Nonlinear Equations with Small Parameter: Volume 2 Waves and Boundary Problems 9783110534979, 9783110533835

Table of contents :
Preface
Contents
Introduction
1. The Solitary Waves Generation due to Passage through the Local Resonance
2. Regular Perturbation of Ill-Posed Problems
3. Asymptotics at Characteristic Points
4. Asymptotic Expansions of Singular Perturbation Theory
5. Asymptotic Solution of the Schrödinger Equation
6. The Kelvin–Helmholtz Instability
7. Nonlinear Cauchy Problems for Elliptic Equations
Bibliography
Index

Citation preview

Sergey G. Glebov, Oleg M. Kiselev, Nikolai N. Tarkhanov Nonlinear Equations with Small Parameter

De Gruyter Series in Nonlinear Analysis and Applications

| Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Mikio Kato, Nagano, Japan Wojciech Kryszewski, Torun, Poland Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Simeon Reich, Haifa, Israel Alfonso Vignoli, Rome, Italy Katrin Wendland, Freiburg, Germany

Volume 23/2

Sergey G. Glebov Oleg M. Kiselev Nikolai N. Tarkhanov

Nonlinear Equations with Small Parameter | Volume 2: Waves and Boundary Problems

Mathematics Subject Classification 2010 34E, 35C Authors Prof. Dr. Sergey G. Glebov Chair of Mathematics Ufa State Petroleum Technological University Faculty of General Scientific Discipline Kosmonavtov St. 1 Ufa 450062 Russian Federation [email protected]

Prof. Dr. Nikolai N. Tarkhanov Universität Potsdam Institut für Mathematik Am Neuen Palais 10 14469 Potsdam Germany [email protected]

Prof. Dr. Oleg M. Kiselev Russian Academy of Sciences UFA Scientific Centre Institute of Mathematics Chernyshevsky str. 112 Ufa 450008 Russian Federation [email protected]

ISBN 978-3-11-053383-5 e-ISBN (PDF) 978-3-11-053497-9 e-ISBN (EPUB) 978-3-11-053390-3 ISSN 0941-813X Library of Congress Control Number: 2018934811 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck www.degruyter.com

| To Valentina

Preface Prerequisites for reading the second volume of the book are somewhat stronger than those for the first volume. Apart from acquaintance with the theory of partial differential equations we suppose some knowledge of basics of pseudodifferential operators. Still we tried to give a self-contained exposition. The main concepts and methods will be comprehensible for graduate students of physics and engineering sciences. The authors gratefully acknowledge the enduring encouragement from the publishing house “De Gruyter” during the preparation of the book, which actually took 3 years.

https://doi.org/10.1515/9783110534979-201

Contents Preface | VII Introduction | XV 1

1.4.1 1.4.2 1.4.3 1.4.4

The Solitary Waves Generation due to Passage through the Local Resonance | 1 The Nonlinear Schrödinger Equation. Scattering of Solitons on Resonance | 1 Problem Statement and Result | 3 Incident Waves | 4 Scattering | 6 Scattered Waves | 8 Numerical Justification of Asymptotic Analysis | 10 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation | 11 Main Result | 11 Pre-Resonance Expansion | 14 Internal Asymptotics | 16 Post-Resonance Expansion | 23 The Perturbed KDV Equation and Passage through the Resonance | 28 Forced Oscillations | 29 Inside the Resonance | 31 Post-Resonance Expansion | 37 Numerical Simulations | 39 Auto-Resonant Soliton and Perturbation with Decaying Amplitude | 40 Justification | 41 Derivation of the Model Equation for Auto-Resonance | 42 An Asymptotic Solution of the Model Equation | 43 Effect of Dissipation | 46

2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6

Regular Perturbation of Ill-Posed Problems | 48 Mixed Problems with a Parameter | 49 Preliminaries | 49 The Cauchy Problem | 53 A Perturbation | 58 The Main Theorem | 62 The Well-Posed Case | 65 On Finding the Solution | 67

1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4

X | Contents

2.1.7 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2 2.4.3 2.4.4

Dirac Operators | 72 Kernel Spikes of Singular Problems | 76 Soft Expansions | 77 Harmonic Extension | 79 Auxiliary Results | 80 Formulas for Coefficients | 81 Laurent Series | 84 Expansion of the Poisson Kernel | 85 An Asymptotic Expansion of the Martinelli–Bochner Integral | 86 Asymptotic Expansion | 87 The Bochner–Martinelli Integral | 88 Regularization | 94 Proof of the Theorem | 95 A Formula for the Number of Lattice Points in a Domain | 97 Logarithmic Residue Formula | 97 The Integral Formula | 98 The One-Dimensional Case | 101 Some Comments | 102

3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8 3.2.9 3.3

Asymptotics at Characteristic Points | 103 Asymptotic Solutions of the 1D Heat Equation | 103 Preliminaries | 103 On the Heat Equation | 104 Blow-Up Techniques | 106 Further Reduction | 109 The Unperturbed Problem | 111 Asymptotic Solutions | 113 Local Solvability at a Cusp | 117 Euler Theory on a Spindle | 119 Pseudodifferential Operators on Manifolds with Conical Points | 120 Meromorphic Families | 122 Characteristic Values | 123 Factorization | 126 Resolvent | 129 Unitary Reduction | 130 Inhomogeneous Equation | 133 Transposed Equations | 140 Index | 143 The Laplace–Beltrami Operator on a Rotationally Symmetric Surface | 145 Calculus on Singular Varieties | 145 Geometry | 146

3.3.1 3.3.2

Contents | XI

3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.3.8 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 3.4.7 3.4.8 3.4.9

Laplace–Beltrami Operator | 148 Weighted Spaces | 149 Resolvent | 151 Fredholm Theory | 153 Asymptotics | 155 Index Formula | 156 Boundary Value Problems for Parabolic Equations | 158 Anisotropic Ellipticity | 161 Parabolicity after Petrovskii | 162 Characteristic Points | 163 Weighted Spaces | 166 Solution in a Special Domain | 174 Local Parametrices | 182 The Global Parametrix | 192 Regularity of Solutions | 206 Some Particular Cases | 210

4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6 4.2.7 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 4.3.7 4.3.8

Asymptotic Expansions of Singular Perturbation Theory | 214 Small Random Perturbations of Dynamical Systems | 214 White Noise Perturbation of Dynamical Systems | 214 The Case of the Homogeneous Differential Equation | 216 The Case of the Homogeneous Boundary Condition | 220 The Case of a Right-Hand Side of Zero Average Value | 222 Conclusion | 222 Appendix | 223 Formal Asymptotic Solutions | 225 Asymptotic Phenomena | 225 Blow-Up Techniques | 228 Formal Asymptotic Solution | 230 The Exceptional Case p = 2 | 233 Degenerate Problem | 235 Generalization to Higher Dimensions | 235 Parameter-Dependent Norms | 240 The Shapiro–Lopatinskii Condition | 241 Boundary Value Problems with Small Parameter | 241 Asymptotic Expansion | 242 The Main Spaces | 245 Auxiliary Results | 249 The Main Result | 251 Local Estimates in the Interior | 253 The Case of Boundary Points | 255 Conclusion | 257

XII | Contents

4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 4.4.7

Pseudodifferential Calculus with a Small Parameter | 257 Singular Problems with a Small Parameter | 257 Loss of Initial Data | 258 A Passive Approach to Operator-Valued Symbols | 260 Operators with a Small Parameter | 264 Ellipticity with a Large Parameter | 269 Another Approach to Parameter-Dependent Theory | 269 Regularization of Singularly Perturbed Problems | 275

5 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6 5.4.7 5.4.8

Asymptotic Solution of the Schrödinger Equation | 278 Semiclassical Approximations of Quantum Mechanics | 278 Standard Approximation | 278 Preliminary Results | 280 The Schrödinger Equation for Quadratic Hamiltonians | 281 Exact Solution with a Rapidly Decreasing Initial Symbol | 282 Symmetrized Generating Function | 285 Asymptotic Solution of the Schrödinger Equation | 286 Symbol Classes | 286 Construction of a Formal Expansion | 288 Asymptotic Solution | 289 One More Asymptotic Decomposition | 291 The Trace of the Schrödinger Operator | 292 Anti-Wick Symbols | 292 The Trace Formula | 293 Trace Asymptotics for ℏ → 0 | 293 A Lefschetz Fixed Point Formula | 294 Quantum Dynamics in the Fermi–Pasta–Ulam Problem | 295 Wave Decay Processes | 296 Classical Limit | 297 Quantum Equations of Decay | 299 Analysis of Quantum Equations | 302 Existence of Solutions | 303 Successive Approximations | 307 Asymptotic under Large Time | 313 Conclusion | 314

6 6.1 6.1.1 6.1.2 6.1.3 6.1.4

The Kelvin–Helmholtz Instability | 316 Derivation of the Fundamental Equation | 318 Setting of the Problem | 318 Conditions on the Unknown Boundary | 319 Derivation of an Equation for the Curve | 320 A Hamiltonian Form of the Equation of Tangential Discontinuity | 322

Contents |

6.1.5 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.3.6 6.3.7 6.3.8 6.3.9 6.4 6.4.1 6.4.2 6.4.3 6.4.4 6.4.5 7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6 7.1.7 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.2.6 7.3 7.3.1

Conservation Laws | 323 Small Perturbation of Tangential Discontinuity | 324 Linearization of the Equation of Tangential Discontinuity | 324 On the Ellipticity of the System (6.18) | 326 Small Perturbations of Rectilinear Tangential Discontinuity | 327 The Linearization of Equation (6.17) | 328 Remarks on the Linearized System | 329 Analytic Continuation from a Boundary Subset | 331 The Riemann Mapping Theorem | 333 Hardy Spaces | 334 The Cauchy Formula | 335 Quenching Functions | 336 The Goluzin–Krylov Formula | 338 A Uniqueness Theorem | 339 Approximation through Holomorphic Functions | 340 Expansion in a Fourier Series | 341 Approximation through Legendre Polynomials | 343 A Numerical Approach to the Riemann Hypothesis | 344 The Riemann Zeta Function | 345 Analytic Continuation in a Lune | 346 A Carleman Formula for a Half-Disk | 349 Reduction of the Riemann Hypothesis | 352 Numerical Experiments | 355 Nonlinear Cauchy Problems for Elliptic Equations | 357 A Variational Approach to the Cauchy Problem | 357 Relaxations of Ill-Posed Problems | 357 The Cauchy Problem | 359 Variational Setting | 361 Euler’s Equations | 364 Examples | 366 Mixed Problems | 367 Inverse Problem Approach | 369 The Cauchy Problem for Chaplygin’s System | 371 Preliminaries | 371 Chaplygin’s System | 372 Variational Setting | 373 Existence of Solutions | 375 Stable Cauchy Problems | 377 Approximate Solutions | 379 Hyperbolic Formulas in Elliptic Cauchy Problems | 380 The Cauchy Problem | 384

XIII

XIV | Contents

7.3.2 7.3.3 7.3.4 7.3.5 7.3.6 7.3.7 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5

Hyperbolic Reduction | 385 The Planar Case | 387 The Carleman Formula | 389 Poisson’s Formula | 391 The Kirchhoff Formula | 393 Concluding Remarks | 395 A WKB Solution to the Navier–Stokes Equations | 396 Basic Equations of the Dynamics of an Incompressible Viscous Fluid | 396 Generalized Navier–Stokes Equations | 398 Energy Estimates | 401 First Steps towards the Solution | 403 A WKB Solution | 405

Bibliography | 407 Index | 421

Introduction In the second volume we focus on partial differential equations with small parameter in domains of Rn . The nature of the small parameter may be diverse. It can be caused by characteristic points of the differential operator on the boundary, singular points of of the boundary itself, the discrepancy of an asymptotic solution to the problem, etc. Instead of explaining in general terms what we mean by asymptotic phenomena, we prefer to single out at first one class of such phenomena, to wit the asymptotics of solutions to elliptic boundary value problems. Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the boundary, and in this form asymptotics can be treated already on smooth configurations. Let B be the unit ball in Rn with centre at the origin, and y0 be a fixed point on the boundary of B. Consider the Dirichlet problem of finding a harmonic function u in B with prescribed limit values u 0 on ∂B = S. Were y0 a singular point of S, one had to take care of interpreting the equality u = u 0 at y0 . In the analysis on manifolds with singularities one copes with this problem in an evident and soft way. Namely, one requires the condition u = u 0 away from the point y0 on S. This results in an infinite dimensional null-space of the problem, which consists of all harmonic functions in B vanishing on S \ {y0 }. From the structure theorem for distributions with a point support it follows that every such harmonic function is a finite linear combination of the derivatives of the Poisson kernel ℘(x, y) for B. One can thus control the null-space by considering solutions in weighted spaces with weight functions being powers of the distance |x − y0 |. The data of the problem are forced to be in weighted functions spaces, too, which makes difficult the use of the Poisson kernel. To apply this latter, the Dirichlet data should be regularised at y0 to define a distribution on S coinciding with u 0 outside of y0 . In fact the regularisation consists of subtracting a finite number of terms of the Taylor expansion of ℘(x, y) at y = y0 . Hence the cokernel of the problem is also spanned by harmonic functions in B which vanish on S \ {y0 }. Summarising we deduce that the variation of the index of the Dirichlet problem in weighted spaces under changing the weight exponent is easily evaluated from the structure theorem for harmonic functions in B vanishing outside y0 on S. In fact one can derive a canonical representation for harmonic functions in B vanishing on S \ {y0 }. Suppose u is a harmonic function in B of finite order of growth near the boundary S. Then u has weak limit values u 0 ∈ D󸀠 (S) on the sphere in the sense that lim ∫ u((1 − ε)y)𝑣(y) ds = ⟨u 0 , 𝑣⟩

ε→0

S

for each 𝑣 ∈ C∞ (S), where ds is the induced Lebesgue measure on S. Moreover, u can be reconstructed from its weak boundary values by the Poisson formula u(x) = https://doi.org/10.1515/9783110534979-202

XVI | Introduction ⟨u 0 , ℘(x, ⋅)⟩ for x ∈ B. This formula shows in particular that the behaviour of u close to a boundary point y ∈ S in B is completely determined by the behaviour of u0 near y on S. Conversely, given any distribution u 0 on S, the function u(x) = ⟨u 0 , ℘(x, ⋅)⟩, x ∈ B, is harmonic in B and its weak limit values on S coincide with u 0 . For these and more general results we refer the reader to [269]. What happens when u 0 fails to be a distribution near a compact set K of zero measure on S? Consider the simplest case where K = {y0 } is a fixed point of S and u 0 (y) = |y − y0 |z , with ℜz ≤ −(n −1). This function u 0 can be extended to a distribution on all of S in many ways. Pick e.g., a local chart U around y0 on S with coordinates y󸀠 = (y1 , . . . , y n−1 ) and a function χ ∈ C∞ (S) supported in U, which is equal to 1 near y0 . If N is a natural number with ℜz + N > −(n − 1), then ∂ αy󸀠 𝑣(y0 )

󸀠

⟨u 0,N , 𝑣⟩ = ⟨u 0 , χ(𝑣(y ) − ∑ |α| 0. Then S near y0 is the graph of y n = √1 − |y󸀠 |2 , where y󸀠 = (y1 , . . . , y n−1 ), hence we may take y󸀠 as local coordinates on S. For j = 1, . . . , n − 1, the full derivative in y j is ∂ j − (y j /y n )∂ n . The tangential space to S is also spanned by the system of vector fields ∂ yj ∂ − , j |y| ∂|y| ∂y j = 1, . . . , n, although these latter are not independent. Substituting this into (1) leads to ∞ h j (x − y0 ) ℘(x, y0 ) (3) u(x) = ⟨u 0,R , ℘(x, ⋅)⟩ + ∑ |x − y0 |2j j=0 for x ∈ B, where (h j (z))j=0,1,... is a sequence of homogeneous harmonic polynomials of degree j in Rn whose coefficients satisfy growth estimates similar to (2). This formula refines upon the construction of Laurent series for harmonic functions in B vanishing on ∂B \ {y0 }, cf. [270]. Under certain factorisation the polynomials h j are uniquely determined from u and u 0,R and they are given by explicit formulas, see [225]. Obviously, formula (1) goes far beyond the Dirichlet problem for the Laplace equation in a ball. If one relaxes the boundary condition at only one boundary point y0 ∈ S, there arise singular solutions of the problem like the summands on the right-hand side of (3). The structure of singular solutions may be intricate and they stand up to analysis in non-weighted Sobolev spaces. Commonly the weight function is chosen to be a power of the distance |x − y0 | which plays the role of a small parameter. The use of a series which does not necessarily converge is a typical instance of a “formal procedure.” The present book is solely concerned with the formal expansions used in the analysis of asymptotic phenomena. Our purpose is to clarify the role of formal procedures in various concrete problems of analysis and mathematical physics. Those who employ mathematics as a tool have rarely been inhibited by the fear of divergence. A mathematician will try to show that formal procedures used by good physicists indeed are valid if only the meaning of validity is properly interpreted. Chapter 1 deals with a generation of solitary waves for several model equations. We study a passage through a local and autoresonance and discuss this phenomenon in detail. Chapter 2 is devoted to regular perturbations of ill-posed problems of analysis and geometry. Chapter 3 is an exposition of boundary value problems for parabolic equations in bounded domains. They fail to be regular, for there are characteristic points at the boundary. In case the contact degree at which outer characteristics meet

XVIII | Introduction

the boundary coincides with the order of equation, the analysis reveals many common features with analysis of elliptic equations on manifolds with conical singularities. Chapter 4 introduces asymptotic expansions of singular perturbation theory. An interesting aspect in the theory is the question on possibility to reduce a singular perturbation to a regular one. It is handled within the framework of algebras of pseudodifferential operators containing a small parameter. The operators in question possess symbols whose invertibility specifies the class of those singular perturbations which are called elliptic. Under the ellipticity assumption a singular perturbation can be reduced to a regular one. Chapter 5 is written for more general representation of solutions than asymptotic expansions, to wit those of deformation quantisation. In Chapter 6 we revisit those expansions which arise in analytic function theory of one complex variable. Finally, in Chapter 7 we show that the asymptotic formulas arising in the Cauchy problem for solutions of elliptic equations are of much the same character as those of analytic continuation.

1 The Solitary Waves Generation due to Passage through the Local Resonance In this chapter we present our results related to the generation of solitary waves due to a passage through the local resonance and auto-resonance. We study some model equations and present detailed evaluations. The main aims of this chapter are the following: – To show a changing of number of solitons in the nonlinear Schrödinger equation. – To study a problem on a generation of solitary packets of waves by a small perturbation for the nonlinear Klein–Gordon equation. – To investigate the influence of perturbation on a small forced solution of a perturbed KDV equation. – To construct a soliton version for a special growing solution for the analogue of the principal resonance equation.

1.1 The Nonlinear Schrödinger Equation. Scattering of Solitons on Resonance In this section, we investigate a propagation of solitons for the nonlinear Schrödinger equation under a small driving force. The driving force passes through the resonance. The process of scattering on the resonance leads to a change in the number of solitons. After the resonance the number of solitons depends on the amplitude of the driving force. The nonlinear Schrödinger equation (NLSE) is a mathematical model for a wide class of wave phenomena from signal propagation to optical fibers [129, 268] to surface wave propagation [304]. This equation is integrable by an inverse scattering transform method [303] and can be considered as an ideal model equation. The perturbations of this ideal model lead to nonintegrable equations. Here, we consider a nonintegrable example that is NLSE perturbed by the small driving force, 2

i∂ t Ψ + ∂2x Ψ + |Ψ|2 Ψ = ϵ2 fe iS/ϵ ,

0 0 and u1 (x1 , t2 ) satisfies i∂ t 2 u 1 + ∂2x1 u 1 + |u 1 |2 u 1 = 0

4 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

and the initial condition u 1 |t 2 =t 0 = h1 (x1 ) ,

t0 = const < 0 .

Then, in the domain 0 < t2 < c the asymptotic solution of (1.1) has the form Ψ(x, t, ϵ) = ϵ𝑣1 (x1 , t2 ) + O(ϵ2 ) ,

(1.3)

where 𝑣1 (x1 , t2 ) is a solution of NLSE with the initial condition 𝑣1 |t 2=0 = u 1 (x1 , 0) + (1 − i)√πf(x1 ) .

(1.4)

Formula (1.4) is the main result. This formula connects the main order term u 1 of the asymptotic solution before the scattering, the external driving force f , and the initial condition 𝑣1 for the main order term of the asymptotic solution after the scattering. Let us explain the result for the soliton solution. If in the domain −c < t2 < 0 the solution has an N-soliton form, then in the domain 0 < t2 < c the number of solitons is defined by the initial condition (1.4). The given analysis is valid at |t2 | ≤ C, for ∀C = const. When |ϵ2 t| ≫ 1, the perturbation force will modulate the parameters of the soliton solution, see [125, 126].

1.1.2 Incident Waves In this section, we construct the asymptotic solution of equation (1.1) in the preresonance domain. This solution contains two parts. The first part is a specific solution of the nonhomogeneous equation. This solution oscillates with the frequency of the driving force. The amplitudes are determined by algebraic equations. The second part of the solution is a solution of the homogeneous equation. The solution contains an undefined function due to integration. This undefined function is usually determined by the initial condition for the Cauchy problem. We construct the formal asymptotic solution in a WKB-like form Ψ(x, t, ϵ) = ϵu 1 (x1 , t2 ) + ϵ3 u 3 (x1 , t2 ) + ϵ2 B2 (x1 , t2 ) exp(iS/ϵ2 ) + ϵ4 (B4,1 (x1 , t2 ) exp(iS/ϵ2 ) + B4,−1 (x1 , t2 ) exp(−iS/ϵ2 ))

(1.5)

+ ϵ5 B5,2 (x1 , t2 ) exp(2iS/ϵ2 ) . To determine the coefficients of the asymptotics substitute (1.5) into equation (1.1). This yields ϵ2 ( − S󸀠 B2 − f ) exp(iS/ϵ2 ) + ϵ3 (i∂ t 2 u 1 + ∂2x1 u 1 + |u 1 |2 u 1 ) + ϵ4 (( − S󸀠 B4,1 + i∂ t 2 B2 + ∂2x1 B2 + 2|u 1 |2 B2 ) exp(iS/ϵ2 ) + (S󸀠 B4,−1 + u 21 B∗2 ) exp(−iS/ϵ2 ))

1.1 The Nonlinear Schrödinger Equation. Scattering of Solitons on Resonance |

5

+ ϵ5 (i∂ t 2 u 3 + ∂2x1 u 3 + 2|u 1 |2 u 3 + u 21 u ∗3 + 2u 1 |B2 |2 + ( − 2S󸀠 B5,2 + u ∗1 B22 ) exp(2iS/ϵ2 )) = ϵ6 R(t2 , x1 ) . The residue part of the asymptotics has the form R(t2 , x1 ) = O(|B2 |3 + ϵ3 |u 3 |3 + ϵ6 |B4,1 |3 + ϵ6 |B4,−1 |3 + ϵ9 |B5,2 |3 ) .

(1.6)

ϵ5

Collect the terms with the same order of ϵ up to the order of and reduce similar terms. This yields differential equations for u 1 , u 3 and algebraic equations for B2 , B4,±1 , and B5,2 ; i∂ t 2 u 1 + ∂2x1 u 1 + |u 1 |2 u 1 = 0 , i∂ t 2 u 3 +

∂2x1 u 3

2

+ 2|u 1 | u 3 +

u 21 u ∗3

(1.7) 2

= −2|B2 | u 1 ,

−S󸀠 B2 = f ,

(1.9)

󸀠

−S B4,1 = i∂ t 2 B2 + 󸀠

S B4,−1 = 󸀠

−2S B5,2 =

(1.8)

−u 21 B∗2 u ∗1 B22 .

∂2x1 B2

2

+ |u 1 | B2 ,

,

(1.10) (1.11) (1.12)

The functions u 1 , u 3 are uniquely determined by initial conditions at the moment t2 = t0 . We suppose that t0 = const < 0 and u 1 |t 2=t 0 = h1 (x1 ) ;

u 3 |t 2=t 0 = h3 (x1 ) ;

where functions h1 and h3 are smooth and rapidly vanish as |x1 | → ±∞. It is known that the solutions u 1 and u 3 exist for bounded values of t2 , see [57, 127]. Remark. The solution of the equation for u 3 contains growing terms as t2 → ∞, see, for example, [127]. These terms are secular as t2 ∼ ϵ−1 . However, we do not consider such long terms in this work. The coefficients of the representation (1.5) have a singularity as S󸀠 → 0. The order of singularity of B j,k is easily calculated, B2 = O(t−1 ) ,

B4,1 = O(t−3 ) .

To determine the asymptotics of u 3 as t2 → −0 we construct the solution of the form (−1,0) u 3 = t−1 (x1 , t2 ) + ln |t2 |u 3 (0,1) (x1 , t2 ) + t2 ln |t2 |u 3 (1,1) (x1 , t2 ) + ̂ u 3 (x1 , t2 ) . 2 u3 (1.13) Substitute this representation into equation (1.8) and collect the terms of the same order with respect to t2 . This yields equations for coefficients of the asymptotics (1.13)

u 3 (−1,0) = i2|f|2 u 1 , (−1,0)

u 3 (0,1) = −iL(u 3 u3

(1,1)

=

(0,1) −iL(u 3 )

), ,

L(̂ u 3 ) = it2 ln |t2 |L(u 3 (1,1) ) + iu 3 (1,1) .

(1.14)

6 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

Here L(u) is a linear operator of the form L(u) = i∂ t 2 u + ∂2x1 u + 2|u 1 |2 u + u 1 2 u ∗ . Functions u 3 (−1,0) , u 3 (0,1) and u 3 (1,1) are determined from algebraic equations. These functions are bounded as −const. < t2 ≤ 0, const. > 0. The function ̂ u 3 is a solution of the nonhomogeneous linearized Schrödinger equation. The right-hand side of the equation is a smooth function as −const. < t2 ≤ 0, const. > 0. The solution of this equation can be obtained using results of [127]. In particular, if u 1 is an N-solitons solution of NLSE, then there exists the bounded solution of the nonhomogeneous linearized Schrödinger equation (1.14) as −const. < t2 ≤ 0, const. > 0. The asymptotic form (1.5) allows us to solve equation (1.1) up to the order ϵ6 . To obtain a more accurate approximation one needs to include terms without fast oscillations of the order o(ϵ3 ) into the asymptotic solution (1.5). Therefore, we define the domain of validity of (1.5) by the following relation: ϵ6 R(t2 , x1 ) = o(ϵ3 ) ,

ϵ→0.

Coefficients of (1.5) have singularity at t2 = 0. The residue part increases as t2 → 0. From formulas (1.6) and (1.7)–(1.12) one can easily obtain the behavior of the residue part: 6 −9 R(t2 , x1 ) = O(t−3 2 + ϵ t2 ) ,

t2 → −0 .

This yields the domain of validity of (1.5) −t2 ≫ ϵ

or

− t ≫ ϵ−1 .

1.1.3 Scattering In the neighborhood of the point t2 = 0 the frequency of the driving force becomes resonant. Formally, this means that representation (1.5) is not valid. In this section, we construct another representation for the solution of equation (1.1). This representation is valid in the neighborhood of the resonance line t2 = 0, Ψ(x, t, ϵ) = ϵw1 (x1 , t1 )) + ϵ2 w2 (x1 , t1 ) + ϵ3 ln ϵw3,1 (x1 , t1 ) + ϵ3 w3 (x1 , t1 )

ϵ→0.

(1.15)

Here we use a new scaled variable t1 = t2 /ϵ. Representation (1.15) is matched with (1.5). This means that these formulas are equivalent up to a value o(ϵ5 ) as t2 → −0. The coefficients of (1.15) are determined by the ordinary differential equations (1.16), (1.18), (1.20), and matching conditions.

1.1 The Nonlinear Schrödinger Equation. Scattering of Solitons on Resonance |

7

To obtain the behavior of the coefficients of (1.15) as t1 → −∞ match (1.15) with (1.5). Write (1.5) in terms of t1 −3 2 Ψ(x, t, ϵ) = ϵ(u 1 (x1 , 0) − (t−1 1 f + it1 f ) exp(iS/ϵ )) 2 −2 + ϵ2 (∂ t 2 u 1 (x1 , t2 )|t 2 =0 t1 + t−1 1 i|f| u 1 (x 1 , 0) + O(t1 ))

+ ϵ3 ln ϵ( − iL(2i|f|2 u 1 )|t 2 =0 + o(1)) 1 u 3 (x1 , 0) + o(1)) , + ϵ3 ( ∂2t 2 u 1 (x1 , t2 )|t 2 =0 t21 + ̂ 2

1 ≪ −t1 ≪ ϵ−1 , ϵ → 0 .

To obtain equations for coefficients of (1.15) substitute (1.15) into equation (1.1). This yields ϵ2 ((∂ t 1 w1 − f exp(iS/ϵ2 )) + ϵ3 (∂ t 1 w2 + ∂2x1 w1 + γ|w1 |w1 ) + ϵ4 (∂ t 1 w3 + ∂2x1 w2 + +w1 2 w2 ∗ + 2γ|w1 |w2 ) = ϵ5 ρ(t1 , x1 , ϵ) . The function ρ(t1 , x1 , ϵ) can be represented in the form ρ(t1 , x1 , ϵ) = O(|w1 |2 w3 + ∂2x1 w3 + ϵ|w2 |3 + ϵ4 |w3 |3 ) . Collect the terms of the same order with respect to ϵ. As a result, we obtain the equations for coefficients of (1.15), i∂ t 1 w1 = f exp(it21 /2) .

(1.16)

The matching conditions give w1 = u 1 (x1 , 0) t1 → −∞. The solution of this problem is represented in terms of the Fresnel integral t1

w1 = u 1 (x1 , 0) − if(x1 ) ∫ exp(iθ2 /2)dθ .

(1.17)

−∞

Equations for higher-order terms are i∂ t 1 w2 = −∂2x1 w1 + |w1 |2 w1 ,

(1.18)

i∂ t 1 w3,1 = 0 , i∂ t 1 w3 =

−∂2x1 w2

(1.19) 2

− 2|w1 | w2 −

w21 w∗2

.

(1.20)

The higher-order terms satisfy fist-order ordinary differential equations with respect to t1 . The spatial variable x1 is a parameter in these equations. The solutions of these equations are uniquely defined by terms of the order of 1 in asymptotics as t1 → −∞. The asymptotics as t1 → −∞ is obtained by matching w2 = ∂ t 2 u 1 (x1 , 0)t1 + o(1) ; w3,1 = −iL(2i|f|2 u 1 (x1 , 0)) + o(1) , 1 u 3 (x1 , 0) + o(1) . w3 = ∂2t 2 u 1 (x1 , 0)t21 + ̂ 2

8 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

To determine the behavior of the solution after resonance we need to calculate the asymptotics as t1 → +∞ of the coefficients for representation (1.15). Calculations give w1 (x1 , t1 ) = u 1 (x1 , 0) − if(x1 )[ic1 +

exp(it21 /2) + O(t−3 1 )] , it1

where c1 = (1 − i)√π. Denote by w1 (x1 , t1 )|t 1 →∞ = w1,0 (x1 ) . The function w2 (x1 , t1 ) has asymptotics of the form w2 (x1 , t1 ) = t1 w2,1 (x1 ) + w2,0 (x1 ) + g1 (x1 )

exp(it21 /2) it21

+ O(t−4 1 ),

where w2,1 = −∂2x1 w1,0 + |w1,0 |2 w1 ; t1

w2,0 (x1 ) = lim ( ∫ [∂2x1 w0 (x1 , θ) + |w0 (x1 , θ)|2 w0 (x1 , θ)]dθ − w2,1 t1 ) , t 1 →∞

g1 (x1 ) = k 1 ∂ x1 f + k 2

−∞

|f|2 f ,

k 1 and k 2 are constants;

w3 (x1 , t1 ) = t21 w3,2 (x1 ) + o(t21 ) ,

t1 → ∞ ,

where w3,2 (x1 ) = i (∂2x1 w2,1 + 2|w1,0 |2 w2,1 + w21,0 w∗2,1 ) . The domain of validity for (1.15) is defined in the same way as in the previous section. We require the following relation to be valid: ϵ5 ρ(t1 , x1 , ϵ) = o(ϵ2 ) ,

t1 → ∞ .

The above determined behavior of coefficients of asymptotics (1.15) gives the domain of validity for (1.15) |t1 | ≪ ϵ−1 or |t| ≪ ϵ−2 .

1.1.4 Scattered Waves In this section, we construct the asymptotic solution of equation (1.1) after passage through the resonance. The leading-order term of the solution satisfies NLSE and depends on x1 , t2 before resonance. However, this leading-order term is determined by another solution of NLSE, which generally speaking contains another number of solitons.

1.1 The Nonlinear Schrödinger Equation. Scattering of Solitons on Resonance | 9

We construct the asymptotic solution of the form Ψ(x, t, ϵ) = ϵ𝑣1 (x1 , t2 ) + ϵ2 𝑣2 (x1 , t2 ) + ϵ2 A2 (t2 , x1 ) exp(iS/ϵ2 ) + ϵ4 (A4,1 (t2 , x1 ) exp(iS/ϵ2 ) + A4,−1 (t2 , x1 ) exp(−iS/ϵ2 )) .

(1.21)

Substitute this representation into (1.1): ϵ2 (−S󸀠 A2 − f) exp(iS/ϵ 2 ) + ϵ3 (∂ t 2 𝑣1 + ∂2x1 𝑣1 + |𝑣1 |2 𝑣1 ) + ϵ4 (∂ t 2 𝑣2 + ∂2x1 𝑣2 + 2|𝑣1 |2 𝑣2 + 𝑣12 𝑣2∗ + (−S󸀠 A4,1 + ∂ t 2 A2 + ∂2x1 A2 + 2|𝑣1 |2 A2 ) exp(iS/ϵ2 ) + (S󸀠 A4,−1 + 𝑣12 A∗2 ) exp(−iS/ϵ2 )) = ϵ5 r(t2 , x1 , ϵ) . Here r(t2 , x1 , ϵ) depends on coefficients of the asymptotics (1.21). This dependence is easy calculated. The coefficients A2 , A4,1 , and A4,−1 have singularity on the resonance curve. To determine the domain of validity of (1.21) we need to derive the explicit formula for r r(t2 , x1 , ϵ) = O(1 + ϵ|A2 |3 + ϵ ln |ϵ| + ϵ7 (|A4,1 |3 + |A4,−1 |3 )) . Collect the terms of the same order of small parameter and the same exponents. This yields the equations for coefficients of representation (1.21). i∂ t 2 𝑣1 + ∂2x1 𝑣1 + |𝑣1 |2 𝑣1 = 0 ; i∂ t 2 𝑣2 +

∂2x1 𝑣2

+ 2|𝑣1 |

2

𝑣2 + 𝑣12 𝑣2∗

(1.22) =0

(1.23)

− S󸀠 A2 = f ; − S󸀠 A4,1 = −∂ t 2 A2 − ∂2x1 A2 − 2|𝑣1 |2 A2 ; S󸀠 A4,−1 = −𝑣12 A∗2 . Initial conditions for differential equations for 𝑣1 are obtained by matching. These conditions are evaluated on the resonance curve t0 = 0. 𝑣1 |t 2 =0 = u 1 (x1 , 0) + (1 − i)√πf(x1 ) ;

(1.24)

𝑣2 |t 2 =0 = w2,0 (x1 ) .

(1.25)

The residue part ϵ5 r(t2 , x1 , ϵ) = o(ϵ2 ) as t2 ≫ ϵ. This condition is determined the domain of validity for (1.21). Formula (1.24) is the connection formula for the leading-order term of the asymptotic solution before and after resonance. An additional term (1 − i)√πf(x1 ) leads to changing of the solution after passage through the resonance.

