unique and closed in L2 (Ω), as a consequence of inequality (5.16) 1,p combined with Lemma 5.2(b). Notice that the embedding W0 (Ω) !→ Dϕ1 is continuous, as ϕ1 ∈ C 1 (Ω). We denote by Aϕ1 the Friedrichs representation of the quadratic form 2Q0 in L2 (Ω); see [42], Theorem VI.2.1, p. 322. This means that Aϕ1 is a positive semideﬁnite, selfadjoint linear operator on L2 (Ω) with domain dom(Aϕ1 ), such that dom(Aϕ1 ) is dense in Dϕ1 and Aϕ1 v, w! = 2 · Q0 (v, w)

for all v, w ∈ dom(Aϕ1 ).

Notice that our deﬁnition of Q0 yields Aϕ1 ϕ1 = 0. Since the embedding Dϕ1 !→ L2 (Ω) is compact, the null space of Aϕ1 denoted by ker(Aϕ1 ) = v ∈ dom(Aϕ1 ): Aϕ1 v = 0 is ﬁnite-dimensional, by the Riesz–Schauder theorem [42], Theorem III.6.29, p. 187. 5.4. Simplicity of the ﬁrst eigenvalue for the linearization Also throughout this paragraph we keep our assumption 2 < p < ∞. In addition to (H1), we impose the following technical hypothesis on the domain Ω.

418

P. Takáˇc

H YPOTHESIS (H2). If N 2 and ∂Ω is not connected, then there is no function v ∈ Dϕ1 , Q0 (v) = 0, with the following four properties: (i) v = ϕ1 · χS a.e. in Ω, where S ⊂ Ω is Lebesgue measurable with 0 < |S|N < |Ω|N ; (ii) S is connected and S ∩ ∂Ω = ∅; (iii) every connected component of the set U is entirely contained either in S or in Ω \S; (iv) (∂S) ∩ Ω ⊂ Ω \ U . It has been conjectured in [57], §2.1, that Hypothesis (H2) always holds true provided (H1) is satisﬁed. The cases, when Ω is either an interval in R1 or else ∂Ω is connected if N 2, will be covered within the proof of Proposition 5.4 ([57], Proposition 4.4, pp. 202–205): We will show that there is no function v ∈ Dϕ1 , Q0 (v) = 0, with properties (i)–(iv). Lemma 5.1 provides another variational formula for λ1 , namely, "

λ1 = inf

' A(∇ϕ1 )∇u, ∇u!RN dx : 0 ≡ u ∈ Dϕ1 ;

p−2 (p − 1) Ω ϕ1 |u|2 dx

Ω

(5.29)

cf. (1.8). This is a generalized Rayleigh quotient formula for the ﬁrst (smallest) eigenvalue p−2 of the selfadjoint operator (p − 1)−1 Aϕ1 + λ1 ϕ1 on L2 (Ω). The following result determines all minimizers: A minimizer u ∈ Dϕ1 for λ1 in (5.29) is unique up to a constant multiple of ϕ1 . This statement is equivalent to: P ROPOSITION 5.4. Let 2 < p < ∞ and assume that both Hypotheses (H1) and (H2) are satisﬁed. Then a function u ∈ Dϕ1 satisﬁes Q0 (u, u) = 0 if and only if u = κϕ1 for some constant κ ∈ R. This result is due to [57], Proposition 4.4, pp. 202–205; its proof given below is quite technical. We stress that this proposition is the only place where Hypothesis (H2) is needed explicitly. All other results in this article depend solely on the conclusion of the proposition, that is, dim(ker(Aϕ1 )) = 1, which in turn implies (H2). P ROOF OF P ROPOSITION 5.4. Step 1. Recall that the embedding Dϕ1 !→ L2 (Ω) is compact, by Lemma 5.2(b). Let u be any (nontrivial) minimizer for λ1 in (5.29). First, suppose that u changes sign in Ω. Denote u+ = max{u, 0} and u− = max{−u, 0}. Then we have, using [39], Theorem 7.8, p. 153,

λ1 =

ϕ1

(u+ )2

A(∇ϕ1 )∇u+ , ∇u+ !RN dx

p−2 (p − 1) Ω ϕ1 (u+ )2 dx

p−2 − 2

ϕ (u ) A(∇ϕ1 )∇u− , ∇u− !RN dx + Ω 1 p−2 · Ω

p−2 u2 (p − 1) Ω ϕ1 (u− )2 dx Ω ϕ1 p−2

Ω

p−2 2 u Ω ϕ1

·

Ω

Nonlinear spectral problems

p−2

Ω

ϕ1

Ω

(u+ )2

p−2 2 u

ϕ1

+

p−2

Ω

ϕ1

Ω

(u− )2

p−2 2 u

ϕ1

419

λ1

= λ1 . Consequently, both u+ and u− are (nontrivial) minimizers for λ1 . Step 2. Let V denote the set of all connected components of the open set U = {x ∈ Ω: ∇ϕ1 (x) = 0}. We show that if u ∈ ker(Aϕ1 ) then u is a constant multiple of ϕ1 in each set V ∈ V. Since ϕ1 satisﬁes (5.3), it is of class C ∞ in U , by classical regularity theory [39], def Theorem 8.10, p. 186. Now, for each γ ∈ R ﬁxed, consider the function vγ = u − γ ϕ1 in Ω. Then both vγ+ and vγ− belong to ker(Aϕ1 ) and thus satisfy the equation p−2 −∇ · A(∇ϕ1 )∇vγ± = λ1 (p − 1)ϕ1 vγ± 0 in U.

(5.30)

Again, by classical regularity theory [39], Theorem 8.10, p. 186, we have vγ± ∈ C ∞ (U ). So we may apply the strong maximum principle [39], Theorem 3.5, p. 35, to (5.30) in every set V ∈ V to conclude that either vγ+ ≡ 0 in V , or else vγ+ > 0 throughout V , and similarly for vγ− . This means that sgn(u − γ ϕ1 ) ≡ const in V . Moving γ from −∞ to +∞, we get u ≡ κV ϕ1 in V for some constant κV ∈ R, as claimed. Step 3. Let u ∈ ker(Aϕ1 ). Next we show that the fraction u/ϕ1 : Ω → R takes only ﬁnitely many values after u has been suitably adjusted on a set of zero Lebesgue measure. As above, for each γ ∈ R ﬁxed, consider the function v˜γ = (u/ϕ1 ) − γ in Ω. We move γ from −∞ to +∞ and use the fact that vγ± = v˜γ± ϕ1 ∈ ker(Aϕ1 ) to conclude that v˜0 = u/ϕ1 must coincide with a ﬁnitely-valued function almost everywhere in Ω, because ker(Aϕ1 ) is ﬁnite-dimensional and contains ϕ1 . Step 4. By contradiction, suppose that ker(Aϕ1 ) has dimension 2. From ϕ1 ∈ ker(Aϕ1 ) and u ∈ ker(Aϕ1 ) ⇒ u± ∈ ker(Aϕ1 ) we deduce that there exists a function v ∈ ker(Aϕ1 ) with the following four properties: (i) v = ϕ1 · χS a.e. in Ω, where S ⊂ Ω is a Lebesgue measurable set such that both S and Ω \ S have positive measure; (ii) the closure S is connected and S ∩ ∂Ω = ∅; (iii) for every V ∈ V we have either V ⊂ S or V ⊂ Ω \ S; (iv) x ∈ (∂S) ∩ Ω ⇒ ∇ϕ1 (x) = 0. Indeed, such a function v can be easily constructed, starting from an arbitrary function u ∈ ker(Aϕ1 ), u ≡ γ ϕ1 for any γ ∈ R: u(x)/ϕ1 (x) =

m

κi · χSi (x),

x ∈ Ω,

i=1

where κ1 < κ2 < · · · < κm , m 2, and every Si ⊂ Ω is Lebesgue measurable of positive measure, Si ∩ Sj = ∅ for i = j . Let us ﬁx any γ such that κ1 < γ < κ2 . For vγ = u − γ ϕ1 we have vγ− ∈ ker(Aϕ1 ) and the fraction v˜γ− (x) =

vγ− (x) ϕ1 (x)

= (γ − κ1 )χS1 (x),

x ∈ Ω,

420

P. Takáˇc

takes precisely two possible values a.e. on Ω, namely, 0 and γ − κ1 . Clearly, ϕ1 · χS1 ∈ ker(Aϕ1 ) and so ϕ1 · χΩ\S1 = ϕ1 (1 − χS1 ) ∈ ker(Aϕ1 ) as well; also, |S1 |N > 0 and |Ω \ S1 ∩ ∂Ω = ∅. In particular, S1 |N > 0. Replacing S1 by Ω \ S1 if necessary, we may assume S1 possesses a connected component K such that K ∩ ∂Ω = ∅. The set S1 being compact, def def we have K = S where S = S1 ∩ K. It is obvious that the function v = ϕ1 · χS is in ker(Aϕ1 ) and satisﬁes (i) and (ii); property (iii) follows from Step 2, whereas (iv) can be deduced from (iii) and inequalities (5.13). Step 5. Next, suppose also ∂Ω ⊂ S. With a help from (5.13), this is equivalent to Ω \ S ⊂ Ω. Choose any number k such that 0 < k < minx∈Ω\S ϕ1 (x), and deﬁne the functions def

ϕ1(k) = min{ϕ1 , k}

def and v (k) = max v, ϕ1(k)

in Ω.

(k)

Recalling ϕ1 , v ∈ ker(Aϕ1 ), we have ϕ1 , v (k) ∈ Dϕ1 together with v (k) = v · χS + k · χΩ\S 1,p in Ω. The Hilbert space Dϕ1 being the completion of W0 (Ω) in the norm · Dϕ1 , there is a sequence {wn }∞ n=1 ⊂ W0 (Ω) such that wn − vDϕ1 → 0 and therefore also 1,p

wn(k) − v (k) Dϕ1 → 0 as n → ∞, where def wn(k) = max wn , ϕ1(k)

in Ω, n = 1, 2, . . . .

Here we have used the continuity of the mapping u → u+ : Dϕ1 → Dϕ1 which is a version of Stampacchia’s theorem; see [64], Theorem 1.56, p. 79. Set G = {x ∈ Ω: ϕ1 (x) > k} and observe that G ⊃ Ω \ S and Ω \ G ⊂ int(S). In view (k) of wn = max{wn , k} in G, we have the inequality

% Ω

A(∇ϕ1 )∇wn(k) , ∇wn(k)

RN

dx

=

+ Ω\G

&

%

Ω\G

· · · dx G

A(∇ϕ1 )∇wn(k) , ∇wn(k)

& RN

%

dx +

A(∇ϕ1 )∇wn , ∇wn

G

& RN

dx.

Letting n → ∞, we arrive at

%

A(∇ϕ1 )∇v (k) , ∇v (k)

Ω

%

& RN

dx

A(∇ϕ1 )∇v (k) , ∇v (k)

Ω\G

= Ω

& % A(∇ϕ1 )∇v, ∇v RN dx.

& RN

%

dx + G

A(∇ϕ1 )∇v, ∇v

& RN

dx (5.31)

Nonlinear spectral problems

421

Furthermore, in view of v (k) = v · χS + k · χΩ\S in Ω and v = 0 < k in Ω \ S, we have Ω

p−2 (k) 2 ϕ1 dx v

=

+ S

p−2 2

=

Ω\S

p−2 (k) 2 ϕ1 dx v

S

ϕ1

p−2

v dx + k 2 Ω\S

ϕ1

dx

p−2 2

> Ω

ϕ1

v dx.

(5.32)

We combine inequalities (5.31) and (5.32) with (5.29) to conclude that

λ1

A(∇ϕ1 )∇v (k) , ∇v (k) !RN dx

0 such that the inequalities y 2 (p−2)/(p−1) |u(y) − u(x)|2 u (t) t (p − 1) dt y 1/(p−1) − x 1/(p−1) x a 2 p−2 u (t) ϕ (t) C dt, 0 x < y a, 1

(5.33)

0

hold for every function u ∈ Dϕ1 ; in particular, the limit u(0+) = limx→0+ u(x) exists. An analogous result is valid for the interval (−a, 0). It follows that every function u ∈ Dϕ1 is Hölder-continuous in [−a, a]. R EMARK 5.6. Unfortunately, no comparable result about the trace of a function u ∈ Dϕ1 on the set Ω \ U = {x ∈ Ω: ∇ϕ1 (x) = 0} is available for N 2 as yet. This is the main reason why one needs to assume Hypothesis (H2) in Proposition 5.4. 1,p

P ROOF OF L EMMA 5.5. The Sobolev space W0 (−a, a) being dense in Dϕ1 , it sufﬁces 1,p to verify (5.33) for u ∈ W0 (−a, a). Employing Cauchy’s inequality, we compute for

422

P. Takáˇc

all 0 x < y a: u(y) − u(x) y = u (t) dt x

y

u (t)2 t (p−2)/(p−1) dt

1/2

x

y

t −(p−2)/(p−1) dt

1/2

x

1/2 = (p − 1)1/2 y 1/(p−1) − x 1/(p−1)

u (t)2 t (p−2)/(p−1) dt

x

1/2 ,

0

which yields the ﬁrst inequality in (5.33). The second inequality is obtained from the fact that x ϕ1 (x) < 0 for all 0 < |x| a, and the following asymptotic formula: p−2 ϕ (x) as |x| → 0, ϕ1 (x) = −cx 1 + O |x|1+b 1

(5.34)

with b = 1/(p − 1) and a constant c ≡ c(p, a) > 0. This formula can be obtained directly by integrating (5.3); see, e.g., [48], Eqs. (2.6) and (2.7), [37], Proof of Lemma 1.3, p. 238, or [47], Eq. (33), for details.

5.5. Another compact embedding for 1 < p < 2 In this paragraph, we switch to the case 1 < p < 2 and further require only (H1). In fact, for some β with Hypothesis (H2) always holds true in this case. Owing to ϕ1 ∈ C 1,β (Ω), 0 < β < α < 1, this can be seen as follows. The Hilbert space Dϕ1 endowed with the norm (5.15) is continuously embedded into W01,2 (Ω). A function v described in Hypothesis (H2) cannot belong to W01,2 (Ω), by an equivalent characterization of a Sobolev space due to Beppo Levi; see, e.g., [44], Theorem 5.6.5, p. 276. R EMARK 5.7. It is not difﬁcult to verify that the conclusion of Proposition 5.4 remains valid also for the ramiﬁcation 1 < p < 2: A function u ∈ Dϕ1 satisﬁes Q0 (u, u) = 0 if and only if u = κϕ1 for some constant κ ∈ R. However, in its proof one has to work ϕL2 of Dϕ1 in L2 (Ω). One shows that with the selfadjoint operator Aϕ1 on the closure D 1 dim(ker(Aϕ1 )) = 1 in much the same way as for p > 2, making use of Beppo Levi’s equivalent characterization of W01,2 (Ω) quoted above. Notice that, by (5.7) for 1 < p < 2, (5.16) becomes & % (p − 1)v2Dϕ (5.35) A(∇ϕ1 )∇v, ∇v RN dx v2Dϕ for v ∈ Dϕ1 , 1

Ω

and so Lemma 5.1 applies with no change.

1

Nonlinear spectral problems

423

Next we highlight a couple of places at which the technique we use for p < 2 differs from that for p > 2. The most substantial difference between the two techniques is that the role of the compact embedding Dϕ1 !→ L2 (Ω) needs to be replaced by that of W01,2 (Ω) !→ Hϕ1 , where Hϕ1 is the Hilbert space deﬁned below, Hϕ1 !→ L2 (Ω). Let us deﬁne another norm on W01,2 (Ω) by def

vHϕ1 =

1/2

p−2 2

Ω

ϕ1

v dx

for v ∈ W01,2 (Ω),

(5.36)

and denote by Hϕ1 the completion of W01,2 (Ω) with respect to this norm. Embeddings that involve Hϕ1 are established next. They are taken from [57], Lemma 8.2, p. 226. L EMMA 5.8. Let 1 < p < 2 and let hypothesis (H1) be satisﬁed. Then we have: (a) The embedding Hϕ1 !→ L2 (Ω) is continuous. (b) The embedding W01,2 (Ω) !→ Hϕ1 is compact. P ROOF. Part (a) follows immediately from (5.13). To prove (b), ﬁrst notice that there exist constants 0 < c1 c2 < ∞ such that c1 ϕ1 (x)/d(x) c2 for all x ∈ Ω, where the function def

d(x) = dist(x, ∂Ω) = inf |x − x0 |, x0 ∈∂Ω

x ∈ Ω,

denotes the distance from x to ∂Ω. By well-known results taken from [43], §8.8, or [63], §3.5.2, or simply by an inequality similar to (5.18), the Sobolev space W01,2 (Ω) is continuously embedded in the weighted Lebesgue space L2 (Ω; d(x)−2 dx) endowed with the norm def

vL2 (Ω;d(x)−2 dx) =

v 2 d(x)−2 dx

1/2 < ∞.

Ω

Notice that Hϕ1 = L2 (Ω; d(x)p−2 dx). Consequently, using again the splitting Ω = Ωδ ∪ Ωδ from the proof of Lemma 5.2, we conclude that the embedding W01,2 (Ω) !→ Hϕ1 is compact. 5.6. A few geometric inequalities In Sections 5.2–5.5 we have shown the most relevant properties of the quadratic form Q0 (v) = Q0 (v, v) deﬁned in (5.10) and those of its domain, the Hilbert space Dϕ1 . In the sections to follow, we often need to compare the quadratic form Qφ (v) = Qφ (v, v) 1,p A natural deﬁned in (5.11) with Q0 (v) = Q0 (v, v), at least for φ ∈ W0 (Ω) ∩ C 1 (Ω). way to do this is to compare the kernels of these quadratic forms, so that we can use

424

P. Takáˇc

the Hilbert space Dϕ1 not only for Q0 but also for Qφ . To this end, we will use the following elementary, but important geometric inequalities due to Takáˇc [57], Appendix A, pp. 233–235. Recall that R+ = [0, ∞). We begin with the following auxiliary inequalities [57], Lemma A.1, p. 233: L EMMA 5.9. Let 1 < p < ∞ and p = 2. Assume that Θ ∈ L∞ (0, 1) satisﬁes Θ 0

1 in (0, 1) and T = 0 Θ(s) ds > 0. Then there exists a constant cp (Θ) > 0 such that the following inequalities hold true for all a, b ∈ RN : If p > 2 then /p−2 p−2 . cp (Θ) max |a + sb| 0s1

1 0

T ·

|a + sb|p−2 Θ(s) ds .

max |a + sb|

/p−2 ,

(5.37)

/p−2 p−2 . cp (Θ) . max |a + sb|

(5.38)

0s1

and if 1 < p < 2 and |a| + |b| > 0 then .

T·

max |a + sb|

/p−2

0s1

1

|a + sb|p−2 Θ(s) ds

0

0s1

P ROOF. Only the inequalities involving the constant cp (Θ) are nontrivial. Set q = p − 2; hence −1 < q < ∞, q = 0. We prove the following weaker inequality ﬁrst, with some constant κ > 0:

1

1/q |a + sb|q Θ(s) ds

κ|a| for all a, b ∈ RN .

(5.39)

0

The case a = 0 is trivial; so we will always assume a ∈ RN \ {0}. Owing to the rotational invariance of the Euclidean norm in RN , we may restrict our attention to the plane R2 (N = 2) with a = (a1 , 0) ∈ R2 and b = (b1 , b2 ) ∈ R2 . Moreover, the homogeneity of both def

sides in (5.39) allows us to assume a = e1 = (1, 0) ∈ R2 . We need to distinguish between the cases q > 0 and −1 < q < 0. Case q > 0. Consider the function F : R2 → R+ deﬁned by def

1

F (b) =

|e1 + sb|q Θ(s) ds

for b ∈ R2 .

0

This is a continuous function which satisﬁes q 1 1 F (b) b s q Θ(s) ds 2 σ

for 0 < σ 1 and |b| 2/σ.

(5.40)

Nonlinear spectral problems

425

This follows from 1 |e1 + sb| s|b| 2

1 whenever 0 < σ s 1 and σ |b| 1. 2

Taking into account 0 Θ ≡ 0 in (0, 1), we ﬁnd and ﬁx a number σ ∈ (0, 1) such that

1 q σ s Θ(s) ds > 0. Consequently, by (5.40), F possesses a global minimum which is atdef

tained at some b0 ∈ R2 . It remains to show that κ = F (b0 )1/q > 0. Indeed, F (b0 ) = 0 would force e1 + sb0 = 0 ∈ R2 for almost every s ∈ (0, 1) such that Θ(s) > 0, which is impossible. This proves (5.39) for q > 0. Case −1 < q < 0. We observe that inequality (5.39) is valid if and only if

1

def

|e1 + sb|q Θ(s) ds C = κ q

for all b ∈ R2 .

0

Since Θ ∈ L∞ (0, 1), it is obvious that it sufﬁces to show this estimate for Θ ≡ 1 in (0, 1) and for all b = (b1 , 0) ∈ R2 , that is, C = sup

b1 ∈R 0

1

|1 + sb1 |q ds < ∞.

Indeed, for b1 −1 we have

1

|1 + sb1 | ds q

0

1

(1 − s)q ds =

0

1 . 1+q

For b1 < −1 we estimate

1

|1 + sb1 |q ds = |b1 |−1

0

|b1 |

|1 − t|q dt

0

= |b1 |−1

1+q 2 2 1 < 1 + |b1 | − 1 |b1|q < . 1+q 1+q 1+q

Hence, inequality (5.39) is valid also for −1 < q < 0. def = Θ(1 − s) for 0 s 1 to get Next, in (5.39) we replace the function Θ(s) by Θ(s) the following new inequality, with some other constant κ¯ > 0:

1

ds |a + sb| Θ(s)

1/q κ|a| ¯ for all a, b ∈ RN .

q

0

by Θ(s) = Θ(1 − s), we Replacing the pair (a, b) by (a + b, −b), and the function Θ(s) have also

1 0

1/q |a + sb| Θ(s) ds q

κ|a ¯ + b|

for all a, b ∈ RN .

(5.41)

426

P. Takáˇc

Finally, we take advantage of the convexity of a norm to deduce the desired inequality

1

1/q |a + sb| Θ(s) ds q

c2+q (Θ)1/q · max |a + sb| 0s1

0 def

from (5.39) and (5.41), where c2+q (Θ) = min{κ, κ} ¯ > 0.

√ For instance, one gets optimal constants c2 (Θ0 ) = 1/2 and c2 (Θ1 ) = (3 2)−1 for Θ0 (s) ≡ 1 and Θ1 (s) ≡ 1 − s, respectively. We are now able to estimate the quadratic form associated with the symmetric matrix A(a) deﬁned in (5.6). The inequalities below follow from a combination of (5.7) with (5.37) for p > 2 and (5.38) for 1 < p < 2, respectively. We omit the index RN for the Euclidean inner product ·, ·! in RN . L EMMA 5.10. In the situation of Lemma 5.9, we have for all a, b, v ∈ RN : If p > 2 then

cp (Θ)

p−2 .

/p−2

0s1 1%

max |a + sb|

|v|2

& A(a + sb)v, v Θ(s) ds

0

(p − 1)T ·

.

max |a + sb|

/p−2

0s1

|v|2 ,

(5.42)

and if 1 < p < 2 and |a| + |b| > 0 then (p − 1)T ·

1%

.

max |a + sb|

/p−2

0s1

|v|2

& A(a + sb)v, v Θ(s) ds

0

/p−2 p−2 . max |a + sb| |v|2 . cp (Θ) 0s1

(5.43)

Finally, to estimate the quadratic form Qφ (v) = Qφ (v, v) deﬁned in (5.11), take 1,p Θ1 (s) ≡ 1 − s (0 s 1) and any φ ∈ W0 (Ω). If p > 2 then, by (5.42), for any v ∈ Dϕ1 inequality (5.16) is replaced by

cp (Θ1 )

p−2

:#

1

Ω

v2Dϕ

0

1

$ ; A ∇(ϕ1 + sφ) (1 − s) ds ∇v, ∇v dx ∞.

(5.44)

Nonlinear spectral problems

427

On the other hand, if 1 < p < 2 then, by (5.43), for any v ∈ Dϕ1 inequality (5.35) is replaced by :# Ω

1

$ ; A ∇(ϕ1 + sφ) (1 − s) ds ∇v, ∇v dx

0

p−2 v2Dϕ . cp (Θ1 )

(5.45)

1

6. An improved Poincaré inequality for p > 2 We are now equipped with most of the technical tools we need to establish the existence of a 1,p weak solution u ∈ W0 (Ω) to problem (1.7) in the “resonant” case λ = λ1 for 2 < p < ∞ and under the condition that f ∈ L2 (Ω) satisﬁes f, ϕ1 ! = 0. 6.1. Statement and proof of Poincaré’s inequality Recall the decomposition (5.1) of a function u ∈ L2 (Ω) into the orthogonal sum where u = ϕ1 −2 u, ϕ1 ! and L2 (Ω)

u = u · ϕ 1 + u)

def

%

& u) , ϕ1 = 0.

(6.1)

We motivate our approach by the following well-known inequality that follows directly from the spectral decomposition of the Dirichlet Laplacian in L2 (Ω): 2 2 |∇u| dx − λ1 |u| dx (λ2 − λ1 ) |u) |2 dx (6.2) Ω

Ω

Ω

for every u ∈ W01,2 (Ω). Here, it sufﬁces that Ω be a bounded domain in RN (N 1), and λ1 and λ2 stand for the ﬁrst (smallest) and second eigenvalues of −, respectively, so that 0 < λ1 < λ2 . As a consequence of this inequality, one obtains immediately the “existence” part of the Fredholm alternative for the positive Dirichlet Laplacian − at the ﬁrst eigenvalue λ1 . In the work of Fleckinger and Takáˇc [34], Theorem 3.1, p. 957, the power p = 2 was replaced by any power p 2; the Poincaré inequality (6.2) was thus extended to the “degenerate” case 2 < p < ∞. T HEOREM 6.1. Assume that both Hypotheses (H1) and (H2) are satisﬁed. Then there 1,p exists a constant c ≡ c(p, Ω) > 0 such that for all u ∈ W0 (Ω),

|∇u| dx − λ1

|u|p dx

p

Ω

Ω

p−2 2 p c u |∇ϕ1 |p−2 ∇u) dx + ∇u) dx . Ω

Ω

(6.3)

428

P. Takáˇc

If the constant c in (6.3) is replaced by zero, one obtains the classical Poincaré inequality; see, e.g., [39], Ineq. (7.44), p. 164. We call our inequality (6.3) an improved Poincaré inequality. Estimating both integrals on the right-hand side in (6.3) from below, we obtain [34], Corollary 1.2, p. 953: C OROLLARY 6.2. Let Ω be as in Theorem 6.1. Then there is another constant c ≡ 1,p c (p, Ω) > 0 such that for all u ∈ W0 (Ω),

|∇u|p dx − λ1

|u|p dx

Ω

Ω

p−2 ) 2 u dx + u) p dx . c u Ω

(6.4)

Ω

This inequality trivializes to (6.2) in the “regular” case p = 2, with the optimal (largest possible) constant λ2 − λ1 on the right-hand side being replaced by another positive constant 2c λ2 − λ1 . R EMARK 6.3. Except when u = 0, we may replace u ∈ W0 (Ω) by v = u/u in in1,p equality (6.3) and thus restate it equivalently as follows, for all v ) ∈ W0 (Ω) with v ) , ϕ1 ! = 0: 1,p

p c 2 . Qv ) v ) v ) Dϕ + v ) 1,p W (Ω) 1 p 0

(6.5)

This remark indicates that our proof of inequality (6.3) should distinguish between the cases when the ratio u) W 1,p (Ω) /|u | is bounded away from zero by a constant γ > 0, 0 say, u) W 1,p (Ω) 0

|u |

γ,

and when it is sufﬁciently small, say, u) W 1,p (Ω) 0

|u |

γ,

where γ > 0 is small enough. The former case is treated in a standard way analogous to (1.8), whereas the latter case requires a more sophisticated approach based on the second-order Taylor formula (5.10) applied to the expression Qv ) (v ) ) on the left-hand side in (6.5) where v = u/u . For either of these cases we need a separate auxiliary result: We derive two formulas for Rayleigh quotients outside and inside an arbitrarily small cone around the axis spanned by ϕ1 , respectively.