10 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

1.1.5 Numerical Justification of Asymptotic Analysis In this section we justify our asymptotic formula (1.4). Let us consider the pure soliton initial condition for equation (1.1): Ψ(x, t, ϵ)|t=t 0 =

2√2ϵη exp{−i2cϵx − 4(c2 − η2 )t0 ϵ2 } cosh(2ηϵx + 8cηϵ2 t0 + s)

(1.26)

According of our analytical results this initial condition leads to one soliton solution as the leading-order term of the asymptotic solution: u 1 (x1 , t2 ) =

2√2η exp{−i2cx1 − 4i(c2 − η2 )t2 } . cosh(2ηx1 + 8cηt2 + s)

This soliton propagates up to the resonance curve t = 0. To annihilate this soliton on the resonance curve one may choose the specific form of the amplitude of the perturbation such that the left-hand side of relation (1.4) equals zero: 0 = u 1 (x1 , 0) + (1 − i)√πf(x1 ) . Hence, f(x1 ) =

−(1 + i) −(1 + i) √2η exp{−i2cx1 } u 1 (x1 , 0) = . cosh(2ηx1 + s) √π 2√π

0.4 0.2

200

0 100

–0.2 –0.4 –200

0 –100 –100

0 100 200 –200 Fig. 1.2: Annihilation of soliton

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation

|

11

To illustrate this by numerical simulations we choose ϵ = 0.1, η = 1, s = 0, c = 0, and t0 = −200. Then the original equation (1.1) has the form i∂ t Ψ + ∂2x Ψ + |Ψ|2 Ψ = 0.01f exp{i0.005t2 } . The initial condition is Ψ|t=−200 =

0.2√2 , cosh(0.2x)

and the amplitude of the perturbation is f =

√2 −(1 + i) . √π cosh(0.2x)

The numerical simulations of the annihilation process for the soliton of NLSE are presented in Fig. 1.2. This justifies the formulas obtained above by the matching method.

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation This section is devoted to the problem of the generation of solitary packets of waves by a small external driving force. We study this problem for the nonlinear Klein–Gordon equation. The wave packets appear due to the passing of the external driving force through resonance. After the resonance the envelope function of the wave packet is determined by the nonlinear Schrödinger equation. In the most important cases the envelope function is a sequence of solitary waves, which are called solitons. The wave packets with the solitons as the envelope function are propagated without deformation. The parameters of the solitons are obviously defined by the value of the driving force on a resonance curve.

1.2.1 Main Result Let us consider the Klein–Gordon equation with cubic nonlinearity ∂2t U − ∂2x U + U + γU 3 = ϵ2 f(ϵx) exp {i

S(ϵ 2 t, ϵ2 x) } + c.c. , ϵ2

0 1 the leading-order term of the asymptotics satisfies the Cauchy problem 2i∂ t 2 Ψ1,0,Φ + ∂2ξξ Ψ1,0,Φ + γ|Ψ1,0,Φ |2 Ψ1,0,Φ = 0 , Ψ1,0,Φ |t 2=1 = f(ξ)(1 + i)√ π . The solution of this Cauchy problem contains solitary waves if the initial data are sufficiently large [303].

14 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

Remark on WKB asymptotics. Theorem 1.2.1 describes the special asymptotic solution of equation (1.27). It is defined by the driving force. One can add any solution of the WKB type [116] of the order ϵ2 to this solution. This leads to an asymptotic solution of the form N

Ũ = U(t, x, ϵ) + ∑ ϵ n U n (t, x, ϵ) . n≥2

The coefficients U n (t, x, ϵ) of the asymptotics are calculated by standard methods of WKB theory. This additional term leads to ponderous formulas and does not change the leading-order term of the asymptotics constructed in theorem 1.2.1.

1.2.2 Pre-Resonance Expansion In this section, the formal asymptotic solution is constructed in the domain before resonance. The asymptotic expansion has the form of WKB type. The leading-order term of the asymptotics has the order of the driving force and oscillates with its frequency. The constructed asymptotics is valid as −l ≫ ϵ. The result of this section is formulated below. Let us construct the formal asymptotic solution for equation (1.27) in the form N

U = ∑ ϵ n U n (t, x, ϵ) ,

(1.28)

n≥2

where U n = ∑ U n,k (t2 , x2 , ϵx) exp {ik k∈Ω n

S(t2 , x2 ) } . ϵ2

Set Ω n for the higher-order term with the number n is described by the formula {{±1} , Ωn = { {±1, ±3, . . . , ±(2l + 3)} , {

n ≤5; l = [(n − 6)/4] , n ≥ 6 .

The functions U n,k and U n,−k are complex conjugated. Let us substitute (1.28) in equation (1.27) and collect the terms of the same order of ϵ. As a result, we obtain a recurrent sequence of algebraic equations, f U2,1 = − , l ∂ x f∂ x S U3,1 = 2i 1 2 2 , l 2if[∂ t 2 S∂ t 2 l − ∂ x2 S∂ x2 l] − 4(∂ x2 S)2 ∂2x1 f U4,1 = l3 2i∂ t 2 f∂ t 2 S + ∂2x1 f + i∂2t 2 Sf − , l2

(1.29) (1.30)

(1.31)

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation

| 15

where l = (∂ t 2 S)2 − (∂ x2 S)2 − 1 . The curve where the phase function S satisfies the eikonal equation is called the resonance curve, (1.32) l[S] = (∂ t 2 S)2 − (∂ x2 S)2 − 1 = 0 . The amplitude U n,1 has a singularity on this curve. The formula for the n-th order term has the form U n,k =

1 2 [∂ U n−4,k + 2ik∂ t 2 S∂ t 2 U n−2,k + ikS t 2 t 2 U n−2,k − 2ik∂ x2 S∂ x2 U n−2,k l t2 t2 − ik∂2x2 SU n−2,k − ∂2x1 x1 U n−2,k − 2∂2x1 x2 U n−3,k − ∂2x2 x2 U n−4,k − 2ik∂ x2 S∂ x1 U n−1,k + γ

∑

U n1 ,k1 U n2 ,k2 U n3 ,k3 ] .

(1.33)

n1 +n 2 +n 3 =n , k1 +k2 +k3 =k k∈Ω n

Lemma 1.2.2. The coefficient U n,k has the following behavior U n,k = O(l−(n−k) ), k > 0 ,

l → −0 .

(1.34)

Proof. Let us prove this lemma as k = 1. The validity of formula (1.34) for n = 2, 3, 4 is directly obtained from (1.29), (1.30), and (1.31). Suppose now that this formula is valid for the term U n−1,1 . The increase of the order of the singularity as l → 0 takes place due to differentiation with respect to x2 , t2 and the nonlinear term in formula (1.33). Differentiation of the terms in formula (1.33) leads to formula (1.34). Let us consider U n,k for k > 1. The validity of formula (1.34) for small values of n and k is obtained by direct calculations. Consider the n-th order term. It contains the terms with different values of k. The higher-order terms with k = 3 have the greatest order of singularity, U n,3 = O(l−(n−3) ) , l → −0 . (1.35) This takes place because the right-hand side of (1.33) contains the term U n−4,±1 U2,±1 U2,±1 . The calculation of the order of singularity for this term leads to formula (1.35). The terms of the type of U n3 ,±3 U n1 ,∓k1 U n2 ,±k1 , n1 + n2 + n3 = n lead to weak singularities, for example for k 1 = 3 we obtain the order of singularity equal to n − 9. Consider the nonlinear terms U n1 ,k1 U n2 ,k2 U n3 ,k3 from the right-hand side of (1.33) when the number of the higher-order term is equal to n. Calculate the order of singularity for this term using the (n − 1)-th step of induction. Indexes of the amplitudes are connected by formulas n1 + n2 + n3 = n ,

k1 + k2 + k3 = k .

Using (1.34) for n1 , n2 , n3 < n we obtain that the order of the singularity for this term is equal to (n − k).

16 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

The right-hand side of (1.33) contains derivatives of previous terms with respect to x2 , t2 . It leads to an increase of the order of the singularity but nevertheless, we obtain the leading order from nonlinear terms. The lemma is proved. The domain of validity as l → −0 for the formal asymptotic solution in the form (1.28) is defined by ϵ n+1 U n+1 ≪1. ϵn Un It yields −l ≫ ϵ . Using these lemmas we obtain the asymptotic representation for (1.28) as l → −0 N

U = ∑ ϵ n ∑ exp{ikS/ϵ2 } n=2

k∈Ω n

∞

∑

j

Un k lj ,

l → −0 .

(1.36)

j=−(n−k)

The following theorem is proved. Theorem 1.2.3. In the domain −l ≫ ϵ the formal asymptotic solution of equation (1.27) modulo O(ϵ N+1 l1−N ) has the form (1.28). The coefficients of the asymptotics U n+1,k are defined from algebraic equations (1.29), (1.30), (1.31), and (1.33).

1.2.3 Internal Asymptotics This part of the work contains the asymptotic construction of the solution for equation (1.27) in the neighborhood of the curve l = 0. The domain of validity of this asymptotics intersects with domain of validity of expansion (1.28). These expansions are matched. Theorem 1.2.4. In the domain |l| ≪ 1 the formal asymptotic solution for equation (1.27) modulo O(ϵ N+1 λ N lnN−1 |λ|) has the form N

U = ∑ ϵ n W n (t1 , x1 , t2 , x2 , ϵ) ,

(1.37)

n≥1

where W n = ∑ W n,k (x2 , t2 , x1 , t1 ) exp {ik k∈Ω n

S(t2 , x2 ) } , ϵ2

(1.38)

The function W n,k , k = 1 is the solution of the problem for equation (1.43) with zero condition as λ → −∞ and solutions of algebraic equations (1.45) in the case k ≠ 1. The functions W n,k and W n,−k are complex conjugated. There is an essential difference between asymptotics (1.37) and external pre-resonance asymptotics (1.28). First, the leading-order term in (1.37) has an order ϵ in contrast the leading-order term in (1.28) has an order ϵ2 . Second, the coefficients of asymptotics (1.37) depend on fast variables x1 = x2 /ϵ and t1 = t2 /ϵ.

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation

| 17

The proof of theorem 1.2.4 consists of three steps. First, we derive equations for coefficients of the asymptotics. Second, we solve the problems for asymptotic coefficients. Third, we determine the domain of the validity for expansion (1.37). 1.2.3.1 The Equations for Coefficients Let us construct the internal asymptotic expansion in the domain |l| ≪ 1. Denote 1 λ(x1 , t1 , ϵ) = l(ϵx1 , ϵt1 ) . (1.39) ϵ In the domain 1 ≪ λ ≪ ϵ−1 both asymptotics (1.28) and (1.37) are valid. This fact allows us to obtain the asymptotic representation for coefficients of the internal asymptotics. Substitute l = ϵλ in formula (1.36) and expand the obtained expression with respect to powers of small parameter ϵ. This yields W n,k =

∞

∑

λ−j U n+1 k (x2 , t2 , x1 ) , j

k ∈ Ωn ,

λ → −∞ .

(1.40)

j=(n−k+1)

Let us obtain the differential equations for the coefficients of asymptotics (1.37). Substitute (1.37) and (1.38) in equation (1.27) and collect the terms with equal powers of small parameters and exponents. This yields the equations for coefficients W n,k . In particularly, the terms of the order ϵ2 give us the equations for the leading-order terms of the asympotics 2i∂ t 2 S∂ t 1 W1,1 − 2i∂ x2 S∂ x1 W1,1 − λW1,1 = f ,

(1.41)

and a complex conjugated equation for W1,−1 . The relation of the order ϵ3 in equation (1.27) gives four equations. Two of them are complex conjugate differential equations for W2,1 and W2,−1 : 2i∂ t 2 S∂ t 1 W2,1 − 2i∂ x2 S∂ x1 W2,1 − λW2,1 = ∂2x1 W1,1 − ∂2t 1 W1,1 − i[∂2t 2 S − ∂2x2 S]W1,1 − 2i∂ t 2 S∂ t 2 W1,1 + 2i∂ x2 S∂ x2 W1,1 − 3γ|W1,1 |2 W1,1 ,

(1.42)

two other equations are algebraic. These last equations allow us to determine the functions W3,3 and W 3,−3 γ W3,3 = (W1,1 )3 . 8 The higher-order terms are calculated in the same way. In particularly, the terms in the case of the lower index being equal to 1 are determined by differential equations, 2i∂ t 2 S∂ t 1 W n,1 − 2i∂ x2 S∂ x1 W n,1 − λW n,1 = F n,1 .

(1.43)

The right-hand side of equation (1.43) has the form F n,1 = −2i∂ t 2 S∂ t 2 W n−1,1 + 2i∂ x2 S∂ x2 W n−1,1 + (∂ t 2 S)2 W n−1,1 − (∂ x2 S)2 W n−1,1 − ∂2t 1 W n−1,1 + ∂2x1 W n−1,1 − ∂ t 2 ∂ t 1 W n−2,1 + ∂ x2 ∂ x1 W n−2,1 − ∂2t 2 W n−3,1 + ∂2x2 W n−3,1 − γ

∑ n1 +n 2 +n 3 =n+1 , k1 +k2 +k3 =1 k j ∈Ω nj , j=1,2,3

W n1 ,k1 W n2 ,k2 W n3 ,k3 .

(1.44)

18 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

The higher-order terms in the case of the lower index not being equal to 1 are determined by algebraic equations γ (−2i∂ t 2 S∂ t 2 W n−2,k + 2i∂ x2 S∂ x2 W n−2,k + (∂ t 2 S)2 W n−2,k k2 − 1 − (∂ x2 S)2 W n−2,k − ∂2t 1 W n−2,k + ∂2x1 W n−2,k − ∂ t 2 ∂ t 1 W n−3,k + ∂ x2 ∂ x1 W n−3,k −

W n,k =

− ∂2t 2 W n−4,k + ∂2x2 W n−4,k −

∑

W n1 ,k1 W n2 ,k2 W n3 ,k3 ) .

(1.45)

n1 +n 2 +n 3 =n+1 , k1 +k2 +k3 =k k j ∈Ω nj , j=1,2,3

1.2.3.2 The Solvability of Equations for Higher-Order Terms In this section, we present the explicit form for the higher-order term W n,1 and investigate the asymptotic behavior as λ → ±∞. 1.2.3.3 Characteristic Variables The function W n,1 satisfies equation (1.43). The solution is constructed by the characteristic method. Define the characteristic variables σ, ξ . We choose a point (x01 , t01 ) such that ∂ x2 l|(x01 ,t 01) ≠ 0 as the origin and denote by σ the variable along the characteristic family for equation (1.43). We suppose σ = 0 on the curve λ = 0. The variable ξ mensurates the distance along the curve λ = 0 from the point (x01 , t01 ). This point (x01 , t01 ) corresponds to ξ = 0. Let the positive direction for parameter ξ coincide with positive direction of x2 in the neighborhood of (x01 , t01 ). The characteristic equations for (1.43) have the form dt1 = 2∂ t 2 S(ϵx1 , ϵt1 ) , dσ

dx1 = −2∂ x2 S(ϵx1 , ϵt1 ) . dσ

(1.46)

The initial conditions for the equations are x1 |σ=0 = x01 ,

t1 |σ=0 = t01 .

(1.47)

Lemma 1.2.5. The Cauchy problem for characteristics has a solutions as |σ| < c1 ϵ−1 , c1 = const. > 0. Proof. The Cauchy problem (1.46) and (1.47) is equivalent to the system of the integral equations σ

t1 =

t01

+ 2 ∫ ∂ t 2 S(ϵx1 , ϵt1 )dζ , 0

σ

x1 =

x01

− 2 ∫ ∂ x2 S(ϵx1 , ϵt1 )dζ .

(1.48)

0

Substitute t2̃ = (t1 − t01 )ϵ, x̃ 2 = (x1 − x01 )ϵ. This yields ϵσ

t2̃ = 2 ∫ ∂ t 2 S(x̃ 2 − ϵx01 , t2̃ − ϵt01 )dζ , 0

ϵσ

x̃ 2 = −2 ∫ ∂ x2 S(x̃ 2 − ϵx01 , t2̃ − ϵt01 )dζ . 0

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation

| 19

The integrands are smooth and bounded functions on the plane x2 , t2 . There exists the constant c1 = const. > 0 such that the integral operator is a contraction operator as ϵ|σ| < c1 . Lemma 1.2.5 is proved. Assumption. We assume that the change of variables (x1 , t1 ) → (σ, ξ) is unique in the neighborhood of the curve λ = 0. This assumption means that the characteristics for equation (1.43) do not touch the curve λ = 0. This means ∂ x2 l∂ x2 S − ∂ t 2 l∂ t 2 S ≠ 0 . It is convenient to use the following asymptotic formulas for the change of variables (x1 , t1 ) → (σ, ξ). Lemma 1.2.6. In the domain |σ| ≪ ϵ−1 the asymptotics as ϵ → 0 of the solutions for the Cauchy problem (1.46) and (1.47) have the form N

x1 (σ, ξ, ϵ) − x01 (ξ) = −2σ∂ x2 S + 2 ∑ ϵ n σ n+1 g n (ϵx1 , ϵt1 ) + O(ϵ N+1 σ N+2 ) ,

(1.49)

n=1 N

t1 (σ, ξ, ϵ) − t01 (ξ) = 2σ∂ t 2 S + 2 ∑ ϵ n σ n+1 h n (ϵx1 , ϵt1 ) + O(ϵ N+1 σ N+2 ) ,

(1.50)

n=1

where gn = −

󵄨󵄨 dn 󵄨󵄨 (∂ S) , x 󵄨󵄨 2 dσ n 󵄨 σ=0

hn =

󵄨󵄨 dn 󵄨󵄨 (∂ S) . t 󵄨󵄨 2 dσ n 󵄨 σ=0

The lemma is proved by integration by parts of equations (1.48). The next proposition gives us the asymptotic formula that connects variables σ and λ as σ, λ → ±∞. Lemma 1.2.7. Let be σ ≪ ϵ−1 , then: λ = ϕ(ξ)σ + O(ϵσ 2 ) ,

ϕ(ξ) =

dλ 󵄨󵄨󵄨 󵄨󵄨 dσ 󵄨󵄨σ=0

σ→∞.

Proof. From formula (1.39) we obtain the representation in the form ∞

λ = ∑ λ j (x1 , t1 , ϵ)σ j ϵ j−1 , j=1

where λ j (x1 , t1 , ϵ) =

1 dj λ(x1 , t1 , ϵ)|σ=0 . j! dσ j

This yields λ= Let

dλ 󵄨󵄨 d2 λ 󵄨󵄨σ=0 σ + O(ϵσ 2 2 ) . dσ dσ

󵄨󵄨 2 󵄨󵄨 󵄨󵄨 d l 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 dσ 2 󵄨󵄨󵄨 ≥ const , ξ ∈ R . 󵄨 󵄨

20 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

The function dλ/dσ is not equal to zero, dλ 1 = ( − ∂ x2 λ∂ x2 S + ∂ t 2 λ∂ t 2 S) ≠ 0 . dσ 2 Let us suppose dλ/dσ > 0. This yields λ = ϕ(ξ)σ + O(ϵσ 2 ) ,

ϕ(ξ) =

dλ 󵄨󵄨󵄨 󵄨 . dσ 󵄨󵄨󵄨σ=0

The lemma is proved. 1.2.3.4 Solutions of Equations for Higher-Order Terms The higher-order terms W n,±1 are solutions of equation (1.43) with the given asymptotic behavior λ → −∞. Equation (1.43) can be written in characteristic variables as i

d W n,1 − λW n,1 = F n,1 . dσ

(1.51)

Lemma 1.2.8. The solution of equation (1.43) with the asymptotic behavior (1.40) as λ → −∞ has the form σ

σ

ζ

W n,1 = exp (−i ∫ dζλ(x1 , t1 , ϵ)) ∫ dζF n,1 (x1 , t1 , ϵ) exp (−i ∫ dχλ(x1 , t1 , ϵ)) . −∞

0

0

(1.52) Proof. By direct substitution we see that expression (1.52) is the solution of (1.51). The asymptotics of this solution as λ → −∞ can be obtained by integration by parts and substitution d = 2∂ t 2 S∂ t 1 − 2∂ x2 S∂ x1 . dσ This yields ∞

W n,1 = ∑ ( j=0

2∂ t 2 S∂ t 1 − 2∂ x2 S∂ x1 j F n,1 )[ ], iλ iλ

λ → −∞ .

(1.53)

From formula (1.44) we obtain that formulas (1.53) and (1.40) are equivalent. The lemma is proved. 1.2.3.5 Asymptotics as λ → ∞ and Domain of Validity of the Internal Asymptotics The domain of validity of the internal expansion is determined by the asymptotics of higher-order terms. In this section, we show that the n-th order term of the asymptotic solution increases as λ n−1 when λ → ∞. This increase of higher-order terms allows us to determine the domain of validity for internal asymptotics (1.37) as λ → ∞.

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation

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1.2.3.6 Asymptotics of Higher-Order Terms This section contains two propositions concerning asymptotic behavior as λ → ∞ for higher-order terms in (1.37). The first lemma describes the asymptotic behavior of higher-order terms as λ → ∞ and the second one contains a result about the asymptotics of the phase function. Lemma 1.2.9. The asymptotic behavior of W n,1 as 1 ≪ λ ≪ ϵ−1 has the form n−1 j−1 j

W n,1 = ∑ ∑ (λ ln

k

σ (j,k) |λ|W n 1 (ξ)) exp (−i ∫ dζλ(x1 , t1 , ϵ))

j=0 k=0

0 ∞

+∑( j=0

2∂ t 2 S∂ t 1 − 2∂ x2 S∂ x1 j F n,1 )[ ]. iλ iλ

(1.54)

Proof. Let us calculate the asymptotics of the leading-order term ζ

σ

σ

W1,1 = exp (−i ∫ dζλ(x1 , t1 , ϵ)) ∫ dζf(x1 ) exp (i ∫ dχλ(x1 , t1 , ϵ)) 0

−∞

0

σ

∞

ζ

= exp (−i ∫ dζλ(x1 , t1 , ϵ)) ∫ dζf(x1 ) exp (i ∫ dχλ(x1 , t1 , ϵ)) −∞

0

0

∞

σ

ζ

− exp (−i ∫ dζλ(x1 , t1 , ϵ)) ∫ dζf(x1 ) exp (i ∫ dχλ(x1 , t1 , ϵ)) . −σ

0

0

Further, by integration by parts of the last term we obtain formula (1.54) as n = 1, where ∞

σ

(0,0)

W1,1 (ξ) = ∫ dσf(x1 ) exp (i ∫ dχλ(x1 , t1 , ϵ)) , −∞

0

F1,1 = f(x1 ) . To calculate the asymptotics of W2,1 in formula (1.52) we use the asymptotics with respect to σ of the leading-order term. Integral (1.52) contains the term with linear increase with respect to σ when n = 2. We eliminate this growing part from the integral explicitly. The residuary integral converges as σ → ∞. It can be calculated in the same manner as it was calculated for W1,1 . This yields formula (1.54) as n = 2, where (1,0)

(0,0)

W2,1 (ξ) = W1,1 (ξ) . The same direct calculations are realized for the n-th order term. The lemma is proved.

22 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

To complete the proof of theorem 1.2.4 we need to obtain the domain of validity of asymptotics (1.37). The formal series (1.37) is asymptotic when ϵ n+1 W n+1 ≪1, ϵn W n

ϵ→0.

Lemma 1.2.9 gives λ ≪ ϵ−1 . After substitution λ = ϵl we obtain l ≪ 1. Theorem 1.2.4 is proved. 1.2.3.7 Asymptotics of the Phase Function as λ → ∞ To obtain the asymptotics as λ → ∞ we need to derive the asymptotics of the phase function in formula (1.54). Lemma 1.2.10. As λ → ∞: σ

∫ dξλ = 0

S 1 + (∂ x2 S(x1 − x01 ) + ∂ t 2 S(t1 − t01 )) + O(ϵλ3 ) . ϵ ϵ2

(1.55)

Proof. Substitute the asymptotics of λ from Lemma 1.2.9. Calculate the asymptotics of the integral in formula (1.55) σ

σ

∫ dζλ(x1 , t1 , ϵ) = ∫ 0

dζ [(−∂ x2 l∂ x2 S + ∂ t 2 l∂ t 2 S)ζ + O(ϵζ 2 )] = 2

0

σ2 + O(ϵσ 3 ) . 4 The asymptotics of the phase function S(x2 , t2 ) in the neighborhood of the curve l = 0 is represented by a segment of the Taylor series. This yields (−∂ x2 l∂ x2 S + ∂ t 2 l∂ t 2 S)

S 1 = (∂ x2 S(x1 − x01 ) + ∂ t 2 S(t1 − t01 )) ϵ ϵ2 1 + (S x2 x2 (x1 − x01 )2 + 2S x2 t 2 (x1 − x01 )(t1 − t01 ) + S t 2 t 2 (t1 − t01 )2 ) 2 + O(ϵ(|t1 − t01 | + |t1 − t01 |)3 ) . Instead of (x1 − x01 ) and (t1 − t01 ) substitute their asymptotic behavior with respect to ϵ from Lemma 1.2.6. This substitution and the result of Lemma 1.2.7 complete the proof of Lemma 1.2.10. The asymptotics as λ → −∞ contains fast oscillating terms with phase functions kS, k ∈ Z. The leading-order term of the asymptotics as λ → ∞ contains the oscillations with an additional phase function. We obtain this result from Lemma 1.2.9. Denote this new phase function by Φ(x2 , t2 )/ϵ2 . The asymptotics of this function is obtained in Lemma 1.2.10. The nonlinearity and additional phase function lead to a more complicated structure of the phase set for higher-order terms of the asymptotics as λ → ∞.

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation

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23

Lemma 1.2.11. The phase set K n for the n-th order term of the asymptotics as λ → ∞ is determined by formula K1 = ±Φ ,

K2 = ±Φ, ±S ,

K n = ∪j1 +j2 +j3 =n χ j1 + χ j2 + χ j3 ,

χ jk ∈ K jk .

The proof of this lemma follows from the asymptotic formula for n-th order term. Representation (1.37), formula (1.54), and Lemma 1.2.9 allow us to construct the asymptotics as λ → ∞ of the internal expansion in an explicit form N

n−1 n−2

n=1

j=0 k=0

(j,k)

U = ∑ ϵ n ( ∑ ∑ λ j ln k |λ|W n 1

(ξ))

1 × exp [ − i( (∂ x2 S(x1 − x01 ) + ∂ t 2 S(t1 − t01 )) + O(ϵλ3 ))] ϵ ∞

N

+ ∑ ϵn ( ∑ ( n=1

j=0

2∂ t 2 S∂ t 1 − 2∂ x2 S∂ x1 j F n,1 S(t2 , x2 ) } )[ ]) exp {i iλ iλ ϵ2

N

+ ∑ ϵn ( n=2

∑

W n,k exp {ik

k∈Ω,k=±1 ̸

S(t2 , x2 ) } ) + c.c. ϵ2

(1.56)

This representation and formula (1.45) complete the proof of the lemma.

1.2.4 Post-Resonance Expansion This section contains the construction of the asymptotics of the solution for (1.27) after passage through resonance. The constructed solution has the order ϵ and oscillates. The envelope function of these oscillations satisfies the nonlinear Schrödinger equation. This section consists of two parts. The first part contains the construction of the formal asymptotic solution. We obtain the equations for higher-order terms of the asymptotics. The asymptotic behavior for higher-order terms as l → −0 follows from Section 1.2.3.7. In the second part of this section, we determine the domain of validity for this external asymptotics near resonance curve l(x2 , t2 ) = 0. The matching method gives us the initial conditions for the higher-order terms of the asymptotics. The main result of this section is formulated in the following theorem. Theorem 1.2.12. In the domain ϵ ≪ l ≪ ϵ−1 the formal asymptotic solution of equation (1.27) modulo O(ϵ N+1 [1 + l1−N ln N−2 l]) has the form N

n−2

1

k=0

U(x, t, ϵ) = ∑ ϵ n ∑ ln k (ϵ)( ∑ exp{±iΦ(x2 , t2 )/ϵ2 }Ψ n,k,±Φ (x1 , t1 , t2 )+ ±Φ

∑ exp{iχ(x2 , t2 )/ϵ2 }Ψ n,k,χ (x1 , t1 , t2 )) .

(1.57)

χ∈K 󸀠n,k

Here the function Φ(x2 , t2 ) satisfies the eikonal equation (∂ t 2 Φ)2 − (∂ x2 Φ)2 − 1 = 0

(1.58)

24 | 1 The Solitary Waves Generation due to Passage through the Local Resonance and initial condition on the curve l = 0: Φ|l=0 = S|l=0 ,

∂ t 2 Φ l=0 = ∂ t 2 S|l=0 .

The leading-order term of the asymptotics is a solution of the Cauchy problem for the nonlinear Schrödinger equation 2i∂ t 2 Φ∂ t 2 Ψ1,0,Φ + ∂2ξ Ψ1,0,Φ + i[∂2t 2 Φ − ∂2x2 Φ]Ψ1,0,Φ + γ|Ψ1,0,Φ |2 Ψ1,0,Φ = 0 , ∞

σ

Ψ1,0,Φ |l=0 = ∫ dσf(x1 ) exp (i ∫ dχλ(x1 , t1 , ϵ)) , −∞

0

where ξ is defined from dx1 = ∂ t2 Φ , dξ

dt1 = ∂ x2 Φ . dξ

The coefficients Ψ n,k,±Φ are determined from the Cauchy problems for the linearized Schrödinger equation (1.64). The coefficients Ψ n,k,χ and χ ∈ K 󸀠n,k are determined from the algebraic equations (1.65). The set K 󸀠n,k = K n,k \{±Φ}. Theorem 1.2.1 follows from theorems 1.2.3, 1.2.4, and 1.2.12. 1.2.4.1 Structure of the Second External Asymptotics Let us construct the formal asymptotic solution from theorem 1.2.12. Substitute (1.57) in the original equation and collect the terms of the same order with respect to ϵ. This yields N + 1 equations and residuals of the order ϵ N+1 . After collecting the terms with the same phase functions we obtain the recurrent system of equations for coefficients of (1.57). Let us consider equations under exp(iΦ/ϵ2 ). The terms of the order ϵ1 give us equation (1.58) for the phase function of eigenoscillations. The initial data is determined by matching conditions and represented by the value of the driven phase S on the resonance curve l = 0, Φ|l=0 = S|l=0 ,

∂ t 2 Φ|l=0 = ∂ t 2 S|l=0 .

The terms of the order ϵ2 2i (∂ t 2 Φ∂ t 1 Ψ1,0,Φ − ∂ x2 Φ∂ x1 Ψ1,0,Φ ) = 0 give us the homogeneous transport equation ∂ t 2 Φ∂ t 1 Ψ1,0,Φ − ∂ x2 Φ∂ x1 Ψ1,0,Φ = 0 .

(1.59)

This equation allows us to determine the dependence of the leading-order term on the characteristic variable ζ . Equation (1.59) along the characteristics dx1 = −∂ x2 Φ , dζ

dt1 = ∂ t2 Φ dζ

(1.60)

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation

|

25

can be written in the form of ordinary differential equation dΨ1,0,Φ =0. dζ

(1.61)

This yields Ψ1,0,Φ depending on ξ , where ξ is defined by dx1 = ∂ t2 Φ , dξ

dt1 = ∂ x2 Φ . dξ

The terms of the order ϵ3 , which oscillates as exp(iΦ/ϵ2 ), are 2i (∂ t 2 Φ∂ t 1 Ψ2,0,Φ − ∂ x2 Φ∂ x1 Ψ2,0,Φ ) + 2i∂ t 2 Φ∂ t 2 Ψ1,0,Φ + [(∂ t 1 ξ)2 − (∂ x1 ξ)2 ]∂2ξξ Ψ1,0,Φ + i[∂2t 2 Φ − ∂2x2 Φ]Ψ1,0,Φ + γ|Ψ1,0,Φ |2 Ψ1,0,Φ = 0 . It is convenient to write this equation in the form of the ordinary differential equation in terms of characteristic variables dΨ2,0,Φ = −2i∂ t 2 Φ∂ t 2 Ψ1,0,Φ − [(∂ t 1 ξ)2 − (∂ x1 ξ)2 ]∂2ξξ Ψ1,0,Φ dζ ± 2 − i[∂2t 2 Φ − ∂2x2 Φ]Ψ1,0,Φ − γ|Ψ1,0 | Ψ1,0,Φ .

(1.62)

Equation (1.61) shows that the right-hand side of equation (1.62) does not depend on ζ . To avoid secularities in the asymptotics we demand the right-hand side of equation is equal to zero. It allows us to determine the dependence of the leading-order term on the slow variable t2 , 2i∂ t 2 Φ∂ t 2 Ψ1,0,Φ + [(∂ t 1 ξ)2 − (∂ x1 ξ)2 ]∂2ξξ Ψ1,0,Φ + i[∂2t 2 Φ − ∂2x2 Φ]Ψ1,0,Φ + γ|Ψ1,0,Φ |2 Ψ1,0,Φ = 0 .

(1.63)

The equations for the higher-order terms are obtained by the same manner 2i (∂ t 2 Φ∂ t 1 Ψ n+1,k,Φ − ∂ x2 Φ∂ x1 Ψ n+1,k,Φ ) = 2i∂ t 2 Φ∂ t 2 Ψ n,k,Φ − ∂2ξξ Ψ n,k,Φ − i[∂2t 2 Φ − ∂2x2 Φ]Ψ n,k,Φ + ∂ t 1 ξ∂2ξt 2 Ψ n−1,k,Φ − γ

∑

Ψ k1 ,k2 ,α Ψ l 1 ,l 2 ,β Ψ m1 ,m2 ,δ ,

k1 ,k2 ,l 1 ,l 2 , m1 ,m2 ,α,β,δ

where k 1 + l1 + m1 = n + 2, k 2 + l2 + m2 = k, α + β + δ = Φ, α ∈ K k1 ,k2 , β ∈ K l 1 ,l 2 , δ ∈ K m1 ,m2 . To construct the uniform asymptotic expansion with respect to ζ we obtain the linearized Schrödinger equation for higher-order terms 2i∂ t 2 Φ∂ t 2 Ψ n,k,Φ + ∂2ξξ Ψ n,k,Φ + i[∂2t 2 Φ − ∂2x2 Φ]Ψ n,k,Φ = −∂ t 1 ξ∂2ξt 2 Ψ n−1,k,Φ − γ

∑

Ψ k1 ,k2 ,α Ψ l 1 ,l 2 ,β Ψ m1 ,m2 ,δ ,

k1 ,k2 ,l 1 ,l 2 , m1 ,m2 ,α,β,δ

(1.64)

26 | 1 The Solitary Waves Generation due to Passage through the Local Resonance where k 1 + l1 + m1 = n + 2, k 2 + l2 + m2 = k, α + β + δ = Φ, α ∈ K k1 ,k2 , β ∈ K l 1 ,l 2 , δ ∈ K m1 ,m2 . The amplitudes Ψ n,χ as χ ≠ ±Φ are determined by the algebraic equations [−(χ t 2 )2 + (χ x2 )2 + 1] Ψ n,k,χ = F n,k,χ ,

χ ≠ ±Φ .

(1.65)

Here, the right-hand side of the equation depends on previous terms and their derivatives F n,k,χ = −2iχ t 2 ∂ t 1 Ψ n−1,k,χ + 2iχ x2 ∂ x1 Ψ n−1,k,χ − 2iχ t 2 ∂ t 2 Ψ n−2,k,χ − i [χ t 2 t 2 − χ x2 x2 ] Ψ n−2,k,χ − ∂2t 1 t 2 Ψ n−3,k,χ − ∂2t 2 t 2 Ψ n−4,k,χ −γ

∑

Ψ k1 ,k2 ,α Ψ l 1 ,l 2 ,β Ψ m1 ,m2 ,δ ,

k1 ,k2 ,l 1 ,l 2 , m1 ,m2 ,α,β,δ

where k 1 + l1 + m1 = n − 4, k 2 + l2 + m2 = k, α + β + δ = χ, α ∈ K k1 ,k2 , β ∈ K l 1 ,l 2 , δ ∈ K m1 ,m2 . These equations are similar to the equations for amplitudes from the pre-resonance section. The result obtained is formulated below. Lemma 1.2.13. The coefficients of the formal asymptotic solution (1.57) satisfy the recurrent system of equations (1.58), (1.63), (1.64), and (1.65). The right-hand side of equation (1.64) has a singularity as l → 0. The singularity appears due to Ψ n,k,χ as χ ≠ ±Φ. The analysis of the right-hand side of the equation allows us to calculate the order of singularity as l → 0. It is equal to O(l−(n−1) ). Below we prove the solvability of equation (1.64) with the given asymptotics as l → 0. Lemma 1.2.14. The asymptotics as l → 0 of the solution of equation (1.64) has the form 1

Ψ n,k,Φ (x1 , t1 , t2 ) =

∑

−(j−1)

j,m

∑ Ψ n,k,Φ (x1 , t1 ) l j (ln l)m + O(1) ,

l→0.