Nonlinear spectral problems

429

Given any number 0 < γ < ∞, we set

Minimization outside a cone around ϕ1 .

def 1,p Cγ = u ∈ W0 (Ω): u) W 1,p (Ω) γ u , def 1,p Cγ = u ∈ W0 (Ω):

0

) u 1,p γ u . W (Ω) 0

Notice that Cγ is a closed cone in W0 (Ω) and Cγ is the closure of Cγc , the complement 1,p

1,p

of Cγ in W0 (Ω). We consider also the hyperplane

0 independent from t and φ such that c1 Ni (t, φ) Pi (t, φ) c2 Ni (t, φ),

i = 0, 1.

(6.8)

λ1 (p − 1). Pick a minimizing P ROOF OF L EMMA 6.5. On the contrary, assume that Λ 1,p ∞ sequence {φn }n=1 in W0 (Ω) such that φn ≡ 0 in Ω, φn , ϕ1 ! = 0, φn W 1,p (Ω) → 0, and 0

P1 (1, φn )

λ1 (p − 1) →Λ P0 (1, φn )

as n → ∞.

Next, set tn = P0 (1, φn )1/2 and Vn = φn /tn for n = 1, 2, . . . . Hence, we have tn → 0,

as n → ∞. Inequalities (6.8) guarantee that both P0 (tn , Vn ) = 1 and P1 (tn , Vn ) → Λ 1−(2/p) Vn W 1,p (Ω) are bounded, and so we may extract a subsequences Vn Dϕ1 and tn 0

sequence denoted again by {Vn }∞ Vn $ z n=1 such that Vn $ V weakly in Dϕ1 and tn 1,p 1,p weakly in W0 (Ω) as n → ∞. Using the embedding W0 (Ω) !→ Dϕ1 , we get z ≡ 0 1,p in Ω. Furthermore, both embeddings Dϕ1 !→ L2 (Ω) and W0 (Ω) !→ Lp (Ω) being compact by Lemma 5.2(b), and Rellich’s theorem, respectively, we have also Vn → V strongly 1−(2/p) Vn → 0 strongly in Lp (Ω). It follows that V , ϕ1 ! = 0 and in L2 (Ω) and tn 1−(2/p)

1 p−2 2 P0 (0, V ) = ϕ V dx = 1, 2 Ω 1 & 1%

λ1 (p − 1). P1 (0, V ) = A(∇ϕ1 )∇V , ∇V Λ 2 Consequently, Proposition 5.4 forces V = κϕ1 in Ω, where κ ∈ R is a constant, κ = 0 by P0 (0, V ) = 1. But this is a contradiction to V , ϕ1 ! = 0.

> λ1 (p − 1) as claimed. We conclude that Λ 1,p

P ROOF OF T HEOREM 6.1. If u ∈ W0 (Ω) satisﬁes u, ϕ1 ! = 0, then (6.6) implies

|∇u|p dx − λ1 Ω

|u|p dx Ω

) p λ1 λ1 p ∇u dx, |∇u| dx = 1 − 1− Λ∞ Λ ∞ Ω Ω

(6.9)

432

P. Takáˇc

where λ1 /Λ∞ < 1 by Lemma 6.4. Thus, we may assume u, ϕ1 ! = 0 and so we need to prove only inequality (6.5). We will apply Lemmas 6.4 and 6.5 to the following two cases, respectively. Case v ) W 1,p (Ω) γ . Here, γ > 0 is an arbitrary, but ﬁxed number. In analogy with 0 inequality (6.9) above, we have

∇ϕ1 + ∇v ) p dx − λ1 Ω

ϕ1 + v ) p dx Ω

) p λ1 ) p ∇v dx ∇ϕ1 + ∇v dx cγ 1− Λγ Ω Ω

(6.10)

for all v ) ∈ W0 (Ω) such that v ) , ϕ1 ! = 0 and v ) W 1,p (Ω) γ , where cγ > 0 is 1,p

0

a constant independent from v ) . The last inequality follows from the boundedness of 1,p the orthogonal projections u → u · ϕ1 and u → u) in W0 (Ω). Recalling the embed1,p ding W0 (Ω) !→ Dϕ1 , we deduce from (6.10) that inequality (6.5) is valid provided v ) W 1,p (Ω) γ . 0

Case v ) W 1,p (Ω) γ . Here, γ > 0 is sufﬁciently small. According to (6.7) and 0 Lemma 6.5 we have Qv ) v ) = P1 1, v ) − λ1 (p − 1)P0 1, v ) λ1 (p − 1) 1− P1 1, v )

Λ c˜ · N1 1, v )

(6.11)

for all v ) ∈ W0 (Ω) such that v ) , ϕ1 ! = 0 and v ) W 1,p (Ω) γ , where γ > 0 is suf1,p

0

ﬁciently small and c˜ > 0 is a constant independent from v ) . Recall that the expressions Pi (1, v ) ) and Ni (1, v ) ) (i = 0, 1) have been deﬁned after Lemma 6.5. From (6.11) we deduce that inequality (6.5) is valid also when v ) W 1,p (Ω) γ . 0 Finally, in order to prove inequality (6.4), we make use of the embeddings Dϕ1 !→ 1,p L2 (Ω) and W0 (Ω) !→ Lp (Ω) to estimate the right-hand side in (6.4). This ﬁnishes our proof of Theorem 6.1. 1,p

R EMARK 6.6. Recall that p > 2 and W0 (Ω) !→ Dϕ1 !→ L2 (Ω). Let f (x, u) ≡ f (x) be independent from u ∈ R where f ∈ L2 (Ω) satisﬁes f, ϕ1 ! = 0. Although the func1,p tional Jλ1 deﬁned in (1.4) is no longer coercive on W0 (Ω), it is still not only bounded from below, but also “very close” to being coercive on the weighted Sobolev space Dϕ1 , as a direct consequence of improved Poincaré’s inequality (6.3). This property of Jλ1 will be used in the next paragraph to derive an existence theorem for problem (1.7) when λ = λ1 .

Nonlinear spectral problems

433

6.2. Fredholm alternative at λ1 In analogy with the case p = 2, inequality (6.3) guarantees the solvability of the Dirichlet boundary value problem −p u = λ1 |u|p−2 u + f (x) in Ω;

u=0

on ∂Ω,

(6.12)

in the following special case: T HEOREM 6.7. If f ∈ Dϕ 1 satisﬁes f, ϕ1 ! = 0, then problem (6.12) possesses a weak 1,p

solution u ∈ W0 (Ω). This theorem is due to Fleckinger and Takáˇc [34], Theorem 3.3, p. 958. Here we have denoted by Dϕ 1 the dual space of Dϕ1 , with the duality induced by the inner product ·, ·! in L2 (Ω). We have taken advantage of the fact that the Hilbert space Dϕ1 is continuously and densely embedded in L2 (Ω); see Lemma 5.2(b). Hence, also the embedding L2 (Ω) !→ Dϕ 1 is continuous. Notice that a sufﬁcient condition for f ∈ Dϕ 1 is f ∈ W −1,2 (Ω) and f |G ∈ L2 (G) in some open set G ⊃ Ω \ U . The orthogonality condition f, ϕ1 ! = 0 is sufﬁcient, but not necessary to obtain existence for problem (6.12) provided p = 2 (1 < p < ∞), according to recent results obtained in [22], Theorem 1.3, for N = 1, in [24], Theorem 1.1, for any N 1 and 1 < p < 2, and in [58], Theorems 3.1 and 3.5, for any N 1; see also [21], Theorems 1.1–1.3. P ROOF OF T HEOREM 6.7. Our proof of this theorem combines the improved Poincaré inequality (6.3) with a generalized Rayleigh quotient formula. To this end, we may assume that f ∈ Dϕ 1 satisﬁes f ≡ 0 in Ω and f, ϕ1 ! = 0. Deﬁne the number Mf (0 Mf ∞) by def

Mf =

sup

1,p v∈W0 (Ω) Ω v∈ / {κϕ1 : κ∈R}

| f, v!|p

. − λ1 Ω |v|p dx

|∇v|p dx

(6.13)

Clearly, Mf > 0. Moreover, (6.3) entails f, v!p f p −1,p W

(Ω)

) p v 1,p

W0 (Ω)

|∇v|p dx − λ1 Ω

|v|p dx Ω

for all v ∈ W0 (Ω), where Cf = c−1 f 1,p

Cf

p W −1,p (Ω)

is a constant. This shows that

434

P. Takáˇc

Mf Cf < ∞. In a similar way we arrive at p−2 f, v!2 v p−2 2 v f 2D v ) Dϕ ϕ1 1 1,p Cf |∇v|p dx − λ1 |v|p dx for all v ∈ W0 (Ω), Ω

(6.14)

Ω

where Cf = c−1 f 2D

ϕ1

is a constant, and · Dϕ stands for the dual norm on Dϕ 1 . 1

1,p

From (6.13) and inequality (6.14) we can draw the following conclusion: If v ∈ W0 (Ω) is such that v ) ≡ 0 in Ω and

Ω

| f, v!|p 1

Mf , p − λ1 Ω |v| dx 2

|∇v|p dx

then f, v! = 0 and Cf p−2 v f, v!p−2 C p−2 v ) p−2 2 , 1,p f W0 (Ω) Mf where Cf = [2(Cf /Mf )]1/(p−2)f W −1,p (Ω) is a constant, i.e., v C v ) f

1,p

W0 (Ω)

(6.15)

.

Next, take any maximizing sequence {vn }∞ n=1 in W0 (Ω) for the generalized Rayleigh quotient (6.13), that is, vn) ≡ 0 in Ω and 1,p

Ω

|∇vn

| f, vn !|p

→ Mf − λ1 Ω |vn |p dx

|p dx

as n → ∞.

(6.16)

Since both, the numerator and the denominator are p-homogeneous, we may assume 1,p vn W 1,p (Ω) = 1 for all n 1. The Sobolev space W0 (Ω) being reﬂexive, we may pass 0

1,p

to a convergent subsequence vn $ w weakly in W0 (Ω); hence, also vn → w strongly in Lp (Ω), by Rellich’s theorem, and f, vn ! → f, w! as n → ∞. We insert these limits into (6.16) to obtain p |∇w|p dx − λ1 |w|p dx 1 − λ1 |w|p dx = Mf−1 f, w! . (6.17) Ω

Ω

Ω

In particular, we have w ≡ 0 in Ω, therefore also w) ≡ 0 by (6.15), and consequently | f, w!| = 0 by (6.17). We combine (6.13) with (6.17) to get Ω |∇w|p dx = 1. Hence, the supremum Mf in (6.13) is attained at w in place of v.

Nonlinear spectral problems

435

Finally, we can apply the calculus of variations to the inequality p 1,p |∇v|p dx − λ1 |v|p dx − Mf−1 f, v! 0 for v ∈ W0 (Ω) Ω

Ω

to derive

p−2 f, w!f (x) in Ω, −p w − λ1 |w|p−2 w = Mf−1 f, w! w = 0 on ∂Ω. def

1/(p−1)

It follows that u = Mf Theorem 6.7 is proved.

f, w!−1 · w is a weak solution of problem (6.12).

1,p

6.3. Application to the embedding W0

!→ Lp 1,p

The “geometry” of the Sobolev embedding W0 (Ω) !→ Lp (Ω) for p = 2 is easily described by the eigenvalues {λk }∞ k=1 , with 0 < λ1 < λ2 λ3 · · · , and the associated eigenfunctions {ϕk }∞ , with ϕ k , ϕk ! = 1 and ϕk , ϕ ! = 0 if k = , for the positive k=1 Dirichlet Laplace operator − in L2 (Ω). Simply, the unit sphere from W01,2 (Ω), after having been embedded into L2 (Ω), becomes an (inﬁnite-dimensional) ellipsoid with −1/2 the axes of length λk in direction ϕk (k = 1, 2, . . . ). Such a clear geometric picture is unknown for p = 2. Only the ﬁrst two eigenvalues of the nonlinear operator −p are known to have a variational characterization: λ1 by formula (1.8) and λ2 by a minimax formula [3], Remarques 2.2, pp. 15–16, and 0 < λ1 < λ2 . A divergent sequence of eigenvalues 0 < λ1 < λ2 λ3 · · · of −p , characterized by a minimax formula, has been obtained in [3], Remarques 2.2, pp. 15–16, as well, but it is unknown if these are all eigenvalues of −p . It is shown in [4], Proposition 2, p. 5, that there is no eigenvalue in the open interval (λ1 , λ2 ). A weaker result, namely, that there is no eigenvalue in some open interval (λ1 , λ1 + δ), δ > 0, was obtained earlier by Anane [2], Théorème 2, p. 727. Using the results and methods from this section, we would like to address the problem of 1,p geometry of the Sobolev embedding W0 (Ω) !→ Lp (Ω) for p > 2: a kind of “stability” and nonsimplicity of the ﬁrst eigenvalue λ1 . More precisely, let us “squeeze” (deform) the unit sphere in Lp (Ω) along a ﬁxed vector f ∈ W −1,p (Ω) orthogonal to ϕ1 , that is, 1,p consider a new norm on W0 (Ω) deﬁned by p 1/p def p uLp (Ω);f = uLp (Ω) + u, f !

1,p

for u ∈ W0 (Ω),

(6.18)

where f ∈ W −1,p (Ω) is a given distribution from the dual space W −1,p (Ω) of W0 (Ω), with f, ϕ1 ! = 0. Of course, if f ∈ Lp (Ω) then · Lp (Ω);f is an equivalent norm on Lp (Ω). Clearly ϕ1 Lp (Ω);f = ϕ1 Lp (Ω) = 1. Next, deﬁne a Rayleigh quotient analogous to (1.8), def 1,p p p |∇u| dx: u ∈ W0 (Ω) with uL (Ω);f = 1 . (6.19) μf = inf Ω

1,p

436

P. Takáˇc

Observe that μf λ1 . On the other hand, if f ≡ 0 in Ω then improved Poincaré’s inequal1,p ity (6.3) guarantees for all u ∈ W0 (Ω),

|∇u|p dx − λ1

) p ∇u dx

|u|p dx c

Ω

Ω

Ω −p W −1,p (Ω)

cf

p · u, f ! .

Recall that c ≡ c(p, Ω) > 0 is a constant. Thus, we have proved the following result:

L EMMA 6.8. If f ∈ W −1,p (Ω) satisﬁes f W −1,p (Ω) (c/λ1 )1/p and f, ϕ1 ! = 0, then μf = λ1 . Clearly, the inﬁmum in (6.19) is attained at u = ±ϕ1 . In addition, our proof of Theorem 6.7 guarantees that this inﬁmum is attained also at a point different from ±ϕ1 , provided f is restricted to Dϕ 1 \ {0}. P ROPOSITION 6.9. Assume that f ∈ Dϕ 1 satisﬁes 0 < f W −1,p (Ω) (c/λ1 )1/p and f, ϕ1 ! = 0. Then μf = λ1 and the inﬁmum in (6.19) is attained at ±ϕ1 and another point 1,p u0 ∈ W0 (Ω), u0 ≡ ±ϕ1 in Ω. P ROOF. According to the proof of Theorem 6.7, the supremum Mf (0 < Mf < ∞) de1,p / {κϕ1 : κ ∈ R}. Of ﬁned in (6.13) is attained at some w ∈ W0 (Ω) in place of v, with w ∈ −1 course, in (6.13) we may replace w by u0 = wLp (Ω);f w; hence u0 Lp (Ω);f = 1. We combine formulas (6.13) and (6.19) with Lemma 6.8 to conclude that Mf = 1/λ1 provided 0 < f W −1,p (Ω) (c/λ1 )1/p . Hence, u0 is another minimizer for μf = λ1 in (6.19) which is not co-linear to ϕ1 . To summarize our results from this paragraph, we have shown that even if μf = λ1 holds for 0 < f W −1,p (Ω) (c/λ1 )1/p , there are two eigenfunctions ϕ1 and u0 associated with μf which are not co-linear. This nonuniqueness (as opposed to the uniqueness in Corollary 3.5) is due to the fact that the arguments with u+ and u− presented for λ1 after formula (3.1) can no longer be applied to μf in (6.19).

7. A saddle point method for p < 2 Similarly as in the previous section, for the sake of simplicity also in this section we restrict ourselves to the case F (x, u) = f (x)u, i.e., f (x, u) ≡ f (x) is independent from u ∈ R. Hence, the functional Jλ introduced in (1.4) takes the form 1 Jλ (u) = Jλ (u; f ) = p def

λ |∇u| dx − p Ω

|u| dx −

p

p

Ω

f (x)u dx Ω

(7.1)

Nonlinear spectral problems

437

1,p

for u ∈ W0 (Ω). In contrast to the case p > 2 in Section 6, Remark 6.6 (and under similar assumptions), 1,p for 1 < p < 2 the functional Jλ1 will turn out to be unbounded from below on W0 (Ω) along curves “close” to ±τ ϕ1 as τ → +∞, even though it still remains coercive on the 1,p 1,p orthogonal complement W0 (Ω)) of lin{ϕ1 } in W0 (Ω), def 1,p 1,p W0 (Ω)) = u ∈ W0 (Ω): u, ϕ1 ! = 0 .

(7.2) 1,p

Hence, we take advantage of the orthogonal decomposition W0 (Ω) = lin{ϕ1 } ⊕ 1,p W0 (Ω)) deﬁned in (6.1) again. This picture shows that the functional Jλ1 has a simple “saddle point” geometry. Such a scenario is typically suitable for a saddle point theorem ([51], Theorem 4.6, p. 24) which guarantees the existence of a critical point for Jλ1 by means of a minimax formula for a critical value of Jλ1 . This observation was used in the work of Drábek and Holubová [24], Theorem 1.1, to establish an existence and nonexistence result for problem (6.12) when 1 < p < 2. In this section we present their method.

7.1. Simple saddle point geometry The following notion is crucial. 1,p

D EFINITION 7.1. We say that a continuous functional E : W0 (Ω) → R has a simple 1,p saddle point geometry if we can ﬁnd u, v ∈ W0 (Ω) such that v, ϕ1 ! < 0 < u, ϕ1 ! and max E(u), E(v)

λ1 in formula (6.6). This shows that the = W 1,p (Ω)) . Hence, being also weakly lower semiconfunctional Jλ1 is coercive on C∞ 0 1,p ) tinuous, Jλ1 possesses a global minimizer u) 0 over W0 (Ω) , Jλ1 u) 0 =

inf 1,p

w∈W0 (Ω))

Jλ1 (w) > −∞.

438

P. Takáˇc

Now let us look for the functions u and v, respectively, in Deﬁnition 7.1 in the forms of u± = ±τ ϕ1 + τ 1−(p/2)φ

with τ ∈ (0, ∞) sufﬁciently large,

(7.3)

where φ ∈ C01 (Ω) is a function chosen as follows: First, recall our notation U = x ∈ Ω: ∇ϕ1 (x) = 0 and U = Ω \ U = x ∈ Ω: ∇ϕ1 (x) = 0 . Then U is a compact subset of Ω with empty interior, by (5.3) combined with (5.13). satisﬁes f ≡ 0 in Ω, we must have f ≡ 0 in U as well. In particular, Since f ∈ C 0 (Ω) U contains the closure of an open ball G ⊂ RN such that either f > 0 in G, or else f < 0 in G. In either case we can easily ﬁnd a function φ ∈ C01 (Ω) that vanishes outside the ball G and satisﬁes f, φ! = 1. For τ ∈ (0, ∞) we compute u± , ϕ1 ! = ±τ ϕ1 2L2 (Ω) + τ 1−(p/2) φ, ϕ1 !.

(7.4)

It follows that u− , ϕ1 ! < 0 < u+ , ϕ1 ! for all τ > 0 large enough. Next we use (5.9) and (5.10) to obtain Jλ1 (u± ) = Jλ1 ±τ ϕ1 + τ 1−(p/2)φ = Q±τ −p/2 φ (φ, φ) − τ 1−(p/2) f, φ! = Q±τ −p/2 φ (φ, φ) − τ 1−(p/2).

(7.5)

We recall that the quadratic forms Q±τ −p/2 φ are given by formula (5.10). Since infG |∇ϕ1 | > 0, infG ϕ1 > 0, and φ is supported in G by our choice of G and φ, we conclude that both summands in Q±τ −p/2 φ (φ, φ) are bounded independently from τ τ0 , provided τ0 ∈ (0, ∞) is large enough. Finally, from (7.5) we deduce that Jλ1 (u± ) → −∞ as τ → +∞. The conclusion of the lemma follows.

7.2. A Palais–Smale condition In order to be able to apply Rabinowitz’s saddle point theorem [51], Theorem 4.6, p. 24, we need another lemma.

L EMMA 7.3 ([24], Lemma 2.2, p. 188). Let 1 < p < 2. Assume f ∈ W −1,p (Ω) with f, ϕ1 ! = 0. Then the functional Jλ1 satisﬁes the Palais–Smale (P.–S.) condition, i.e., 1,p every sequence {un }∞ n=1 in W0 (Ω), such that Jλ1 (un ) → c ∈ R and Jλ1 (un ) → 0 in

W −1,p (Ω) as n → ∞, contains a strongly convergent subsequence in W0 (Ω). 1,p

Nonlinear spectral problems

439

P ROOF. Let {un }∞ n=1 be an arbitrary (P.–S.) sequence in W0 (Ω). As usual, we ﬁrst show 1,p that it is bounded in W0 (Ω). On the contrary, suppose that a subsequence, denoted again ∞ by {un }n=1 , satisﬁes un W 1,p (Ω) → ∞ as n → ∞. The P.–S. condition implies 1,p

0

Jλ1 (un ) =

1 p

|∇un |p dx − Ω

λ1 p

|un |p dx − Ω

f (x)un dx → c

(7.6)

Ω

and %

& Jλ1 (un ), vn = un −11,p

W0 (Ω)

|∇un |p dx − λ1 Ω

|un |p dx Ω

f (x)vn dx → 0

−

(7.7)

Ω

where vn = un /un W 1,p (Ω) . We combine these two facts to get 0

& % Jλ1 (un ), vn − un −11,p

W0 (Ω)

Jλ1 (un )

1 p−1 p p |∇vn | dx − λ1 |vn | dx → 0. = 1− un 1,p W0 (Ω) p Ω Ω −1/p

It follows that vn W 1,p (Ω) = 1 and vn Lp (Ω) → λ1 0

(7.8)

. Now we can argue similarly as

−1/p ±λ1 ϕ1

1,p

in the proof of Lemma 6.4 to conclude that vn → holds strongly in W0 (Ω) as n → ∞ for a suitable subsequence. Applying this result and (7.8) to (7.7) we arrive at −1/p f vn dx → ±λ1 f ϕ1 dx = 0, Ω

Ω

a contradiction to our assumption f, ϕ1 ! = 0. Thus, we have proved that {un }∞ n=1 must be 1,p bounded in W0 (Ω). 1,p Next, W0 (Ω) being reﬂexive, we extract a weakly convergent subsequence, denoted 1,p again by {un }∞ n=1 , i.e., un $ u weakly in W0 (Ω) as n → ∞. Hence, un → u strongly in 1,p Lp (Ω) by Rellich’s theorem. From the deﬁnition of a P.–S. sequence in W0 (Ω) we have p Jλ1 (un ), un − u! → 0 as n → ∞. Using un → u strongly in L (Ω) we observe that the last limit is equivalent to |∇un |p dx − |∇un |p−2 ∇un · ∇u dx → 0. Ω

Ω

We apply Young’s inequality to the second integral to get 1 1 p p lim inf un 1,p + u 1,p lim sup un 1,p n→∞ p W0 (Ω) W0 (Ω) W0 (Ω) p n→∞ p

440

P. Takáˇc

which entails lim sup un W 1,p (Ω) uW 1,p (Ω) . 0

n→∞

0

1,p

On the other hand, un $ u weakly in W0 (Ω) yields uW 1,p (Ω) lim inf un W 1,p (Ω) . n→∞

0

0

From the last two inequalities we deduce lim un W 1,p (Ω) = uW 1,p (Ω)

n→∞

0

0

1,p

which, when combined with un $ u again, guarantees un → u strongly in W0 (Ω). The lemma is proved.

7.3. Fredholm alternative at λ1 Now we are ready to apply the saddle point theorem to the energy functional Jλ1 deﬁned in (7.1) in order to establish the following existence result for problem (6.12). This result complements Theorem 6.7, not only because 1 < p < 2, but also f, ϕ1 ! = 0. P ROPOSITION 7.4 ([24], Proposition 2.1, p. 189). Let 1 < p < 2. Assume g ) ∈ C 0 (Ω) ) ) ) with g , ϕ1 ! = 0 and g ≡ 0 in Ω. Then there exists a constant ρ ≡ ρ(g ) > 0 such that, for any f ∈ W −1,p (Ω) with f, ϕ1 ! = 0 and f − g ) W −1,p (Ω) < ρ, problem (6.12) has at least one weak solution. P ROOF. Since we keep λ = λ1 constant, but vary the function f ∈ L∞ (Ω) in probdef lem (6.12), it will be convenient for us to use the notation Ef (u) = Jλ1 (u; f ) for 1,p u ∈ W0 (Ω); cf. (7.1). By Lemma 7.2, the functional Eg ) has a simple saddle point geometry. But this property clearly remains preserved for Ef for all sufﬁciently small perturbations of g ) , that is, also for f ∈ W −1,p (Ω) with f − g ) W −1,p (Ω) < ρ. Here, ρ ≡ ρ(g ) ) > 0 is a sufﬁciently

small number. Indeed, notice that fn → g ) in W −1,p (Ω) as n → ∞ implies inf 1,p

w∈W0 (Ω))

Efn (w) →

inf 1,p

w∈W0 (Ω))

Eg ) (w),

by arguments used in the proof of Lemma 6.4. According to Lemma 7.3 the functional Ef satisﬁes the P.–S. condition for any f ∈ W −1,p (Ω) with f, ϕ1 ! = 0. Next we take such f with f − g ) W −1,p (Ω) < ρ. Hence, by a standard variational argument (a saddle point theorem [51], Theorem 4.6, p. 24), the

Nonlinear spectral problems

441

functional Ef has at least one critical point which corresponds to a weak solution of the original problem (6.12). To cover also the case ζ = 0, excluded in Proposition 7.4, yet another method was applied in [24], Section 2, pp. 189–193, based on well-ordered and unordered pairs of sub- and supersolutions for problem (6.12). We will present this method in detail in Section 10. Therefore, here we only state the main result from [24], Theorem 1.1, p. 184; its proof will be given in Section 10. However, part (i) is a special case of Corollary 8.15 which will be established already in Section 8.6. with f ) , ϕ1 ! = 0 and f ) ≡ 0 T HEOREM 7.5. Let 1 < p < 2. Assume f ) ∈ C 0 (Ω) ) in Ω. Then there exist two numbers ζ∗ ≡ ζ∗ (f ) and ζ ∗ ≡ ζ ∗ (f ) ) with −∞ < ζ∗ < 0 < ζ ∗ < ∞, such that problem (6.12) with f = f ) + ζ ϕ1 has (i) no solution for ζ ∈ R \ [ζ∗ , ζ ∗ ]; (ii) at least one solution for ζ ∈ [ζ∗ , ζ ∗ ]. with g ) , ϕ1 ! = 0 and g ) ≡ 0 in Ω, there exists a Moreover, given any g ) ∈ C 0 (Ω) ) number ρ ≡ ρ(g ) > 0 such that problem (6.12) has at least one solution whenever f ∈ L∞ (Ω) satisﬁes f − g ) L∞ (Ω) < ρ. In Section 10 we will establish a much stronger result for any p = 2, namely, that there are also two other numbers ζ# and ζ # with ζ∗ ζ# < 0 < ζ # ζ ∗ , such that problem (6.12) with f = f ) + ζ ϕ1 has at least two distinct solutions provided ζ# < ζ < ζ # and ζ = 0; cf. [58], Theorems 3.1 and 3.5.