(1.66)

j=−(n−2) m=0

Proof. Determine the order of the singularity of the right-hand side of the equation as l → 0. First consider equation (1.64) for n = 3, k = 0. The solution of this equation gives us the coefficient Ψ3,0,Φ . The nonlinearity contains the term |Ψ2,0,S |2 Ψ1,0,Φ . The function Ψ2,0,S has singularity of the order l−1 as l → 0. It determines the order of singularity for the right-hand side l−2 . We construct the asymptotics of Ψ3,0,χ in the form −1,0 0,1 1,1 ̂ 3,0,Φ , Ψ 3,0,Φ = Ψ 3,0,Φ l−1 + Ψ 3,0,Φ ln(l) + Ψ 3,0,Φ l ln(l) + Ψ (1.67) Substitute (1.67) in equation for n = 3. It leads to recurrent system of equations for (j,k) coefficients Ψ3,0,Φ (−1,0)

−2i∂ t 2 Φ∂ t 2 lΨ3,0,Φ = −Ψ1,0,Φ |Ψ2,0,S |2 l2 , (0,1)

(−1,0)

(1,1)

(0,1)

2i∂ t 2 Φ∂ t 2 lΨ3,0,Φ = L[Ψ3,0,Φ ] , 2i∂ t 2 Φ∂ t 2 lΨ3,0,Φ = L[Ψ3,0,Φ ] .

1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein–Gordon Equation |

27

Here, we denote the linear operator by L[Ψ] = 2i∂ t 2 Φ∂ t 2 Ψ + ∂2ξ Ψ + i[∂2t 2 Φ − ∂2x2 Φ]Ψ + γ(2|Ψ1,0,Φ |2 Ψ + (Ψ1,0,Φ )2 Ψ ∗ ) . ̂3,0,Φ of the asymptotics satisfies the nonhomogeneous linear The regular part Ψ Schrödinger equation. The right-hand side of the equation is smooth (1,1)

(1,1)

̂3,0,Φ ] = −l ln |l|L[Ψ L[Ψ 3,0,Φ ] − 2i∂ t 2 Φ∂ t 2 lΨ 3,0,Φ . The initial condition for the regular part of the asymptotics is determined below by matching with the internal asymptotic expansion. The structure of the terms Ψ n,k,±Φ for n > 3 has a similar form. The right-hand side of equation (1.64) depends on junior terms. These singularities can be eliminated −(n−2) −j+1

F n,k,Φ = ∑

j=0

(j,m) ̂ n,k,Φ . ∑ l j lnm |l|f n,k,Φ + F

m=0

(j,m)

The coefficients f n,k,Φ do not contain singularities as l → 0. These coefficients are easy calculated. The direct substitution of (1.66) in equation and collecting the terms with the same order of l complete the proof of Lemma 1.2.14. 1.2.4.2 The Domain of Validity of the Second External Asymptotics and Matching Procedure The domain of validity of the second external asymptotics is determined by ϵV n+1 ≪1. Vn Formulas (1.57) and (1.66) give the condition l≪ϵ. The domain |l| ≪ 1 of validity of the internal asymptotics and domain of validity of the second external asymptotics are intersected. This fact allows us to complete the construction of the second external asymptotics by the matching method [113]. The structure of singular parts of the internal asymptotics as λ → +∞ and external asymptotics as l → 0 are equivalent. The coefficients coincide due to our constructions. The matching of regular parts of these asymptotics takes place due to (0,0)

Ψ n,0,Φ |l=0 = W n

(ξ) .

(0,0)

The function W n (ξ) is determined in Lemma 1.2.9. In particular, the initial condition for the leading-order term has the form ∞

σ

Ψ1,0,Φ |l=0 = ∫ dσf(x1 ) exp (i ∫ dχλ(x1 , t1 , ϵ)) . −∞

0

28 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

The soliton theory for nonlinear Schrödinger equation leads us to the fact that the function Ψ1,0,Φ contains the solitary waves when f(x1 ) is sufficiently large. Theorem 1.2.12 is proved.

1.3 The Perturbed KDV Equation and Passage through the Resonance Originally the KDV equation was derived as fundamental equation governing propagation of waves in shallow water [145]. The same equation appeared in the theory of nonlinear ion acoustic waves in cold plasmas [290, 300]. Later it became clear that the KDV equation is a basic model for the description of internal waves in stratified fluids, propagating in waveguides which exist naturally in the ocean, or can be created in a laboratory [185],[276],[102]. In all mentioned above applications the KDV equation arises as an asymptotic limit. Taking account of additional physical factors it is easy to obtain some kinds of small perturbations for the KDV equation u t + αu x + 6uu x + u xxx = G .

(1.68)

The explicit asymptotic derivations for the forced KDV equation have been carried for water waves in a channel [10] – [295], internal waves in a shallow fluid [103],[44], inertial waves in a narrow tube [104], etc. In this section we study the such kind of the derived KDV equation U t + 6UU x + U xxx = ϵ2 f(ϵx) cos (

S(ϵ 2 x, ϵ2 t) ) , ϵ2

(1.69)

the right-hand side of (1.69) is the representation of the external force. In the frame of the shallow water model this external force corresponds to the atmospheric disturbance [214]. We suppose that external force is small and has an order of ϵ 2 , where ϵ is a small parameter. The function f(z) is smooth and rapidly vanishes as |z| → ∞. The phase function S(x, t) of the perturbation and all its derivatives are bounded. Our goal is to investigate the influence of the perturbation on a small forced solution. This special solution corresponds to the forced oscillations and has the order of ϵ 2 . The distinct feature of the considered problem is slowly varied frequency of the perturbation. There is a moment when the frequency of the perturbation is close to the eigenfrequency of the non perturbed KDV equation and the force becomes resonant. It leads to the exchange of the solution behavior and change the order of our special asymptotic solution. Here we present the detailed asymptotic description of the process.

1.3 The Perturbed KDV Equation and Passage through the Resonance | 29

1.3.1 Forced Oscillations 1.3.1.1 Formal Constructions In this section we construct the asymptotic solution related to forced oscillations. The solution has the order of ϵ2 and oscillates with the frequency of the perturbation. It is more convenient for us to use the complex form representation of the solution for (1.69). The solution has the WKB-form [116] ∞

U(x, t, ϵ) = ∑ ϵ n U n (x, t, ϵ) + c.c.,

(1.70)

n=2

where U n (x, t, ϵ) = ∑ u n,ψ (x2 , t2 , x1 ) exp{iψ(x2 , t2 )/ϵ2 } .

(1.71)

ψ∈Ω n

Variables x m = ϵ m x, t m = ϵ m t, m = 1, 2 are slow. The set Ω n of the phase functions depends on the number of the correction term. For example, Ω2 = {±S}, Ω3 = {±S}, Ω4 = {±S, ±2S} and the general formula for the phase set is n Ω n = {±S, ±2S, . . . , ± [ ] S} . 2 The validity of this formula can be obtained by direct calculations and taking into account the nonlinearity of (1.69). Substitute expansion (1.70) into (1.69) and collect the terms of the equal order with respect to small parameter ϵ and exponents. It leads to equations for amplitudes u n,ψ , ψ ∈ Ω n . All equations are algebraic. The terms of the order of ϵ2 give the equation for the leading-order term u 2,S =

−f . [∂ t 2 S − (∂ x2 S)3 ]

(1.72)

The terms of the order of ϵ3 give equation for u n,S u 3,S =

−3(∂ x2 S)2 ∂ x1 u 2,S . [∂ t 2 S − (∂ x2 S)3 ]

(1.73)

Relations of the order ϵ4 give two equations for the amplitudes. On this step the solution contains the related to the forced oscillations original phase S and the generated by nonlinearity phase 2S. Amplitudes are determined as follows u 4,S = − (−∂ t 2 u 2,S − 3i∂ x2 S∂2x1 u 2,S + 3(∂ x2 S)2 ∂ x2 u 2,S + 3(∂ x2 S)2 ∂ x1 u 3,S +2∂2x2 S∂ x2 Su 2,S ) / (∂ t 2 S − (∂ x2 S)3 ) . u 4,2S =

−6∂ x2 Su 22,S [∂ t 2 S − 4(∂ x2 S)3 ]

.

(1.74) (1.75)

30 | 1 The Solitary Waves Generation due to Passage through the Local Resonance The general formula for the n-th correction term u n,ψ , ψ ∈ Ω n has a form u n,ψ = −[ − ∂ t 2 u n−2 − 3i∂ x2 S∂2x1 u n−2 + 3(∂ x2 S)2 ∂ x2 u n−2 + 3(∂ x2 )2 ∂ x1 u n−1 + 2∂2x2 S∂ x2 Su n−2 + ∂3x1 u n−3 + ∂3x1 x1 x2 u n−4 + ∂3x1 x2 x2 u n−5 + ∂3x2 x2 x2 u n−6 − 3i∂2x2 S∂ x1 u n−3 − 3i∂2x2 S∂ x2 u n−4 − 6i∂ x2 S∂2x1 x2 u n−3 − 3i∂ x2 S∂2x2 u n−4 − i∂3x2 Su n−6 + 6 +6

∑

∑

∂ x1 u k1 ,ψ 1 u k2 ,ψ 2

k1 +k2 =n−1, ψ 1 +ψ 2 =ψ

∂ x2 u k1 ,ψ 1 u k2 ,ψ 2

k1 +k2 =n−2, ψ 1 +ψ 2 =ψ

+ 6i

∑

−1

∂ x2 ψ1 u k1 ,ψ 1 u k2 ,ψ 2 ] × [∂ t 2 ψ − (∂ x2 ψ)3 ]

.

(1.76)

k1 +k2 =n, ψ 1 +ψ 2 =ψ

It is easy to see that the power of the expression [∂ t 2 ψ − (∂ x2 ψ)3 ] in the denominator in (1.76) increases along with the number n. The representation (1.70) of the solution for (1.69) becomes invalid when [∂ t 2 ψ − (∂ x2 ψ)3 ] equals zero. We consider the resonance with the external force. The equation l[S] ≡ ∂ t 2 S − (∂ x2 S)3 = 0 determines the curve l in the space of the independent variables (x2 , t2 ). The frequencies of the eigenmode and external force equal on this curve. The curve of the such type is usually called the resonant one. We have to note that the expression [∂ t 2 ψ−(∂ x2 ψ)3 ] for ψ ≠ S does not vanish on it. 1.3.1.2 The Domain of Validity for the Forced Oscillations Solution (1.70) In this subsection we describe the asymptotic behavior of coefficients of (1.70) as [∂ t 2 S − (∂ x2 S)3 ] → 0 and determine the domain of validity for this asymptotic solution. One can see that the solution (1.70) loses the asymptotic property in the neighborhood of the curve l = 0. The order of the singularity as [∂ t 2 S − (∂ x2 S)3 ] → 0 grows along with the number n of the correction term. The terms under the phase function S of the forced oscillations have the maximal order of the singularity as l → −0. Lemma 1.3.1. The terms u n,S have the following asymptotic behavior u n,S = O(l−(n−1) ) ,

as

l → −0 .

Proof of this Lemma can be obtained by induction on the number n of the correction term. The validity of this formula for small values of n follows from formulas (1.72), (1.73), (1.74), (1.75).

1.3 The Perturbed KDV Equation and Passage through the Resonance | 31

The asymptotics of the u n,S as l → −0 can be observed from relation (1.76). We suppose that the order of singularity for u n−1,S be l−(n−2) and then analyze the terms from right-hand side of (1.76). The increase of the order of the singularities relates to the differentiation with respect to x2 and t2 . The accordance between the order of the differentiation with respect to x2 and t2 and the number of the correction terms gives the validity of the Lemma. This lemma allows to obtain the asymptotics for u n,S as l → −0 u n,S =

∞

u kn l−k ,

∑

l → −0 .

(1.77)

k=−(n−1)

The order of the singularity as l → −0 for the amplitudes u n,ψ , ψ ≠ S is smaller than for u n,S . This order can be calculated as well as was shown in Lemma 1.3.1. Asymptotic solution (1.70) is valid when ϵ max |U n+1 | = o ( max |U n |) , (x 2 ,t 2 )

(x 2 ,t 2 )

ϵ→0.

The Lemma 1.3.1 yields −l ≫ ϵ .

1.3.2 Inside the Resonance In this section we construct the formal asymptotic solution of equation (1.69) in the neighborhood of the resonance curve l = 0. We show the solution bifurcation under resonant forcing. 1.3.2.1 Asymptotic Solution. Derivation of Equations In previous section we showed that the solution of the form (1.70) is not valid when |l| ∼ ϵ. It make us to change the scale of the independent variables and construct the solution of another form. Here we use the scaled variable λ = ϵl . We construct the solution of the form ∞

U(x, t, ϵ) = ∑ ϵ n W n (x1 , x2 , t1 , t2 , ϵ) ,

(1.78)

n=1

where W n = ∑ w n,ψ (x1 , x2 , t1 , t2 ) exp{iψ(x2 , t2 )/ϵ2 } , ψ∈Ψ n

the set Ψ n depends on the number n of the correction term of (1.78). For example, Ψ1 = {±S}, Ψ2 = {±S, ±2S} Substitute (1.78) into original equation and collect the terms of the same order with respect to ϵ and exponents.

32 | 1 The Solitary Waves Generation due to Passage through the Local Resonance Terms of the order ϵ2 give us four equations. Two of them are equations for the complex conjugate functions [∂ t 2 S − 4(∂ x2 S)3 ] w2,2S = −3∂ x2 S(w1,S )2 , ∂ t 1 w1,S − 3(∂ x2 S)2 ∂ x1 w1,S + iλw1,S = f .

(1.79)

It is easy to obtain the equations for the higher-order corrections terms of the asymptotic solution. Equations for amplitudes w n,S are differential. ∂ t 1 w n,S − 3(∂ x2 S)2 ∂ x1 w n,S + λw n,S = f n,S .

(1.80)

Here f n,S = −∂ t 2 w n−1,S − 6

∑

w k1 ,ψ 1 [∂ x1 w k2 −1,ψ 2 + ∂ x2 w k2 −2,ψ 2 + i∂ x2 ψ2 w k2 ,ψ 2 ]

k1 +k2 =n+1, ψ 1 +ψ 2 =S

− ∂3x1 w n−2,S − 3∂3x1 x1 x2 w n−3,S − 3∂3x1 x2 x2 w n−4,S − 6i∂ x2 S∂2x1 x2 w n−2,S − 3i∂2x2 S∂ x1 w n−2,S − 3i∂ x2 S∂2x1 w n−2,S + 3(∂ x2 S)2 ∂ x1 w n,S − ∂3x2 w n−5,S − 3i∂2x2 S∂ x2 w n−3,S − 3i∂ x2 S∂2x2 w n−3,S + 3(∂ x2 S)2 ∂ x2 w n−1,S − i∂3x2 Sw n−3,S + 3∂ x2 S∂2x2 Sw n−1,S .

(1.81)

Amplitudes w n,ψ , ψ ≠ S are determined from algebraic equations i [∂ t 2 ψ − (∂ x2 ψ)3 ] w n,ψ = f n,ψ where f n,ψ has the same structure as was shown in (1.81) 1.3.2.2 Equations for the Coefficients. Solutions In this subsection we obtain the solutions for the equations for coefficients of the asymptotics. First, we consider equation (1.80) for the coefficients of (1.78) that correspond to the phase function S of the perturbation. This equation can be solved by method of characteristics. The characteristics for (1.80) are determined by dt1 =1, dσ

dx1 = −3(∂ x2 S)2 . dσ

(1.82)

and initial conditions t1 |σ=0 = t01 ,

x1 |σ=0 = x01 .

(1.83)

Here we use the notation ξ and σ for characteristic variables. We choose the point (x01 , t01 ) such that ∂ x2 l|(x01 ,t 01) ≠ 0 as an origin. The variable σ is the variable along the characteristics. We suppose that σ = 0 on the curve λ = 0. The variable ξ mensurates the distance along the curve λ = 0 and the value ξ = 0 relates to the point (x01 , t01 ). Lemma 1.3.2. The Cauchy problem for characteristics has a solution when |σ| < c1 ϵ−1 , c1 = const > 0.

1.3 The Perturbed KDV Equation and Passage through the Resonance | 33

Proof. The Cauchy problem is equivalent to the following system σ

t1 =

t01

+σ,

x1 =

x01

2

− 3 ∫ dζ (∂ x2 S(ϵx1 , ϵt1 )) .

(1.84)

0

After substitution t2̃ = (t1 − t01 )ϵ, x̃ 2 = (x1 − x01 )ϵ we obtain ϵσ

t2̃ = ϵσ ,

x̃ 2 = −3 ∫ dζ (∂ x2 S(x̃ 2 + ϵx01 , ϵζ + ϵt01 ))

2

0

The integrand is a smooth and bounded function with respect to x2 , t2 . There is a constant c1 > 0 such that the integral operator is a construction operator when ϵ|σ| < c1 . Lemma is proved. To describe the internal solution it is convenient to use the characteristic variables σ, ξ . Lemma 1.3.3. In the domain |σ| ≪ ϵ−1 the asymptotics as ϵ → 0 of the solutions for Cauchy problem (1.82), (1.83) have the form N

2

x1 (σ, ξ, ϵ) − x01 (ξ) = −3σ (∂ x2 S) + 3 ∑ ϵ n σ n+1 g n (ϵx1 , ϵt1 ) + O (ϵ N+1 σ N+2 ) , (1.85) 1

t1 (σ, ξ, ϵ) − t01 (ξ) = σ , where gn = −

(1.86)

dn 2 (∂ x2 S) |σ=0 dσ n

Proof. This Lemma can be obtained by integration by parts of (1.84). The differentiation along (1.82) has a form d = ∂ t 1 − 3(∂ x2 S)2 ∂ x1 dσ In order to obtain the asymptotics of the solution (1.78) across the resonance layer we should connect the transversal variable λ and variable σ along the characteristics. Lemma 1.3.4. Let be σ ≪ ϵ−1 , then λ = φ(ξ)σ + O(ϵσ 2 ) ,

φ(ξ) =

dλ |σ=0 dσ

Proof. For variable λ we can obtain the representation ∞

λ = ∑ λ j (x1 , t1 , ϵ)σ j ϵ j−1 , j=1

σ→∞.

34 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

here λ j (x1 , t1 , ϵ) = It yields λ=

󵄨󵄨 1 dj 󵄨󵄨 󵄨󵄨 λ(x , t , ϵ) 1 1 󵄨󵄨 j! dσ j 󵄨 σ=0

d2 λ dλ 󵄨󵄨󵄨󵄨 σ + O (ϵσ 2 2 ) 󵄨󵄨 dσ 󵄨󵄨σ=0 dσ

Let be

󵄨󵄨 2 󵄨󵄨 󵄨󵄨 d λ 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 dσ 2 󵄨󵄨󵄨 ≥ const , 󵄨 󵄨 The function dλ/dσ is not equal zero

ξ∈R

dλ = ∂ t 1 λ − 3(∂ x2 S)2 ∂ x1 λ ≠ 0 dσ It yields λ = φ(ξ)σ + O(ϵσ 2 ) ,

φ(ξ) =

dλ 󵄨󵄨󵄨󵄨 󵄨 dσ 󵄨󵄨󵄨σ=0

Lemma is proved. Along these characteristics the differential equation (1.80) for the amplitudes w n,S that correspond to the phase function S becomes ordinary differential equation dw n,S + iλw n,S = f n,S . dσ

(1.87)

Solution of this equation with the asymptotics (1.77) has a form ζ σ σ { } { } { } w n,S = exp {−i ∫ dζλ(x1 , t1 , ϵ)} ∫ dζf n,S (x1 , t1 , ϵ) exp { i ∫ dμλ(x1 , t1 , ϵ)} . { } { 0 } −∞ { 0 } (1.88) The asymptotics of this solution when λ → −∞ can be obtained by integration by parts j ∞ ∂ t − 3(∂ x2 S)2 ∂ x1 f n,S w n,S = ∑ ( 1 ) [ ] , λ → −∞ . (1.89) iλ iλ j=0

It is easy to see that this internal representation is matched with the part (1.77) of the external solution that corresponds forced oscillations. 1.3.2.3 Going Out of the Resonance. Asymptotics as λ → +∞ In subsection we describe the going out of the resonance and obtain the asymptotic behavior of internal resonant expansion (1.78) as λ → +∞. Here we give two Lemmas about the asymptotic behavior of the coefficients of σ (1.78) and the phase function ∫0 dξλ(x1 , t1 , ϵ) as λ → +∞. The first Lemma allows us to determine the domain of validity for (1.78) and the second one shows the oscillations of the leading-order term after the passage through the resonant layer.

1.3 The Perturbed KDV Equation and Passage through the Resonance | 35

Lemma 1.3.5. The asymptotic behavior of w n,S when 1 ≪ λ ≪ ϵ−1 has a form w1,S ∼ O(1) and for n ≥ 2 2n−3 j−1

w n,S = ∑ ∑ (λ ln j

+∑( j=0

σ { } (j,k) |λ|w n,S (ξ)) exp {−i ∫ dζλ(x1 , t1 , ϵ)}

{

j=0 k=0 ∞

k

∂ t 1 − 3(∂ x2 iλ

S)2 ∂

j x1

) [

0

f n,S ] . iλ

} (1.90)

Proof. The asymptotics of the coefficients w n,S is calculated recurrently. First, we obtain the asymptotics for the leading-order term w1,S w1,S

ζ σ σ { } { } { } = exp {−i ∫ dζλ(x1 , t1 , ϵ)} ∫ dζf1,S (x1 , t1 , ϵ) exp {i ∫ dμλ(x1 , t1 , ϵ)} { } { 0 } −∞ { 0 } ζ σ ∞ { } { } { } = exp {−i ∫ dζλ(x1 , t1 , ϵ)} ∫ dζf1,S (x1 , t1 , ϵ) exp {i ∫ dμλ(x1 , t1 , ϵ)} { } { 0 } −∞ { 0 } ζ σ ∞ { } { { } } − exp {−i ∫ dζλ(x1 , t1 , ϵ)} ∫ dζf1,S (x1 , t1 , ϵ) exp { i ∫ dμλ(x1 , t1 , ϵ)} { } { 0 }σ { 0 }

By integration by parts we obtain formula (1.90) when n = 1, where ∞

ζ { } { } = ∫ dζf(x1 ) exp {i ∫ dμλ(x1 , t1 , ϵ)} { } −∞ { 0 } To calculate the asymptotics of the higher-order term w n,S we use the asymptotics with respect to σ that was obtained on previous steps. In this case the representation (1.88) contains the increasing terms with respect to σ. To obtain the asymptotics for w n,S we eliminate this growing part from the integral explicitly. The residual integral converges as σ → ∞. It can be calculated in the same manner as it was shown for the leading-order term of the asymptotics. To estimate the order of λ as λ → +∞ it is necessary to note that the highest order of the integrand consist in nonlinear term related to UU x . It is easy to trace the procedure of the increasing of singularity for integrand. The first integration when n = 1 yields w1,S ∼ const , λ → +∞ . (0,0) w1,S

The next integration gives w2,S ∼ λ ,

λ → +∞ .

After next substitution and n − 2 integration of the eliminated part of the asymptotics we obtain the representation (1.90). Lemma is proved.

36 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

This Lemma allows us to determine the domain of validity for (1.78). Representation (1.78) is asymptotic when ϵ

max |W n+1 | = o ( max |W n |) ,

x 2 ,t 2 ,x 1 ,t 1

x 2 ,t 2 ,x 1 ,t 1

ϵ→0.

(1.91)

Lemma 1.3.5 and condition (1.91) give λ ≪ ϵ −1/2 . In terms of the external variable l we obtain l ≪ ϵ1/2 . 1.3.2.4 Asymptotics of the Phase Function as λ → +∞ To determine the oscillations after the passage through the resonance we should investigate the asymptotic behavior of the phase function as λ → ∞. Lemma 1.3.6. As λ → ∞ σ

∫ dζλ = 0

S 1 + [∂ x2 S(x1 − x01 ) + ∂ t 2 S(t1 − t01 )] + O (ϵλ3 ) . 2 ϵ ϵ

(1.92)

Proof. Let us use the asymptotics from Lemma 1.3.4 for calculation (1.92). σ

σ

∫ dζλ = ∫ dζ [(∂ t 2 l − 3(∂ x2 S)2 ∂ x2 l) ζ + O (ϵζ 2 )] 0

0

= (∂ t 2 l − 3(∂ x2 S)2 ∂ x2 l)

σ2 + O (ϵσ 3 ) . 2

The asymptotics of the phase function S(x2 , t2 )/ϵ2 in the neighborhood of the curve l = 0 is represented by a segment of the Taylor series: S 1 + (∂ x2 S(x1 − x01 ) + ∂ t 2 S(t1 − t01 )) 2 ϵ ϵ 1 2 + (∂ x2 S(x1 − x01 )2 + 2∂2x2 t 2 S(x1 − x01 )(t1 − t01 ) + ∂2t 2 S(t1 − t01 )2 ) 2 + O (ϵ(|x1 − x01 | + |t1 − t01 |)3 ) . Substitute the asymptotic representation of |x1 − x01 | and |t1 − t01 | with respect to ϵ from Lemma 1.3.3 and combine it with the result of Lemma 1.3.4. It complete the proof of the Lemma. The asymptotic representation of (1.78) as λ → +∞ contains the different modes. The leading-order term contains the oscillations with an additional phase. This result follows from Lemma 1.3.5. We denote this new phase by φ(x2 , t2 )/ϵ2 . The asymptotics of this phase function is obtained in Lemma 1.3.6. The additional mode and nonlinearity of the original equation leads to complicated structure of the phase set for the higher-order terms of the solution.

1.3 The Perturbed KDV Equation and Passage through the Resonance | 37

1.3.3 Post-Resonance Expansion In this section we construct the asymptotics of the solution after the passage through the resonance. This constructed solution has the order ϵ and oscillates. The equations for the coefficients write out by multiscale method. Initial conditions for these equations are obtained by matching method. 1.3.3.1 Formal Constructions. Equations for Coefficients The domain of validity of the internal solution shows that it is necessary to introduce the new external variable l3 = ϵ−1/2 l. We construct the solution of the form ∞

U(x, t, ϵ) = ∑ ϵ

1+n 2

Vn ,

(1.93)

n=1

where Vn =

(n−1)/2

∑ ln k (ϵ) [∑ exp{±iφ/ϵ2 }Ψ n,k,±φ + ∑ exp{iχ/ϵ2 }Ψ n,k,χ ] .

k=0

±φ

χ

The amplitudes Ψ n,k,χ = Ψ n,k,χ (t2 , x3 , t3 ) depend on the new scaled variables x3 = ϵ3/2 x and t3 = ϵ3/2 t. Substitute (1.93) in (1.69) and collect the terms of the same order with respect to ϵ. Terms of the order ϵ give us the problem for φ ∂ t 2 φ − (∂ t 2 φ)3 = 0

(1.94)

φ|l=0 = S|l=0 .

(1.95)

and Terms of the order ϵ 3/2 does not give any equations. The relation of this order with respect to ϵ is satisfied due to (1.94). Terms of the order of ϵ2 allow to obtain the relations for Ψ3,0,χ for χ = S, 2φ. The equation for Ψ3,0,S is coincided with (1.72) from Section 1.3.1. The coefficient Ψ3,0,2φ is determined by Ψ1,0,φ 2 (1.96) ) . Ψ3,0,2φ = ( ∂ x2 φ Note that equation (1.96) is not solvable yet. The coefficient Ψ1,0,φ is not determined. The terms of the order ϵ5/2 give the homogeneous transport equation for Ψ1,0,±φ ∂ t 3 Ψ1,0,±φ − 3(∂ x2 φ)2 ∂ x3 Ψ1,0,±φ = 0 .

(1.97)

This equation allows to determine the dependence of the leading-order term with respect to character variable ζ . Equation (1.97) along the characteristics dt3 =1, dζ

dx3 = −3(∂ x2 φ)2 dζ

(1.98)

38 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

looks as follow

dΨ1,0,±φ =0. dζ

(1.99)

Equation (1.99) shows that Ψ1,0,±φ does not depend on ζ . Here we denote the variable along the characteristics by ζ and the transversal variable by ξ . The variable ξ is defined by dt3 dx3 = 3(∂ x2 φ)2 , =1 (1.100) dξ dξ Terms of the order of ϵ3 give us the non homogeneous equation for Ψ2,0,φ ∗ ∂ t 3 Ψ2,0,φ − 3(∂ x2 φ)2 ∂ x3 Ψ2,0,φ = −∂ t 2 Ψ1,0,φ + 3∂ x2 φ∂2x2 φΨ1,0,φ + i∂ x2 φΨ1,0,φ Ψ3,0,2φ . (1.101) As we showed above the leading-order term Ψ1,0,φ does not depend on variable ζ . To construct the bounded solution of (1.101) we demand that the right-hand side of this equation equals zero. Using formula (1.96) we obtain

∂ t 2 Ψ1,0,φ − 3∂ x2 φ∂2x2 φΨ1,0,φ +

i |Ψ1,0,φ |2 Ψ1,0,φ = 0 . ∂ x2 φ

(1.102)

It allows us to determine the leading-order term of the asymptotics. The initial condition for this equation is obtained by the matching method, ∞

Ψ1,0,φ |l=0

ζ { } { } = ∫ dζf(x1 ) exp {i ∫ dμλ(x1 , t1 , ϵ)} . { } −∞ { 0 }

(1.103)

The higher-order correction terms are determined by the same manner. The amplitude Ψ n,k,φ satisfies to the ordinary differential equations and Ψ n,k,χ satisfies to algebraic equations. The differential equations for Ψ n,k,φ have the form ∂ t 2 Ψ n,k,φ − 3∂ x2 φ∂2x2 φΨ n,k,φ = Fn,k,φ ,

(1.104)

where Fn,k,φ = −∂3x2 φΨ n−4,k,φ − 3i∂2x2 φ∂ x3 Ψ n−3,k,φ − 3i∂ x2 φ∂2x3 Ψ n−2,k,φ − ∂3x3 Ψ n−5,k,φ −6

∑

n1 +n 2 =n, k1 +k2 =k, α1 +α2 =φ

i∂ x2 α 1 Ψ n1 ,k1 ,α1 Ψ n2 ,k2 ,α2 − 6

∑ n1 +n 2 =n−1, k1 +k2 =k, α1 +α2 =φ

∂ x3 Ψ n1 ,k1 ,α1 Ψ n2 ,k2 ,α2 . (1.105)

Algebraic equations look like the equation from Section 1.3.1 i [∂ t 2 χ − (∂ x2 χ)3 ] Ψ n,k,χ = Fn,k,χ ,

(1.106)

where the right-hand side Fn,k,χ depends on previous correction terms. The structure of Fn,k,chi is similar to (1.105) with the change of the index from φ to χ. When χ = S the corresponding amplitudes Ψ n,k,χ have singularities on the resonant curve l = 0. The analysis of these singularities is realized as is shown below in Section 1.3.1.

1.3 The Perturbed KDV Equation and Passage through the Resonance |

39

1.3.3.2 The Domain of Validity for Post-Resonant Expansion (1.93) As it was mentioned above, the amplitudes of (1.93) have singularities on the resonant curve l = 0. These singularities appear primarily due to the structure of the right-hand side of equations (1.104) and (1.106). Here, we determine the asymptotic behavior of the coefficient of (1.93) and the domain of the validity of this representation of the solution. First of all we calculate the order of the singularity for amplitudes Ψ n,k,χ for χ = ±S. It is easy to see that the first appearance of this forced amplitudes take place when n = 3, k = 0. They are determined from algebraic equations l[S]Ψ 3,0,S = f . This yields Ψ3,0,S = O(l−1 ) ,

l → +0 .

The direct calculations give Ψ n,0,S = O(l−[(n+1)/4] ) ,

l → +0 .

(1.107)

It is necessary to note that the singularities for Ψ n,0,S have the strongest order and these terms determine the domain of the validity for (1.93). The asymptotic behavior for Ψ n,0,χ , χ ≠ S, and Ψ n,k,S , k > 0 obtained either from differential equations (1.104) or algebraic equations (1.106) shows that the order of singularity is weaker for them. This gives the domain of the validity for (1.93) l≫ϵ.

1.3.4 Numerical Simulations In this section, we present a result of the numerical simulations for (1.69) and explain the behavior of the solution. We get S(x2 , t2 ) = t22 , f(y) = exp{−y2 }. In this case, the resonance takes place in the neighborhood of the line t = 0. Calculations were made under ϵ = 0.1 and t0 = −50. Here we present the third picture of the solution and his profile as x = 0. We see that the solution has the order of ϵ2 and oscillates with the frequency of the perturbation in the pre-resonant domain. The solution grows up to the value O(ϵ) in the neighborhood of the curve t = 0. After the passage through the resonance domain it also oscillates. These oscillations are realized with the frequency related to the new generated phase φ. In this case, the leading-order term Ψ1,0,φ is the constant and oscillations take place around the constant value defined by formula (1.103).

40 | 1 The Solitary Waves Generation due to Passage through the Local Resonance

0.05 0.04 0.03 0.02 0.01 0 -0.01

-10

-50 -5

0 t

0

50

x

5

100 150

10

Fig. 1.3: Third picture of the numerical solution. Calculations were made under ϵ = 0.1 and t 0 = −50. 0.05

0.04

0.03

0.02

0.01

0 –50

0 –0.01

50

100

150

t

Fig. 1.4: Profile of numerical solution as x = 0. Calculations were made under ϵ = 0.1 and t 0 = −50.

1.4 Auto-Resonant Soliton and Perturbation with Decaying Amplitude In this section, we study the asymptotic solution for the equation ν − i∂ τ Ψ + ∂2ζ Ψ + (|Ψ|2 − τ) Ψ + F − i Ψ = 0 . 2

(1.108)

1.4 Auto-Resonant Soliton and Perturbation with Decaying Amplitude | 41

This equation determines an envelope curve for a single mode of the plane wave with small amplitude in weakly nonlinear media with an external oscillating perturbation. The frequency of perturbation changes in the linear way. This equation can be considered as a generalization of the nonlinear Schrödinger equation [129, 268, 304] and an analogue of a well-known principal resonance equation [66, 117]. It is a nonlinear generalization of a local resonance equation [206]. From a physics perspective it is important to study asymptotic properties of solutions for this equation for large values τ and find growing solutions with respect to τ and localized solutions with respect to ζ , if they exist. The existence of growing solutions leads to the generation of solitary waves with a finite amplitude by autoresonance. Earlier asymptotic formulae for an envelope in the form of a soliton of the nonlinear Schrödinger equation by a local resonance were obtained [94, 95]. In the auto-resonance case there exist some numerical results about a generation of cnoidal waves that are similar to solitons [83, 244]. Results about control of parameters for solitons of finite and small amplitudes by an external perturbation have been published in [89, 118, 182]. We use the sine-Gordon equation to derive equation (1.108). The sine-Gordon equation is a popular model in studies of solitary waves in nonlinear media with dispersion. Taking into account external perturbation gives different perturbations of the sine-Gordon equation. In this section, we study the generation of a solitary wave (breather) in a perturbed equation: u tt − u xx + sin(u) = ε3 f cos(S) − ε3 μu t ,

0 0. The original equation for the function ϕ and variables σ, z becomes the form i∂ σ ϕ + ∂ zz ϕ + |ϕ|2 ϕ +

F iσ ν ϕ =0. e +i ϕ+i 3/4 4σ σ 2√σ

(1.126)

In this case, the dissipative term with the parameter ν and the external force F give the equation for soliton parameters 2νη0 1 π|F| cos(α) ∼0. + 3/4 πκ0 √σ σ cosh ( 2η ) 0 This relation gives that parameter η0 decays as σ −1/4 when σ → ∞ and F = const. It follows that the constructed asymptotic expansion is Ψ = O(1) as τ → ∞.

1.4 Auto-Resonant Soliton and Perturbation with Decaying Amplitude | 47

The growth of the asymptotic solution continues when the amplitude of perturbation grows as F = O(νσ 1/4 ) or as O(√τ) for the original variable. The necessity for the growth of the perturbation amplitude for the auto-resonance phenomenon in other cases was known earlier [119, 245].