8. Asymptotic behavior of large solutions A priori estimates play a crucial role in establishing existence results for various types of ordinary and partial differential equations and their systems. While deriving an a priori estimate, one usually attempts to estimate a suitable norm of an arbitrary solution (or an approximation thereof) directly. In [57], Section 5, pp. 206–215, a somewhat different approach to deriving a priori estimates has been introduced for problem (1.7) where the spectral parameter λ ∈ R takes values near the ﬁrst eigenvalue λ1 of −p . This approach is based on a very thorough investigation of the asymptotic behavior of an unbounded 1,p sequence of possible large solutions u = un ≡ uλ1 +μn ∈ W0 (Ω) of problem (1.7) with λ = λ1 + μn λ2 − δ (n = 1, 2, . . . ) as n → ∞. (Of course, δ > 0 is an arbitrarily small number.) Recall from Section 6.3 that λ2 stands for the second eigenvalue of the positive Dirichlet p-Laplacian −p and there is no other eigenvalue in the open interval (λ1 , λ2 ), by [3], Remarques 2.2, pp. 15–16. In particular, we will see soon that μn → 0 and un = tn−1 (ϕ1 + vn) ) must hold with tn → 0 (tn = 0) and vn) C 1,β (Ω) → 0 as n → ∞. We view t = tn = 0 as an independent bifurcation parameter and look for possible triples 1,p (t, μ, v) ) = (tn , μn , vn) ) ∈ R × R × W0 (Ω)) near the bifurcation point (0, 0, 0) such −1 ) that u = t (ϕ1 + v ) veriﬁes (1.7) with λ = λ1 + μ. Recall that the orthogonal comple1,p ment W0 (Ω)) has been deﬁned in (7.2). The investigation of the asymptotic behavior

442

P. Takáˇc

of vn) as n → ∞ was continued in [58], Proposition 6.1, p. 331 (see (5.2)), from which a stronger version of Theorem 7.5 was derived for any p = 2 ([58], Theorems 3.1 and 3.5). Finally, even more precise, higher-order asymptotic results were obtained recently in [23], Theorem 4.1. We present these asymptotic results in this section; they are summarized in Theorem 8.7 (Section 8.4). Here, also the number ζ = ζn in f = f ) + ζ ϕ1 is a parameter depending on t = tn . Of course, one may ﬁx either μ or ζ , or ﬁx their interdependence, in general. Finally, if the asymptotic dependence of μn , ζn or vn) on tn as n → ∞, obtained in the manner just described, can be excluded by a hypothesis imposed on μn , ζn or vn) , then, by a contradiction argument, we cannot have large solutions of problem (1.7). Consequently, we obtain the boundedness of the solution set indirectly rather than from an a priori estimate directly.

8.1. An approximation scheme In this paragraph we investigate an approximation scheme for a weak solution to the Dirichlet boundary value problem (1.7) provided f ∈ L∞ (Ω) satisﬁes f ≡ 0. Among other things we compute the asymptotic behavior of large solutions. We emphasize that the orthogonality condition f, ϕ1 ! = 0 is not required in this paragraph. We study the following sequence of Dirichlet boundary value problems for n = 1, 2, . . . : −p un = (λ1 + μn )|un |p−2 un + fn (x) in Ω;

un = 0 on ∂Ω.

(8.1)

We often take advantage of the weak formulation of problem (8.1): For each n ∈ N and for 1,p all φ ∈ W0 (Ω), |∇un |p−2 ∇un , ∇φ! dx Ω

= (λ1 + μn )

|un |

Ω

p−2

un φ dx +

fn φ dx.

(8.2)

Ω

∞ ∞ Here, {μn }∞ n=1 is a sequence of real numbers, {fn }n=1 are given functions from L (Ω), 1,p and {un }∞ n=1 are corresponding weak solutions to problem (8.1) in W0 (Ω) which are assumed to exist. We assume that these sequences satisfy the following hypotheses: (S1) λ1 + μn λ2 − δ for n = 1, 2, . . . , where 0 < δ < λ2 − λ1 . (S2) fn converges to some function f in the weak-star topology on L∞ (Ω), i.e., ∗ fn $ f in L∞ (Ω) as n → ∞. We require f ≡ 0 in Ω. (S3) un W 1,p (Ω) → ∞ as n → ∞. 0

We identify L∞ (Ω) with the dual space of L1 (Ω) in a standard way by means of the inner product ·, ·! from L2 (Ω). This duality induces the weak-star topology on L∞ (Ω). Any closed bounded ball in L∞ (Ω) is weakly-star compact; the weak-star topology restricted to this ball is metrizable since L1 (Ω) is separable ([66], Chapter V, §1).

Nonlinear spectral problems

443

By the regularity result in Lemma 2.2 ([3], Théorème A.1, p. 96), hypothesis (S3) is equivalent to (S3 ) un L∞ (Ω) → ∞ as n → ∞. Furthermore, since ∂Ω is assumed to be of class C 1,α , for some 0 < α < 1, we can apply another regularity result, Lemma 2.3 ([18], Theorem 2, p. 829, [45], Theorem 1, p. 1203, for some β ∈ (0, α). Finally, and [62], Theorem 1, p. 127), to conclude that un ∈ C 1,β (Ω), if {μn }∞ is bounded also from below, say, n=1 (S1 ) −λ¯ λ1 + μn ( λ2 − δ) for n = 1, 2, . . . , where 0 λ¯ < ∞, then hypothesis (S3) is equivalent to (S3 ) un C 1,β (Ω) → ∞ as n → ∞. In what follows we often work with a chain of subsequences of {(μn , fn , un )}∞ n=1 by passing from the current one to the next. Nevertheless, we keep the index n unchanged with the understanding that no confusion may arise. def

We commence with the asymptotic behavior of the normalized sequence u˜ n = un −1 ˜ n satisﬁes u˜ n L∞ (Ω) = 1 and L∞ (Ω) un as n → ∞. Observe that each u

p−2 1−p u˜ n + un L∞ (Ω) fn (x) in Ω, −p u˜ n = (λ1 + μn )u˜ n u˜ n = 0 on ∂Ω.

(8.3)

Since ∂Ω is assumed to be of class C 1,α , for some 0 < α < 1, we conclude that u˜ n ∈ for some β ∈ (0, α), and the sequence {u˜ n }∞ is bounded in C 1,β (Ω), by the C 1,β (Ω), n=1 regularity result mentioned above (Lemma 2.3). We allow 1 < p < ∞. L EMMA 8.1. Let β ∈ (0, β). We have μn → 0 and the sequence {u˜ n }∞ n=1 contains a as n → ∞, where κ ∈ R is a constant, convergent subsequence u˜ n → κϕ1 in C 1,β (Ω) |κ| · ϕ1 L∞ (Ω) = 1. In particular, we have un = tn−1 (ϕ1 + vn) ), where {tn }∞ n=1 is a se1 quence of real numbers such that κtn > 0 and tn un 2 ϕ1 in Ω for all n large enough; as n → ∞, with vn) , ϕ1 ! = 0 for n = 1, 2, . . . . moreover, tn → 0 and vn) → 0 in C 1,β (Ω) This lemma generalizes [57], Lemma 5.1, p. 207, where μn = 0 is assumed for all n ∈ N; recall N = {1, 2, 3, . . .}. P ROOF OF L EMMA 8.1. First, we show that the sequence {μn }∞ n=1 is bounded also from below. Let us take φ = un in (8.2):

|∇un | dx = (λ1 + μn )

|un | dx +

p

Ω

p

Ω

fn un dx. Ω

We apply the standard Poincaré inequality (cf. (1.8) for λ1 > 0) to the integral on the left and the Hölder inequality to the second integral on the right to obtain

|un | dx (λ1 + μn ) p

λ1 Ω

Ω

|un |p dx + fn Lp (Ω) un Lp (Ω) .

(8.4)

444

P. Takáˇc

By hypotheses (S2) and (S3), respectively, the sequence fn L∞ (Ω) is bounded whereas un W 1,p (Ω) → ∞ as n → ∞. This forces also fn Lp (Ω) bounded and un Lp (Ω) → ∞ 0 as n → ∞, by hypothesis (S1). Hence, we deduce from (8.4) that the sequence p−1 −μn un Lp (Ω) must be bounded from above for all n ∈ N. This forces lim infn→∞ μn 0. In particular, the sequence {μn }∞ n=1 is bounded. Consequently, in the rest of this proof we may extract a convergent subsequence μn → μ∗ as n → ∞. We have 0 μ∗ λ2 − λ1 − δ for every n = 1, 2, . . . . Now we to the sequence {u˜ n }∞ to obtain another can apply Arzelà–Ascoli’s theorem in C 1,β (Ω) n=1 1,β as n → ∞. Letting n → ∞ in the weak convergent subsequence u˜ n → w˜ in C (Ω) formulation of problem (8.3), we arrive at p−2 w˜ −p w˜ = (λ1 + μ∗ )w˜

in Ω;

w˜ = 0

on ∂Ω.

(8.5)

Since λ1 is the only eigenvalue of −p in the open interval (−∞, λ2 ), we get μ∗ = 0. The eigenvalue λ1 being simple (Corollary 3.5), we conclude that w˜ = κϕ1 in Ω, where κ ∈ R is a constant, κ = 0 by w ˜ L∞ (Ω) = 1. Let {μnk }∞ be a subsequence of {μn }∞ k=1 k=1 such that |μnk | η for some η > 0. Applying the same argument as above we get a contradiction by obtaining a subsequence of {μnk }∞ k=1 that converges to zero. So, indeed, the sequence μn itself converges to 0, and not just a subsequence of it. The remaining statements are deduced from the identity u˜ n − w˜ =

1 − κϕ1 + vn) L∞ (Ω) vn) ϕ + . 1 ϕ1 + vn) L∞ (Ω) ϕ1 + vn) L∞ (Ω)

We combine vn) C 1,β (Ω) → 0 with the Hopf maximum principle (5.13) for ϕ1 to ﬁnd out

that |vn) | 12 ϕ1 in Ω provided n is sufﬁciently large, say, n n0 . In particular, we get ϕ1 + vn) 12 ϕ1 > 0 in Ω for every n n0 . As a consequence of Lemma 8.1, for each n = 1, 2, . . . , we can rewrite problem (8.3) as ⎧ −p ϕ1 + vn) ⎪ ⎪ ⎪ ⎨ = (λ + μ )ϕ + v ) p−2 ϕ + v ) + |t |p−2 t f (x) in Ω, 1 n 1 1 n n n n n ) ⎪ v = 0 on ∂Ω, ⎪ ⎪ & ⎩ % n) vn , ϕ1 = 0,

(8.6)

with all tn = 0, tn → 0 as n → ∞. Furthermore, if κ < 0, we can take advantage of the (p − 1)-homogeneity of problem (8.1) and replace all functions fn , f and un by −fn , −f and −un , respectively, thus switching to the case κ > 0. Hence, without loss of generality and whenever convenient, we may assume tn > 0 and tn un = ϕ1 + vn) 12 ϕ1 > 0 in Ω for all n 1, with tn 0 as n → ∞.

Nonlinear spectral problems

445

8.2. Convergence of approximate solutions A very useful equivalent form of problem (8.1) is the following one obtained by subtracting (5.3) from (8.6) and using the integral Taylor formula with a help from identity (5.5): ⎧ − div An ∇vn) ⎪ ⎪ ⎪ ⎨ p−1 = (p − 1)(λ1 + μn )an vn) + μn ϕ1 + |tn |p−2 tn fn (x) in Ω, ) ⎪ vn = 0 on ∂Ω, ⎪ ⎪ & ⎩% ) vn , ϕ1 = 0,

(8.7)

with the abbreviations def

1

An =

0

A ∇ϕ1 + s∇vn) ds

def

1

ϕ1 + sv ) p−2 ds.

and an =

n

0

(8.8)

Recall that the matrix A(a) is deﬁned in (5.6). We abbreviate also def

Aϕ1 = A(∇ϕ1 )

and write Aϕ1/2 = 1

Aϕ1 .

(8.9)

(n 1) satisﬁes the linThis means that each function Vn = (|tn |p−2 tn )−1 vn) ∈ C 1,β (Ω) ear boundary value problem def

⎧ − div(An ∇Vn ) ⎪ ⎪ μn ⎪ p−1 ⎨ = (p − 1)(λ1 + μn )an Vn + ϕ + fn (x) in Ω, |tn |p−2 tn 1 ⎪ ⎪ V = 0 on ∂Ω, ⎪ ⎩ n Vn , ϕ1 ! = 0.

(8.10)

In order to determine the limits of Vn and μn /(|tn |p−2 tn ) as n → ∞, we need the following two “universal lemmas” for p > 2 and 1 < p < 2, respectively. We keep our Hypothesis (H1) for any 1 < p < ∞ and (H2) for p > 2 throughout the remaining part of the present section. Recall that (H2) holds always true for 1 < p < 2; see Section 5.5. 1,p L EMMA 8.2. Let 2 < p < ∞. Assume that 0 < αn < ∞ and vn) ∈ W0 (Ω) ∩ C 1 (Ω) ) 1 2 satisfy vn = αn Vn → 0 strongly in C (Ω), and Vn $ V weakly in L (Ω) as n → ∞. In addition, assume that Rn $ R weakly in L2 (Ω) and

An ∇Vn , ∇φ! dx = Ω

Rn φ dx Ω

1,p

for all φ ∈ W0 (Ω).

(8.11)

1/2

Then also V ∈ Dϕ1 , which implies Aϕ1 ∇V ∈ [L2 (Ω)]N , and

Ω

Aϕ1 ∇V , ∇φ! dx =

Rφ dx Ω

for all φ ∈ Dϕ1 .

(8.12)

446

P. Takáˇc 1/2

1/2

Moreover, we have Vn → V strongly in Dϕ1 and An ∇Vn → Aϕ1 ∇V strongly in [L2 (Ω)]N as well. A complete proof of this lemma, based on inequalities (5.37) and (5.42), is quite technical and is given in [23], Lemma B.1. It is derived from the proofs of Lemmas 5.3 and 5.4 in [57], pp. 210–213. This lemma will be needed several times later, with a more general right-hand side in (8.10). For 1 < p < 2 we need to employ “improper” integrals of type An and an deﬁned in (8.8). In analogy with Dϕ 1 being the dual space of Dϕ1 (cf. Section 6.2), here we denote by Hϕ 1 the dual space of Hϕ1 (cf. Section 5.5), with the duality induced by the inner product ·, ·! in L2 (Ω). Recall that the Hilbert space Hϕ1 is continuously embedded in L2 (Ω); see Lemma 5.8(a). Hence, also the embedding L2 (Ω) !→ Hϕ 1 is continuous. L EMMA 8.3. Let 1 < p < 2. Assume that 0 < αn < ∞ and vn) ∈ W0 (Ω) ∩ C 1 (Ω) ) 1 and Vn $ V weakly in Hϕ1 as n → ∞. In satisfy vn = αn Vn → 0 strongly in C (Ω), addition, assume that Rn $ R weakly in Hϕ 1 and 1,p

An ∇Vn , ∇φ! dx = Ω

Rn φ dx Ω

for all φ ∈ W01,2 (Ω).

(8.13)

1/2

Then also V ∈ Dϕ1 , which implies Aϕ1 ∇V ∈ [L2 (Ω)]N , and Aϕ1 ∇V , ∇φ! dx = Rφ dx for all φ ∈ Dϕ1 . Ω

(8.14)

Ω

1/2

1/2

Moreover, we have Vn → V strongly in W01,2 (Ω) and An ∇Vn → Aϕ1 ∇V strongly in [L2 (Ω)]N as well. Again, a complete proof of this lemma, based on (5.38) and (5.43), is given in [23], Lemma B.2. It is derived from the proofs of Lemmas 8.4 and 8.5 in [57], pp. 227–228. R EMARK 8.4. Although we have formulated the auxiliary results in Lemmas 8.2 and 8.3 for An only, analogous claims remain valid (with the same proofs) also for def

A(2) n =

1 0

A ∇ϕ1 + s∇vn) (1 − s) ds.

(8.15)

8.3. First-order estimates In this and the next paragraphs we present the asymptotic formulas obtained recently in [23], Section 4. They improve an earlier result from [58], Proposition 6.1, p. 331. From Lemma 8.1 we know that tn → 0 implies vn) C 1,β (Ω) → 0 and μn → 0 as n → ∞. Next p−2 we compute the limits of Vn and μn /(|tn | tn ) as n → ∞.

Nonlinear spectral problems

447

∞ ∞ P ROPOSITION 8.5. Let 1 < p < ∞, p = 2, and let {μn }∞ n=1 ⊂ R, {fn }n=1 ⊂ L (Ω), and 1,p {un }∞ n=1 ⊂ W0 (Ω) be sequences satisfying hypotheses (S1), (S2) and (S3), respectively. 1,p In addition, assume that they satisfy (8.2) for all φ ∈ W0 (Ω) and for each n ∈ N. Then, 1,p writing un = tn−1 (ϕ1 + vn) ) with tn ∈ R, tn = 0, and vn) ∈ W0 (Ω)) , we have tn → 0 as n → ∞, Vn = (|tn |p−2 tn )−1 vn) → V ) strongly in Dϕ1 if p > 2 and in W01,2 (Ω) if 1 < p < 2, and

μn =− n→∞ |tn |p−2 tn

lim

f ϕ1 dx.

(8.16)

Ω

Moreover, the limit function V ) ∈ Dϕ1 ∩ {ϕ1 }⊥,L is the (unique) solution to 2

2 · Q0 V ) , φ =

f † φ dx Ω

for all φ ∈ Dϕ1 ,

(8.17)

where the symmetric bilinear form Q0 is given by (5.11) and f † = f − (

Ω

p−1

f ϕ1 dx)ϕ1

.

Formula (8.16) provides an asymptotic estimate for μn of the ﬁrst-order relative to |tn |p−2 tn as tn → 0, i.e., p−2 (8.18) tn f ϕ1 dx + o |tn |p−1 . μn = −|tn | Ω

We will improve this estimate to a second-order one in the next paragraph. R EMARK 8.6. The linear equation (8.17) represents the weak form of the “limiting” Dirichlet boundary value problem for the limit function Vn = (|tn |p−2 tn )−1 vn) → V ) in the approximation scheme with un = tn−1 (ϕ1 + vn) ). This is a resonant problem to which a standard version of the Fredholm alternative for a selfadjoint linear operator in a Hilbert space applies. More precisely, given a function f ∈ L2 (Ω), a weak solution V ∈ Dϕ1 to the equation 2 · Q0 (V , φ) =

f φ dx Ω

for all φ ∈ Dϕ1 ,

(8.19)

exists in Dϕ1 if and only

if Ω f ϕ1 dx = 0. Such a solution is always unique under the orthogonality condition Ω V ϕ1 dx = 0. Formally, (8.19) is equivalent to the following linear degenerate boundary value problem obtained by linearizing (1.7) with λ = λ1 about ϕ1 ,

p−2 − div A(∇ϕ1 )∇V = λ1 (p − 1)ϕ1 V + f (x) V = 0 on ∂Ω.

in Ω,

(8.20)

We stress that the observations just made remain valid also for 1 < p < 2 when the ϕL2 , the corresponding selfadjoint linear operator has to be considered in the Hilbert space D 1

448

P. Takáˇc

closure of Dϕ1 in L2 (Ω); see Section 5.2. Then, of course, only the orthogonal projection ϕL2 matters in (8.19), according to the orthogonal sum of f to D 1 ϕL2 ⊕ Dϕ⊥,L2 . L2 (Ω) = D 1 1

(8.21)

Consequently, given f ) ∈ {ϕ1 }⊥,L ⊂ L2 (Ω), we denote by 2

2 V ) ≡ V ) f ) ∈ Dϕ1 ∩ {ϕ1 }⊥,L the unique weak solution to problem (8.19) with f ) in place of f . It is easy to see that 2 f ) → V ) : {ϕ1}⊥,L → Dϕ1 is a compact linear mapping. Clearly, this mapping is linear ⊥,L2 be any weakly convergent and bounded. To show that it is compact, let {fn }∞ n=1 ⊂ {ϕ1 } sequence, fn $ f in L2 (Ω) as n → ∞. Hence, {V ) (fn )}∞ n=1 is a weakly convergent ) ) sequence as well, V (fn ) $ V (f ) in Dϕ1 as n → ∞. The embedding Dϕ1 !→ L2 (Ω) being compact, we have also V ) (fn ) → V ) (f ) strongly in L2 (Ω), and

fn φ dx → Ω

f φ dx Ω

uniformly for φ ∈ Dϕ1 with φDϕ1 1. Inserting these results into (8.19) we deduce

% Ω

)

&

Aϕ1 ∇V (fn ), ∇φ dx →

Ω

& % Aϕ1 ∇V ) (f ), ∇φ dx

uniformly for φ ∈ Dϕ1 with φDϕ1 1. We have shown V ) (fn ) → V ) (f ) strongly in Dϕ1 , and thus the desired compactness. P ROOF OF P ROPOSITION 8.5. We have already shown in Lemma 8.1 that tn = 0, tn → 0, μn → 0, and vn) C 1,β (Ω) → 0 as n → ∞. def

Step 1. We now claim that Vn) = (|tn |p−2 tn )−1 vn) is a bounded sequence in L2 (Ω) if p > 2 and in Hϕ1 if 1 < p < 2, and that μn /|tn |p−1 is a bounded sequence in R as well. By contradiction, let us suppose that this is not the case. We set |μn | def Nn = Vn) + |tn |p−1

for n = 1, 2, . . . ,

(8.22)

where Vn) denotes either Vn) L2 (Ω) if p > 2 or Vn) Hϕ1 if 1 < p < 2. Thus, we def

may assume without loss of generality that Nn → ∞. We set Wn) = Vn) /Nn . Then Wn) L2 (Ω) 1 if p > 2 and Wn) Hϕ1 1 if 1 < p < 2. In addition, from (8.10) we

Nonlinear spectral problems

449

obtain the corresponding equation for Wn) ,

%

Ω

& An ∇Wn) , ∇φ dx

= λ1 (p − 1) +

an

μn Nn |tn |p−2 tn

Wn) ϕ1

p−2

p−2 Ω ϕ1

p−1

Ω

ϕ1

φ dx

φ dx + μn (p − 1) Ω

an Wn) φ dx +

1 Nn

fn φ dx Ω

for all φ ∈ W01,2 (Ω). Set an

def

Rn = λ1 (p − 1)

Wn) ϕ1

p−2

p−2 ϕ1

+ μn (p − 1)an Wn) +

μn p−1 ϕ Nn |tn |p−2 tn 1

+

1 fn . Nn

We consider the case 1 < p < 2 ﬁrst. There exist constants c1 > 0 and c2 > 0 such that, for every n sufﬁciently large, we have p−2

c1 an (x)ϕ1

(x) c2

for all x ∈ Ω,

and moreover, an /ϕ1 → 1 as n → ∞ uniformly in Ω. Since Wn) Hϕ1 1, it follows that an Wn) Hϕ c2 . Passing to a subsequence if necessary we may assume Wn) $ W ) p−2

1

weakly in Hϕ1 and an Wn) $ ϕ1 W ) weakly in Hϕ 1 for some W ) ∈ Hϕ1 . Note that fn /Nn → 0 strongly in L∞ (Ω) and μn /(Nn |tn |p−2 tn ) → θ with some θ ∈ [0, 1]. Then Rn $ R weakly in Hϕ 1 , where p−2

def

p−2

R = λ1 (p − 1)ϕ1

W ) + θ ϕ1

p−1

.

Now let us consider p > 2. Since Wn) L2 (Ω) 1 and an → ϕ1 as n → ∞ uniformly in Ω, passing to a subsequence if necessary, we deduce that Wn) $ W ) and Rn $ R weakly in L2 (Ω). ) ) By Lemmas 8.2 and 8.3, there exists a subsequence of {Wn) }∞ n=1 such that Wn → W 1,2 ) strongly in Dϕ1 if p > 2, in W0 (Ω) if 1 < p < 2, and W ∈ Dϕ1 satisﬁes the equation p−2

% Ω

)

&

Aϕ1 ∇W , ∇φ dx = λ1 (p − 1)

Ω

p−2 ϕ1 W ) φ dx

p−1

+θ Ω

ϕ1

φ dx

for every φ ∈ Dϕ1 . Taking φ = ϕ1 in (8.23) we get Ω

% & Aϕ1 ∇ϕ1 , ∇W ) dx − λ1 (p − 1)

Ω

p−1 ϕ1 W ) dx

p

=θ Ω

ϕ1 dx.