2 Regular Perturbation of Ill-Posed Problems This chapter is based on the following simple observation. Consider an operator equation Tu = f with a bounded operator T : H → H̃ in Hilbert spaces. If there is a u ∈ H satisfying Tu = f , then f is orthogonal to the null space of the adjoint operator T ∗ in H.̃ On the other hand, for f ∈ (ker T ∗ )⊥ the equation Tu = f is obviously equivalent to T ∗ Tu = T ∗ f . The latter need not have any solution, however, the slightly perturbed equation T ∗ Tu + εu = 𝑣 is uniquely solvable for any 𝑣 ∈ H, provided that ε > 0. Note that the solution of the equation can be effectively constructed, for the operator T ∗ T + ε is positive definite. We thus get a family u ε = (T ∗ T + ε)−1 T ∗ f in H, whose limit is a good candidate for the solution of Tu = f that is orthogonal to the null space of T. Indeed, if 𝑣 ∈ H satisfies T𝑣 = 0, then by Lemma 12.1.25 of [269] we get (u ε , 𝑣)H = (f, T(T ∗ T + ε)−1 𝑣)H̃ = (f, (TT ∗ + ε)−1 T𝑣)H̃ =0, as desired. If f = Tu for some u ∈ H, then u ε = u − ε(T ∗ T + ε)−1 u is obviously bounded in H. Conversely, if the norm ‖u ε ‖H is bounded uniformly in ε ≪ 1, then u ε converges for ε ↘ 0 to the only solution u ∈ H of Tu = f that is orthogonal to ker T. In this way, we derive a solvability condition and an approximate solution to the equation Tu = f in H. We refer the reader to Section 12.1.5 of [269] for an extremal property of u ε . When applying the approach in the study of the Cauchy problem for solutions of an elliptic equation Au = f , one needs to complete it by refined analysis. By the above, the calculus of Cauchy problems that are ill-posed by their very nature can be elaborated in the framework of the calculus of operators T ∗ T + εI depending on a parameter ε > 0. In order to avoid sophisticated adjoint operators one uses L2 -scalar products, which necessarily leads to unbounded closed operators with dense domains. Hence, it requires much more effort to make use of the construction described above. The operator T is given the domain consisting of those functions u in D which are square integrable along with Au and whose Cauchy data with respect to A vanish on a closed set S ⊂ ∂D. Then the domain of the adjoint operator T ∗ is proved to consist of square integrable functions g on D, such that the Cauchy data of g with respect to A∗ vanish in the complement of S. It follows that the natural domain of the Laplacian T ∗ T is a subspace of square integrable functions u on D, such that the Cauchy data of u with respect to A vanish on S and the Cauchy data of Tu with respect to A∗ vanish https://doi.org/10.1515/9783110534979-002

2.1 Mixed Problems with a Parameter |

49

on ∂D \ S. This gives rise to a mixed boundary value problem for the elliptic operator A∗ A in D similar to the classical Zaremba problem [305]. Our study demonstrates rather strikingly that the calculus of Cauchy problems for solutions of elliptic equations just amounts to the calculus of mixed boundary value problems for elliptic equations with a parameter, cf. [253]. While this observation seems to be of purely mathematical interest, the explicit solutions we construct by the classical Fourier method may be of practical importance in applications.

2.1 Mixed Problems with a Parameter 2.1.1 Preliminaries Let X be a C∞ manifold of dimension n with a smooth boundary ∂X. We tacitly assume that it is embedded into a smooth closed manifold X̃ of the same dimension. For any smooth C -vector bundles E and F over X, we write Diff m (X; E, F) for the space of all linear partial differential operators of order ≤ m between sections of E and F. We denote by E∗ the conjugate bundle of E. Any Hermitian metric (., .)x on E gives rise to a sesquilinear bundle isomorphism ∗E : E → E∗ by the equality ⟨∗E 𝑣, u⟩x = (u, 𝑣)x for all sections u and 𝑣 of E. Pick a volume form dx on X, thus identifying dual and conjugate bundles. For A ∈ Diff m (X; E, F), denote by A󸀠 ∈ Diff m (X; F ∗ , E∗ ) the transposed operator and by 󸀠 A∗ ∈ Diff m (X; F, E) the formal adjoint operator. We have A∗ = ∗−1 E A ∗ F , cf. [269, 4.1.4] and elsewhere. For an open set O ⊂ X, we write L2 (O, E) for the Hilbert space of all measurable sections of E over O with a finite norm (u, u)L2 (O,E) = ∫O (u, u)x dx. We also denote by H s (O, E) the Sobolev space of distribution sections of E over O, whose weak derivatives up to order s belong to L2 (O, E). Given any open set O in X ∘ , the interior of X, we let SA (O) stand for the space of weak solutions to the equation Au = 0 in O. Obviously, the subspace of H s (O, E) consisting of all weak solutions to Au = 0 is closed. Write σ m (A) for the principal homogeneous symbol of the operator A, σ m (A) living on the (real) cotangent bundle T ∗ X of X. From now on we assume that σ m (A) is injective away from the zero section of T ∗ X. Hence, it follows that the Laplacian A∗ A is an elliptic differential operator of order 2m on X. If the dimensions of E and F are equal, then A is elliptic, too. Otherwise we will call it an overdetermined elliptic operator. We can assume without restriction of generality that A is included into a compatibility complex of differential operators A i ∈ Diff m i (X; E i , E i+1 ) over X, where i = 0, 1, . . . , N and A0 = A. This complex is elliptic in a natural way, see for instance, [269]). If A is elliptic, then the compatibility complex is trivial, i.e., A i = 0 for all i > 0.

50 | 2 Regular Perturbation of Ill-Posed Problems Let D be a relatively compact domain in X ∘ with a smooth boundary ∂D. For u ∈ we always regard Au as a distribution section of F over D. A large class of operators A possess the following property, which is usually referred to as the unique continuation property. (U)s Given any domain D ⊂ X ∘ , if u ∈ SA (D) vanishes on a nonempty open subset of D, then u ≡ 0 in all of D. L2 (D, E)

This property implies, in particular, the existence of a left fundamental solution for A in the interior of X. Consider the Hermitian form D(u, 𝑣) = (u, 𝑣)L2 (D,E) + (Au, A𝑣)L2 (D,F) on the space C∞ (D, E) of all smooth sections of E over the closure of D. The functional D(u) = √D(u, u) is usually called the graph norm related to the unbounded operator A : L2 (D, E) → L2 (D, F). Write DA for the completion of C∞ (D, E) with respect to D(⋅). Then DA is a Hilbert space with the scalar product D(., .), and A maps DA continuously to L2 (D, F). Note that if A = ∇ is the gradient operator in Rn , then DA = H 1 (D). Let us clarify what kind of elements are in this space in the general case. To this end we fix a Dirichlet system B j , j = 0, 1, . . . , m − 1, of order m − 1 on ∂D. More precisely, each B j is a differential operator of type E → F j and order m j ≤ m − 1 in a neighborhood U of ∂D, where m i ≠ m j for i ≠ j. Moreover, the symbols σ m j (B j ), if restricted to the conormal bundle of ∂D, have ranks equal to the dimensions of F j . Set t(u) = ⊕m−1 j=0 B j u , for u ∈ H m (D, E). For s > 0, we denote by H −s (∂D, F j ) the dual of the space H s (∂D, F j ) with respect to the pairing in L2 (∂D, F j ). m (D, E). Moreover t(u) has weak boundLemma 2.1.1. For every u ∈ DA , we have u ∈ Hloc m−1 −m j −1/2 ary values on ∂D belonging to ⊕j=0 H (∂D, F j ).

Proof. Fix an element u ∈ DA . Since A is elliptic we deduce from Au ∈ L2 (D, F) that m u ∈ Hloc (D, E). As usual, we denote by H −m (D, E) the completion of C∞ (D, E) with respect to the norm |(u, 𝑣)L2 (D,E) | |u|−m = sup . ‖𝑣‖H m (D,E) ∞ 𝑣∈C (D,E) t(𝑣)=0

Then we easily verify that itly,

A∗

extends to a map of L2 (D, F) to H −m (D, E), more explic(A∗ f, 𝑣) := (f, A𝑣)L2 (D,F) ,

2.1 Mixed Problems with a Parameter |

51

for each f ∈ L2 (D, F) and 𝑣 ∈ H0m (D, E). By this latter space we mean the closure m m of C∞ comp (D, E) in H (D, E), which in turn just amounts to H (X, E), the space of all D

m with support in D. sections of E of Sobolev class Hloc By the very definition, the distribution A∗ f is always orthogonal under the pairing in L2 (D, E) to the null space of the Dirichlet problem for A∗ A. Therefore, for every f ∈ L2 (D, F) there exists a section Gf ∈ H m (D, E) satisfying A∗ A Gf = A∗ f in D and t(Gf) = 0 on ∂D. Any u ∈ D A can be thus presented in the form

u = G Au + (u − G Au) . By the construction, we get G Au ∈ H0m (D, E) and u−G Au ∈ DA ∩SA ∗ A (D). As u−G Au ∈ L2 (D, E) is of finite order growth near ∂D, we conclude by Lemma 9.4.4 of [269] that −m j −1/2 (∂D, F ). t(u − G Au) has weak boundary values on ∂D belonging to ⊕m−1 j j=0 H m−1 m−m j −1/2 As t(G Au) ∈ ⊕j=0 H (∂D, F j ) vanishes on the boundary even in the usual sense for Sobolev spaces, the proof is complete. m−1 Let {C j }m−1 j=0 be the adjoint Dirichlet system for {B j } j=0 with respect to the Green formula for A (see for instance [269, Remark 9.2.6]). For g ∈ H m (D, F), we set

n(g) = ⊕m−1 j=0 C j g . Suppose S is a closed subset of ∂D. The cases S = 0 and S = ∂D are permitted, too. We write S∘ for the interior of S in the relative topology of ∂D. Given any u ∈ L2 (D, E) with Au ∈ L2 (D, F), we say that t(u) = 0 on the set S if ∫ ((Au, g)x − (u, A∗ g)x ) dx = 0

(2.1)

D

for all sections g ∈ C∞ (D, F) satisfying n(g) = 0 on ∂D \ S∘ . m Lemma 2.1.2. If u ∈ DA and t(u) = 0 on S then u ∈ Hloc (D ∪ S∘ , E).

In particular, t(u) has zero boundary values on S∘ in the usual sense of Sobolev spaces. Proof. The case S = 0 has been already treated in Lemma 2.1.1. Assume that S is nonempty. Choose a smooth real-valued function ϱ on X with the property that D = {x ∈ X : ϱ(x) < 0}

(2.2)

and ∇ϱ(x) ≠ 0 for all x ∈ ∂D. Set D ε = {x ∈ X : ϱ(x) < ε}, then D−ε ⋐ D ⋐ D ε for all sufficiently small ε > 0, and the boundary of D±ε is as smooth as the boundary of D. We first show that the weak boundary values of t(u) vanish on S in the sense that m−1

lim ∫ ∑ (B j u, g j )x ds = 0

ε→0+

∂D−ε

j=0

52 | 2 Regular Perturbation of Ill-Posed Problems for all g j ∈ C∞ (U, F j ), j = 0, 1, . . . , m − 1, satisfying (suppg j ) ∩ ∂D ⊂ S. To this end, choose a function g ∈ C∞ (D, F), such that n(g) = ⊕m−1 j=0 g j on ∂D, cf. Lemma 9.3.5 in [269]. Since u ∈ L2 (D, E) and Au ∈ L2 (D, F), we obtain by the Green formula m−1

lim ∫ ∑ (B j u, g j )x ds = lim ∫ ((Au, g)x − (u, A∗ g)x ) dx

ε→0+

∂D−ε

ε→0+

j=0

D−ε

= ∫ ((Au, g)x − (u, A∗ g)x ) dx D

=0 because t(u) = 0 on S in the sense of (2.1) and g ∈ C ∞ (D, F) satisfies n(g) = 0 on ∂D \ S∘ . We thus have A∗ Au ∈ H −m (D, E) and the weak boundary values of t(u) vanish on S. As A∗ A is an elliptic operator of order 2m and u 󳨃→ t(u) is a Dirichlet system of order m − 1, we conclude using the local regularity theorem for solutions of the Dirichlet m problem for A∗ A that u ∈ Hloc (D ∪ S∘ ) (see for instance Theorem 9.3.17 of [269]), as desired. The proof actually shows that for sections u ∈ L2 (D, E) with Au ∈ L2 (D, F) the equality (2.1) just amounts to saying that the weak boundary values of t(u) vanish on S∘ . Let DT stand for the completion of the space of all sections u in C∞ (D, E), satisfying t(u) = 0 on S, with respect to the norm u 󳨃→ D(u). By the very definition, D T is a closed subspace in DA , and it is a Hilbert space itself with the induced Hilbert structure. It is well known that if S is the whole boundary then DT = H0m (D, E). Lemma 2.1.3. If u ∈ DT then t(u) = 0 on S in the sense of (2.1). Proof. If u ∈ DT then there exists a sequence {u k }k∈N in C∞ (D, E) satisfying t(u k ) = 0 on S, such that lim D(u k − u) = 0 . k→∞

Hence, ∫ ((Au, g)x − (u, A∗ g)x ) dx = lim ∫ ((Au k , g)x − (u k , A∗ g)x ) dx k→∞

D

D m−1

= lim ∫ ∑ (B j u k , C j g)x ds k→∞

∂D

j=0

=0 for all g ∈ C∞ (D, F) satisfying n(g) = 0 on ∂D \ S∘ , because t(u k ) = 0 on S. Therefore, t(u) = 0 on ∂S. We are now in a position to characterize the space DT in a much more convenient way.

2.1 Mixed Problems with a Parameter

| 53

Theorem 2.1.4. As defined above, DT is a closed subspace of DA consisting of all u ∈ DA satisfying t(u) = 0 on S. Proof. Write H for the subspace of DA consisting of all u ∈ DA satisfying t(u) = 0 on S. It is easy to see that H is a closed subspace of D A . Lemma 2.1.3 states that DT is a subspace of H. Since DT is complete by the very definition, we shall have established the theorem if we prove that the orthogonal complement D⊥T of DT in H is zero. To this end, pick a section u ∈ H satisfying D(u, 𝑣) = 0 for all 𝑣 ∈ C∞ (D, E), such that t(𝑣) = 0 on S. If moreover 𝑣 fulfills n(A𝑣) = 0 on ∂D \ S∘ then we readily get (u, (A∗ A + I)𝑣)L2 (D,E) = 0 ,

(2.3)

which is due to (2.1). We now observe that every w ∈ C∞ (D, E) can be approximated in the L2 (D, E) norm by sections of the form (A∗ A + 1)𝑣, where 𝑣 ∈ C∞ (D, E) satisfies t(𝑣) = 0 on S and n(A𝑣) = 0 on ∂D \ S∘ . This latter is a consequence of the fact that the unbounded operator T ∗ T + 1 in L2 (D, E) with domain DT ∗ T is positive, and so invertible, see Section 2.1.3 below. We thus deduce from (2.3) that u = 0. It follows that D⊥T = {0}, as desired.

2.1.2 The Cauchy Problem A rough formulation of the Cauchy problem for the operator A in the domain D reads as follows: Given any sections f of F over D and u 0 of ⊕m−1 j=0 F j over S, find a section u of E over D, such that Au = f in D and t(u) has suitable limit values on S coinciding with u 0 . Note that some regularity of u up to S is needed for t(u) to possess limit values on S. Moreover, we are going to use Hilbert space methods for the study of the Cauchy problem. Hence, the space D A seems to be a natural choice for posing the problem. What is still lacking is a proper function space B(S) for the Cauchy data u 0 on S. It is not difficult to introduce such a space in the case where S is the entire boundary, namely B(∂D) = D A /H0m (D, E) . Using Lemma 2.1.1 one sees that this quotient space can be specified within ⊕m−1 j=0 H −m j −1/2 (∂D, F j ) under t, although the norm of the former is essentially stronger than the norm of the latter. Theorem 2.1.4 suggests us to set B(S) =

DA DT

in general. Using the approach of [269, Ch. 1] one can specify B(S) within ⊕m−1 j=0 H −m j −1/2 (S, F j ) under t. Of course, it is difficult to explicitly describe the elements

54 | 2 Regular Perturbation of Ill-Posed Problems

of B(S). However, for applications it suffices to know that there is a natural embedding m−m j −1/2 (S, F j ) 󳨅→ B(S) . ⊕m−1 j=0 H Using the spaces B(S) allows one to reduce the Cauchy problem with nonzero Cauchy data on S to the Cauchy problem with homogeneous boundary data. Indeed, given f ∈ L2 (D, F) and u 0 ∈ B(S), we look for a section u ∈ DA satisfying Au = f in D and t(u) = u 0 on S. By the very definition of the space B(S) there is a U0 ∈ DA with the property that τ(U0 ) = u 0 on S. This latter equality just amounts to saying that U0 − u 0 ∈ DT . Set u = U0 + U, then u ∈ DA is equivalent to U ∈ DA . Furthermore, t(u) = u 0 on S is equivalent to t(U) = 0. Since AU = f − AU0 and AU0 ∈ L2 (D, F), substituting u = U0 + U into the problem leads to the Cauchy problem with u 0 = 0. Namely, let f ∈ L2 (D, F) be an arbitrary section. Find u ∈ DT such that Au = f

(2.4)

in D. If S∘ ≠ 0 and the unique continuation property (U)s holds for A then problem (2.4) has at most one solution, cf. Theorem 10.3.5 of [269]. Otherwise we can not guarantee that the null space SA (D)∩DT of this problem is trivial. It is well known that the Cauchy problem for elliptic equations is ill posed in general. Moreover, if A is overdetermined, then additional necessary conditions arise for the problem to be solvable. In fact, these conditions reflect the fact that the image of DT by A may be not dense in L2 (D, F). Let us formulate this more precisely. To this end, we invoke, as usual, the boundary conditions which are adjoint for t with respect to the Green formula in D. Similarly to (2.1), for g ∈ L2 (D, F) with A∗ g ∈ L2 (D, E), we say that n(g) = 0 on the set ∂D \ S∘ if ∫ ((Au, g)x − (u, A∗ g)x ) dx = 0 ,

(2.5)

D

for all sections u ∈ C∞ (D, E) satisfying t(u) = 0 on S. Recall that A1 ∈ Diff m1 (X; F, E2 ) stands for a compatibility operator for A over X, i.e., A1 is in a sense “smallest” differential operator with the property that A1 A ≡ 0 on X. We make use of the Green formula for A1 in the same way as above to introduce the relations “n(𝑣) = 0 on ∂D \ S∘ ”, for all sections 𝑣 ∈ L2 (D, E2 ) with A1∗ 𝑣 ∈ L2 (D, F), and “t(f) = 0 on S”, for all sections f ∈ L2 (D, F) with A1 f ∈ L2 (D, E2 ). The boundary equations n(𝑣) = 0 for sections of E2 and t(f) = 0 for sections of F are no longer induced by any Dirichlet system on ∂D as those at steps 1 and 0, respectively. Lemma 2.1.5. Assume that f ∈ L2 (D, F) belongs to the closure of A DT in L2 (D, F). Then 1) A1 f = 0 in D in the sense of distributions; 2) t(f) = 0 on S; 3) (f, g)L2 (D,F) = 0 for all g ∈ L2 (D, F) satisfying A∗ g = 0 in D and n(g) = 0 on ∂D \ S∘ .

2.1 Mixed Problems with a Parameter |

55

Proof. 1) Let f belong to the closure of A DT in L2 (D, F). Then there is a sequence {u k }k∈N in DT , such that {Au k }k∈N converges to f in L2 (D, F). Without loss of generality we may assume that each u k is of class C∞ (D, E), for such functions are dense in DT . As A1 A ≡ 0, we get (f, A1∗ 𝑣)L2 (D,F) = lim (Au k , A1∗ 𝑣)L2 (D,F) k→∞

= lim (u k , (A1 A)∗ 𝑣)L2 (D,E) k→∞

= lim 0 k→∞

=0 for all 𝑣 ∈ C∞ (D, E2 ) satisfying n(A1∗ 𝑣) = 0 on ∂D \ S∘ . In particular, this equality is fulfilled for all sections 𝑣 ∈ C∞ (D, E2 ) of compact supports in D, which implies A1 f = 0 in D. 2) Suppose 𝑣 ∈ C∞ (D, E2 ) is any section satisfying n(𝑣) = 0 on ∂D \ S∘ . Then 1∗ n(A 𝑣) = 0 holds on ∂D \ S∘ , too, which is a consequence of A∗ A1∗ = 0 and Stokes’ formula. By 1), we get − (f, A1∗ 𝑣)L2 (D,F) = ∫ ((A1 f, 𝑣)x − (f, A1∗ 𝑣)x ) dx D

=0, the first equality being a consequence of the fact that A1 f = 0 in D. Hence, it follows that t(f) = 0 on S. 3) Finally, (f, g)L2 (D,F) = lim (Au k , g)L2 (D,F) k→∞

= lim ∫ ((Au k , g)x − (u k , A∗ g)x ) dx k→∞

D

= lim 0 k→∞

=0 provided that g ∈ L2 (D, F) satisfies A∗ g = 0 in D and n(g) = 0 on ∂D \ S∘ . This proves 3). The condition 3) is not only necessary but also sufficient in order that f belong to the closure of A DT in L2 (D, F). Lemma 2.1.6. If f satisfies the condition 3) of Lemma 2.1.5, then f lies in the closure of A DT in L2 (D, F). Proof. Write V for the space of all g ∈ L2 (D, F) satisfying A∗ g = 0 in D and n(g) = 0 on ∂D \ S∘ . We shall have established the lemma if we show that V coincides with the

56 | 2 Regular Perturbation of Ill-Posed Problems orthogonal complement of the image A DT in L2 (D, F). By definition, g ∈ (A D T )⊥ if (g, Au)L2 (D,F) = 0 ,

(2.6)

for all u ∈ DT . Since DT contains all smooth functions of compact support in D, we conclude that (A DT )⊥ ⊂ SA ∗ (D). Then equality (2.6) implies that (A DT )⊥ ⊂ V because (g, Au)L2 (D,F) = − ∫ ((A∗ g, u)x − (g, Au)x ) dx , D

for all g ∈ V. On the other hand, the inclusion V ⊂ (A DT )⊥ follows from (2.1) because each u ∈ DT can be approximated in the norm D(⋅) by sections u k ∈ C∞ (D, E) satisfying t(u k ) = 0 on S. Denote by H1 (D, S) the space of all g ∈ L2 (D, F) satisfying A∗ g = A1 g = 0 in D and n(g) = 0 on ∂D \ S∘ . We call H1 (D, S) the harmonic space in the Cauchy problem with data on S. This is an analogue of the well-known harmonic spaces in the Neumann problem for the Laplace operator, cf. [269]. Lemma 2.1.7. When combined with 4) (f, g)L2 (D,F) = 0 for all g ∈ H1 (D, S), the condition 1) of Lemma 2.1.5 implies that f belongs to the closure of A DT in L2 (D, F). Proof. Let the conditions 1) and 4) are fulfilled for f ∈ L2 (D, F). The proof of Lemma 2.1.6 shows that f = f1 + f2 , (2.7) where f1 belongs to the closure of A DT in L2 (D, F) and f2 ∈ V. As A1 f = 0 in D, we deduce by Lemma 2.1.5 that A1 f2 = 0 in D. This means f2 ∈ H1 (D, S). Finally, 4) implies 0 = (f, f2 )L2 (D,F) = (f2 , f2 )L2 (D,F) whence f2 = 0, and so f belongs to the closure of A DT in L2 (D, F). Obviously, if f belongs to the closure of A DT in L2 (D, F), then it satisfies 4) by Lemma 2.1.5, 3). It follows that the condition 3) of Lemma 2.1.5 is equivalent to 1) + 4). Lemma 2.1.8. When combined with 5) (f, g)L2 (D,F) = 0 for all g ∈ H1 (D, S) satisfying t(g) = 0 on S, the conditions 1) and 2) of Lemma 2.1.5 imply that f belongs to the closure of A D T in L2 (D, F). Proof. Let the conditions 1), 2), and 5) hold true for f ∈ L2 (D, F). Taking into account Lemma 2.1.5 and decomposition (2.7) we readily conclude that A1 f2 = 0 in D

2.1 Mixed Problems with a Parameter

| 57

and t(f2 ) = 0 on S. Finally, 5) implies 0 = (f, f2 )L2 (D,F) = (f2 , f2 )L2 (D,F) whence f2 = 0. Thus, f = f1 belongs to the closure of A DT in L2 (D, F), as desired. Remark 2.1.9. Of course, if A is elliptic, then A1 = 0 and the conditions 1) and 2) are always fulfilled. As for the condition 3), one easily proves that each g ∈ L2 (D, F) satisfying A∗ g = 0 in D and n(g) = 0 on ∂D\S∘ vanishes identically in all of D, provided that A∗ is elliptic, S ≠ ∂D and A∗ possesses the unique continuation property (U)s in a neighborhood of D (see, for instance, [269, Theorem 10.3.5]). If A is overdetermined elliptic, then the domain D should possess some convexity property relative to A, in order that H1 (D, S) or {g ∈ H1 (D, S) : t(g) = 0 on S} might be trivial. In the case S = 0, we refer the reader to [269] for more details. We have thus described the closure of A DT in L2 (D, F). It is a more difficult task to describe the image A DT itself. The following lemma is the first step in this direction. Lemma 2.1.10. Let f ∈ L2 (D, F) belong to the closure of A D T in L2 (D, F). Then a section u ∈ DT is a solution to problem (2.4) if and only if (Au, A𝑣)L2 (D,F) = (f, A𝑣)L2 (D,F) ,

(2.8)

for all 𝑣 ∈ DT . Proof. If problem (2.4) is solvable and u is one of its solutions, then (2.8) is obviously satisfied. Conversely, if (2.8) holds for an element u ∈ DT , then A∗ (Au − f) = 0 in D because the space D T contains all smooth functions of compact support in D. It follows that ∫ ((A∗ (Au − f), 𝑣)x − (Au − f, A𝑣)x ) dx = −(Au − f, A𝑣)L2 (D,F) D

=0 for all 𝑣 ∈ C∞ (D, E) satisfying t(𝑣) = 0 on S, which is due to (2.8). Hence, n(Au − f) = 0 on ∂D \ S∘ . Finally, since both Au and f belong to the closure of A DT in L2 (D, F), Lemma 2.1.5, 3) shows that (Au − f, Au − f)L2 (D,F) = 0 , i.e., Au = f in D. In conclusion of this section let us clarify the meaning of (2.8). Namely, this equality amounts to saying that a solution u ∈ DT of the Cauchy problem Au = f is actually a

58 | 2 Regular Perturbation of Ill-Posed Problems

solution to the mixed problem A∗ Au = A∗ f { { { t(u) = 0 { { { {n(Au) = n(f)

in

D;

on

S,

on

∂D \

(2.9) S∘

.

Indeed, the proof of Lemma 2.1.10 shows that A∗ Au = A∗ f in D in the sense of distributions and n(Au) = n(f) in the sense that n(Au − f) = 0 on ∂D \ S∘ . In particular, if n(f) is well defined on ∂D \ S∘ , then also n(Au) is well defined on ∂D \ S∘ . Of course, the mixed problem (2.9) considered in appropriate spaces gives nothing but (2.8). In the next sections, we will systematically use the generalized setting (2.8) of problem (2.4) in order to derive its solvability conditions.

2.1.3 A Perturbation Equation (2.8) surprisingly shows that problem (2.4) may be well posed in many cases. Namely, this is the case if the Hermitian form (A⋅, A⋅)L2 (D,F) is actually a scalar product on DT inducing the same topology as the original scalar product D(⋅, ⋅). For example, not only the gradient operator ∇ in Rn meets this latter condition but also many other overdetermined elliptic operators A with finite-dimensional kernel SA (D). Of course, (A⋅, A⋅)L2 (D,F) is always a scalar product on DT if S ≠ 0 and A possesses the property (U)s . However, the completion of DT with respect to (A⋅, A⋅)L2 (D,F) may lead to a space with elements of arbitrary order of growth near ∂D. This observation suggests that we perturb the Hermitian form (A⋅, A⋅)L2 (D,F), thus obtaining a “good” scalar product on DT . For this purpose, let us introduce a family of Hermitian forms (u, 𝑣)ε = (Au, A𝑣)L2 (D,F) + ε (u, 𝑣)L2 (D,E) on DT , parameterized by ε > 0. For each fixed ε > 0, the corresponding norm ‖u‖ε = √(u, u)ε is equivalent to the graph norm D(u) on D T . More precisely, we get min{1, √ε} D(u) ≤ ‖u‖ε ≤ max{1, √ε} D(u)

(2.10)

for all u ∈ D A . Taking into account Lemma 2.1.10 we now consider the following perturbed Cauchy problem. Given any f ∈ L2 (D, F) and h ∈ L2 (D, E), find an element u ε ∈ DT satisfying (Au ε , A𝑣)L2 (D,F) + ε (u ε , 𝑣)L2 (D,E) = (f, A𝑣)L2 (D,F) + ε (h, 𝑣)L2 (D,E) for all 𝑣 ∈ DT .

(2.11)

2.1 Mixed Problems with a Parameter |

59

Note that the equation (2.11) leads to a perturbation of mixed problem (2.9), more precisely, A∗ Au ε + ε u ε = A∗ f + ε h in D ; { { { (2.12) t(u ) = 0 on S , { { ε { ∘ on ∂D \ S . { n(Au ε ) = n(f) Indeed, since the space DT contains all smooth functions with compact support in D, (2.11) implies A∗ Au ε + ε u ε = A∗ f + ε h in D in the sense of distributions. The boundary condition t(u ε ) = 0 on S follows from Lemma 2.1.3. Finally, n(Au ε ) = n(f) holds in the sense that n(Au ε − f) on ∂D \ S∘ because A∗ (Au ε − f) = ε(h − u ε ) ∈ L2 (D, E) in D and ∫ ((A∗ (Au ε − f), 𝑣)x − (Au ε − f, A𝑣)x ) dx D

= ε (h − u ε , 𝑣)L2 (D,E) − (Au ε − f, A𝑣)L2 (D,F) = 0 for all 𝑣 ∈ C∞ (D, E) satisfying t(𝑣) = 0 on S, the latter equality being due to (2.11). If the restriction of n(f) to ∂D \ S∘ makes sense, then the restriction of n(Au) does so. If considered in appropriate function spaces, the mixed problem (2.12) certainly gives nothing but (2.11). In general, mixed problems (2.9) and (2.12) have noncoercive boundary conditions on ∂D \ S∘ . Hence, they fail to be well-posed in the relevant weighted Sobolev spaces. The principal difference between problems (2.4) and (2.11) is that the last one is wellposed in DT . Lemma 2.1.11. For every ε > 0, f ∈ L2 (D, F) and h ∈ L2 (D, E) there exists a unique solution u ε (f, h) ∈ DT to problem (2.11). Moreover, it satisfies ‖u ε (f, h)‖ε ≤ ‖f‖L2 (D,F) + √ε ‖h‖L2 (D,E) . Proof. Really, the estimates (2.10) imply that the vector space DT endowed with the scalar product (⋅, ⋅)ε ) is a Hilbert space. The Schwarz inequality yields 󵄨󵄨󵄨(f, A𝑣)L2 (D,F) + ε (h, 𝑣)L2 (D,E) 󵄨󵄨󵄨 󵄨 󵄨 ≤ ‖f‖L2 (D,F)‖A𝑣‖L2 (D,F) + ε ‖h‖L2 (D,F) ‖𝑣‖L2 (D,E) ≤ ‖f‖L2 (D,F)‖𝑣‖ε + √ε ‖h‖L2 (D,F) √ε‖𝑣‖2L2 (D,E) ≤ c ε (f, h) ‖𝑣‖ε with c ε (f, h) = ‖f‖L2 (D,F) + √ε ‖h‖L2 (D,E) .

60 | 2 Regular Perturbation of Ill-Posed Problems

Hence, the map 𝑣 󳨃→ (f, A𝑣)L2 (D,F) + ε (h, 𝑣)L2 (D,E) defines a continuous linear functional Ff,h on DT , whose norm is majorized by ‖Ff,h‖ ≤ c ε (f, h). We now use the Riesz theorem to conclude that there exists a unique element u ε (f, h) ∈ DT with Ff,h (𝑣) = (u ε (f, h), 𝑣)ε for every 𝑣 ∈ DT . Clearly, u ε (f, h) is a solution to problem (2.11). Finally, by the Riesz theorem we get ‖u ε (f, h)‖ε ≤ c ε (f, h) , as desired. The equations (2.12) show that Lemma 2.1.11 gives information on the solvability of a mixed problem for the elliptic operator A∗ A + ε with very special data on D, S and ∂D \ S∘ . Let us clarify what kind solvability theorems can be obtained for arbitrary data. For a triple w ∈ L2 (D, E) and 2m−m j −1/2 (S, F j ) , u 0 ∈ ⊕m−1 j=0 H

(2.13)

m−m j −1/2 (∂D \ S∘ , F j ) , u 1 ∈ ⊕m−1 j=0 H

we investigate the problem of finding a section u of the bundle E over D which satisfies A∗ Au + ε u = w { { { t(u) = u 0 { { { {n(Au) = u 1

in

D;

on

S,

on

∂D

(2.14) \ S∘

,

the equations in D and on the boundary of D being understood in a proper sense. From what has already been proved it is clear what we mean by this proper sense, namely (Au, g)L2 (D,F) − (u, A∗ g)L2 (D,E) = (u 0 , n(g))⊕L2 (S,F j ) , (u, 𝑣)ε = (w, 𝑣)L2 (D,E) − (u 1 , t(𝑣))⊕L2 (∂D\S∘ ,F j ) (2.15) for all g ∈ C∞ (D, F) satisfying n(g) = 0 on ∂D \ S∘ , and for all 𝑣 ∈ C∞ (D, E) satisfying t(𝑣) = 0 on S, respectively. Theorem 2.1.12. Let (A∗ A)2 possess the unique continuation property (U)s . Then, for 2m (D ∪ S∘ , E) to Problem 2.14. any triple (w, u 0 , u 1 ) there is a unique solution u ∈ DA ∩ Hloc Moreover, there is a constant C(ε) > 0 which does not depend on (w, u 0 , u 1 ), such that ‖u‖2ε ≤ C(ε)(‖w‖2L2 (D,E) + ‖u 0 ‖2

⊕H 2m−m j −1/2 (S,F j )

+ ‖u 1 ‖2

⊕H m−m j −1/2 (∂D\S∘ ,F j )

).

(2.16)

2.1 Mixed Problems with a Parameter |

61

Proof. Choose arbitrary u 0 and u 1 as in (2.13). Obviously, there are sections 2m−m j −1/2 (∂D, F j ) , U0 ∈ ⊕m−1 j=0 H m−m j −1/2 (∂D, F j ) , U1 ∈ ⊕m−1 j=0 H

such that U0 = u 0 on S, U1 = u 1 on ∂D \ S∘ and ‖U0 ‖2

⊕H 2m−m j −1/2 (∂D,F j )

≤ 2(‖u 0 ‖2

+ ‖U1 ‖2

⊕H 2m−m j −1/2 (S,F j )

⊕H m−m j −1/2 (∂D,F j )

+ ‖u 1 ‖2

⊕H m−m j −1/2 (∂D\S∘ ,F j )

). (2.17)

As the pair {t, n ∘ A} is a Dirichlet system of order 2m − 1 on ∂D, solving the Dirichlet problem for (A∗ A)2 yields a section U 󸀠 ∈ H 2m (D, E) with the following properties (A∗ A)2 U 󸀠 = 0 in { { { t(U 󸀠 ) = U0 on { { { 󸀠 on { n(AU ) = U1

D; (2.18)

∂D , ∂D .

Moreover, there exists a positive constant C > 0 which is independent of U, such that ‖U 󸀠 ‖2H 2m (D,E) ≤ C(‖U0 ‖2

⊕H 2m−m j −1/2 (∂D,F j )

+ ‖U1 ‖2

⊕H m−m j −1/2 (∂D,F j )

),

(2.19)

see for instance [269]. According to Lemma 2.1.11 there exists a solution U 󸀠󸀠 ∈ DT to problem (2.11) with f = 0 and 1 (w − A∗ AU 󸀠 ) − U 󸀠 ε ∈ L2 (D, E) .

h=

Set u = U 󸀠 + U 󸀠󸀠 . Then, integrating by parts and using Lemma 2.1.11 we easily obtain (u, 𝑣)ε = ((A∗ A + ε)U 󸀠 , 𝑣)L2 (D,E) − (n(AU 󸀠 ), t(𝑣))⊕L2 (∂D,F j ) + (w, 𝑣)L2 (D,E) − ((A∗ A + ε)U 󸀠 , 𝑣)L2 (D,E) = (w, 𝑣)L2 (D,E) − (u 1 , t(𝑣))⊕L2 (∂D\S∘ ,F j ) for every 𝑣 ∈ C∞ (D, E) satisfying t(𝑣) = 0 on S, i.e., the second equality of (2.15) holds true. On the other hand, for every g ∈ C∞ (D, F) satisfying n(g) = 0 on ∂D \ S∘ , we get (Au, g)L2 (D,F) − (u, A∗ g)L2 (D,E) = (AU 󸀠 , g)L2 (D,F) − (U 󸀠 , A∗ g)L2 (D,E) + (AU 󸀠󸀠 , g)L2 (D,F) − (U 󸀠󸀠 , A∗ g)L2 (D,E) = (AU 󸀠 , g)L2 (D,F) − (U 󸀠 , A∗ g)L2 (D,E)

62 | 2 Regular Perturbation of Ill-Posed Problems because U 󸀠󸀠 ∈ DT . Once again integrating by parts we obtain (AU 󸀠 , g)L2 (D,F) − (U 󸀠 , A∗ g)L2 (D,E) = (t(U 󸀠 ), n(g))⊕L2 (∂D,F j ) = (u 0 , n(g))⊕L2 (S,F j ) , i.e., the first equality of (2.15) is fulfilled. By the elliptic regularity of the Dirichlet problem for the operator A∗ A + ε we de2m duce that u ∈ Hloc (D ∪ S∘ , E). If all of w and u 0 , u 1 vanish then (2.15) and Theorem 2.1.4 imply that the corresponding solution u lies in DT . On the other hand, the second equality of (2.15) means that u is orthogonal to DT with respect to (⋅, ⋅)ε , i.e., u ≡ 0 which proves the uniqueness. Finally, according to Lemma 2.1.11 we get ‖u‖ε ≤ ‖U 󸀠 ‖ε + ‖U 󸀠󸀠 ‖ε ≤ c ‖U 󸀠 ‖H 2m (D,E) +

1 (‖w‖L2 (D,E) + ‖A∗ AU 󸀠 ‖L2 (D,E) ) + √ε‖U 󸀠 ‖L2 (D,E) . √ε

Combining this estimate with (2.17) and (2.19) we arrive at (2.16), as desired. One sees that the regularity up to ∂D of the solution u in Theorem 2.1.12 fails to correspond to the smoothness of the data w and u 0 , u 1 . To justify this we recall that the boundary conditions n ∘ A on ∂D \ S∘ are not coercive in general. Were n ∘ A coercive we 2m would have u ∈ Hloc (D\∂S, E). However, we could not guarantee even in this case that s u ∈ H (D, E) for some s > m unless certain additional conditions were imposed on the triple (w, u 0 , u 1 ) on ∂S. This is typical for the mixed problems, cf. [64] and elsewhere.

2.1.4 The Main Theorem Set u ε (f) = u ε (f, 0). The inequalities (2.10) and Lemma 2.1.11 give us a rough estimate for the family {u ε (f)} ε>0 , namely D(u ε (f)) ≤

1 ‖f‖L2 (D,F) . √ε

Thus, it might be unbounded while ε → 0+. Let us see how the behavior of the family {u ε (f)} ε>0 reflects on the solvability of problem (2.4). Theorem 2.1.13. The family {u ε (f)} ε>0 is bounded in DT if and only if there exists u ∈ DT satisfying (2.8). Proof. We first prove the following lemma. Lemma 2.1.14. Let there be a set ∆ ⊂ (0, +∞), such that

2.1 Mixed Problems with a Parameter

| 63

1) zero is an accumulation point of ∆; 2) the family {u δ (f)} δ∈∆ is bounded in DT . Then there exists u ∈ DT satisfying (2.8). Proof. Suppose zero is an accumulation point of ∆ and the family {u δ (f)} δ∈∆ is bounded in DT . By (2.11), we have (Au δ (f), A𝑣)L2 (D,F) + δ (u δ (f), 𝑣)L2 (D,E) = (f, A𝑣)L2 (D,F) for all 𝑣 ∈ DT . Passing to the limit, when ∆ ∋ δ → 0, in the last equality and using the fact that {u δ (f)} δ∈∆ is bounded, we obtain lim (Au δ (f), A𝑣)L2 (D,F) = (f, A𝑣)L2 (D,F)

δ→0+

(2.20)

for all 𝑣 ∈ DT . It is well known that every bounded set in a Hilbert space is weakly compact. Hence, there is a subsequence {u δ j (f)} ⊂ DT weakly convergent in DT to an element u ∈ DT . Here, {δ j } converges to 0 when j → ∞. Note that (2.11) implies (u ε (f), 𝑣)L2 (D,E) = 0 for all 𝑣 ∈ DT ∩ SA (D), i.e., both {u δ j (f)} and u are L2 (D, E) -orthogonal to DT ∩ SA (D). Let us show that {u δ j (f)} converges weakly to u in L2 (D, E) when j → ∞. Given any 𝑣 ∈ L2 (D, E), the map u 󳨃→ (u, 𝑣)L2 (D,E) defines a continuous linear functional F𝑣 on DT with ‖F𝑣 ‖ ≤ ‖𝑣‖L2 (D,E). We now invoke the Riesz representation theorem to conclude that there exists a unique element 𝑣̃ ∈ DT with D(u, 𝑣)̃ = F𝑣 (u) for every u ∈ D T . Hence, lim (u δ j (f), 𝑣)L2 (D,E) = lim D(u δ j (f), 𝑣)̃

j→∞

j→∞

= D(u, 𝑣)̃ = (u, 𝑣)L2 (D,E) . This exactly means that {u δ j (f)} converges weakly in L2 (D, E). Now we easily calculate lim (Au δ (f), A𝑣)L2 (D,F) =

∆∋δ→0+

lim (D(u δ (f), 𝑣) − (u δ (f), 𝑣)L2 (D,E) )

∆∋δ→0+

= D(u, 𝑣) − (u, 𝑣)L2 (D,E) = (Au, A𝑣)L2 (D,F) (2.21) for all 𝑣 ∈ DT . Combining (2.20) and (2.21) we see that (2.8) holds true for u. Note that if (2.8) is solvable then there exists a solution u which is L2 (D, E) -orthogonal to DT ∩ SA (D). We will have a stronger statement than Theorem 2.1.13 if we prove the following lemma.

64 | 2 Regular Perturbation of Ill-Posed Problems Lemma 2.1.15. If there exists u ∈ DT satisfying (2.8) then the family {u ε (f)} ε>0 is bounded in DT and lim ‖A(u ε − u)‖L2 (D,F) = 0 . ε→0+

Moreover, {u ε (f)} ε>0 converges weakly to u ∈ DT as ε → 0+, if u is L2 (D, E) -orthogonal to DT ∩ SA (D). Proof. Let there exist u ∈ DT satisfying (2.8). Set R ε = u ε (f) − u. Then (2.8) and (2.11) imply (AR ε , A𝑣)L2 (D,F) + ε (R ε , 𝑣)L2 (D,E) = −ε (u, 𝑣)L2 (D,E) (2.22) for all 𝑣 ∈ DT , i.e., R ε = u ε (0, −u) is the solution to problem (2.11) with f = 0 and h = −u. According to (2.10) and Lemma 2.1.11 we have 1 ‖R ε ‖ε √ε 1 √ε ‖u‖L2 (D,E) ≤ √ε

D(R ε ) ≤

= ‖u‖L2 (D,E) . Therefore, the family {R ε }ε>0 is bounded in DT , and so the family {u ε (f)} ε>0 is bounded, too. Now (2.22) implies lim ‖A(u ε (f) − u)‖2L2 (D,F) = lim ‖AR ε ‖2L2 (D,F)

ε→0+

ε→0+

= − lim ε (‖R ε ‖2L2 (D,E) + (u, R ε )L2 (D,E) ) ε→0+

=0. Finally, let us prove that {u ε (f)} ε>0 converges weakly to u in DT as ε → 0+, provided that u is L2 (D, E) -orthogonal to DT ∩ SA (D). We argue by contradiction. Indeed, if {u ε (f)} ε>0 does not converge weakly to u in DT then there are 𝑣 ∈ DT , γ > 0 and a sequence {ε j } tending to 0+ as j → ∞, such that |D(u ε j − u, 𝑣)| ≥ γ

(2.23)

for every j ∈ N. But the sequence {u ε j } is bounded in the Hilbert space DT , and so it possesses a subsequence which converges weakly in DT . By abuse of notation we denote it again by {u ε j }. As we have already seen in the proof of Lemma 2.1.14, the weak limit of {u ε j } is u. This contradicts (2.23), and thus the assertion of the lemma is proved. The proof of Theorem 2.1.13 is complete. Note that if problem (2.4) is solvable then there exists a unique solution u which is L2 (D, E) -orthogonal to DT ∩ SA (D).

2.1 Mixed Problems with a Parameter

| 65

Corollary 2.1.16. Suppose f belongs to the closure of A DT in L2 (D, F). Then the family {u ε (f)} ε>0 is bounded in DT if and only if problem (2.4) is solvable. Moreover, lim ‖Au ε (f) − f‖L2 (D,F) = 0

ε→0+

and even {u ε (f)} ε>0 converges weakly, when ε → 0+, to the solution u ∈ DT of problem (2.4) which is L2 (D, E) -orthogonal to DT ∩ SA (D). Proof. This follows from Theorem 2.1.13 and Lemmas 2.1.15 and 2.1.10. m Is it true that {u ε (f)} ε>0 converges to u in the topology of Hloc (D ∪ S∘ , E) if u ∈ DT is the 2 solution to problem (2.4) which is L (D, E) -orthogonal to DT ∩ SA (D)? To answer this question we observe, by Lemma 2.1.15, that the family {u ε (f) − u} ε>0 is bounded in DT and

lim ‖A(u ε (f) − u)‖L2 (D,F) = 0 ,

ε→0+

t(u ε (f) − u) = 0 on S for every ε > 0. Then, applying [270, Theorem 7.2.6] we see that {u ε (f)} ε>0 conm (D ∪ S∘ , E). verges to u in Hloc

2.1.5 The Well-Posed Case It is well known that a linear operator T : H → H̃ in normed spaces has a continuous inverse if and only if ‖u‖H ≤ c ‖Tu‖H̃ for every u ∈ H, the constant c > 0 being independent of u. Hence, the (Cauchy) problem (2.4) is well-posed if and only if there exists a constant c > 0 such that ‖u‖L2 (D,E) ≤ c‖Au‖L2 (D,F)

(2.24)

for all u ∈ DT . Theorem 2.1.17. Let the (Cauchy) problem (2.4) be well posed. Then for every f ∈ L2 (D, F) there exists a limit u = lim u ε (f) ε→0+

in D T . Moreover, u is the solution to problem (2.4) if f belongs to the closure of A DT in L2 (D, F). Proof. Indeed, it follows from (2.24) that the Hermitian form h(u, 𝑣) := (Au, A𝑣)L2 (D,F) defines a scalar product on DT inducing the same topology as the original one. We now use the Riesz representation theorem to see that for every f ∈ L2 (D, F) there is a unique element u ∈ DT satisfying (2.8).

66 | 2 Regular Perturbation of Ill-Posed Problems

Moreover, (2.24) yields D(u ε (f) − u) ≤ √ c + 1 ‖u ε (f) − u‖ε . Then using (2.22) and Lemma 2.1.11 we see that D(u ε (f) − u) ≤ √ c + 1 ‖u ε (0, −u)‖ε ≤ √c + 1 √ε ‖u‖L2 (D,E) . Therefore, we get lim D(u ε (f) − u) = 0 ,

ε→0+

and so Corollary 2.1.16 shows that u is a solution to problem (2.4) provided f belongs to the closure of A D T in L2 (D, F). Apparently, if A is a differential operator with finite-dimensional kernel SA (D) then the (Cauchy) problem (2.4) is well posed for A. Example 2.1.18. Let X = R, A = d/dx, D = (a, b) with −∞ < a < b < ∞, and S = {a}. Then D A = H 1 (D). The Cauchy problem { u 󸀠 (x) = f(x) { u(a) = u 0 , {

for

x ∈ (a, b) ,

with u 0 ∈ R, is known to be well posed in Sobolev spaces as well as in spaces of smooth functions on [a, b]. Its solution can be easily found by the formula x

u(x) = u 0 + ∫ f(y) dy . a

Let us look at the corresponding family of mixed problems. In this case we have A∗ = −d/dx and ∂D \ S∘ = {b}, hence the mixed problems are u 󸀠󸀠ε (x) − ε u ε (x) = f 󸀠 (x) { { { u ε (a) = u 0 , { { { 󸀠 {u ε (b) = f(b),

for

x ∈ (a, b) ,

where u 0 ∈ R is arbitrary. One easily calculates that x

b

u ε (x) = u 0 + ∫f(y) cosh(√ε(x − y)) dy + a

sinh(√ε(x − a)) ∫f(y) sinh(√ε(b − y)) dy cosh(√ε(b − a)) a

and lim u ε = u

ε→0+

even in the norm of C1 [a, b], if f ∈ C[a, b].

2.1 Mixed Problems with a Parameter | 67

2.1.6 On Finding the Solution Let us discuss the very important question of how to find the solution of problem (2.11), and hence a solution to problem (2.4). Of course, if an explicit orthonormal basis {e i }i∈N in the space DT with the scalar product (⋅, ⋅)ε is available, then one easily obtains ∞

u ε (f, h) = ∑ (u ε (f, h), e i )ε e i .

(2.25)

j=1

According to (2.11) we have (u ε (f, h), e i )ε = (f, Ae i )L2 (D,F) + ε (h, e i )L2 (D,E) ,

(2.26)

hence (2.25) and (2.26) give us a complete description of the solution u ε (f, h) to problem (2.11). Unfortunately, it is not an easy task to construct an explicit basis {e i }i∈N . Example 2.1.19. Let S = ∂D ∩ S where S is a sufficiently smooth hypersurface near ∂D. Choose a defining function δ(x) for S. Then we can start with a linearly independent system of the form {(δ(x))m−1 P i (x)} in DT , where P i (x) are polynomials of increasing degree taking their values in E x . Orthogonalizing it by the standard Gram–Schmidt procedure we arrive at an orthonormal system in DT . In order to obtain a basis we have certainly to guarantee that the system {(δ(x))m−1 P i (x)} be dense in DT . However, for applications it suffices to have merely a finite number of basis elements. Let us describe an alternative way of finding the solution. Assume that the operator A∗ A + ε possesses the unique continuation property (U)s in a neighborhood of D. Then it has a two-sided fundamental solution there (see for instance [269]). Fix such a fundamental solution Φ ε (x, y) for A∗ A + ε. For each s ≥ 0, it induces a continuous linear map Φ ε : H s (D, E) → H s+2m (D, E) by u 󳨃→ r+ Φ ε (e+ u) where e+ means the extension by zero to all of X and r+ the restriction to D. This map actually extends to a continuous map Φ ε : H s (D, E) → H s+2m (D, E) for all s ∈ R, being a right inverse of A∗ A + ε. Every element u ∈ D A may be thus written in the form u = U + Φ ε ((A∗ A + ε)u) ,

(2.27)

where U ∈ D A ∩ SA ∗ A+ε (D). Indeed, fix u ∈ DA . Since Au ∈ L2 (D, F) we deduce that A∗ Au ∈ H −m (D, E). It follows that Φ ε ((A∗ A + ε)u) ∈ H m (D, E) ⊂ DA . Setting U = u − Φ ε ((A∗ A + ε)u) yields readily (2.27) with U ∈ D A ∩ SA ∗ A+ε (D), as desired. In practice one usually has only a complete linearly independent system {U i }i∈N of solutions to (A∗ A + ε)U = 0 on neighborhoods of D, or even on all of X ∘ .

68 | 2 Regular Perturbation of Ill-Posed Problems Lemma 2.1.20. Assume that A∗ A+ε possesses the unique continuation property (U) s . If M ⊂ SA ∗ A+ε (D) is a dense set in C m−1 (D, E)∩SA ∗ A+ε (D) then it is dense in DA ∩SA ∗ A+ε (D). Proof. When endowed with the scalar product (⋅, ⋅)ε , DA ∩SA ∗ A+ε (D) is a Hilbert space. Hence, it suffices to prove that the orthogonal complement of M in this space is zero. To this end, pick u ∈ D A ∩ SA ∗ A+ε (D). Since u belongs to L2 (D, E) it has a finite order of growth near ∂D, cf. [269]. It follows that the expressions t(u) and n(Au) have weak boundary values u 0 and u 1 in the space of distributions on ∂D. ∞ Let 𝑣0 ∈ ⊕m−1 j=0 C (∂D, F j ). As t is a Dirichlet system of order m − 1 on ∂D, there is a section 𝑣 ∈ C∞ (D, E) satisfying t(𝑣) = 𝑣0 . Then ⟨u 1 , 𝑣0 ⟩ =: lim ∫ (n(Au), 𝑣) x ds δ (x) δ→0−

∂D δ

and the definition does not depend on the particular choice of 𝑣. Since the Dirichlet problem for A∗ A + ε in D is uniquely solvable over the whole scale of Sobolev spaces, we can take 𝑣 ∈ C∞ (D, E) ∩ SA ∗ A+ε (D). If u is orthogonal to M ⊂ SA ∗ A+ε (D) with respect to the scalar product (⋅, ⋅)ε then 0 = (u, 𝑣)ε = lim ∫ (Au, A𝑣)x dx + ε (u, 𝑣)L2 (D,E) δ→0−

Dδ

= lim ( ∫ (n(Au), t(𝑣))x ds δ (x) + ∫ (A∗ Au, 𝑣)x dx) + ε (u, 𝑣)L2 (D,E) δ→0−

Dδ

∂D δ

= lim ∫ (n(Au), t(𝑣))x ds δ (x) δ→0−

∂D δ

for all 𝑣 ∈ M. As M is dense in C m−1 (D, E) ∩ SA ∗ A+ε (D) it follows that n(Au) = 0 on ∂D. On the other hand, since u ∈ DA it can be approximated in the norm D(⋅) by a sequence {u k } ⊂ C∞ (D, E). Then (u, u)ε = lim (u, u k )ε k→∞

= lim lim ∫ (n(Au), u k )x ds δ (x) k→∞ δ→0−

∂D δ

=0 whence u ≡ 0 in D. For M = SA ∗ A+ε (X ∘ ), the hypothesis of Lemma 2.1.20 is not too restrictive. It is fulfilled, e.g., if the complement of D has no compact components in X ∘ , see [269]. In particular, this is the case if ∂D is connected. Applying to {U i }i∈N the Gram–Schmidt orthogonalization procedure with respect to the scalar product (⋅, ⋅)ε , we obtain an orthonormal basis {b i = b i (ε)} i∈N in DA ∩ SA ∗ A+ε (D).

2.1 Mixed Problems with a Parameter

| 69

The equality (2.27) suggests us to look for solutions to mixed problem (2.11) of the form ∞

u ε (f, h) = Φ ε (A∗ f + εh) + ∑ c i (ε)b i (ε)

(2.28)

i=1

where the series on the right-hand side converges in DA . The point is to find the coefficients c i (ε) through f and h. For this purpose, we denote by ΠS,ε the orthogonal projection DA ∩ SA ∗ A+ε (D) → DT ∩ SA ∗ A+ε (D) with respect to the scalar product (⋅, ⋅)ε . Lemma 2.1.21. Each solution u ε (f, h) ∈ DT of problem (2.11) may be written as the series (2.28) where c i (ε) = (f, AΠS,ε b i )L2 (D,F) + ε (h, ΠS,ε b i )L2 (D,E) − (Φ ε (A∗ f + εh), b i )ε . Proof. Indeed, let u ε ∈ DT be a solution of problem (2.11). As we have seen in Section 2.1.3, (A∗ A + ε)u ε = A∗ f + εh in D. Using (2.27) we easily arrive at (2.28) with some uniquely defined coefficients c i (ε). Write Π̃ S,ε for the orthogonal projection DA → DT with respect to (⋅, ⋅)ε . Since ̃ ΠS,ε is self-adjoint in DA , we get (u ε , b i )ε = (Π̃ S,ε u ε , b i )ε = (u ε , Π̃ S,ε b i )ε

(2.29)

= (f, A Π̃ S,ε b i )L2 (D,F) + ε (h, Π̃ S,ε b i )L2 (D,E) , the last equality being a consequence of (2.11). Now (2.28) implies (u ε , b i )ε = (Φ ε (A∗ f + εh), b i )ε + c i (ε) . Combining this with (2.29) yields c i (ε) = (f, A Π̃ S,ε b i )L2 (D,F) + ε (h, Π̃ S,ε b i )L2 (D,E) − (Φ ε (A∗ f + εh), b i )ε . Finally, for every 𝑣 ∈ C∞ comp (D, E) we get (Π̃ S,ε b i , (A∗ A + ε)𝑣)L2 (D,E) = (A Π̃ S,ε b i , A𝑣)L2 (D,F) + ε (Π̃ S,ε b i , 𝑣)L2 (D,E) = (b i , Π̃ S,ε 𝑣)ε = (b i , 𝑣)ε = ((A∗ A + ε)b i , 𝑣)L2 (D,E) =0. This means that Π̃ S,ε b i belongs to DT ∩ SA ∗ A+ε (D) whence Π̃ S,ε b i = ΠS,ε b i , showing the lemma.

70 | 2 Regular Perturbation of Ill-Posed Problems We have thus derived expressions for the coefficients c i (ε) through f and h. However, it is not an easy task to explicitly construct the family of projections {ΠS,ε }. Lemma 2.1.22. For every u ∈ DA ∩ SA ∗ A+ε (D), the projection ΠS,ε u just amounts to the solution of problem (2.11) with f = Au and h = u. Proof. By the very definition, ΠS,ε u ∈ DT ∩ SA ∗ A+ε (D) and (u − ΠS,ε u, 𝑣)ε = 0 for all 𝑣 ∈ DT satisfying (A∗ A + ε)𝑣 = 0 in D. Further, the solution u ε = u ε (Au, u) of problem (2.11) with f = Au and h = u belongs to DT ∩ SA ∗ A+ε (D) because A∗ f + εh = (A∗ A + ε)u = 0. Moreover, (2.12) gives (u − u ε , 𝑣)ε = 0 for all 𝑣 ∈ DT . We wish to show that ΠS,ε u = u ε , which is equivalent to ‖ΠS,ε u − u ε ‖ε = 0. To this end, write (ΠS,ε u − u ε , ΠS,ε u − u ε )ε = − ((u − ΠS,ε u) − (u − u ε ), ΠS,ε u − u ε )ε = − (u − ΠS,ε u, ΠS,ε u − u ε )ε − (u − u ε , ΠS,ε u − u ε )ε . By the above, both summands on the right-hand side vanish because ΠS,ε u − u ε belongs to DT ∩ SA ∗ A+ε (D). Of course, the lemma does not allow one to effectively determine the Fourier coefficients c i . On the one hand, to find c i we only need to know ΠS,ε b i . On the other hand, this requires, by Lemma 2.1.22, a solution of problem (2.11) with very special data f and h. Let us now describe how to find solutions to problem (2.11) for “good” data. For this purpose we introduce for s ≥ 2m the Hermitian form h(u, 𝑣) = (t(u), t(𝑣))⊕H s−mj −1/2 (S,F j ) + (n(Au), n(A𝑣))⊕H s−m−mj −1/2 (∂D\S∘ ,F j ) on the space H of all u ∈ DA ∩ SA ∗ A+ε (D) with the property that s−m j −1/2 t(u) ∈ ⊕m−1 (S, F j ) , j=0 H s−m−m j −1/2 n(Au) ∈ ⊕m−1 (∂D \ S∘ , F j ) , j=0 H

the expressions t(u) and n(Au) being understood in the sense of weak boundary values. Lemma 2.1.23. Suppose s ≥ 2m. When endowed with the scalar product h(⋅, ⋅), H is a Hilbert space.

2.1 Mixed Problems with a Parameter | 71

Proof. Indeed, (2.16) implies that h(⋅, ⋅) is a scalar product on H. Moreover, if {u k } is a Cauchy sequence in H then it a Cauchy sequence in DA ∩ SA ∗ A+ε (D). Since this latter is a Hilbert space, {u k } has a limit u in this space. Moreover, both {t(u k )} and {n(Au k )} converge to t(u) and n(Au) in the space of distributions on ∂D, or, more precisely, −m j −1/2 (∂D, F ) and ⊕ m−1 H −m−m j −1/2 (∂D, F ), respectively. By assumption, in ⊕ m−1 j j j=0 H j=0 s−m j −1/2 (S, F ) {t(u k )} and {n(Au k )} are Cauchy sequences in the Hilbert spaces ⊕m−1 j j=0 H s−m−m j −1/2 (∂D \ S∘ , F ), respectively. Hence, they converge to elements u H and ⊕m−1 j 0 j=0 and u 1 in these spaces. Finally, the uniqueness of a limit yields t(u) = u 0 on S and n(Au) = u 1 on ∂D \ S∘ , i.e., u ∈ H, which completes the proof. Let {U i }i∈N be a complete linearly independent system in H. Applying the Gram– Schmidt orthogonalization to {U i }i∈N we get an orthonormal basis {B i }i∈N in H. Theorem 2.1.24. Let s ≥ 2m. Then, for every w ∈ H s−2m (D, E) and s−m j −1/2 (S, F j ) , u 0 ∈ ⊕m−1 j=0 H s−m−m j −1/2 u 1 ∈ ⊕m−1 (∂D \ S∘ , F j ) , j=0 H

the series

∞

u = Φ ε (w) + ∑ k i B i i=1

converges in DA and satisfies (2.14), provided that k i = h(u − Φ ε (w), B i ) . Proof. This is a direct consequence of Theorem 2.1.12. Recall that the boundary equations t(u) = u 0 on S and n(Au) = u 1 on ∂D\S∘ are interpreted in the sense of (2.15). From this theorem we deduce, in particular, that ∞

ΠS,ε b i = ∑ k i B i , q=1

with the coefficients k i = (n(Ab i )), n(AB i ))⊕H s−m−mj −1/2 (∂D\S∘ ,F j ) . Knowing ΠS,ε b i we can find, by Lemma 2.1.21, the solution of problem (2.11) for any data f ∈ L2 (D, F) and h ∈ L2 (D, F). Of course, if both f and h are smooth enough, namely f ∈ H s−m (D, F) and h ∈ H s−2m (D, E) with s ≥ 2m, then we can determine the solution of problem (2.11) directly by Theorem 2.1.24. One question still unanswered is whether a complete system {U i }i∈N in H may be chosen to consists of solutions to (A∗ A + ε)u = 0 on neighborhoods of D. Analysis similar to that in the proof of Lemma 2.1.20 shows that this is always the case if ∂D is smooth enough, e.g., of class C2m−1 .

72 | 2 Regular Perturbation of Ill-Posed Problems

2.1.7 Dirac Operators Let X = Rn , where n ≥ 2, and E = Rn × Ck , F = Rn × Cl . The sections of E are functions of n real variables with values in Ck , and similarly for F. Let A be a Dirac operator, i.e., a homogeneous first-order differential operator with constant coefficients in Rn , n ∂ , A = ∑ Aj ∂x j j=1 such that (σ(A)(ξ))∗ σ(A)(ξ) = |ξ|2 E k

(2.30)

for all ξ ∈ Rn . Here, A j are (l × k) -matrices of complex numbers and E k is the identity (k × k) -matrix. The Dirac operators satisfy A∗ A = −E k ∆, where ∆ is the usual Laplace operator n in R . The perturbed mixed problem (2.12) reads as ∗ {(−∆ + ε)u ε = A f { { t(u ε ) = 0 { { { {n(Au ε ) = n(f)

in

D;

on

S,

on

∂D \ S∘ ,

where n(f) = (σ(A)(∇ϱ))∗ f and ϱ is a defining function of D in the sense of (2.2). Thus, this is a family of mixed problems for the Helmholtz equation. We are going to study the (Cauchy) problem (2.4) on the unit ball D = B in Rn . To this end, we pass to spherical coordinates x = r S(φ) where φ are coordinates on the unit sphere ∂D = S in Rn . The Laplace operator ∆ in the spherical coordinates takes the form 1 ∂ 2 ∂ ∆ = 2 ((r ) + (n − 2)(r ) − ∆S ) , (2.31) ∂r ∂r r where ∆S is the Laplace–Beltrami operator on the unit sphere. To solve the homogeneous equation (−∆ + ε)u ε = 0 we make use of the Fourier method of separation of variables. Writing u ε (r, φ) = g(r, ε)h(φ) we get two separate equations for g and h, namely ((r

∂ 2 ∂ ) + (n − 2)(r ) − εr2 ) g = c g ∂r ∂r ∆S h = c h ,

c being an arbitrary constant.

2.1 Mixed Problems with a Parameter

| 73

The second equation has nonzero solutions if and only if c is an eigenvalue of ∆S . These are well known to be c = i(n + i − 2), for i = 0, 1, . . . (see for instance [273]). The corresponding eigenfunctions of ∆S are spherical harmonics h i (φ) of degree i, i.e., ∆S h i = i(n + i − 2) h i .

(2.32)

Consider now the following ordinary differential equation with respect to the variable r>0 ∂ 2 ∂ (2.33) ((r ) + (n − 2)(r ) − (i(n + i − 2) + εr2 )) g(r, ε) = 0 . ∂r ∂r This is a version of the Bessel equation, and the space of its solutions is two-dimensional. For example, if ε = 0 then g(r, 0) = ar i + br2−i−n with arbitrary constants a and b is a general solution to (2.33). In this situation any function r i h i (φ) is a homogeneous harmonic polynomial. In the general case the space of solutions to (2.33) contains a one-dimensional subspace of functions bounded at the point r = 0, cf. [273]. For i = 0, 1, . . ., fix a nonzero solution g i (r, ε) of (2.33) which is bounded at r = 0. Then (2.34) (−∆ + ε) (g i (r, ε)h i (φ)) = 0 on all of Rn . Indeed, by (2.31), (2.32) and (2.33) we conclude that this equality holds in Rn \ {0}. We now use the fact that g i (r, ε)h i (φ) is bounded at the origin to see that (2.34) holds. It is known that there are exactly J(i) =

(n + 2i − 2)(n + i − 3)! i!(n − 2)!

linearly independent spherical harmonics of degree i. In [249] a system (j)

{H i (φ)}

i=0,1,... j=1,...,k J(i)

of Ck -valued functions is constructed, such that (j) 1) the components of H i (φ) are spherical harmonics of degree i; (j)

2) {H i (φ)} is an orthonormal basis in L2 (S, E); (j)

3) {A (r i H i (φ))} is an orthogonal system in L2 (B, F). (j)

More precisely, this system {H i (φ)} consists of eigenfunctions of the operator n ∘ A, (j)

(j)

(j)

(σ(A)(rS(φ)))∗ A (r i H i (φ)) = λ i (r i H i (φ)) , (j)

where λ i ≥ 0. Lemma 2.1.25. The system (j)

(j)

{b i (r, φ, ε) := g i (r, ε) H i (φ)}

i=0,1,... j=1,...,k J(i)

is orthogonal with respect to both Hermitian forms (⋅, ⋅)L2 (B,E) and (A⋅, A⋅)L2 (B,F).

(2.35)

74 | 2 Regular Perturbation of Ill-Posed Problems (j)

Proof. Indeed, as {H i } is an orthonormal basis in the space L2 (S, E) on the unit (j)

sphere, the system {b i } is orthogonal in L2 (B, E) because (j) (q) (b i , b p )L2 (B,E)

=

1 (j) (q) (H i , H p )L2 (S,E) ∫ r n−1 g i (r, ε)g p (r, ε) dr 0

=0 for i ≠ p or j ≠ q. Further, integrating by parts we get (j)

(j)

(q)

(j)

(q)

(q)

(Ab i , Ab p )L2 (B,F) = −(b i , ∆b p )L2 (B,E) + g i (1, ε) (H i , n(Ab p ))L2 (S,E) .

(2.36)

On the other hand, (2.34) implies (j)

(q)

(j)

(q)

− (b i , ∆b p )L2 (B,E) + ε (b i , b p )L2 (B,E) = 0

(2.37)

for i ≠ p or j ≠ q. Let us write the expression n ∘ A in spherical coordinates. Denote by S󸀠 (φ) the Jacobi matrix of S(φ). Set (S󸀠 (φ))

−1

T

:= ((S󸀠 (φ)) S󸀠 (φ))

−1

T

(S󸀠 (φ)) . T

Since the rank of S󸀠 (φ) is equal to n − 1, the inverse matrix of (S󸀠 (φ)) S󸀠 (φ) exists and −1 is smooth. Moreover, (S󸀠 (φ)) is a left inverse for S󸀠 (φ). An easy calculation shows that ∂ 1 n−1 󸀠 ∂ −1 ∂ + = S j (φ) ∑ (S (φ)) i,j ∂x j ∂r r i=1 ∂φ i −1

−1

where (S󸀠 (φ)) i,j is the (i, j) -entry of (S󸀠 (φ)) . Now (2.30) implies n

n

k=1

j=1

n ∘ A = ∑ A∗k rS k (φ) ∑ A j where

∂ ∂ = r + R(φ, ∂ φ ) ∂x j ∂r

n

n

n−1

k=1

j=1

i=1

−1

R(φ, ∂ φ ) = ∑ A∗k S k (φ) ∑ A j ∑ (S󸀠 (φ)) i,j

∂ . ∂φ i

Using (2.35) and (2.38) we conclude that (j)

(j)

(j)

λ i (r i H i (φ)) = n(A(r i H i (φ))) (j)

(j)

= i r i H i (φ) + r i R(φ, ∂ φ )H i (φ) . Hence,

(j)

(j)

(j)

R(φ, ∂ φ )H i (φ) = (λ i − i) H i (φ) ,

(2.38)

2.1 Mixed Problems with a Parameter

| 75

and so (2.38) yields (j)

(j)

(j)

n(Ab i ) = r g󸀠i H i + g i R(φ, ∂ φ )H i = (rg󸀠i +

(j) (λ i

−

(j) i)g i ) H i

(2.39)

.

Therefore, (j)

(q)

(H i , n(Ab p ))L2 (S,E) = 0

(2.40)

for i ≠ p or j ≠ q. (j) Combining (2.36) (2.37) and (2.40) we see that the system {b i } is orthogonal with respect to (A⋅, A⋅)L2 (B,F). (j)

Remark 2.1.26. Note that g󸀠i (1, ε) + (λ i − i)g i (1, ε) ≠ 0 for all ε > 0. Indeed, otherwise (j) n(Ab i )

(j)

= 0 on S and (2.36), (2.37) would imply b i ≡ 0, which is wrong.