(8.23)

450

P. Takáˇc

The left-hand side of this equation equals to 2 · Q0 (ϕ1 , W ) ) = Aϕ1 ϕ1 , W ) ! = 0 and thus yields θ = 0. But this and taking φ = W ) in (8.23) show that Q0 (W ) , W ) ) = 0, and thus W ) = κϕ1 for some constant κ ∈ R; see Proposition 5.4 if p > 2 and Remark 5.7 if 1 < p < 2. Due to Ω W ) ϕ1 dx = 0 we have W ) = 0. Summarizing these convergence results, we ﬁnd Wn) = Vn) /Nn → 0 strongly in L2 (Ω) if p > 2, in Hϕ1 if 1 < p < 2, and μn /(Nn |tn |p−1 ) → 0. Therefore, 1=

Nn Vn) + (|tn |p−1 )−1 |μn | = → 0 as n → ∞ Nn Nn

which is a contradiction. We have veriﬁed that both Vn) and μn /|tn |p−1 are bounded. Step 2. Now we prove (8.16) together with vn) /(|tn |p−2 tn ) → V ) strongly in Dϕ1 if p > 2 and in W01,2 (Ω) if 1 < p < 2. We make use of similar arguments as in Step 1. Again, from (8.10) we deduce % & An ∇Vn) , ∇φ dx Ω

= λ1 (p − 1) Ω

μn + |tn |p−2 tn

an Vn) φ dx

Ω

p−1 ϕ1 φ dx

+ μn (p − 1) Ω

an Vn) φ dx

+

fn φ dx Ω

(8.24) for all φ ∈ W01,2 (Ω). Since the sequence {Nn }∞ n=1 deﬁned in (8.22) is bounded, by Step 1, we may assume (by passing to a subsequence if necessary) that Vn) $ V ) weakly in L2 (Ω) if p > 2 (in Hϕ1 if 1 < p < 2, respectively) and μn /(|tn |p−2 tn ) → θ for some θ ∈ R. Now we apply Lemma 8.2 (8.3, respectively), with a new Rn , namely, μn p−1 ϕ + μn (p − 1)an Vn) + fn . |tn |p−2 tn 1

Rn = λ1 (p − 1)an Vn) + def

The computations above imply that def

p−2

Rn $ R = λ1 (p − 1)ϕ1

V ) + θ ϕ1

p−1

+f

weakly in L2 (Ω) (in Hϕ1 , respectively). Therefore, the limit equation reads as follows: & % p−2 ) ϕ1 V ) φ dx Aϕ1 ∇V , ∇φ dx = λ1 (p − 1) Ω

Ω

p−1

+θ Ω

ϕ1

φ dx +

f φ dx

(8.25)

Ω

for all φ ∈ Dϕ1 , and vn) /(|tn |p−2 tn ) → V ) strongly in Dϕ1 if p > 2 (by Lemma 8.2) and in W01,2 (Ω) if 1 < p < 2 (by Lemma 8.3).

Nonlinear spectral problems

451

In particular, for φ = ϕ1 we get

% Ω

& Aϕ1 ∇ϕ1 , ∇V ) dx − λ1 (p − 1) Ω

ϕ1 dx +

p−1

Ω

p

=θ

ϕ1

V ) dx

f ϕ1 dx, Ω

p

that is, 0 = θ + Ω f ϕ1 dx, by Ω ϕ1 dx = 1. This proves (8.16). Using θ = − Ω f ϕ1 dx

p−1 and deﬁning f † = f − ( Ω f ϕ1 dx)ϕ1 , we can rewrite (8.25) as follows: Ω

% & Aϕ1 ∇V ) , ∇φ dx − λ1 (p − 1)

Ω

p−2

ϕ1

V ) φ dx =

f † φ dx Ω

for all φ ∈ Dϕ1 , which is (8.17). The orthogonality condition Ω V ) ϕ1 dx = 0 follows from

) the fact that Ω vn ϕ1 dx = 0 for all n ∈ N. Note that, by Remark 8.6, there is precisely one function V ) satisfying (8.17). Thus, the strong convergence of the whole sequence vn) /(|tn |p−2 tn ) → V ) follows by the standard argument used towards the end of the proof of Lemma 8.1. The proof of the claim of Step 2 and of the entire proposition is now ﬁnished.

8.4. Second-order estimates The following improvement of Proposition 8.5 is due to [23], Theorem 4.1. Its onedimensional “relatives” (but not analogues), for Ω = (0, a) with 0 < a < ∞, can be found in [48], Eq. (2.5), p. 393, for f ∈ C 1 [0, a] and in [47], Eq. (46), p. 335, for f ∈ L1 (0, a) and at any eigenvalue λk (k 1). T HEOREM 8.7 ([23], Theorem 4.1). In the situation of Proposition 8.5 and under the same hypotheses, the asymptotic formula (8.18) has the following improvement as tn → 0:

fn ϕ1 dx + (p − 2)|tn |2(p−1)Q0 V ) , V )

μn = −|tn |p−2 tn Ω

+ (p − 1) |tn |

+ o |tn |2(p−1) . In particular, if

Ω

2(p−1)

f ϕ1 dx Ω

Ω

p−1 ϕ1 V ) dx

(8.26)

fn ϕ1 dx = 0 for all n ∈ N, then

μn = (p − 2) |tn |2(p−1)Q0 V ) , V ) + o |tn |2(p−1) .

(8.27)

452

P. Takáˇc

On the other hand, if μn = 0 for all n ∈ N, then 1 lim fn ϕ1 dx n→∞ |tn |p−2 tn Ω ) ) p−1 ) f ϕ1 dx ϕ1 V dx . = (p − 2)Q0 V , V + (p − 1) Ω

(8.28)

Ω

It is now quite clear how to obtain “indirect” a priori estimates for weak solutions of problem (1.7) provided λ takes values near λ1 , p = 2, and 2 p−1 † f ϕ1 dx ϕ1 ∈ / Dϕ⊥,L in Ω. (8.29) f =f − 1 Ω

For the (unique) solution V ) ∈ Dϕ1 ∩ {ϕ1 }⊥,L to (8.17), condition (8.29) entails V ) ≡ 0 in Ω and therefore also Q0 (V ) , V ) ) > 0, by Proposition 5.4 if p > 2 and Remark 5.7 if 1 < p < 2. Then, for instance, if μn = 0 for all n ∈ N, formula (8.27) leads to a contra1,β (Ω), by hypothdiction. In other words, the sequence {un }∞ n=1 must be bounded in C esis (S3 ) which is equivalent to (S3). We postpone the details until the next subsection (Section 8.5). 2

P ROOF OF T HEOREM 8.7. We take the inner product of (8.6) with φ = ϕ1 + vn) to get ∇ϕ1 + ∇v ) p dx − λ1 ϕ1 + v ) p dx n n Ω Ω p fn ϕ1 + vn) dx. = μn ϕ1 + vn) dx + |tn |p−2 tn Ω

Ω

Next we apply (5.9) and (5.10) to obtain :#

1

p 0

Ω

; $ A ∇ϕ1 + s∇vn) (1 − s) ds ∇vn) , ∇vn) dx # Ω

= μn Ω

p ϕ1 dx

$ ϕ1 + sv ) p−2 (1 − s) ds v ) 2 dx n n

1

− p(p − 1)λ1 0

#

+p

+ |tn |p−2 tn

Ω

0

$ ϕ1 + sv ) p−2 ϕ1 + sv ) ds v ) dx n n n

1

fn ϕ1 dx + |tn |2(p−1) Ω

Ω (2)

fn Vn) dx.

(8.30)

Let us recall the abbreviations An , an , and An introduced in (8.8) and (8.15), and introduce also 1 (1) def ϕ1 + sv ) p−2 ϕ1 + sv ) ds. (8.31) an = n n 0

Nonlinear spectral problems

453

(2)

Also note that An v, v! An v, v! for all v ∈ RN pointwise in Ω. Dividing (8.30) by |tn |2(p−2) and using vn) = |tn |p−2 tn Vn) we arrive at

%

p Ω

=

) ) A(2) n ∇Vn , ∇Vn

&

dx − p(p − 1)λ1

Ω

2 an(2) Vn) dx

1 μn p ϕ dx + p an(1)Vn) dx |tn |p−2 tn |tn |p−2 tn Ω 1 Ω 1 fn ϕ1 dx + fn Vn) dx. + |tn |p−2 tn Ω Ω

(8.32)

Set def

%

Qn (v, w) =

Ω

A(2) n ∇v, ∇w

&

dx − λ1 (p − 1) Ω

an(2)vw dx,

and recall that the symmetric bilinear form Q0 (v, w) is given by (5.11). We wish to pass to the limit as n → ∞ in (8.32). We have Vn) → V ) strongly in Dϕ1 !→ L2 (Ω) if p > 2 1/2 1/2 (in W01,2 (Ω) !→ L2 (Ω) if 1 < p < 2, respectively) and An ∇Vn) → Aϕ1 ∇V ) strongly in [L2 (Ω)]N . If we pass to a subsequence (denoted again by {Vn) }∞ n=1 ), we can assume 1/2 1/2 also Vn) → V ) and An ∇Vn) → Aϕ1 ∇V ) pointwise a.e. in Ω, and there are functions h1 , h2 ∈ L1 (Ω) such that ) 2 V (x) h1 (x) and A(2) 1/2 ∇V ) (x)2 An1/2 ∇V ) (x)2 h2 (x) n n n n for a.e. x ∈ Ω (see, e.g., [44], Theorem 2.8.1, p. 74). Then, by the Lebesgue dominated convergence theorem, Qn Vn) , Vn) → Q0 V ) , V ) as n → ∞. ∗

strongly in Since fn $ f weakly-star in L∞ (Ω) by hypothesis (S2), and an(1) → ϕ1

(1)

p−1

L∞ (Ω), we have also Ω an Vn) dx → Ω ϕ1 V ) dx and Ω fn Vn) dx → Ω f V ) dx. Hence, (8.32) yields p−1

p · Q0 V ) , V ) −

f V ) dx Ω

μn p = lim ϕ dx + fn ϕ1 dx n→∞ |tn |p−2 tn |tn |p−2 tn Ω 1 Ω μn p−1 +p ϕ1 V ) dx lim . n→∞ |tn |p−2 tn Ω 1

454

Recall

P. Takáˇc

Ω

p

ϕ1 dx = 1. Taking into account the limit (8.16), we arrive at p · Q0 V ) , V ) − f V ) dx Ω

μn = lim + fn ϕ1 dx n→∞ |tn |p−2 tn |tn |p−2 tn Ω p−1 ) −p f ϕ1 dx ϕ1 V dx .

1

Ω

(8.33)

Ω

On the other hand, choose φ = Vn) in (8.24) to get % & ) 2 ) ) An ∇Vn , ∇Vn dx − λ1 (p − 1) an Vn dx − fn Vn) dx Ω

μn = |tn |p−2 tn

Ω

Ω

p−1

ϕ1

Vn) dx + μn (p − 1)

Ω

Ω

) 2

a n Vn

dx.

We pass to the limit for n → ∞ to get ) ) p−1 ) ) 2 · Q0 V , V − f V dx = − f ϕ1 dx ϕ1 V dx . Ω

Ω

(8.34)

Ω

Now we subtract (8.34) from (8.33), thus arriving at ) ) p−1 ) (p − 2) · Q0 V , V = −(p − 1) f ϕ1 dx ϕ1 V dx Ω

+ lim

n→∞

which means μn + |tn |p−2 tn

1 |tn |p−2 tn

Ω

μn + |tn |p−2 tn

fn ϕ1 dx Ω

fn ϕ1 dx Ω

# p−2 = |tn | tn (p − 2) · Q0 V ) , V )

+ (p − 1)

p−1

f ϕ1 dx Ω

Ω

ϕ1

V ) dx

$

+ o |tn |p−1 .

From this equation we ﬁnally derive (8.26). Due to the uniqueness of the limit, we use a standard argument to conclude that this asymptotic behavior holds for the original sequence {μn }∞ n=1 as well. Theorem 8.7 is proved. Theorem 8.7 has a very useful consequence for λ = λ1 and p = 2, namely, (5.2) established in [58], Proposition 6.1, p. 331, cf. (8.35). More precisely, for n = 1, 2, . . . we take

Nonlinear spectral problems

455

μn = 0, fn = fn) + ζn ϕ1 where ζn = ϕ1 −2 f ϕ dx, and instead of (S2) assume L2 (Ω) Ω n 1 only (S2) ) fn) converges to some function f ) in the weak-star topology on L∞ (Ω), i.e., ∗

/ Dϕ⊥,L . fn) $ f ) in L∞ (Ω) as n → ∞. We require f ) ∈ 1 Hence, the sequence {ζn }∞ ⊂ R is not assumed to be a priori bounded. n=1 2

C OROLLARY 8.8. In the situation of Proposition 8.5, with μn = 0 (n 1) and with hypothesis (S2) replaced by (S2) ), we have ζn → 0 as n → ∞, and moreover, ζn n→∞ |tn |p−2 tn lim

= (p − 2)ϕ1 −2 · Q0 V ) , V ) = 0. L2 (Ω)

(8.35)

P ROOF. Without loss of generality, we may assume tn > 0 for all n 1 and tn → 0 as n → ∞. Indeed, if tn < 0 for an index n, we take advantage of the (p − 1)-homogeneity of problem (8.1) and replace the functions fn , f ) , ζn and un by −fn , −f ) , −ζn and −un , respectively, thus switching to the case tn > 0. So tn > 0 and hence also tn un = ϕ1 + vn) 1 2 ϕ1 > 0 in Ω for all n 1. By contradiction, suppose ﬁrst that {ζn }∞ n=1 is unbounded. Keeping the same notation for a suitable subsequence, let |ζn | → ∞ as n → ∞. For each n 1, let us replace def def fn = f ) + ζn ϕ1 and un by f˜n = ζn−1 f ) + ϕ1 and u˜ n = |ζn |−p/(p−1)ζn un , respectively. Consequently, each pair (u˜ n , f˜n ) satisﬁes (8.1) in place of (un , fn ), with μn = 0. Furthermore, we have f˜n − ϕ1 L∞ (Ω) → 0 as n → ∞. If the sequence {u˜ n }∞ n=1 contains a subsequence that is unbounded in L∞ (Ω), we can apply Proposition 8.5 (formula (8.16)) with f˜ = ϕ1 in place of f to conclude that f˜, ϕ1 ! = 0, a contradiction. So {u˜ n }∞ n=1 is for some β ∈ (0, α), by regularity bounded in L∞ (Ω), and consequently, also in C 1,β (Ω) (Lemma 2.3). Fix any β ∈ (0, β) and invoke Arzelà–Ascoli’s theorem in order to pass to Thus, letting n → ∞ in the weak formua convergent subsequence u˜ n → u˜ in C 1,β (Ω). lation of problem (8.1) with (u˜ n , f˜n ) and μn = 0, we arrive at −p u˜ = λ1 |u| ˜ p−2 u˜ + ϕ1 (x) in Ω;

u˜ = 0

on ∂Ω.