Theorem 2.1.27. For every δ > 0, the system (j)

{b i (r, φ, ε)}

i=0,1,... j=1,...,k J(i)

is an orthogonal basis in the space DA ∩ S−∆+εE k (B) with the scalar product (⋅, ⋅)δ . Proof. The orthogonality follows immediately from Lemma 2.1.25. As for the complete(j) ness of the system {b i } in DA ∩ S−∆+εE k (B), we observe that the estimates (2.10) guarantee that every scalar product (⋅, ⋅)δ with δ > 0 induces in DA the same topology as D(⋅, ⋅). Hence, it is sufficient to prove the completeness for δ = 1. Finally, since (j) (j) the system of harmonics {H i } is dense in C m−1 (S, E) we see that {b i } is dense in C m−1 (S, E)∩S−∆+εE k (B). Then the completeness is a consequence of Lemma 2.1.20. As a fundamental solution Φ ε (x, y) of the operator −∆ + ε in R3 we may choose one of the standard kernels Φ ε (x, y) = e±√ε|x−y| . In R2 we can take as Φ ε (x, y) a Hankel function, see for instance [273]. Example 2.1.28. Let A = ∇ be the gradient operator in Rn . For every domain D ⋐ Rn , we have DA = H 1 (D). Since the estimate (2.24) holds true for ∇ (see [196]), the (Cauchy) problem (2.4) is well posed in DT . In this case k = 1, l = n, A∗ = − div is a multiple of the divergence operator in Rn and n ∘ A = |x|

∂ ∂ =r ∂n ∂r

where ∂/∂n is the derivative along the outward unit normal vector to ∂D. In particular, this means that every homogeneous harmonic polynomial r i h i is an eigenfunction of

76 | 2 Regular Perturbation of Ill-Posed Problems n ∘ A corresponding to the eigenvalue λ i = i. For example, in R2 we can take (1)

1 g0 (r, ε) , √2π 1 g i (r, ε) cos(iφ) , = √π 1 g i (r, ε) sin(iφ) , = √π

b0 = (1)

bi

(2)

bi

where g i are Hankel’s functions. In the case s = 5/2 and S = {r = 1, φ ∈ [0, π]} the Gram–Schmidt orthogonalization in H gives g0 (r, ε)

(1)

B0 =

√π√|g0 (1, ε)|2 + |g󸀠0 (1, ε)|2 2 g1 (r, ε) cos φ

(1)

B1 = (2)

B1 =

√π√|g1 (1, ε)|2 + |g󸀠1 (1, ε)|2

,

,

2ag0 (r, ε) + √πg1 (r, ε) sin φ , √b

with a = g0 (1, ε)g1 (1, ε) − g󸀠0 (1, ε)g󸀠1 (1, ε) , 3 b = π2 + 4a(1 + |g0 (1, ε)|2 + |g󸀠0 (1, ε)|2 ) , 2 and so on. Example 2.1.29. Let A := ∂1 + √−1∂2 be (2 -multiple of) the Cauchy–Riemann operator in C. Then the (Cauchy) problem (2.4) is ill posed in DT . In this case k = l = 1, A∗ = −∂1 + √−1∂2 and ∂ √ ∂ n ∘ A = z̄ ∂̄ = r + −1 ∂r ∂φ (j)

hold. The system {b i } may be chosen as follows (1)

1 g0 (r, ε) , √2π 1 √ g i (r, ε)e −1 iφ , = √π 1 √ g i (r, ε)e− −1 iφ , = √π

b0 = (1)

bi

(2)

bi (1)

(1)

with λ0 = 0, λ i

(2)

= 0 and λ i

= 2i.

2.2 Kernel Spikes of Singular Problems Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodiffer-

2.2 Kernel Spikes of Singular Problems

| 77

ential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This section is aimed at describing refined asymptotics in the Dirichlet problem in a ball. The beauty of explicit formulas actually highlights the structure of asymptotic expansions in the calculi on singular varieties.

2.2.1 Soft Expansions Suppose u is a harmonic function in B of finite order of growth near the boundary S. Then u has weak limit values u 0 ∈ D󸀠 (S) on the sphere in the sense that lim ∫ u((1 − ε)y)𝑣(y) dσ = ⟨u 0 , 𝑣⟩

ε→0

S

for each 𝑣 ∈ C∞ (S), where dσ is the Lebesgue measure on S. Moreover, the function u can be reconstructed from its weak boundary values by the Poisson formula u(x) = ⟨u 0 , ℘(x, ⋅)⟩ for x ∈ B. This formula shows in particular that the behavior of u close to a boundary point y ∈ S in B is completely determined by the behavior of u 0 near y on S. Conversely, given any distribution u 0 on S, the function u(x) = ⟨u 0 , ℘(x, ⋅)⟩, x ∈ B, is harmonic in B and its weak limit values on S coincide with u 0 . For these and more general results we refer the reader to [229]. What happens when u 0 fails to be a distribution near a compact set K of zero measure on S? Consider the simplest case where K = {y0 } is a fixed point of S and u 0 (y) = |y − y0 |z , with ℜz ≤ −(n −1). This function u 0 can be extended to a distribution on all of S in many ways. Pick e.g., a local chart U around y0 on S with coordinates y󸀠 = (y1 , . . . , y n−1 ) and a function χ ∈ C∞ (S) supported in U, which is equal to 1 near y0 . If N is a natural number with ℜz + N > −(n − 1), then β

∂ y󸀠 𝑣(y0 )

⟨u 0,N , 𝑣⟩ = ⟨u 0 , χ (𝑣(y󸀠 ) − ∑ |β| 2. Set { u(x) if x ∈ B ∪ O , U(x) = { −Ku(x) if x ∈ Rn \ B . { This function is continuous away from the closed set S \ O in Rn because u vanishes on O. Moreover, n xj ∂ ∂ 1 x Ku(x) = ∑ ( n−2 u ( 2 )) ∂|x| |x| ∂x |x| |x| j j=1

=

x 1 ∂u x 2−n u( 2)− n ( 2) n−1 |x| ∂|x| |x| |x| |x|

for any |x| > 2. Hence, it follows that the derivatives of u and −Ku in the outward normal vector to S coincide on O. By Theorem 3.2 of [270] we conclude that U(x) is a harmonic function on B ∪ O ∪ (Rn \ B), which completes the proof. Given any harmonic function u in B vanishing on S \ {y0 }, Lemma 2.2.3 shows that u extends to a harmonic function U on Rn \ {y0 }, which is given by {u(x) U(x) = { −Ku(x) {

for

x ∈ B \ {y0 } ,

for

x ∈ Rn \ B .

(2.43)

Recall that the Poisson kernel for the ball B is ℘(x, y) =

1 1 − |x|2 , σ n |x − y|n

where σ n is the area of the unit sphere in Rn . It is actually defined away from the diagonal in Rn × Rn . Lemma 2.2.1 prompts that ℘(x, y0 ) survives under the transformation (2.43). n−1 Lemma 2.2.4. Given any β ∈ Z≥0 , the equality

−K ((−∂ y󸀠 )β ℘) (x, y0 ) = (−∂ y󸀠 )β ℘(x, y0 ) holds for all x ∈ Rn \ B.

80 | 2 Regular Perturbation of Ill-Posed Problems

Proof. Since K is applied in x one can assume without restriction of generality that β = 0. In this particular case the verification is fairly straightforward.

2.2.3 Auxiliary Results Expansion (2.41) is not invariant because it contains the derivatives (−∂ y󸀠 )β in local coordinates near y0 on S. To get rid of the noninvariance we can express the local derivatives on S through the derivatives in the coordinates of the surrounding space Rn . Assume, e.g., that y0 lies in the upper half-space x n > 0. Then S near y0 is the graph of y n = √1 − |y󸀠 |2 where y󸀠 = (y1 , . . . , y n−1 ), hence we may take y󸀠 as local coordinates on S. For j = 1, . . . , n − 1, the full derivative in y j is ∂ j − (y j /y n )∂ n . The tangential space to S is also spanned by the system of vector fields yj ∂ ∂ − , ∂y j |y| ∂|y| j = 1, . . . , n, although these latter are not independent. Substituting this into (2.41) leads to ∞ h j (x − y0 ) ) ℘(x, y0 ), x ∈ B , (2.44) u(x) = ⟨u 0,R , ℘(x, ⋅)⟩ + ( ∑ |x − y0 |2j j=0 where h j (z) are homogeneous polynomials of degree j whose coefficients satisfy growth estimates 2j lim √

j→∞

1 |h j (D)∗ h j (z)| = 0 , j!

(2.45)

cf. Corollary 8.11 in [270]. Lemma 2.2.5. Let h(z) be a homogeneous polynomial of degree j on Rn . In order that h(x − y0 ) ℘(x, y0 ) |x − y0 |2j be harmonic in x ∈ B it is necessary and sufficient that (1 − |x|2 ) ∆h(x − y0 ) − 4 ⟨∇h(x − y0 ), y0 ⟩ ≡ 0 .

(2.46)

Proof. Indeed, a trivial verification shows that ∆(D x ) (

h(x−y0 ) 1 (1−|x|2 )∆h(x−y0 ) − 4⟨∇h(x−y0 ), y0 ⟩ ℘(x, y )) = 0 σn |x−y0 |2j |x−y0 |n+2j

for all x ≠ y0 . Since the numerator on the right-hand side is a polynomial in x, it vanishes for all x ∈ B if and only if it vanishes identically on Rn . Hence, the lemma follows.

2.2 Kernel Spikes of Singular Problems |

81

Replacing x − y0 by z in (2.46) yields (|z|2 + 2⟨z, y0 ⟩)∆h(z) + 4 ⟨∇h(z), y0 ⟩ = 0 for all z ∈ Rn . As the left-hand side is the sum of two homogeneous polynomials of orders j and j−1, respectively, we readily conclude that the identity (2.46) is equivalent to {|z|2 ∆h(z) = 0, { ⟨∆h(z) z + 2 ∇h(z), y0 ⟩ = 0 , { or {∆h(z) = 0, (2.47) { ⟨∇h(z), y0 ⟩ = 0 { for all z ∈ Rn . Our next goal is to show that the terms of (2.44) are invariant under the transformation (2.43). Lemma 2.2.6. Given any j ∈ Z≥0 , the equality −K (

h j (x − y0 ) h j (x − y0 ) ℘(x, y0 )) = ℘(x, y0 ) |x − y0 |2j |x − y0 |2j

holds for all x ∈ Rn \ B. Proof. By Lemma 2.2.4 it suffices to show that h j ( |x|x 2 − y0 ) h j (x − y0 ) 󵄨󵄨 x 󵄨2j = |x − y0 |2j 󵄨󵄨 2 − y0 󵄨󵄨󵄨 󵄨󵄨 |x| 󵄨󵄨 for all x ∈ Rn \ B. This equality easily reduces to h j (x − |x|2 y0 ) = h j (x − y0 ) .

(2.48)

To prove this latter we observe that the polynomial on the left-hand side is harmonic, which is a consequence of (2.47). The polynomial on the right-hand side of (2.48) is harmonic, too. These polynomials coincide on the unit sphere S, hence they are equal on all of Rn , as desired.

2.2.4 Formulas for Coefficients Suppose u is a harmonic function in B vanishing on S\{y0 }. By Lemma 2.2.3, u extends to a harmonic function U on Rn \ {y0 }. The harmonic continuation U is given by (2.43). Combining (2.44) and Lemma 2.2.6 we deduce that U represents by the formula ∞

U(x) = ( ∑ j=0

h j (x − y0 ) ) ℘(x, y0 ) |x − y0 |2j

(2.49)

82 | 2 Regular Perturbation of Ill-Posed Problems for all x ∈ Rn \ {y0 }. Here h j (z) are homogeneous polynomials of degree j satisfying (2.45) and (2.47). The first of these two conditions implies that series (2.49) converges uniformly in x on compact subsets of Rn \ {y0 }. Since the summands are harmonic functions the series actually converges in the space C∞ (Rn \ {y0 }). Hence, we may integrate it termwise over each cycle away from y0 in Rn . Let G ∆ (g, u) stand for the standard Green operator of the Laplace operator in Rn , i.e., n n ∂u ∂g dx[k] − u ∑ (−1)k−1 dx[k] G ∆ (g, u) = g ∑ (−1)k−1 ∂x ∂x k k k=1 k=1 where dx[k] is the wedge product of the differentials dx1 , . . . , dx n excepting dx k . Lemma 2.2.7. Let D be a bounded domain with smooth boundary, such that y0 ∈ D, and F a harmonic function on D. For any sufficiently small ε > 0 holds ∞

−2(n + 2j) ∫ F(y0 + εz)h j (z) dz σn εj j=0

∫ G ∆ (F, U) = ∑

B

∂D

∞

n + 2j +∑ ∫ F(y0 + εz)h j (z) dσ j j=0 σ n ε S

∞

2(n + 2j) ∫ F(y0 + εz)⟨z, y0 ⟩h j (z) dσ . j+1 j=0 σ n ε

+∑

S

(2.50) Proof. Choose any ε > 0 with the property that the ball B(y0 , ε) lies in D, i.e., ε ≤ dist(y0 , ∂D). Since U is harmonic outside of y0 the Stokes formula gives ∫ G ∆ (F, U) = ∂D

∫

G ∆ (F, U)

∂B(y0 ,ε) ∞

=∑ j=0

∫

G ∆ (F(x),

∂B(y0 ,ε)

h j (x − y0 ) ℘(x, y0 )) . |x − y0 |2j

An easy computation shows that the integrands, if restricted to the sphere ∂B(y0 , ε), are equal to G ∆ (F(x), =

h j (x − y0 ) ℘(x, y0 )) |x − y0 |2j

n ∂ ∂ 1 ∑ (−1)k−1 (F(x) g j (x) − F(x) g j (x))dx[k] n+2j ∂x k ∂x k σn ε k=1

−

n + 2j F(x)g j (x) dσ σ n ε n+2j+1

where g j (x) = (1 − |x|2 )h j (x − y0 ).

2.2 Kernel Spikes of Singular Problems | 83

We now apply Stokes’ formula for B(y0 , ε) to the first term on the right-hand side. Since F is harmonic and h j (x − y0 ) satisfies (2.46) the exterior derivative is 1 −2(n + 2j) F(x) ∆g j (x) dx = F(x)h j (x − y0 ) dx . σ n ε n+2j σ n ε n+2j It follows that ∞

−2(n + 2j) n+2j j=0 σ n ε

∫ G ∆ (F, U) = ∑ ∂D

∞

−∑ j=0

F(x)h j (x − y0 ) dx

∫ B(y0 ,ε)

n + 2j σ n ε n+2j+1

F(x)(1 − |x|2 )h j (x − y0 ) dσ .

∫ ∂B(y0 ,ε)

Changing the variables by x = y0 + εz, with z ∈ B, and taking into account the homogeneity of h j (z) we arrive at (2.50), as desired. The equations (2.47) make it obvious that ∆ (⟨z, y0 ⟩h j (z)) = ⟨z, y0 ⟩ ∆h j (z) + 2 ⟨∇h j (z), y0 ⟩ =0, i.e., ⟨z, y0 ⟩h j (z) is a homogeneous harmonic polynomial of degree j + 1 for all j. Lemma 2.2.8. Let D be a bounded domain with smooth boundary, such that y0 ∈ D. Then for every homogeneous harmonic polynomial H(z) of degree k we have ∫ G ∆ (H(x − y0 ), U(x)) ∂D

=

2(n + 2k − 2) n + 2k − 2 ∫ H(z)h k (z) dσ + ∫ H(z)⟨z, y0 ⟩h k−1 (z) dσ . σn σn S

S

(2.51) Proof. Let us apply (2.50) for F(x) = H(x − y0 ). Then ∫ F(y0 + εz)h j (z)dz = ε k ∫ H(z)h j (z) dz B

B 1 k

= ε ∫ r k+j+n−1 dr ∫ H(z)h j (z) dσ 0

=

S

εk k+j+n

∫ H(z)h j (z) dσ S

and ∫ F(y0 + εz)h j (z)dσ = ε k ∫ H(z)h j (z) dσ , S

S k

∫ F(y0 + εz)⟨z, y0 ⟩h j (z) dσ = ε ∫ H(z)⟨z, y0 ⟩h j (z) dσ S

S

84 | 2 Regular Perturbation of Ill-Posed Problems for all j = 0, 1, . . . . As the homogeneous harmonic polynomials of different degrees are orthogonal under integration over the sphere S, the equality (2.51) follows from (2.50). Formula (2.51) uniquely determines the polynomials h j (z) through the function u in B.

2.2.5 Laurent Series Let (Y k,l (z)) be a set of homogeneous harmonic polynomials in Rn whose restrictions to S form an orthonormal basis in L2 (S) (spherical harmonics). Here k is the degree of Y k,l and, given any k ∈ Z≥0 , the index l varies from 1 to σ(n, k). For example, σ(n, k) =

(n + 2k − 2)(n + k − 3)! k!(n − 2)!

if n > 2, cf. [258, 266]. For k = 0 the only homogeneous harmonic polynomial of degree k and of L2 (S) norm 1 is Y0,1 = 1/√σ n . Hence, h0 = ∫ G ∆ ( ∂D

1 , U(x)) . n−2

(2.52)

We now proceed by induction. Given any k ≥ 1, suppose h0 , h1 , . . . , h k−1 have already been defined. Write σ(n,k)

h k (z) = ∑ c k,l Y k,l (z) , l=1

then the coefficients c k,l are uniquely determined from (2.51). More precisely, we get c k,l = ∫ G ∆ ( ∂D

σn Y k,l (x − y0 ), U(x)) − 2 ∫ Y k,l (z)⟨z, y0 ⟩h k−1 (z) dσ n + 2k − 2

(2.53)

S

for l = 1, . . . , σ(n, k). We are now in a position to formulate the main result of this section. It specifies Laurent series expansions for harmonic functions in B vanishing on ∂B\{y0 }, cf. [270]. Theorem 2.2.9. Let u 0 be a hyperfunction on S \ {y0 }, and u 0,R any extension of u 0 to a hyperfunction on all of S. Every harmonic function u in B equal to u 0 on S \ {y0 } has the form ∞ h j (x − y0 ) u(x) = ⟨u 0,R , ℘(x, ⋅)⟩ + ∑ ℘(x, y0 ), x ∈ B , (2.54) |x − y0 |2j j=0 where (h j (z))j=0,1,... is a sequence of homogeneous harmonic polynomials of degree j in Rn , which are uniquely determined by formulas (2.52) and (2.53) with u − ⟨u 0,R , ℘(x, ⋅)⟩ in place of u.

2.2 Kernel Spikes of Singular Problems | 85

Proof. The theorem follows from Lemma 2.2.1 completed by the recurrence relation (2.53). What is left is to show that the polynomials h j are actually independent of the particular choice of the system (Y k,l ). To this end we assume that ∞

∑ j=0

h j (x − y0 ) ℘(x, y0 ) = 0 |x − y0 |2j

for all x ∈ B. Since ℘(x, y0 ) ≠ 0 in B it follows that the series ∞

∑ j=0

h j (x − y0 ) |x − y0 |2j

vanishes in B. A familiar homogeneity argument now shows that all the h j are identically zero, as desired.

2.2.6 Expansion of the Poisson Kernel In this section we sketch a direct approach to formula (2.54), which is based on the expansion of the Poisson kernel ℘(x, y) in spherical harmonics. A similar expansion for the standard fundamental solution of the Laplace operator goes back as far as [55, 249], etc. Lemma 2.2.10. In the cone Cy0 = {(x, y) ∈ Rn × Rn : |x − y0 | > |y − y0 |} the equality holds ∞ σ(n,k) Y k,l (x−y0 ) σn Y (y−y0 ) ℘(x, y0 ) ℘(x, y) = ∑ ∑ |y−y0 | 2 k,l |x−y0 |2k k=0 l=1 1 − ( |x−y0 | ) where the series converges absolutely together with all derivatives uniformly on compact subsets of Cy0 . Proof. Setting x − y0 = w and y − y0 = z and canceling the factor 1 − |x|2 on both sides of the equality, we reduce the expansion to ∞ σ(n,k) Y k,l (w) 1 σn 1 = ∑ ∑ Y k,l (z) |w − z|n k=0 l=1 1 − (|z|/|w|)2 |w|2k |w|n

(2.55)

for all w, z ∈ Rn with |w| > |z|. Let z ∈ B be fixed. We represent |w − z|−n by the Fourier series in L2 (S). Namely, ∞ σ(n,k) 1 = ∑ ∑ c k,l (z)Y k,l (w) n |w − z| k=0 l=1

where c k,l (z) are the Fourier coefficients of |w − z|−n with respect to the system (Y k,l ), i.e., 1 Y k,l (w) dσ . c k,l (z) = ∫ |w − z|n S

86 | 2 Regular Perturbation of Ill-Posed Problems These integrals can be easily evaluated by the Poisson formula, for Y k,l (w) are harmonic. Namely, we get c k,l (z) =

σn ∫ ℘(z, w) Y k,l (w) dσ 1 − |z|2 S

σn Y k,l (z) = 1 − |z|2 whence

∞ σ(n,k) 1 σn = ∑ ∑ Y k,l (z)Y k,l (w) , n |w − z| 1 − |z|2 k=0 l=1

the series converging in the norm of L2 (S) uniformly in z on compact subsets of B. The harmonic extension with respect to w leads us to the equality 1 1 σ n |w|n−2

󵄨󵄨 w 󵄨󵄨2 󵄨󵄨 2 󵄨󵄨 − |z|2 ∞ σ(n,k) 󵄨󵄨 |w| 󵄨󵄨 󵄨󵄨 w 󵄨󵄨 n = ∑ ∑ Y k,l (z)Y k,l (w) k=0 l=1 󵄨󵄨󵄨 |w|2 − z󵄨󵄨󵄨 󵄨 󵄨

for all w ∈ B, where the series converges absolutely and uniformly with respect to w and z on compact subsets of B. Applying to this the Kelvin transformation in w yields ∞ σ(n,k) Y k,l (w) 1 σn 1 = ∑ ∑ Y k,l (z) , n 2 |w − z| 1 − (|z|/|w|) |w|2k |w|n k=0 l=1

which proves (2.55).

2.3 An Asymptotic Expansion of the Martinelli–Bochner Integral Any locally integrable function f on a hypersurface S ⊂ Cn with singular points can be represented in the form f = h+ − h− , where h± are holomorphic functions on the opposite sides of S. The asymptotic behavior of h± close to any singular point ζ 0 ∈ S is completely determined by that of the Bochner–Martinelli integral of f . If one looks for an operator algebra related to the problem of analytic representation one is thus led to the study of the Bochner–Martinelli integral in function spaces with asymptotics on S. While singular points complicate the problem the key question is of how to regularize the divergent Bochner–Martinelli integral. There is no canonical way to do this. At first glance the only essential condition for the regularization is that it should differ from the integral itself by a smoothing operator outside of the singular points. However, one should be able to handle the remainders within the calculus, so the less awkward terms have to be removed, the better the regularization is. The aim of this section is to show a regularization of the Bochner–Martinelli integral in a ball B ⊂ Cn . To this end we put an artificial singularity in a boundary point

2.3 An Asymptotic Expansion of the Martinelli–Bochner Integral |

87

ζ 0 ∈ ∂B. For the regularization we derive an asymptotic expansion near the distinguished point ζ 0 . This particular result seems to be of general interest, for it actually relies on the explicit solution of the Dirichlet problem in a ball. Since the Dirichlet problem is solvable even in “wild” domains, we can use the solution to reduce other problems to the boundary.

2.3.1 Asymptotic Expansion Let B = {z ∈ Cn : |z| < 1} be the unit ball in Cn and S = ∂B be the unit sphere. The Bochner–Martinelli kernel in Cn is U(ζ, z) = where

ζ j̄ − z̄ j (n − 1)! n ∑ (−1)j−1 d ζ ̄ [j] ∧ dζ n (2πi) j=1 |ζ − z|2n

dζ = dζ1 ∧ . . . ∧ dζ n , ̄ ∧ d ζ j+1 ̄ ∧ . . . ∧ d ζ n̄ . d ζ ̄ [j] = d ζ1̄ ∧ . . . ∧ d ζ j−1

Given a fixed point ζ 0 ∈ S, we will consider functions f ∈ C(S \ {ζ 0 }) which admit asymptotic expansions f(ζ) =

1 cN + O( ) 0 2N |ζ − ζ | |ζ − ζ 0 |2N−1

(2.56)

for ζ → ζ 0 , where c N is a complex constant. We are looking for an asymptotic expansion of the Bochner–Martinelli integral (Mf ) (z) = ∫ f(ζ)U(ζ, z), z ∈ B , S

when z → z0 . If N ≥ n then the integral

M |ζ − ζ 0 |−2N

diverges. To evaluate Mf we therefore need a regularization for such integrals unless N < n. This will be done in Section 2.3.3. Denote by Cζ 0 the circular cone of angle α < π/2 whose axis is the inward normal vector to S at ζ 0 . Theorem 2.3.1. If f ∈ C(S \ {ζ 0 }) has asymptotic expansion (2.56), then Mf fulfills (Mf ) (z) =

cN 1 1 + O( ) 2 |z − ζ 0 |2N |z − ζ 0 |2N−1

when z → ζ 0 within the cone Cζ 0 ∩ {z ∈ Cn : ℑ⟨z − ζ 0 , ζ ̄ 0 ⟩ = 0}. For the proof we need a number of lemmas.

(2.57)

88 | 2 Regular Perturbation of Ill-Posed Problems

2.3.2 The Bochner–Martinelli Integral Denote by ℘(ζ, z) the Poisson kernel for the ball, i.e., ℘(ζ, z) =

1 1 − |z|2 dσ(ζ) , σ 2n |ζ − z|2n

where dσ is the Lebesgue measure on S and σ 2n = 2π n /Γ(n) the area of S. Lemma 2.3.2.

1 − ⟨ζ, z⟩̄ 󵄨 U(ζ, z)󵄨󵄨󵄨S = ℘(ζ, z) , 1 − |z|2

where ⟨ζ, z⟩ = ∑nj=1 ζ j z j . Proof. Cf. Chapter 1 in [157]. Denote by n

∂ , ∂z j

E = ∑ zj j=1

n ∂ Ē = ∑ z̄ j ∂ z̄ j j=1

the Euler operator and its conjugate in the variables of Cn . When writing z j = x j + ıx n+j for j = 1, . . . , n, one has n

E + Ē = ∑ (x j j=1

=: |z|

∂ ∂ + x n+j ) ∂x j ∂x n+j

∂ . ∂|z|

Lemma 2.3.3. Assume F is a harmonic function of Hardy class H 1 in B. Then 1

(MF) (z) = F(z) − Ē ∫ F (tz) t n−2 dt . 0

Proof. Let F(z) = ∑ Y k,l (z) k,l≥0

be an expansion of F in homogeneous harmonic polynomials Y k,l , Y k,l being of degree k in z and degree l in z.̄ The series converges in the norm of L1 (S), hence uniformly in B. It follows that 1

1

∫ F (tz) t 0

n−2

dt = ∑ ∫ Y k,l (tz) t n−2 dt k,l≥0 0

2.3 An Asymptotic Expansion of the Martinelli–Bochner Integral

|

89

1

= ∑ ∫ Y k,l (z) t k+l+n−2 dt k,l≥0 0

1 Y k,l (z) , k + l + n−1 k,l≥0

= ∑ and so

1

l Y k,l (z) . k + l + n−1 k,l≥0

Ē ∫ F (tz) t n−2 dt = ∑ 0

By [230], k+n−1 Y k,l k+l+n−1 l = (1 − ) Y k,l k+l+n−1

MY k,l =

whence l Y k,l (z) k+l+n−1 k,l≥0

(MF) (z) = ∑ Y k,l (z) − ∑ k,l≥0 1

= F(z) − Ē ∫ F (tz) t n−2 dt , 0

as desired. Lemma 2.3.4. Let N > 0 be an arbitrary real number and f a smooth function in B. Then 1

f(z) = ∫ ((E + Ē + N)f ) (tz) t N−1 dt

(2.58)

0

for all z ∈ B. Proof. An easy calculation shows that ̄ ) (tz) = t ((E + E)f

∂ (f(tz)) ∂t

whence 1

1

∫ ((E + Ē + N)f ) (tz) t N−1 dt = ∫ 0

1

∂ (f(tz)) t N dt + N ∫ f(tz) t N−1 dt ∂t

0

󵄨󵄨t=1 = f(tz) t 󵄨󵄨󵄨 󵄨t=0 = f(z) , N

as desired.

0

90 | 2 Regular Perturbation of Ill-Posed Problems

For f(z) =

1 formula (2.58) reads |z − ζ 0 |2N 1

1 − |tz|2 1 = N t N−1 dt . ∫ |z − ζ 0 |2N |ζ 0 − tz|2N+2

(2.59)

0

Lemma 2.3.5. If z → ζ 0 within the cone Cζ 0 ∩ {z ∈ Cn : ℑ⟨z − ζ 0 , ζ ̄ 0 ⟩ = 0}, then M(

1 1 1 1 ) (z) = + O( ) . 2 |z − ζ 0 |2n−2 |ζ − ζ 0 |2n−2 |z − ζ 0 |2n−3

Proof. Since F(z) =

1 |z − ζ 0 |2n−2

is a harmonic function of class H 1 in the ball B, Lemma 2.3.3 yields 1

(MF) (z) = F(z) − Ē ∫ F(tz) t n−2 dt 0 1

̄ (tz) t n−2 dt . = F(z) − ∫ ( EF) 0

By (2.58), 1

̄ (tz) t n−2 dt ∫ (EF) 0

=

1 F(z) 2 1

−

n−1 ∫ F(tz) t n−2 dt 2 0 1

−

1 ̄ (tz) t n−2 dt ∫ (EF − EF) 2 0

for all z ∈ B. If F(z) = f(|z − ζ 0 |) for some smooth function f on [0, 1), one easily verifies that EF =

1 f 󸀠 (|z − ζ 0 |) ⟨z, z̄ − ζ ̄ 0 ⟩ , 2 |z − ζ 0 |

󸀠 0 ̄ = 1 ⟨z − ζ 0 , z⟩̄ f (|z − ζ |) , EF 0 2 |z − ζ |

2.3 An Asymptotic Expansion of the Martinelli–Bochner Integral

|

91

and so (MF) (z) =

1 F(z) 2 1

n−1 + ∫ F(tz) t n−2 dt 2 0 1

f 󸀠 (|tz − ζ 0 |) n−1 1 ̄ ∫ dt − (⟨z, ζ ̄0 ⟩ − ⟨ζ 0 , z⟩) t 4 |tz − ζ 0 | 0

for all z ∈ B. Since ⟨z, ζ ̄0 ⟩ − ⟨ζ 0 , z⟩̄ = ⟨z − ζ 0 , ζ ̄ 0 ⟩ − ⟨ζ 0 , z̄ − ζ ̄ 0 ⟩ = 2ı ℑ ⟨z − ζ 0 , ζ ̄ 0 ⟩ the last term on the right in the above formula disappears provided z lies on the hyperplane ℑ⟨z − ζ 0 , ζ ̄ 0 ⟩ = 0. It follows that 1

1 n−1 ∫ F(tz) t n−2 dt (MF) (z) = F(z) + 2 2

(2.60)

0

for all z satisfying ℑ⟨z − ζ 0 , ζ ̄ 0 ⟩ = 0. Let us evaluate the integral 1

∫ 0

|z|

t n−2 s n−2 1 dt = n−1 ∫ z ds 0 2n−2 |tz − ζ | |z| |s |z| − ζ 0 |2n−2 0

for z ∈ B. To this end write 󵄨󵄨 z 󵄨󵄨2 z 󵄨󵄨 󵄨 − ζ 0 󵄨󵄨󵄨 = s2 − 2s ℜ ⟨ , ζ ̄ 0 ⟩ + 1 󵄨󵄨s 󵄨󵄨 |z| 󵄨󵄨 |z| = s2 − 2s cos φ + 1 where φ is the angle between the vectors z and ζ 0 . Since n−2 1 1 s n−2 ∂ = ) ( (s2 − 2s cos φ + 1)n−1 2n−2 (n − 2)! ∂ cos φ s2 − 2s cos φ + 1

we get 1

∫ 0

t n−2 dt |tz − ζ 0 |2n−2 |z|

=

n−2 1 ds 1 ∂ ∫ 2 ) ( n−1 n−2 |z| 2 (n − 2)! ∂ cos φ s − 2s cos φ + 1 0

=

n−2 |z|√1 − cos2 φ 1 1 1 ∂ ) . arctan ( 1 − |z| cos φ |z|n−1 2n−2 (n − 2)! ∂ cos φ √1 − cos2 φ

92 | 2 Regular Perturbation of Ill-Posed Problems Let α(z) stand for the angle between the vectors z − ζ 0 and 0 − ζ 0 . We obviously have |z| sin φ , |z − ζ 0 | 1 − |z| cos φ cos α(z) = |z − ζ 0 | sin α(z) =

whence tan α(z) =

|z|√1 − cos2 φ 1 − |z| cos φ

.

By assumption α(z) does not exceed the opening of Cζ 0 , namely α < |z|√1 − cos2 φ 1 − |z| cos φ

π , which yields 2

< tan α

for any z ∈ Cζ 0 . On the other hand, 1 − |z| cos φ = |z − ζ 0 | cos α(z) ≥ |z − ζ 0 | cos α hence 1 − |z| cos φ is equivalent to |z − ζ 0 | for z ∈ Cζ 0 . Differentiating in cos φ and using the Taylor expansion for arctan t for t ∈ [0, tan α] we thus obtain 1

∫ 0

t n−2 1 dt = O ( ) 0 2n−2 |tz − ζ | |z − ζ 0 |2n−3

which finishes the proof. If z → ζ 0 along the inward normal vector to S at ζ 0 it is easy to derive the explicit expansion 1

∫ 0

t n−2 1 1 n−2 n − 2 (−1)ν ( dt = ∑ ( ) − 1) . 2n − 3 − ν |z − ζ 0 |2n−3−ν ν |tz − ζ 0 |2n−2 |z|n−1 ν=0

Consider the integral M |ζ − ζ 0 |−2ν with 0 < ν ≤ n − 2. The function F(z) = |z − ζ 0 |−2ν satisfies the polyharmonic equation ∆ n−ν

1 =0 |z − ζ 0 |2ν

for z ≠ ζ 0 . If restricted to the ball B it can therefore be represented by the Almanzi formula n−ν−1

F(z) = ∑ |z|2j F j (z) j=0

where F j are harmonic functions in B, cf. [258, p. 529]

2.3 An Asymptotic Expansion of the Martinelli–Bochner Integral |

93

Lemma 2.3.6. If z → ζ 0 within the cone Cζ 0 ∩ {z ∈ Cn : ℑ⟨z − ζ 0 , ζ ̄ 0 ⟩ = 0}, then M(

1 1 1 1 ) (z) = + O( ) . 2 |z − ζ 0 |2ν |ζ − ζ 0 |2ν |z − ζ 0 |2ν−1

Proof. The proof is by induction on ν from n − 2 to 1. If ν = n − 2 then 1 = F0 (z) + |z|2 F1 (z) , |z − ζ 0 |2n−4 with F0 and F1 harmonic functions in B. Since ∆

1 = ∆ (|z|2 F1 (z)) |z − ζ 0 |2n−4 = 4n F1 (z) + 4 |z|

we obtain

∂ F1 (z) ∂|z|

2−n ∂ = n F1 (z) + |z| F1 (z) . ∂|z| |z − ζ 0 |2n−2

Solving this differential equation yields 1

F1 (z) = ∫ 0

2−n t n−1 dt , |tz − ζ 0 |2n−2

cf. Lemma 2.3.4. It follows that F0 (z) = |z − ζ 0 |4−2n − |z|2 F1 (z) is a harmonic function in B. The function F0 (z) + F1 (z) gives a harmonic continuation of |z − ζ 0 |4−2n from S into B. Hence, F0 (z) + F1 (z) = |z − ζ 0 |4−2n − |z|2 F1 (z) + F1 (z) = |z − ζ 0 |4−2n + (1 − |z|2 ) F1 (z) for each z ∈ B. By Lemma 2.3.3, 1

t n−2 1 1 ̄∫ ) (z) = − E dt M( |ζ − ζ 0 |2n−4 |z − ζ 0 |2n−4 |tz − ζ 0 |2n−4 0

1

+ (1 − |z|2 ) F1 (z) − Ē ∫ (1 − |tz|2 ) F1 (tz) t n−2 dt 0

for z ∈ B. Analysis similar to that in the proof of Lemma 2.3.5 actually shows that M(

1 1 1 1 ) (z) = + O( ) . 0 2n−4 0 2n−4 2 |z − ζ | |ζ − ζ | |z − ζ 0 |2n−5

94 | 2 Regular Perturbation of Ill-Posed Problems if z → ζ 0 within the cone Cζ 0 ∩ {z ∈ Cn : ℑ⟨z − ζ 0 , ζ ̄ 0 ⟩ = 0}. This is our claim for ν = n − 2. Since −2ν (2n − 2ν − 2) 1 = ∆ |z − ζ 0 |2ν |z − ζ 0 |2ν+2 similar considerations still apply to the other steps of induction.

2.3.3 Regularization If ν > n − 1, the Bochner–Martinelli integral M(

1 ) (z) |z − ζ 0 |2ν

diverges. To evaluate it we need a regularization procedure. For this purpose we extend F(ζ) = |z − ζ 0 |−2ν to a harmonic function in B. Lemma 2.3.7. Suppose f ∈ C(S \ {ζ 0 }) is of finite order of growth when ζ → ζ 0 . Then f has a harmonic continuation F to the ball B. Proof. Were f a distribution on the sphere S, we would define the harmonic continuation by F0 (z) = ⟨f, ℘(⋅, z)⟩S for z ∈ B. Since S is compact, the right-hand side really gives a harmonic function of finite order of growth in B, whose weak boundary values on S coincide with f . Under the assumption of Lemma 2.3.7 f possesses many extensions to a distribution on S, but a canonical one. To get some extension one proceeds in a standard way. Choose a local chart U around ζ 0 on S with coordinates y = (y1 , . . . , y2n−1 ). Pick a C∞ function χ on S with a support in U, such that χ(y) ≡ 1 near ζ 0 . Given any φ ∈ C∞ (S), set ⟨f N , φ⟩S = ⟨f, χ(φ(y) − ∑

|α| √n and ε > 0 is small enough. The boundary ∂T consists of a piece S r of the (n − 1) -dimensional sphere {t ∈ Rn : |t| = r}, a piece S R of the (n − 1) dimensional sphere {t ∈ Rn : |t| = R}, and pieces H j of hypersurfaces t j = ε parallel to the coordinates hyperplanes t j = 0. According to this structure of the boundary of T, we represent the integral I1 as the sum of integrals I1,S r , I1,S R and I1,H j with j = 1, . . . , n. Let the piece H1 tend to the hyperplane {t1 = 0}. At this hyperplane we obviously get n

n

∑ (t2j − 2t j cos(2πx j )) + n = ∑ (t j − cos(2πx j ))2 + sin2 (2πx j ) + 1 j=1

j=2

≥1. Therefore, the integral I1,H1 tends to zero as H1 tends to the hyperplane {t1 = 0}. Analogously, I1,H j tends to zero as H j tends to the hyperplane {t j = 0}, for each j = 2, . . . , n. It remains to consider the limits of the integrals I1,S r and I1,S R , when r → 0 and n−1 be the part of the unit sphere with center at the origin which lies in R → ∞. Let S≥0 n−1 the cube 0 ≤ t j ≤ 1, j = 1, . . . , n. We endow S≥0 with the usual orientation, and then n−1 n−1 S r = −r S≥0 and S R = R S≥0 . (When we tended H j to the hyperplane {t j = 0} for all j = 1, . . . , n, then S r and S R became one 2n -th spheres.) Hence, it follows readily that I1,S r = − ∫ dx ∫ 2n−1 (n − 1)! X

→0

n−1 S≥0

∑nj=1 (−1)j−1 r2n−1 t1 . . . t n (rt j − cos(2πx j ))dt[j] (r2 − 2 ∑nj=1 rt j cos(2πx j ) + n)

n

2.4 A Formula for the Number of Lattice Points in a Domain

| 101

as r → 0. On the other hand, we get I1,S r = ∫ dx ∫ 2n−1 (n − 1)! X

∑nj=1 (−1)j−1 R2n−1 t1 . . . t n (Rt j − cos(2πx j ))dt[j] (R2 − 2 ∑nj=1 Rt j cos(2πx j ) + n)

n−1 S≥0

n

n

→ ∫ dx ∫ 2n−1 (n − 1)! ∑ (−1)j−1 t1 . . . t n t j dt[j] , X

j=1

n−1 S≥0

as R → ∞. The last integral just amounts to V(X), for n

∫ 2n−1 (n − 1)! ∑ (−1)j−1 t1 . . . t n t j dt[j] = ∫ 2n−1 (n − 1)! t1 . . . t n ds j=1

n−1 S≥0

n−1 S≥0

=1. Thus, if the domain T expands to the nonnegative one 2n -th space as above, the integral I1 tends to V(X). Moreover, the integral I2 converges to the integral on the righthand side of formula (2.64), for ∂Z󸀠 = (∂X × T) ∪ (X × ∂T and (−1)j−1 dx[j] = ν j ds for all j = 1, . . . , n, as desired. For the most practical cases n = 2 and n = 3 Theorem 2.4.1 was first proved in [4].

2.4.3 The One-Dimensional Case In this section, we clarify the structure of formula (2.64) by directly computing the integral on the right-hand side of this formula in the case n = 1. Let X = (a, b), where m < a < m + 1 and M < b < M + 1, m and M being integer numbers satisfying m < M. Then ∞

I = ∫ dt ∫ 0 ∞

=∫ 0

∂X

1 sin 2πx 2π (t − cos 2πx)2 + (sin 2πx)2

1 sin 2πb sin 2πa ( − )dt . 2π (t − cos 2πb)2 + (sin 2πb)2 (t − cos 2πa)2 + (sin 2πa)2

Substituting s = t − cos 2πb and s = t − cos 2πa into the first and second terms on the right-hand side, respectively, we get ∞

I=

∫ − cos 2πb

1 sin 2πb ds − 2π s2 + (sin 2πb)2

∞

∫ − cos 2πa

1 sin 2πa ds 2π s2 + (sin 2πa)2

s s 1 1 arctan |∞ arctan |∞ − . = 2π sin 2πb − cos 2πb 2π sin 2πa − cos 2πa

102 | 2 Regular Perturbation of Ill-Posed Problems

To be specific, we consider the case m + 1/2 < a < m + 1 , M < b < M + 1/2 , then sin 2πa < 0 and sin 2πb > 0. Hence, it follows that 1 π cos 2πb π cos 2πa ( − arctan ( − ) − ( − ) + arctan ( − )) 2π 2 sin 2πb 2 sin 2πa 1 = (π + arctan cot 2πb − arctan cot 2πa) . 2π

I=

Finally, on using the equality arctan x = π/2 − arccot x we deduce 1 (π − arctan cot(2πb − 2πM) + arctan cot(2πa − 2π(m + 1/2)) 2π = (M − m) − (b − a) ,

I=

which just amounts to N(X) − V(X), as desired.

2.4.4 Some Comments It is easy to see that the integrations over t ∈ [0, ∞)n and x ∈ ∂X in formula (2.64) can be exchanged. In this way, we get n

N(X) − V(X) = ∫ ∑ (−1)j−1 F j (x) sin 2πx j dx[j] , ∂X

(2.69)

j=1

where ∞

∞

0

0

t[j] 2n−2 (n − 1)! F j (x) = dt ∫. . .∫ n π (∑k=1 (t k − cos 2πx k )2 + ∑nk=1 (sin 2πx k )2 )n are functions of cos 2πx j and sin 2πx j , for j = 1, . . . , n. The differential form under the integral over ∂X on the right-hand side of (2.69) is smooth away from the lattice of halfinteger points in Rn . As is seen from Section 2.4.3, the differential form is not closed outside this lattice. The coefficients F j bear certain symmetry in variables x1 , . . . , x n , perhaps, it suffices to compute only one of these coefficients in order to determine the others. Moreover, F j can be computed in a closed form, however, the expressions are cumbersome, cf. formula (3) in [4]. It is possible that formula (2.69) can be applied to construct asymptotics of the difference N(X) − V(X) as R → ∞, where X is the ball of radius R with center at 0 or, more generally, an ellipsoid (

x1 2 xn 2 ) + . . . + ( ) < R2 , a1 an

or another expanding domain, cf. [82, 105, 171]. However, we will not develop this point here.

3 Asymptotics at Characteristic Points 3.1 Asymptotic Solutions of the 1D Heat Equation 3.1.1 Preliminaries Boundary value problems for parabolic equations in a bounded domain with smooth boundary fail to be regular in general, for there are characteristic points on the boundary. In his paper [217] Petrovskii found conditions on the behavior of the nearby boundary points of inflection with horizontal tangents, which are necessary and sufficient for the first boundary value problem for the heat equation to be well posed. Most criteria of regularity beginning with that of [217] appeal to the contact degree at which outer characteristics meet the boundary. If the contact degree coincides with the order of equation, the analysis reveals many common features with analysis on manifolds with conical points. This situation was well understood in the 1960s, see [142, 143, 195, 212, 260], etc. If the contact degree is different from the order of the operator, the problem can be handled within the framework of degenerate elliptic equations. In [142] one studies the first boundary value problem for a single second-order equation n

Lu := −

∑

a i,j (x)u 󸀠󸀠x i ,x j + ∑ a i (x)u 󸀠x i + a0 (x)u = f

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

(3.1)

i=1

with σ 2 (L)(x, ξ) :=

∑

a i,j (x)ξ i ξ j ≥ 0 ,

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

for all ξ = (ξ1 , . . . , ξ n ) in Rn . Here, u(x) is a real function defined in a bounded domain G in Rn with C∞ boundary, and x = (x1 , . . . , x n ) represents the coordinates. The coefficients are real and of class C∞ in the closure of G. The first boundary value problem consists in prescribing the values of u on a certain portion of the boundary ∂G. One wishes to obtain unique solutions of the problem that are smooth up to and including the boundary. If the leading part is elliptic, i.e., σ 2 (L)(x, ξ) > 0 for ξ ≠ 0, we have the usual Dirichlet problem. Another well-known example of (3.1) is the heat equation −u 󸀠󸀠x,x +u 󸀠t = 0. For this classical equation, however, certain aspects of the first boundary value problem have never been adequately studied. It is customary to call operators L, with σ 2 (L)(x, ξ) ≥ 0, degenerate elliptic. The systematic study of the general class of such equations was initiated by Fichera [75] who established estimates in L p norms and proved the existence of generalized solutions. Oleynik [212] proved under certain conditions that “weak solutions are strong” and that solutions are actually smooth up to the boundary. Following [75], the boundary is divided into three portions, on two of which the boundary values of u will be given. Let Σ3 be the set of noncharacteristic boundary https://doi.org/10.1515/9783110534979-003

104 | 3 Asymptotics at Characteristic Points points, i.e., those where σ 2 (L)(x, ν) > 0, Σ2 the set of characteristic boundary points where n

∑ (a i (x) + div a i,⋅ (x)) ν i > 0 , i=1

and Σ1 = ∂G \ (Σ2 ∪ Σ3 ). As usual, we use ν = (ν1 , . . . , ν2 ) to denote the unit exterior normal at ∂G. The first boundary problem is that of finding a solution of (3.1) which has given values on Σ2 ∪ Σ3 . After subtraction of a function with the same values, one may assume that the given boundary values on Σ2 ∪ Σ3 are zero. Under certain conditions [142] establishes that this problem has a smooth solution in G. The proof of regularity in [142] is based on a global argument, which cannot prove local regularity. There are simple examples showing that if Σ1 touches Σ2 or Σ3 , then the solution need not be smooth. On the other hand, there are interesting cases, such as the heat equation, in which they do touch and where, nevertheless, the solutions are smooth.

3.1.2 On the Heat Equation Consider the heat equation for one space variable u 󸀠󸀠x,x − u 󸀠t = 0 in the plane domain (with C∞ boundary except at the corners shown) of the type represented in Fig. 3.1.

P1 P2 P4

t

P5 P3 x

Fig. 3.1: Characteristic points.

In the first boundary problem the value of the solution is prescribed on the whole boundary except for the top segment. There is extensive literature devoted to this problem, however, most of the research treats only domains of the kind of horizontal strip limited from the left and from the right by disjoint smooth curves whose angular coefficients never vanish. At the lower corners the boundary values have to satisfy certain compatibility conditions. Concerning domains of the type of Fig. 3.1, Levi in his paper [168] pointed out that the problem of the behavior of the solution at the characteristic points P3 , P4 , and P5 , and the characteristic segment [P1 , P2 ] (all of which belong to Σ2 ) is a very difficult one, and there has been little further study of this problem. In [142] Kohn and Nirenberg proved that if the solution is smooth in the closure of the domain G of Fig. 3.1, then the boundary values of u may have to satisfy compatibility conditions at the point P4 depending on the value of the curvature of the

3.1 Asymptotic Solutions of the 1D Heat Equation

| 105

boundary there. Furthermore, if the curvature is not zero, the solution need not be C∞ there, but the smaller the curvature, the smoother the solution at that point. It is C∞ if the curvature vanishes. It is expected that a solution there might be nonsmooth at P4 = (t0 , x0 ), since for t < t0 on the two sides of P4 the solution is determined by different data, and there may not be matching of smoothness at P4 . At all other points, in particular the points P3 and P5 , where the boundary curve is convex, the solution is C∞ . Kondrat’ev [143] studied boundary value problems for general parabolic equations in domains like Fig. 3.1. In the case of second-order parabolic equations Kondrat’ev noted that at convex boundary points like P3 and P5 , the smaller the curvature the smoother is the solution. In their paper [142], Kohn and Nirenberg said “He informed us in October 1966 that he could prove that for the heat equation the solution is C∞ at convex boundary points P3 = (x0 , t0 ) provided that at these points the boundary curve has the form t − t0 = c (x − x0 )p , where c > 0 and p ≥ 2 is an integer.” To the best of our knowledge, this result has not been published except for the case p = 2. In [142] the solution u is proved to be C∞ at points like P3 and P5 , where the boundary has positive curvature. This is proved for general second-order parabolic equations in n dimensions. The proof applies to a singular transformation of variables not unlike one used by Kondrat’ev in [143], which “blows up” the points P3 and P5 . Both [143] and [142] assume that the boundary is C∞ in a neighborhood of the characteristic points under study. For the heat equation, the existence of a classical solution to the first boundary values problem in noncylindrical domains was first obtained by Gevrey [92]. This result applies in particular to the plane domains G consisting of all (x, t) ∈ R2 , such that |x| < 1 and f(|x|) < t < f(1), where f(r) is a C1 function on (0, 1] with f(r) > 0, f 󸀠 (r) ≠ 0 for all r ∈ (0, 1] and f(0+) = 0. The boundary point (0, 0) is regular if f −1 (t) satisfies the Hölder condition of exponent larger than 1/2. When applied to the function f(r) = r p , this implies 0 < p < 2. In [14] a more intricate situation is treated when the domain G nearby the origin consists of those (x, t) ∈ R2 which satisfy x > 0 and −ax2 < t < bx2 , where a and b are fixed positive numbers. The boundary of G is, therefore, not smooth and it has a cuspidal singularity at the origin, which can actually be thought of as characteristic point. For a recent account of the theory along more classical lines using the concept of (ir) regularity of a boundary point for a partial differential equation, we refer the reader to [87]. The present work is aimed to study the first boundary value problem for secondorder parabolic equations in the case when the contact degree of outer characteristics with the boundary is less than the order of equation. The problem was shown to fit into analysis on compact manifolds with cuspidal points elaborated by Rabinovich et al. [228]. We restrict our discussion to the 1D heat equation. We hope that the methods employed here may prove useful in treating more general systems.

106 | 3 Asymptotics at Characteristic Points

3.1.3 Blow-Up Techniques Consider the first boundary value problem for the heat equation in a domain G ⊂ R2 of the type of Fig. 3.2. The boundary of G is assumed to be C∞ except for a finite number of characteristic points. At points like P1 and P2 the boundary curve possesses a tangent that is horizontal, hence ∂G is characteristic for the heat equation at such points. The characteristic touches the boundary with the degree ≥ 2, which is included in the treatise [143]. At points like P2 the boundary curve is not smooth but it smoothly touches a characteristic from below and above. Such points are, therefore, cuspidal singularities of the boundary; explicit treatable cases have been studied in [14].

P1

G t

P2

P4 P5 P3 = (x0 , t0 ) x

Fig. 3.2: A cusp.

We restrict our discussion to boundary points like P3 and P5 . These are cuspidal singularities of the boundary curve, which smoothly touches a vertical line at P3 and P5 . Thus, the boundary meets a characteristic at P3 and P5 at contact degree < 2. As mentioned, the study of regularity of such points for solutions of the first boundary value problem for the heat equation goes back at least as far as [92]. While the approach of [92] is based on potential theory, we apply the so-called blow-up techniques developed in [228]. This allows us to not only obtain a regularity theorem in a sharper form including asymptotics of solutions but also to prove the Fredholm property in suitable weighted Sobolev spaces for more general cusps. The first boundary value problem for the heat equation in G is formulated as follows. Write Σ1 for the set of all characteristic points P1 , P2 , . . . on the boundary of G. Given functions f in G and u 0 on ∂G \ Σ1 , find a function u on G \ Σ1 that satisfies −u 󸀠󸀠x,x + u 󸀠t = f u = u0

in

G,

on

∂G \ Σ1 ,

(3.2)

cf. Section 3.1.1. By the local principle of Simonenko [254], the Fredholm property of problem (3.2) in suitable function spaces is equivalent to the local invertibility of this problem at each point of the closure of G. Here, we focus upon the points like P3 .

3.1 Asymptotic Solutions of the 1D Heat Equation |

107

Assume that the domain G is described in a neighborhood of the point P3 = (x0 , t0 ) by the inequality t − t0 > f(|x − x0 |) , (3.3) where f is a monotone increasing C∞ function of r ∈ (0, 1] with f(0+) = 0. We take P3 to be the origin. We now blow up the domain G at the point P3 by introducing new coordinates (ω, r) with the aid of x = f −1 (r) ω , (3.4) t=r, where |ω| < 1 and r ∈ (0, f(1)]. By definition, the new coordinates are singular at r = 0, for the entire segment [−1, 1] on the ω -axis is blown down into the origin by (3.4). The rectangle (−1, 1) × (0, f(1)] transforms under the change of coordinates (3.4) into the part of the domain G nearby P3 lying below the line t = f(1). In the domain of coordinates (ω, r) problem (3.2) reduces to an ordinary differential equation with respect to the variable r with operator-valued coefficients. More precisely, under transformation (3.4) the derivatives in t and x change by the formulas ∂u ∂u 1 ∂u = − −1 , ω 󸀠 −1 ∂t ∂r f (r)f (f (r)) ∂ω ∂u 1 ∂u = , ∂x f −1 (r) ∂ω and so (3.2) transforms into 󸀠󸀠 (f −1 (r))2 U r󸀠 − U ω,ω −

f −1 (r) f 󸀠 (f −1 (r))

󸀠 = (f −1 (r))2 F ωU ω

U = U0

in

(−1, 1) × (0, f(1)) ,

on

{±1} × (0, f(1)] ,

(3.5)

where U(ω, r) and F(ω, r) are pullbacks of u(x, t) and f(x, t) under transformation (3.4), respectively. We are interested in the local solvability of problem (3.5) near the edge r = 0 in the rectangle (−1, 1) × (0, f(1)). Note that the ordinary differential equation degenerates at r = 0, since the coefficient (f −1 (r))2 of the higher-order derivative in r vanishes at r = 0. In order to handle this degeneration in an orderly fashion, we find a change of coordinate s = δ(r) in an interval (0, r0 ] with some r0 < f(1), such that (f −1 (r))2

d d = . dr ds

Such a function δ is determined uniquely up to a constant from the equation δ󸀠 (r) = (f −1 (r))−2 and is given by r

δ(r) = δ(r0 ) + ∫ r0

dϑ , (f −1 (ϑ))2

(3.6)

108 | 3 Asymptotics at Characteristic Points for r ∈ (0, f(1)]. The constant δ(r0 ) is not essential, one can choose δ(r0 ) = 0. Problem (3.5) becomes 󸀠󸀠 + U s󸀠 − U ω,ω

d F 󸀠 = 󸀠 −1 log √δ󸀠 (δ−1 (s)) ωU ω ds δ (δ (s))

U = U0

in

(−1, 1) × δ(0, f(1)) ,

on

{±1} × δ(0, f(1)] ,

(3.7)

where we use the same letter to designate U and the push-forward of U under the transformation s = δ(r), and δ(0, f(1)) = (δ(0), δ(f(1))). Example 3.1.1. After [92], consider f(r) = r p with p > 0. Then p−2 p−2 p { (r p − r0 p ) , { δ(r0 ) + p−2 δ(r) = { { δ(r ) + log r , 0 { r0

if

p ≠ 2 ,

if

p=2.

For p > 2, the value δ(0) is obviously finite. For p ∈ (0, 2], we get δ(0+) = −∞. Choosing p−2 p { { r0 p , if p ≠ 2 , δ(r0 ) = { p − 2 { if p = 2 , log r0 , { we arrive at the local boundary value problem 2

󸀠󸀠 + U s󸀠 − U ω,ω

p−2 1 1 p−2 󸀠 =( ωU ω s) F 2−p s p

U = U0

in

(−1, 1) × (δ(0), p/(p − 2)) ,

on

{±1} × (δ(0), p/(p − 2)] ,

(3.8)

for p ≠ 2, and 󸀠󸀠 − U s󸀠 − U ω,ω

1 󸀠 = es F ωU ω 2 U = U0

in

(−1, 1) × (−∞, 0) ,

on

{±1} × (−∞, 0] ,

(3.9)

for p = 2. This example demonstrates rather strikingly that the value δ(0) actually characterizes the threshold of weak singularities in the first boundary value problem for the heat equation. If δ(0+) is finite, one can certainly assume that δ(0+) = 0, for if not, we take r0 = 0 and choose δ(r0 ) = 0. Then δ−1 (0+) = 0 and so δ󸀠 (δ−1 (0+)) = +∞, i.e., the 󸀠 in (3.7) blows up at δ(0+). This manifests singularities of solutions at coefficient of U ω 󸀠 in (3.7) need not s = 0. On the other hand, if δ(0+) = −∞, then the coefficient of U ω blow up at δ(0+), as is seen from (3.8) and (3.9). In the case δ(0) = −∞, the boundary value problem (3.5) can be specified within the calculus of pseudodifferential operators with operator-valued symbols on the real axis s ∈ R developed by Rabinovich et al. in [228]. The symbols under considerations take their values in the space of boundary value problems on the interval ω ∈ [−1, 1].

3.1 Asymptotic Solutions of the 1D Heat Equation

| 109

For those pseudodifferential operators whose symbols are slowly varying at the point s = −∞, the paper [228] gives a criterion of local solvability at −∞. Note that this criterion does not apply directly to problem (3.7), for [228] deals with classical polyhomogeneous symbols while our problem requires quasihomogeneous symbols. However, the approach of [228] still works in the anisotropic case while the derivatives in s are counted with weight factor 2. The symbol of problem (3.7) is slowly varying at s = −∞ if and only if, for each j = 1, 2, . . ., an inequality 󵄨󵄨󵄨 d j 󵄨󵄨󵄨 󵄨󵄨 󵄨 sup (3.10) 󵄨󵄨( ) log √δ󸀠 (δ−1 (s)) 󵄨󵄨󵄨 < ∞ 󵄨󵄨 s∈(−∞,δ(r 0)) 󵄨󵄨󵄨 ds 󵄨 is valid. Inequalities (3.10) can be easily reformulated in terms of the original function f, namely, 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 d j sup 󵄨󵄨󵄨((f −1 (r))2 ) log f −1 (r)󵄨󵄨󵄨 < ∞ , 󵄨󵄨 dr r∈(0,r 0 ) 󵄨󵄨󵄨 󵄨

for all j = 1, 2, . . . . From the Hardy–Littlewood inequality it follows that, under these conditions, and if moreover j > 1, then the left-hand side is arbitrarily small if r0 is small enough.

3.1.4 Further Reduction From now on we will tacitly assume that the coefficients of (3.7) are slowly varying at s = −∞, i.e., (3.10) is fulfilled. Using transformations that are rather standard in Sturm–Liouville’s theory we reduce problem (3.7) to a simpler form. Set d log √δ󸀠 (δ−1 (s)) , ds F , F̃ = 󸀠 −1 δ (δ (s))

b(s) =

then (3.7) rewrites as 󸀠󸀠 󸀠 + b(s) ωU ω = F̃ U s󸀠 − U ω,ω

U = U0

in

(−1, 1) × δ(0, f(1)) ,

on

{±1} × δ(0, f(1)] .

Introduce

1 a(ω, s) = exp (− ω2 b(s)) 2 ∞ which is a bounded C function with positive values on the closed cylinder [−1, 1] × δ(0, f(1)]. An easy computation shows that problem (3.7) transforms to U s󸀠 −

1 󸀠 ) ω = F̃ (aU ω a U = U0

in

(−1, 1) × δ(0, f(1)) ,

on

{±1} × δ(0, f(1)] .

110 | 3 Asymptotics at Characteristic Points On replacing the unknown function by U = value problem ̃ 𝑣󸀠s − 𝑣󸀠󸀠 ω,ω + c𝑣 = √ a F 𝑣 = √ aU0 where c(ω, s) =

1 𝑣 √a

we finally arrive at the boundary

in (−1, 1) × δ(0, f(1)) , on

{±1} × δ(0, f(1)] ,

(√a)󸀠󸀠ω,ω − (√a)󸀠s √a

(3.11)

,

cf. [50, v. I, p. 250]. Example 3.1.2. If f(r) = r p , then 1 1 p − 1 ω2 { { {− 2 b(s) + 4 (p − 2)2 s2 , c(ω, s) = { 2 { {− 1 b(s) + ( 1 ω) , { 2 4

if

p ≠ 2 ,

if

p=2.

In the general case we get 1 1 c(ω, s) = − b(s) + ω2 ((b(s))2 + b 󸀠 (s)) , 2 4

(3.12)

where b is a C∞ function slowly varying at s = −∞. It follows that c inherits this behavior at s = −∞ uniformly in ω ∈ [−1, 1]. Our approach to solving problem (3.11) is fairly standard in the theory of linear equations. On choosing a proper scale of weighted Sobolev spaces in the strip C = [−1, 1] × R and taking the data 𝑣0 = √aU0 in the corresponding trace spaces on the boundary ω = ±1 of C we can assume without loss of generality that 𝑣0 ≡ 0. We think of (3.11) as a perturbation of the problem with homogeneous boundary conditions ̃ 𝑣󸀠s − 𝑣󸀠󸀠 ω,ω = √ a F 𝑣=0

in

C,

on

∂C .

(3.13)

This is exactly the first boundary value problem for the heat equation in the cylinder C which is nowadays well understood, cf. for instance Chapter 3 in [273]. If g = √a F̃ vanishes, problem (3.13) possesses infinitely many linearly independent solutions of the form π 2 π 𝑣n (ω, s) = c n exp (− ( n) s) sin n(ω + 1) , (3.14) 2 2 with n a natural number. In order to eliminate the solutions with n large enough it is necessary to pose growth restrictions on 𝑣(ω, s) for s → −∞. As but one possibility to do that we mention Sobolev spaces with exponential and powerlike weight factors, see [228]. Since the coefficients of the operator are stationary, the Fourier transform in s applies to reduce the problem to a Sturm–Liouville eigenvalue problem on the interval (−1, 1), see Chapter 5 in [50, v. 1]. Instead of the Fourier transform one can

3.1 Asymptotic Solutions of the 1D Heat Equation

| 111

use orthogonal decompositions over the eigenfunctions, which immediately leads to asymptotics of solutions of the unperturbed problem at s = −∞. On returning to problem (3.11) we observe that it differs from the unperturbed problem by the multiplication operator 𝑣 󳨃→ c𝑣. If the unperturbed problem is a Fredholm one and the perturbation 𝑣 󳨃→ c𝑣 is compact, then the perturbed problem is also a Fredholm one. The local version of this assertion states that if the unperturbed problem is invertible and the perturbation 𝑣 󳨃→ c𝑣 is small, then the perturbed problem is also invertible. Since, under our assumptions, c(ω, s) → 0 uniformly in ω ∈ [−1, 1], as s → −∞, the operator 𝑣 󳨃→ c𝑣 is compact in natural scales of weighted Sobolev spaces.

3.1.5 The Unperturbed Problem In this section, we treat problem (3.13) in the infinite strip C = (−1, 1) × R. We are interested in a solution to this problem in a half-strip s ∈ (−∞, S), where S = δ(f(1)). A solution can be found by the Fourier method of separation of variables, see for instance Section 2 of Chapter 3 in [273]. We first look for a solution of the corresponding homogeneous problem of the form 𝑣(ω, s) = 𝑣1 (ω)𝑣2 (s), obtaining two eigenvalue problems for determining the functions 𝑣1 (ω) and 𝑣2 (s). The first of the two looks like 𝑣1󸀠󸀠 = λ 𝑣1

for

ω ∈ (−1, 1) ,

𝑣1 (±1) = 0 .

(3.15)

2

It has a nonzero solution only for the values λ n = − ( 2π n) , where n ∈ N. The solution is 𝑣1,n (ω) = sin √−λ n (ω + 1) (3.16) up to a constant factor. Substituting λ = λ n into the equation for 𝑣2 (s), we readily find 𝑣2,n (s) = exp(λ n s) up to a constant factor. We have thus constructed a sequence of solutions 𝑣n (ω, s) = c n exp(λ n s) sin √−λ n (ω + 1) to the homogeneous problem (3.13), cf. (3.14). Note that each solution 𝑣n is unbounded at s = −∞. It is a general property of Sturm–Liouville eigenvalue problems that system (3.16) is orthogonal and complete in L2 (−1, 1). Moreover, this system is orthonormal, as is easy to check. Now let g be an arbitrary function on C, such that g(⋅, s) ∈ L2 (−1, 1) for each s < S. For any fixed s < S, we represent g as Fourier series over the orthonormal basis (3.16) ∞

g(ω, s) = ∑ g n (s) sin √−λ n (ω + 1) , n=1

112 | 3 Asymptotics at Characteristic Points

where 1

g n (s) = ∫ g(ω, s) sin √−λ n (ω + 1) dω . −1

We seek for a solution 𝑣 of problem (3.13) in the form of Fourier series over the eigenfunctions of problem (3.15), i.e., ∞

𝑣(ω, s) = ∑ 𝑣n (s) sin √−λ n (ω + 1) , n=1

s being thought of as the parameter. The function 𝑣(ω, s) satisfies the boundary conditions of (3.13), since all summands of the series satisfy them. Substituting the series into (3.13) yields ∞

∑ (𝑣󸀠n (s) − λ n 𝑣n (s) − g n (s)) sin √−λ n (ω + 1) = 0 , n=1

for all ω ∈ (−1, 1). This equation is satisfied if and only if all the coefficients vanish, i.e., 𝑣󸀠n (s) − λ n 𝑣n (s) = g n (s) ,

(3.17)

for s < S. It is worth pointing out that no initial conditions for 𝑣n (s) are available, and so 𝑣n (s) is not determined uniquely. On solving this ordinary differential equation we get s 󸀠

𝑣n (s) = ∫ e λ n (s−s ) g n (s󸀠 )ds󸀠 ,

(3.18)

sn

where s n < S is an arbitrary constant. The change of s n results in an additional multiple of e λ n s , for s

∫e

s λ n (s−s 󸀠)

󸀠

󸀠

g n (s )ds −

󸀠

e λ n (s−s ) g n (s󸀠 )ds󸀠 = c n e λ n s ,

s n +∆s n

sn s +∆s

∫

󸀠

with c n = ∫s n n e−λ n s g n (s󸀠 )ds󸀠 . n We have thus proved the following lemma. Lemma 3.1.3. Suppose that g is an arbitrary function on the cylinder C satisfying g(⋅, s) ∈ L2 (−1, 1) for all s < S. Then problem (3.13) has a formal solution of the form ∞

s

n=1

sn

󸀠

𝑣(ω, s) = ∑ (∫ e λ n (s−s ) g n (s󸀠 )ds󸀠 ) sin √−λ n (ω + 1) .

3.1 Asymptotic Solutions of the 1D Heat Equation

| 113

If we pose the additional condition 𝑣(ω, s0 ) = 0 for some s0 < S, then the functions 𝑣n (s) should fulfill the initial condition 𝑣n (s0 ) = 0. In this case, 𝑣n are uniquely determined by formulas (3.18) with s n = s0 for all n ∈ N, which leads to the uniqueness of the formal solution. In our setting the elimination of all nontrivial solutions of the homogeneous problem except for a finite number is achieved by requiring the solution to belong to a scale of Sobolev spaces with exponential weight functions.

3.1.6 Asymptotic Solutions We can now return to the study of perturbed problem (3.11). We write the corresponding equation in the form 𝑣󸀠s + C(s)𝑣 = g ,

(3.19)

where C(s) = − (

d 2 ) + c(ω, s) dω

is a continuous function on (−∞, S) with values in second-order ordinary differential operators on (−1, 1). We think of C(s) as an unbounded operator in L2 (−1, 1) whose domain consists of all 𝑣 ∈ H 2 (−1, 1) satisfying 𝑣(−1) = 𝑣(1) = 0. As but one result of the theory of Sturm–Liouville boundary value problems we mention that C(s) is closed. As usual in the theory of ordinary differential equations with operator-valued coefficients, we associate the operator pencil s(s, σ) = (ıσ) + C(s) with (3.19). It depends on parameters s ∈ (−∞, S) and σ ∈ C. Our basic assumption is that s(s, σ) stabilizes to an operator pencil s(−∞, σ) independent of s, as s → −∞. This just amounts to saying that the coefficient c(ω, s) extends continuously to s = −∞. We tacitly assume that c(−∞, ω) ≡ 0, for we are interested in true cusps, see Fig. 3.2. Lemma 3.1.4. Let k ≥ 1 be an integer. When acting from H 2k (−1, 1) ∩ H̊ 1 (−1, 1) to H 2(k−1) (−1, 1), the operator s(−∞, 0) = C(−∞) is invertible. Proof. See Section 3.1.5. ̊ Moreover, s(−∞, σ) acting from H 2k (−1, 1)∩ H(−1, 1) to H 2(k−1) (−1, 1) has a bounded inverse everywhere in the entire complex plane C except for the discrete set π 2 σ n = −ıλ n = ı ( n) 2 with n ∈ N. It is worth pointing out that s(−∞, σ)−1 = RC(−∞) (−ıσ), the resolvent of C(−∞) at −ıσ.

114 | 3 Asymptotics at Characteristic Points

Lemma 3.1.5. There exists a constant c with the property that, for all complex σ lying away from any angular sector containing the positive imaginary axis, the inequality ‖𝑣‖2H 2k (−1,1) + |σ|2k ‖𝑣‖2L2 (−1,1) ≤ c (‖s(−∞, σ)𝑣‖2H 2(k−1) (−1,1) + |σ|2(k−1) ‖s(−∞, σ)𝑣‖2L2 (−1,1) ) is fulfilled whenever 𝑣 ∈ H 2k (−1, 1) ∩ H̊ 1 (−1, 1) with k ≥ 1. Proof. The operator pencils s(−∞, σ) with this property are said to be anisotropic elliptic. See [3] for a more general estimate. If s(s, λ) stabilizes at s = −∞, then the singularity at s = −∞ gives rise to a finite number of singular solutions. However, an irregular singular point is a complicated conglomeration of singularities, which does not allow one to construct explicit asymptotic formulas. By a solution of (3.19) is meant any function 𝑣(s) with values in H 2 (−1, 1) satisfying 𝑣(−1) = 𝑣(1) = 0, which has a strong derivative in L2 (−1, 1) for almost all s < S, and which fulfills (3.19). Lemma 3.1.3 readily suggests a scale of Hilbert spaces to control the solutions. For any k = 0, 1, . . . and γ ∈ R, we introduce H k,γ (−∞, S) to be the space of all functions on (−∞, S) with values in H 2k (−1, 1), such that the norm S

1/2 k

‖𝑣‖H k,γ (−∞,S) := ( ∫ e−2γs ∑ ‖𝑣(j) (s)‖2H 2(k−j) (−1,1) ds) −∞

j=0

is finite, cf. Slobodetskii [256]. In particular, H 0,γ (−∞, S) consists of all square integrable functions on (−∞, S) with values in L2 (−1, 1) with respect to the measure e−2γs ds. Recall that the numbers σ n are called eigenvalues of the operator pencil s(−∞, σ), for there are nonzero functions φ n = 𝑣1,n in H 2 (−1, 1) vanishing at ±1 and satisfying s(−∞, σ n )φ n = 0. The functions φ n are called eigenfunctions of s(−∞, σ) at σ n . We now bring three theorems on asymptotic behavior of solutions of the homogeneous problem (3.19) as s → −∞. They fit the abstract theory of [187] well. However, [187] is a straightforward generalization of the asymptotic formula of Evgrafov [65] for solutions of first-order equations to equations of an arbitrary order. Our results thus go back at least as far as [65], while we refer to the more available paper [187]. Theorem 3.1.6. Let c(ω, s) → c(ω, −∞) in the L2 (−1, 1) -norm when s → −∞. Suppose that in the strip −μ < ℑσ < −γ there lie exactly N of the eigenvalues σ n , and that there are no eigenvalues σ n on the lines ℑσ = −μ and ℑσ = −γ. Then the solution 𝑣 ∈ H 1,γ (−∞, S) of problem (3.19) with g ∈ H 0,μ (−∞, S) has the form 𝑣(s) = c1 s1 (s) + . . . + c N s N (s) + R(s) ,

3.1 Asymptotic Solutions of the 1D Heat Equation

|

115

where s1 , . . . , s N are solutions of the homogeneous problem in H 1,γ (−∞, S), which do not depend on 𝑣, c1 , . . . , c N constants, and R ∈ H 1,μ (−∞, S). Proof. An easy computation using the continuous embedding H 1 (−1, 1) 󳨅→ C[−1, 1] shows that from the convergence of c(s, ⋅) to c(−∞, ⋅) in the L2 (−1, 1) -norm it follows that C(s) → C(−∞) in the operator norm of L(H 2 (−1, 1), L2 (−1, 1)), as s → −∞. Hence, the desired conclusion is a direct consequence of Theorem 3 in [187] with H0 = L2 (−1, 1) , H1 = H 2 (−1, 1) ∩ H̊ 1 (−1, 1) .