But this equation has no weak solution by the nonexistence result from Theorem 4.2. We have shown that {ζn }∞ n=1 is bounded. Now, again by contradiction, suppose that the sequence {ζn }∞ n=1 does not converge to zero. Hence, it must contain a convergent subsequence ζn → ζ ∈ R \ {0} as n → ∞. ∗

def

It follows that fn $ f in L∞ (Ω) as n → ∞, where f = f ) + ζ ϕ1 . But f, ϕ1 ! = ζ ϕ1 2L2 (Ω) = 0 contradicts Proposition 8.5 (formula (8.16)) again. We have veriﬁed ζn → 0 as n → ∞. In particular, the sequence {fn }∞ n=1 satisﬁes hypothesis (S2) with f, ϕ1 ! = 0. Finally, formula (8.35) follows directly from (8.28). The convergence in Theorem 8.7 above is uniform for fn ≡ f (n = 1, 2, . . . ) from any bounded set in L∞ (Ω). More precisely, we have the following corollary:

456

P. Takáˇc

C OROLLARY 8.9 ([23], Corollary 4.4). Let K be a closed bounded ball in L∞ (Ω). Assume that fn ≡ f (n = 1, 2, . . .) and tn → 0 as n → ∞ in Theorem 8.7. Then there exists a sequence {ηn }∞ n=1 ⊂ (0, 1), ηn → 0 as n → ∞, such that for all f ∈ K and for all n = 1, 2, . . . we have −2(p−1) p−2 |tn | μ − |t | t f ϕ dx − (p − 2) · Q0 V ) , V ) n n n 1 Ω p−1 ) f ϕ1 dx ϕ1 V dx ηn . − (p − 1) Ω

(8.36)

Ω

∞ The sequence {ηn }∞ n=1 depends on K and {tn }n=1 , but neither on the choice of f ∈ K nor on the sequence {μn }∞ n=1 ⊂ (−∞, δ ] where δ = λ2 − λ1 − δ > 0.

P ROOF. Assume the contrary to (8.36), that is, there exists a number η > 0 and a sequence {fn }∞ n=1 ⊂ K such that, for all n = 1, 2, . . . , we have −2(p−1) p−2 |tn | μn − |tn | tn fn ϕ1 dx Ω − (p − 2) · Q0 V ) fn† , V ) fn† p−1 ) † fn ϕ1 dx ϕ1 V fn dx η. − (p − 1) Ω

(8.37)

Ω

Here, the function V ) (fn† ) ∈ Dϕ1 ∩{ϕ1 }⊥,L stands for the weak solution to problem (8.17) with fn† in place of f † , where 2

fn† = fn −

Ω

p−1 fn ϕ1 dx ϕ1

and f † = f − Ω

p−1 f ϕ1 dx ϕ1 ;

see Remark 8.6 above. The ball K being weakly-star compact in L∞ (Ω), hence metrizable in the weak-star ∗ topology, from {fn }∞ n=1 we can extract a weakly-star convergent subsequence fn $ f in L∞ (Ω). We apply Theorem 8.7 to conclude that −2(p−1) p−2 |tn | tn fn ϕ1 dx μn − |tn | Ω − (p − 2) · Q0 V ) f † , V ) f † p−1 ) † − (p − 1) f ϕ1 dx ϕ1 V (f ) dx → 0 Ω

Ω

(8.38)

Nonlinear spectral problems

457

∗

as n → ∞. On the other hand, fn $ f weakly-star in L∞ (Ω) yields fn $ f weakly in L2 (Ω). From Remark 8.6 we infer V ) (fn† ) → V ) (f † ) strongly in Dϕ1 for p > 2 (W01,2 (Ω) for 1 < p < 2, respectively). It follows that ) † ) † Q 0 V f , V f − Q0 V ) f † , V ) f † → 0

(8.39)

p−1 ) † f ϕ dx ϕ V (f ) dx n 1 n 1 Ω Ω p−1 − f ϕ1 dx ϕ1 V ) (f † ) dx → 0

(8.40)

n

n

and

Ω

Ω

as n → ∞. Finally, we combine (8.38)–(8.40) to get a contradiction with inequality (8.37). The uniform convergence (8.36) is proved. 8.5. A priori bounds We recall our Hypotheses (H1) for any 1 < p < ∞ and (H2) for p > 2. Proposition 8.5 and Theorem 8.7 have the following applications to a priori estimates which play a decisive role in obtaining our existence and multiplicity results in the next sections. First, let us consider the spectral problem (1.7) with −∞ < λ λ2 − δ, δ > 0. T HEOREM 8.10. Let 1 < p < ∞, p = 2, and let 0 < δ < λ2 − λ1 and 0 λ¯ < ∞. As2 sume that K is a nonempty, weakly-star compact set in L∞ (Ω) such that K ∩ Dϕ⊥,L =∅ 1 and f, ϕ1 ! = 0 for all f ∈ K. Then there exists a constant C(K) > 0 with the follow1,p ing property: Any weak solution u ∈ W0 (Ω) to problem (1.7) obeys uC 1,β (Ω) C(K), provided f ∈ K and λ ∈ R satisﬁes (a) −λ¯ λ λ1 if p > 2; (b) λ1 λ λ2 − δ if p < 2. If the hypothesis λ −λ¯ in (a) is dropped, then the estimate uC 1,β (Ω) C(K) has to be weakened to uW 1,p (Ω) + uL∞ (Ω) C(K). 0

This theorem is due to Takáˇc [57], Theorem 2.1, p. 194, for p > 2 and 0 λ λ1 , and to [57], Theorem 2.6, p. 196, for 1 < p < 2 and λ = λ1 . 2 of Dϕ1 in L2 (Ω) We recall from Section 5.2 that the orthogonal complement Dϕ⊥,L 1

might be nontrivial if 1 < p < 2, i.e., Dϕ⊥,L = {0}. Of course, if p > 2 then the condition 1 2

K ∩ Dϕ⊥,L = ∅ trivializes to 0 ∈ / K. 1 2

P ROOF OF T HEOREM 8.10. We set δ = λ2 − λ1 − δ > 0 and λ = λ1 + μ. On the contrary to the conclusion, suppose that there are three sequences {μn }∞ n=1 ⊂ 1,p ∞ ∞ (−∞, δ ], {fn }n=1 ⊂ K, and {un }n=1 ⊂ W0 (Ω), such that each λ = λ1 + μn obeys

458

P. Takáˇc

(a) and (b), i.e., (p − 2)μn 0, and un W 1,p (Ω) → ∞ as n → ∞. The set K being 0 weakly-star compact in L∞ (Ω), we may extract a weakly-star convergent subsequence ∗ fn $ f in L∞ (Ω) as n → ∞. Hence, f ∈ K. We observe that now all three hypotheses (S1), (S2), and (S3) from Section 8.1 are satisﬁed. Recall that, under condition (S1 ), (S3) is equivalent to (S3 ): un C 1,β (Ω) → ∞ as n → ∞. But then we can apply Theorem 8.7, (8.27), to get 0 (p − 2)−1 μn = |tn |2(p−1)Q0 V ) , V ) + o |tn |2(p−1) . This forces Q0 (V ) , V ) ) 0, whence V ) ≡ 0 in Ω, by Proposition 5.4 if p > 2 and 2 Remark 5.7 if 1 < p < 2. From (8.17) and (8.29) we get f † = f ∈ Dϕ⊥,L , a contradiction 1 to our assumption K ∩ Dϕ⊥,L = ∅. The theorem is proved. 1 2

The following theorem complements Theorem 8.10; it was established in [23], Theorem 5.5. T HEOREM 8.11. Let 1 < p < ∞, 0 < δ < λ2 − λ1 , and 0 λ¯ < ∞. Assume that K is a nonempty, weakly-star compact set in L∞ (Ω) that satisﬁes f, ϕ1 ! = 0 for every f ∈ K. Then there exists a constant C(K) > 0 with the following property: Any weak solution u ∈ 1,p W0 (Ω) to problem (1.7) obeys uC 1,β (Ω) C(K), provided f ∈ K satisﬁes f, ϕ1 ! > 0, and λ ∈ R and u satisfy either of the following two conditions: (a) −λ¯ λ λ1 and u(x) ˆ 0 for some xˆ ∈ Ω; ˆ 0 for some xˆ ∈ Ω. (b) λ1 λ λ2 − δ and u(x) The corresponding result holds also for f ∈ K satisfying f, ϕ1 ! < 0, with the reversed inequalities for u(x) ˆ in conditions (a) and (b) as well. If the hypothesis λ −λ¯ in (a) is dropped, then the estimate uC 1,β (Ω) C(K) has to be weakened to uW 1,p (Ω) + uL∞ (Ω) C(K). 0

P ROOF. The proof is similar to that of Theorem 8.10. Each triple (μn , fn , un ) (n ∈ N) must satisfy also fn , ϕ1 ! > 0 and either of the conditions (a) or (b) with ∗ λ = λ1 + μn and xˆ = xˆ n ∈ Ω. Hence, fn $ f in K as n → ∞ combined with our assumption f, ϕ1 ! = 0 imply f, ϕ1 ! > 0 as well. Next, instead of using Theorem 8.7, (8.27), one has to apply (the easier) Proposition 8.5, (8.18), to get μn = −|tn |

p−2

f ϕ1 dx + o |tn |p−1 .

tn Ω

Consequently, −

1 μn f, ϕ1 ! > 0 |tn |p−2 tn 2

Nonlinear spectral problems

459

for all n large enough, say, n n0 . Recall that tn un 12 ϕ1 > 0 in Ω for all n n0 , by Lemma 8.1. We conclude that −|tn |−(p−2) μn un

1 f, ϕ1 !ϕ1 > 0 4

in Ω for every n n0 .

But this fact violates both conditions (a) and (b) which require μn u(xˆn ) 0 for some xˆn ∈ Ω. Notice that for λ = λ1 and p = 2, at least one of the conditions (a) or (b) in both, Theorem 8.10 and Theorem 8.11, is automatically satisﬁed. We state this result next as a simple consequence of a combination of Theorems 8.10 and 8.11 for the resonant problem (6.12), vis. −p u = λ1 |u|p−2 u + f ) (x) + ζ · ϕ1 (x) in Ω;

u=0

on ∂Ω,

(8.41)

where f ) ∈ L∞ (Ω)) and ζ ∈ R. It was shown originally in [57], Theorems 2.1 and 2.3, and [57], Theorems 2.6 and 2.8, for p > 2 and 1 < p < 2, respectively; see also [58], Theorems 3.2 and 3.6. We write f ≡ f ) + ζ ϕ1 according to (5.1). C OROLLARY 8.12. Let 1 < p < ∞, p = 2. Assume that K is a nonempty, weakly-star 2 compact set in L∞ (Ω) such that K ∩ Dϕ⊥,L = ∅ and g, ϕ1 ! = 0 for all g ∈ K. Then we 1 have: 1,p (i) There exists a constant C(K) > 0 such that, if f ∈ K and if u ∈ W0 (Ω) is any weak solution to problem (8.41), then uC 1,β (Ω) C(K). (ii) Given a number δ > 0, there exists a constant C(K, δ) > 0 such that, if f = f ) + ζ ϕ1 with f ) ∈ K and |ζ | δ, then any weak solution to problem (8.41) satisﬁes uC 1,β (Ω) C(K, δ). P ROOF. Part (i) follows directly from Theorem 8.10. Part (ii) follows from Theorem 8.11, provided one allows only δ |ζ | δ where 0 < δ δ < ∞ are arbitrary, but ﬁxed numbers. Hence, uC 1,β (Ω) C(K, δ, δ ). However, if we apply Corollary 8.8 instead of Theorem 8.11, we obtain part (ii) as it stands. Theorem 8.11 has another important consequence, namely, the following improvement of the “classical” strong maximum and anti-maximum principles (cf. Theorem 4.4 and Remark 4.5) due to Arcoya and Gámez [5], Theorem 27, p. 1908, for K = {f } = {0}. T HEOREM 8.13. Let 1 < p < ∞. Assume that K is a nonempty, weakly-star compact set in L∞ (Ω) that satisﬁes the following two conditions for each f ∈ K: (i) f, ϕ1 ! = 0; and 1,p (ii) the resonant problem (6.12) has no weak solution u ∈ W0 (Ω). Then there exists a constant δ ≡ δ(K), 0 < δ < λ2 − λ1 , such that for every f ∈ K with

1,p Ω f ϕ1 dx > 0, any weak solution u ∈ W0 (Ω) to problem (1.7) satisﬁes the strong maximum and anti-maximum principles:

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(SMP) u > 0 in Ω whenever λ1 − δ < λ < λ1 ; (AMP) u < 0 in Ω whenever λ1 < λ < λ1 + δ,

respectively. The corresponding result holds also if Ω f ϕ1 dx < 0, with the reversed inequality (λ1 − λ)u < 0 in Ω whenever 0 < |λ − λ1 | < δ. P ROOF. Denote f ϕ1 dx > 0 . K+ = f ∈ K: Ω

In analogy with the proof of Theorem 8.10 above, let us ﬁx a number δ with 0 < δ < λ2 − λ1 , and write λ = λ1 + μ. Next, on the contrary to the conclusion of our theorem