Thus, any solution 𝑣 ∈ H 1,γ (−∞, S) of (3.19) with a “good” right-hand side g can be written as the sum of several singular functions and a “remainder” that behaves better at infinity. The singular functions s1 , . . . , s N are linearly independent and do not depend on the particular solution 𝑣. What is still lacking is that they are not explicit. The concept of stabilization that we have so far used falls outside the framework of “small perturbations.” To meet this heuristic concept, we need some further restrictions on the speed at which C(s) tends to C(−∞) when s → −∞. Let σ n be a fixed eigenvalue of the limit pencil s(−∞, σ). Assume, moreover, that the pencil s(s, σ) stabilizes to s(−∞, σ) as s → −∞. Since σ n is a simple eigenvalue of s(−∞, σ), for s sufficiently large, there exists a simple eigenvalue σ n (s) of the pencil s(s, σ) that tends to σ n as s → −∞. We write φ n (s) for the corresponding eigenfunction with ‖φ n (s)‖L2 (−1,1) = 1. Theorem 3.1.7. Suppose s0

∫ s2 ‖c󸀠 (⋅, s)‖2L2 (−1,1) ds < ∞ , −∞

for some (and so for all) s0 ≤ S. Let 𝑣(s) be a solution of homogeneous equation (3.19) for s < S, such that 𝑣 ∈ H 1,γ (−∞, S) with λ n+1 < γ < λ n . Then, S

𝑣(s) = e−ı ∫s

σ n (s 󸀠 )ds 󸀠

(c φ n (s) + R(s))

(3.20)

where c is a constant and R ∈ H 1,0 (−∞, S). Proof. By Theorem 1 of [187], it suffices to verify if, under the assumption of Theorem 3.1.7, the integral s0

∫ s2 ‖C󸀠 (s)‖2L(H1 ,H0 ) ds −∞

is finite, where H0 and H1 are the same spaces as in the proof of Theorem 3.1.6. To this end, we pick any 𝑣 ∈ H 2 (−1, 1). The Sobolev embedding theorem implies that 𝑣 is

116 | 3 Asymptotics at Characteristic Points actually continuous on the interval [−1, 1], and the C[−1, 1] -norm of 𝑣 is dominated by C ‖𝑣‖H 1 (−1,1) with C a constant independent of 𝑣. By Hölder’s inequality, 1/2

1 󸀠

󸀠

2

‖C (s)𝑣‖H0 = ( ∫ |c (ω, s)𝑣(ω)| dω) −1 󸀠

≤ ‖c (⋅, s)‖L2 (−1,1) ‖𝑣‖C[−1,1] , and so ‖C󸀠 (s)𝑣‖H0 ≤ C ‖c󸀠 (⋅, s)‖L2 (−1,1) ‖𝑣‖H1 . Hence, it follows that ‖C󸀠 (s)‖L(H1 ,H0 ) ≤ C ‖c󸀠 (⋅, s)‖L2 (−1,1) , establishing the desired estimate. Were σ n (s) independent of s, we would deduce under the assumptions of Theorem 3.1.7 that S

e−ı ∫s

σ n (s 󸀠 )ds 󸀠

R(s) = e−λ n (S−s) R(s) ∈ H 1,λ n (−∞, S)

which belongs to H 1,γ (−∞, S). Hence, the remainder in formula (3.20) behaves better than 𝑣(s) itself, as s → −∞, showing the asymptotic character of this formula. If the coefficient c(ω, s) bears a transparent structure close to the point at (minus) infinity, then the asymptotic behavior of solutions can be described more precisely. Suppose J

c(ω, s) = ∑ c j (ω) j=0

1 1 + c J+1 (ω, s) J+1 j s s

(3.21)

on the interval (−∞, S), where c j are smooth functions on [−1, 1] for j ≤ J, and c J+1 a smooth function on [−1, 1] × (−∞, S) satisfying ‖c J+1 (⋅, s)‖L2 (−1,1) ≤ C , ‖c󸀠J+1 (⋅, s)‖L2 (−1,1) ≤

C , s

for s → −∞. Theorem 3.1.8. Under the above assumptions, any solution 𝑣(s) of the homogeneous problem (3.19) that belongs to the space H 1,γ (−∞, S) with λ n+1 < γ < λ n , has the form J−1

𝑣(s) = s ıσ0 e λ n s (c sin √−λ n (ω + 1) + c ∑ ψ j (ω) j=1

1 1 + R(s) J ) , j s s

where c is a constant depending on the solution 𝑣(s), the constant σ 0 and the functions ψ j ∈ H 2 (−1, 1) vanishing at ±1 do not depend on the solution, and R ∈ H 1,0 (−∞, S).

3.1 Asymptotic Solutions of the 1D Heat Equation

| 117

Proof. This follows from Theorem 2 of [187] if one takes into account the computations of Section 3.1.5. The constant σ 0 and the functions ψ j are computed by means of a finite number of algebraic operations.

3.1.7 Local Solvability at a Cusp Changing the coordinates by ω=

x , f −1 (t)

s = δ(t) , we return to the coordinates (x, t) in the domain G close to the boundary point P3 = (0, 0), see Fig. 3.2. Then Theorems 3.1.6 and 3.1.7 are traced back to solutions of the heat equation u 󸀠t − u 󸀠󸀠x,x = f with zero Dirichlet data near the cuspidal point in G. We get 𝑣(ω, s) = √a(ω, s) u(x, t) , g(ω, s) = √a(ω, s) (f −1 (t))2 f(x, t) , where a(ω, s) = exp 14 x2 /(f −1 (t)f 󸀠 (f −1 (t))). Let H k,γ (0, T) consist of all functions u(t, x) defined for 0 < t < T = f(1) and |x| < f −1 (t), such that √a u((δ ∘ f)−1 (s)ω, δ−1 (s)) belongs to H k,γ (−∞, S). We endow H k,γ (0, T) with a norm in an obvious way. This scale of Hilbert spaces fits well to control the solutions of the heat equation near the singular point P3 in the domain G. Since ∂ ∂ f −1 (t) ∂ = (f −1 (t))2 + 󸀠 −1 , x ∂s ∂t f (f (t)) ∂x ∂ ∂ = f −1 (t) , ∂ω ∂x the norm in H k,γ (0, T) under natural assumptions on f proves to be equivalent to the norm ‖u‖H k,γ (0,T) 1/2

dxdt := (∫∫ e−2γδ(t) ∑ |((f −1 (t))2 ∂ t )j (f −1 (t)∂ x )α (√au) |2 −1 3 ) (f (t)) 2j+|α|≤2k

,

G0

where G0 is the part of G nearby P3 lying below the line t = f(1). Theorem 3.1.9. Let c(ω, s) → c(ω, −∞) in the L2 (−1, 1) -norm when s → −∞. Suppose that in the strip −μ < ℑσ < −γ there lie exactly N of the eigenvalues σ n , and that there are

118 | 3 Asymptotics at Characteristic Points no eigenvalues σ n on the lines ℑσ = −μ and ℑσ = −γ. Then the solution u ∈ H 1,γ (0, T) of problem (3.2) with (f −1 (t))2 f ∈ H 0,μ (0, T) has the form u(t) = c1 u 1 (t) + . . . + c N u N (t) + R(t) , where u 1 , . . . , u N are linearly independent solutions of the homogeneous problem in H 1,γ (0, T) which do not depend on u, c1 , . . . , c N constants, and R ∈ H 1,μ (0, T). Proof. This follows from Theorem 3.1.6 with u j (x, t) = s j (

x , δ(t)) , f −1 (t)

for j = 1, . . . , N. Theorem 3.1.9 shows that any solution u ∈ H 1,γ (0, T) of (3.2) with a “good” right-hand side f can be written as the sum of several singular functions and a “remainder” that behaves better at the cuspidal point P3 . The singular functions u 1 , . . . , u N prove to be independent of the particular solution u. Unfortunately, they are not explicit. Let σ n = −ıλ n be a fixed eigenvalue of the limit pencil s(−∞, σ). Suppose the pencil s(s, σ) stabilizes to s(−∞, σ) as s → −∞. For s sufficiently large there exists a simple eigenvalue σ n (s) of the pencil s(s, σ) which tends to σ n as s → −∞. We write φ n (s) for the corresponding eigenfunction normalized by ‖φ n (s)‖L2 (−1,1) = 1. Theorem 3.1.10. Suppose s0

∫ s2 ‖c󸀠 (⋅, s)‖2L2 (−1,1) ds < ∞ −∞

for some (and so for all) s0 ≤ S. Let u(t) be a solution of homogeneous equation (3.2) on the interval (0, T), such that u ∈ H 1,γ (0, T) with λ n+1 < γ < λ n . Then, u(x, t) = e

δ(T)

−ı ∫δ(t) σ n (s 󸀠 )ds 󸀠

(c

1 x φ n ( −1 , δ(t)) + R(t)) , f (t) √a

(3.22)

where c is a constant and R ∈ H 1,0 (0, T). Proof. For the proof it suffices to apply formula (3.20) and pass to the coordinates (x, t). We now look for restrictions on the geometry of the singular point P3 under which Theorem 3.1.10 is applicable. To this end, let f(r) = r p close to r = 0, where p > 0. Then, 1 1 1 1 p − 1 ω2 + , 2 p − 2 s 4 (p − 2)2 s2 1 1 2 c(ω, s) = + ( ω) , 4 4 c(ω, s) =

3.2 Euler Theory on a Spindle |

119

for p ≠ 2 and p = 2, respectively. Hence, the stabilization condition of Theorem 3.1.10 is fulfilled for all p. We finish the section with local solvability of the Dirichlet problem for the heat equation near the boundary point P3 in G. By the local solvability at P3 it is meant that there is a disk B of small radius around P3 , such that for each f in G with (f −1 (t))2 f ∈ H 0,γ (0, T) there is a function u ∈ H 1,γ (0, T) satisfying u 󸀠t − u 󸀠󸀠x,x = f in G ∩ B and u = 0 on ∂G∩B. Yet another designation for the local solvability is the local invertibility from the right at P3 . For a more in-depth discussion of local invertibility we refer the reader to [228]. Recall that local solvability at each point of G is equivalent to the Fredholm property, which is due to the local principle of [254]. Theorem 3.1.11. Suppose that γ ∈ R is different from λ n for all n = 1, 2, . . . . Then the Dirichlet problem for the heat equation is locally solvable at the cuspidal point P3 . Proof. As mentioned above, condition (3.10) just amounts to saying that our problems fits into the framework of analysis of pseudodifferential operators with slowly varying symbols. Hence, the desired result follows in much the same way as Corollary 23.2 of [228]. If u 󸀠 , u 󸀠󸀠 ∈ H 1,γ (0, T) are two solutions to the Dirichlet problem in G ∩ B, then their difference u = u 󸀠 − u 󸀠󸀠 belongs to the space H 1,γ (0, T) and satisfies the Dirichlet problem with right-hand side f being zero. By Theorem 3.1.10, u has the form u = c1 u 1 + . . . + c N u N + R, where N is the greatest number with λ N > γ, u 1 , . . . , u N are linearly independent solutions of the homogeneous Dirichlet problem in H 1,γ (0, T), and R a solution of the homogeneous Dirichlet problem in H 1,∞ (0, T). The regularity theory of [228] gives even more, namely that R ∈ H k,∞ (0, T) for all k ∈ N. In particular, if γ > λ1 := −(π/2)2 , then the solution u of the Dirichlet problem near P3 is determined uniquely up to a solution of the homogeneous Dirichlet problem which belongs to H k,∞ (0, T) for each k = 1, 2, . . . . For f = 0, the solution u itself belongs to H k,∞ (0, T) for all k = 1, 2, . . . . Hence, it follows that u(0, 0+) = 0, i.e., the boundary point P3 is regular in Wiener’s sense, see [294]. This viewpoint very surprisingly sheds some new light on the connection between regularity criteria of boundary points in Dirichlet problems and the concept of differential operators with slowly varying coefficients. For a thorough treatment we refer the reader to [87].

3.2 Euler Theory on a Spindle We consider a homogeneous pseudodifferential equation on a cylinder C = R×X over a smooth compact closed manifold X whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators over X. When assuming the symbol to be independent of the variable t ∈ R, we show an explicit formula for solutions of the equation. Namely, to each nonbijectivity point

120 | 3 Asymptotics at Characteristic Points

of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler’s theorem on the exponential solutions. Our setting is the model for the analysis on manifolds with conical points, since C can be thought of as a “stretched” manifold with conical points at t = −∞ and t = ∞. When compared with the general theory, our approach is constructive while highlighting all the features of this latter.

3.2.1 Pseudodifferential Operators on Manifolds with Conical Points The aim of this section is to bring together two areas in which the Euler theory for ordinary differential equations with constant coefficients is a powerful source of intuition and an important ingredient. One of the two deals with abstract meromorphic functions, taking their values in the space of bounded operators between Banach spaces. This area was intensively studied in the late 1960s by Blekher [32], Krein and Trofimov [150], Eni [62], Sigal [252], Markus and Sigal [181], Gokhberg and Sigal [97], etc., who developed the earlier papers of Keldysh [128] and Gokhberg [96]. The other area is the analysis of pseudodifferential operators on manifolds with conical points. It originated with the paper of Kondrat’ev [144] and was developed by Plamenevskii [220], Schulze [236, 237], Melrose and Mendoza [192], Schrohe and Schulze [234, 235], Maz’ya, Kozlov and Rossmann [190], and other authors. The definition of a pseudodifferential operator close to a conical point relies on the concept of a parameter-dependent pseudodifferential operator on a smooth closed manifold, as is introduced by Agranovich and Vishik [3]. On the other hand, the key result of the theory is an asymptotic expansion of solutions near conical points; the idea goes back at least as far as Evgrafov [65], Agmon and Nirenberg [2], Kondrat’ev [144], and Maz’ya and Plamenevskii [187]. If pulled back to t = ∞ by the diffeomorphism t 󳨃→ e−t , the asymptotics are nothing but Euler solutions to the equation defined by the conormal symbol at the conical point. This latter is a parameter-dependent pseudodifferential operator on a cross-section of the manifold in a neighborhood of the conical point, the parameter being the covariable τ ∈ R of t. Moreover, it extends meromorphically in τ to a neighborhood of the real axis. In the present work we will thus be concerned with an equation 1 󸀠 ∫ dz ∫ e i(t−t )z a(z)u(t󸀠 )dt󸀠 = f(t), t ∈ R , 2π Γγ

(3.23)

R

on the cylinder C = R × X over a C∞ compact closed manifold X. Here, a(z) is a meromorphic function in a strip Ξ = {z ∈ C : ℑz ∈ (a, b)} containing the line Γ γ = R + iγ, which takes its values in the space of classical pseudodifferential operators of order m on X. We assume that Γ γ contains no pole of a(z) and that the restriction of a(z) to each horizontal line within the strip Ξ behaves like a parameter-dependent pseudo-

3.2 Euler Theory on a Spindle |

121

∞ differential operator on X. When first defined on functions u ∈ C∞ comp (R, C (X)), the operator on the left-hand side of (3.23) extends to a mapping of weighted Sobolev spaces on the cylinder, H s,γ (C) → H s−m,γ (C), for every s ∈ R. Thus, we may take H s−m,γ (C) as a domain for f and H s,γ (C) as such for u. Under the natural condition of ellipticity, we give an explicit formula for the resolvent a−1 (z) and prove that equation (3.23) has a unique solution u for each right-hand side f . We then use this result to investigate equation (3.23) on weighted Sobolev spaces H s,w (C), with w = (w− , w+ ) a pair of real numbers controlling the growth of functions at t = ±∞. This scale of Sobolev spaces on the cylinder is better suited for specifying the solutions of (3.23) than the one-parameter scale. In particular, taking w = (−γ, γ) yields H s,w (C) = H s,γ (C) for all s, γ ∈ R. Generally speaking, the operator on the left-hand side of (3.23) cannot be extended to a continuous mapping H s,w (C) → H s−m,w (C), even if w± ∈ (a, b). To give meaning to (3.23) we distinguish between the cases −w− < w+ and −w− > w+ . If −w− < w+ , then in order that a function u belong to H s,w (C) it is necessary and sufficient that u ∈ H s,γ (C) for each −w− < γ < w+ . Moreover, for any u ∈ H s,w (C), 󸀠 the Fourier transform Fu(z) = ∫R e−izt u(t󸀠 )dt󸀠 is a holomorphic function in the strip −w− < ℑz < w+ with values in H s (X). Denote by Dom A the subspace of H s,w (C) consisting of all u with the property that resp e itz a(z)Fu(z) = 0 at each pole p of a(z) in the strip −w− < ℑz < w+ . It is fairly straightforward that Dom A is of finite codimension. For any u ∈ Dom A, the integral on the left-hand side of (3.23) is independent of the particular choice of γ in the interval (−w− , w+ ). Moreover, it gives a function in H s−m,w (C), thus defining an operator A : Dom A → H s−m,w (C). We prove that if a(z) is invertible on both Γ−w− and Γ w+ , then the equation Au = f has a unique solution u ∈ Dom A for every f in a subspace of H s−m,w (C) of finite codimension. In particular, A is a Fredholm operator. Let us now turn to the case −w− > w+ . If A were a continuous linear operator H s,w (C) → H s−m,w (C), the transpose A󸀠 would define an operator H −s+m,−w (C) → H −s,−w (C), where −w = (−w− , −w+ ). Note that the couple −w already meets the condition w− < −w+ , and so we may apply the above arguments again, with a(z) replaced by a󸀠 (−z), to arrive at an operator A󸀠 : Dom A󸀠 → H −s,−w (C). The domain of A󸀠 is a subspace of H −s+m,−w (C) of finite codimension, which is nonzero unless a(z) has no pole in the strip w+ < ℑz < −w− . Hence, it follows for u ∈ H s,w (C) that (A󸀠 )󸀠 u is determined uniquely up to elements of the annihilator of Dom A󸀠 in H s−m,w (C). As this annihilator is finite-dimensional, the transpose of A󸀠 is defined as modulo operators of finite rank. We set A = (A󸀠 )󸀠 for any one choice of the operators on H s,w (C), taking their values in the annihilator of Dom A󸀠 in H s−m,w (C). Then A is well defined as mapping H s,w (C) → H s−m,w (C), and the definition agrees with the usual one in the case of differential operators. We prove that if a(z) is invertible on both Γ−w− and Γ w+ , then the equation Au = f has a solution u ∈ H s,w (C) for each f ∈ H s−m,w (C). Moreover, the space of solutions of the corresponding homogeneous equation is finite-dimensional, i.e., A is a Fredholm operator. In both cases, we show an index formula for A that turns out to be a version of the logarithmic residue theorem of Gokhberg and Sigal [97]. Let us finally remark that our results easily extend to the case where X is a C∞ compact

122 | 3 Asymptotics at Characteristic Points

manifold with boundary. On such a manifold there live parameter-dependent boundary value problems with the transmission property. Hence, we may consider meromorphic functions a(z) in the strip Ξ taking their values in the boundary value problems. The inverse (resolvent) of an elliptic meromorphic function is available in the same class, and so our arguments still hold in this context. The necessary tools were developed in Schrohe and Schulze [234, 235].

3.2.2 Meromorphic Families Let Ψclm (X) stand for the space of classical pseudodifferential operators of order m on X. By a parameter-dependent classical pseudodifferential operator of order m on X, with parameter τ ∈ R is meant any family a(τ) of operators in Ψclm (X) with the property that τ enters into the symbol of a(τ) as an additional covariable. The space of such operators is denoted by Ψclm (X; R). The space Ψclm (X) bears a natural Fréchet topology. Hence, we may consider holomorphic functions in the strip Ξ taking their values in Ψclm (X). Denote by Am (Ξ) the space of all holomorphic functions h(z) in Ξ with values in Ψclm (X), such that h(τ+iγ) ∈ Ψclm (X; R) uniformly in γ on compact segments in (a, b). Lemma 3.2.1. For each a(τ) ∈ Ψclm (X; R) there exists a function h(z) ∈ Am (C) such that h(τ) = a(τ) modulo Ψ −∞ (X; R). Proof. Cf. Theorem 2.2.8 in Schulze [237]. Note that if h ∈ Am (Ξ) and h(τ + iγ) ∈ Ψ −∞ (X; R) for some γ ∈ (a, b), then h ∈ A−∞ (Ξ). Recall that a family a(τ) ∈ Ψclm (X; R) is said to be parameter-dependent elliptic m if σ (a)(x; τ, ξ) ≠ 0 for each x ∈ X and all (τ, ξ) ∈ R × T ∗x (X) different from zero. If h ∈ Am (Ξ) and h(τ + iγ) is parameter-dependent elliptic for some γ ∈ (a, b), then so is the restriction of h(z) to each horizontal line within the strip Ξ. Indeed, the principal symbol σ m (h(τ + iγ)) is independent of γ ∈ (a, b). We will also consider meromorphic functions in the strip Ξ taking their values in m Ψcl (X). We restrict our attention to those having a finite number of poles in each strip α ≤ ℑz ≤ β with a < α ≤ β < b. Let Mm (Ξ) stand for the space of all such functions a(z) fulfilling moreover the following properties: – For each excision function χ(z) for the set of poles of h(z), we have (χa)(τ + iγ) ∈ Ψ clm (X; R) uniformly in γ on compact segments in (a, b). j – Close to a pole p ∈ Ξ, we have a(z) = ∑−1 j=−μ a j (z − p) + h(z) with a j operators −∞ of finite rank in Ψ (X) and h(z) a holomorphic function near p with values in Ψclm (X). Lemma 3.2.2. When topologizing Am (Ξ) and Mm (Ξ) in a natural way, we have Mm (Ξ) = M−∞ (Ξ) + Am (Ξ) in the sense of a nondirect sum of Fréchet spaces.

3.2 Euler Theory on a Spindle | 123

Proof. Cf. Theorem 5 in Schulze [236, 2.1.2] or Theorem 4.1.8 in Schrohe and Schulze [234]. The spaces Mm (Ξ) inherit an “algebra” structure under the pointwise composition of pseudodifferential operators on X. Lemma 3.2.3. If a(z) ∈ Mm (Ξ) and b(z) ∈ Mn (Ξ), then a(z)b(z) ∈ Mm+n (Ξ). Proof. Cf. Proposition 6 in Schulze [236, 2.1.2] or Proposition 4.1.7 in Schrohe and Schulze [234]. Let a(z) ∈ Mm (Ξ). Write a(z) = a s (z) + a r (z) by Lemma 3.2.2, where a s (z) ∈ M−∞ (Ξ) and a r (z) ∈ Am (Ξ). We say that a(z) is parameter-dependent elliptic if so is a r (z). From what has already been said it follows that this definition is correct, i.e., independent of the particular choice of the split of a. Lemma 3.2.4. Suppose a(z) ∈ Mm (Ξ) is parameter-dependent elliptic. Then a(z) is invertible away from a discrete subset of Ξ that meets every strip α ≤ ℑz ≤ β, with a < α ≤ β < b, only at a finite number of points. Moreover, a−1 (z) ∈ M−m (Ξ). Proof. Write a(z) = a s (z) + h(z) with some a s (z) ∈ M−∞ (Ξ) and h(z) ∈ Am (Ξ). As h(z) ∈ Am (Ξ) is parameter-dependent elliptic, there is an h−1 (z) ∈ M−m (Ξ) such that h−1 (z)h(z) = h(z)h−1 (z) = 1 for all z ∈ Ξ (cf. [237, 1.2.4]). By Lemma 3.2.3, we have h−1 (z)a s (z) ∈ M−∞ (Ξ). Hence, the operator 1 + h−1 (z)a s (z) is invertible for all but countably many z ∈ Ξ, and its inverse is of the form 1 + g(z) with g(z) ∈ M−∞ (Ξ) (cf. Lemma 4.3.13 in [234]). Now it is easy to check that a−1 (z) = (1 + g(z))h−1 (z) fills the bill. The operator a−1 (z) is called the resolvent of a(z). Evidently, it is parameter-dependent elliptic along with a(z). In Section 3.2.5 we show an explicit formula for the principal part of a−1 (z).

3.2.3 Characteristic Values In the sequel, an important role is played by the notion of the multiplicity of a characteristic value of a meromorphic operator-valued function. This concept goes as far as Gokhberg and Sigal [97] who extended the work of Krein and Trofimov [150] for analytic operator-valued functions. Let a(z) ∈ Mm (Ξ) be a meromorphic function in the strip Ξ with values in Ψclm (X). Our standing assumption on a(z) is that this function is parameter-dependent elliptic, as is explained in Section 3.2.2. For a fixed z ∈ Ξ away from the set of poles, a(z) can be thought of as an operator H s (X) → H s−m (X) for any one s ∈ R. The particular choice of s is actually not important because the kernel and the cokernel of a(z) consist of C∞ functions on X.

124 | 3 Asymptotics at Characteristic Points A point z0 ∈ Ξ is said to be a characteristic value of a(z) if there exists a holomorphic function u(z) in a neighborhood of z0 with values in H s (X), such that u(z0 ) ≠ 0 but a(z)u(z) is holomorphic at z0 and vanishes at this point. It is worth pointing out that a(z)u(z) is not a priori defined at z0 , however, it is well defined in a punctured neighborhood of z0 . We call u(z) a root function of a(z) at z0 . Suppose z0 is a characteristic value of a(z) and u(z) is a corresponding root function. The order of z0 as a zero of a(z)u(z) is called the multiplicity of u(z), and the function u(z0 ) ∈ H s (X) an eigenfunction of a(z) at z0 . If supplemented by the zero function on X, the eigenfunctions of a(z) at z0 form a linear space. This space is called the kernel of a(z) at z0 , and is denoted by ker a(z0 ). By the rank of an eigenfunction u 0 ∈ H s (X) we mean the supremum of the multiplicities of all root functions u(z) such that u(z0 ) = u 0 . Lemma 3.2.5. For any characteristic value z0 of a(z), the kernel of a(z) at z0 is finitedimensional and consists of C∞ functions on X. Moreover, the rank of each eigenfunction of a(z) at z0 is finite. Proof. We have −1

a(z) = ∑ a j (z − z0 )j + h(z)

(3.24)

j=−μ

in a neighborhood of z0 , where a j are smoothing operators of finite rank on X and h(z) is a holomorphic function near z0 with values in Ψclm (X). Let us observe from the very beginning that h(z0 ) is an elliptic pseudodifferential operator on X. Indeed, write a(z) = a s (z) + a r (z) by Lemma 3.2.2, where a s (z) ∈ M−∞ (Ξ) , a r (z) ∈ Am (Ξ) . Comparing this with (3.24) near z0 , we see that σ m (h(z0 )) = σ m (a r (z0 ). As a r (z0 ) is elliptic, so is h(z0 ), which is our claim. If u(z) is a holomorphic function in a neighborhood of z0 with values in H s (X), then 0

a(z)u(z) = ∑ ( ∑ ν=−μ

j+k=ν

1 a j u (k) (z0 )) (z − z0 )ν + h(z0 )u(z0 ) + O(|z − z0 |) k!

close to z0 . Hence, it follows that in order that u(z) be a root function of a(z) at z0 it is necessary and sufficient that u(z0 ) ≠ 0 and μ+ν

∑ k=0

1 a ν−k u (k) (z0 ) = 0 for all k! μ

ν = −μ, . . . , −1 ;

1 h(z0 )u(z0 ) = − ∑ a−k u (k) (z0 ) . k! k=1

(3.25)

3.2 Euler Theory on a Spindle | 125

Since h(z0 ) is an elliptic operator in Ψclm (X), the second equation of (3.25) shows that u(z0 ) lies in a finite-dimensional subspace of C∞ (X) which is completely determined by h(z0 ) and the operators a−μ , . . . , a−1 in (3.24). This establishes the first part of the lemma. To prove the second part, let u(z) be a root function of a(z) at z0 . This means that f(z) = a(z)u(z) is a holomorphic function near z0 and f(z0 ) = 0. By Lemma 3.2.4, we get u(z) = a−1 (z)f(z) in a punctured neighborhood of z0 . As a−1 (z) ∈ M−m (Ξ) and u(z0 ) ≠ 0, we can assert that the order of z0 as a zero of f(z) does not exceed the order of z0 as a pole of a−1 (z). This latter is finite, which completes the proof. By a canonical system of eigenfunctions of a(z) at z0 we mean any system of eigenfunc(1) (I) (1) tions u 0 , . . . , u 0 with the property that the rank of u 0 is the maximum of the ranks (i) of all eigenfunctions of a(z) at z0 and the rank of u 0 is the maximum of the ranks of all eigenfunctions in a direct complement in ker a(z0 ) of the linear span of the vectors (1) (i−1) (i) u 0 , . . . , u 0 , for i = 2, . . . , I. Let r i be the rank of u 0 , for i = 1, . . . , I. It is a simple matter to see that the rank of any eigenfunction of a(z) at the characteristic value z0 is always equal to one of the r i . Hence, it follows that the numbers r i are determined uniquely by the function a(z). Note that a canonical system of eigenfunctions is not, in general, uniquely determined. The numbers r i are said to be partial null multiplicities of the characteristic value z0 of a(z). Following [97], we call n(a(z0 )) = r1 + . . . + r I the null multiplicity of the characteristic value z0 of a(z). If a(z) has no root function at z0 , we set n(a(z0 )) = 0. We may apply these arguments as well to the inverse family a−1 (z), as is clear from Lemma 3.2.4. By abuse of notation, we call both the characteristic values of a(z) and those of a−1 (z) the singular values of a(z). Suppose that z0 is a characteristic value of a−1 (z) in the strip Ξ. Denote by ϱ 1 , . . . , ϱ J the partial null multiplicities of this characteristic value of a−1 (z). The numbers ϱ ι are also referred to as the partial polar multiplicities of the singular value z0 of a(z). Moreover, we call n(a−1 (z0 )) = ϱ 1 + . . . + ϱ J the polar multiplicity of the singular value z0 of a(z) and denote it by p(a(z0 )) (cf. [97]). If a−1 (z) has no root function at z0 , we set p(a(z0 )) = 0. Definition 3.2.6. The quantity m(a(z0 )) = n(a(z0 ))−p(a(z0 )) is called the multiplicity of a singular value z0 of the family a(z). If a(z) is holomorphic at a point z0 ∈ Ξ and the operator a(z0 ) is invertible, then z0 is said to be a regular point of a(z). Note that the multiplicity of each regular point of a(z) is equal to zero. We will need an auxiliary result concerning the multiplicity of a characteristic value. Lemma 3.2.7. Assume that z0 ∈ Ξ is a characteristic value of a(z) ∈ Mm (Ξ). If b j (z), n j = 1, 2, are invertible holomorphic functions near z0 with values in Ψclj (X), then z0 is a characteristic value of c(z) = b 2 (z)a(z)b 1 (z), and the partial null multiplicities of z0 for c(z) and a(z) coincide.

126 | 3 Asymptotics at Characteristic Points

Proof. Indeed, the multiplicity of any root function u(z) of a(z) at z0 is equal to the multiplicity of the root function b −1 1 (z)u(z) of c(z) at z0 . In particular, the kernels of a(z) and c(z) at z0 are isomorphic, and the desired conclusion follows. Lemma 3.2.7 actually shows that both the partial null multiplicities and the partial polar multiplicities of the singular value z0 for c(z) and a(z) coincide. In particular, we get m(c(z0 )) = m(a(z0 )).

3.2.4 Factorization In this section we briefly sketch a special factorization of a meromorphic operatorvalued function close to a characteristic value, as is given by Gokhberg and Sigal [97]. Lemma 3.2.8. Let a(z) ∈ Mm (Ξ) be parameter-dependent elliptic and z0 ∈ Ξ be a singular value of a(z). Then there are invertible holomorphic functions b1 (z) and b 2 (z) near z0 with values in Ψcl−m (X) and Ψcl0 (X), respectively, such that N

b 2 (z)a(z)b 1 (z) = π0 + ∑ π ν (z − z0 )m ν

(3.26)

ν=1

close to z0 , where m1 ≤ . . . ≤ m N are integers and π0 , π1 , . . . , π N are mutually orthogonal projections, such that π 1 , . . . , π N ∈ Ψ −∞ (X) are of rank 1 and π0 + ∑Nν=1 π ν = 1. Proof. The proof consists in an inspection of the proof of Theorem 3.1 in [97]. For the convenience of the reader we repeat the relevant material from [97] with necessary modifications. Let us expand a(z) as a Laurent series (3.24) in a neighborhood O of the point z0 . By the above, h(z0 ) is an elliptic pseudodifferential operator of order m on X. As σ m (h(z)) = σ m (a(z)) for z in O\{z0 } and a(z) is invertible in a punctured neighborhood of z0 , it follows that the index of the operator h(z0 ) is equal to 0. We can, therefore, assert that there is a smoothing operator s0 of finite rank on X, such that e0 = s0 + h(z0 ) is invertible. By continuity, the operator e(z) = s0 + h(z) is invertible in some neighborhood O󸀠 of z0 . By shrinking O, if necessary, we may assume that O1 = O. Then we get a(z) = g(z) + e(z) = e(z) (1 + e −1 (z)g(z)) , j for all z ∈ O, where g(z) = ∑−1 j=−μ a j (z − z0 ) − s0 . −1 Clearly, s(z) = e (z)g(z) is a holomorphic function in O \ {z0 } whose values are smoothing operators of finite rank on X. In the neighborhood O it admits a representation −1

s(z) = ∑ s j (z − z0 )j + t(z) , j=−μ

3.2 Euler Theory on a Spindle | 127

where s−μ , . . . , s−1 are smoothing operators of finite rank on X and t(z) is a holomorphic function in O with values in smoothing operators of finite rank on X. Let N denote the intersection of the null spaces of the operators a j , j = −μ, . . . , −1, and s0 in D󸀠 (X). Since all these operators are of finite rank, we can see that N is a subspace of D󸀠 (X) of finite codimension. If u ∈ N, then g(z)u = 0, and so s(z)u = 0 for all z ∈ O. In N, we consider the subspace N0 consisting of all functions u ∈ N satisfying s−μ u = . . . = s−1 u = 0. This subspace has a finite codimension in N and, hence, in D󸀠 (X). A familiar argument shows that there exists a direct complement D󸀠 (X) ⊖ N0 of N0 in D󸀠 (X), which is invariant with respect to each of the operators s−μ , . . . , s−1 (as well as a−μ , . . . , a−1 and s0 , but we will not use this latter fact). Moreover, since all the a j , s0 and s j are smoothing operators, it follows that D󸀠 (X) ⊖ N0 is a subspace of C∞ (X). Let π be the projection that projects D󸀠 (X) onto D󸀠 (X) ⊖ N0 parallel to N0 . By the above, π is a smoothing operator. Set π 0 = 1 − π. It is a simple matter to see that π0 s(z)π = π 0 t(z)π. From this we deduce that 1 + s(z) = 1 + πs(z)π + π 0 t(z)π = (1 + πs(z)π)(1 + π0 t(z)π) . The operator-valued function f(z) = 1 + π0 t(z)π is holomorphic in O, and its values are invertible operators, namely f −1 (z) = 1 − π 0 t(z)π. Thus a(z) can be represented in the form a(z) = e(z)d(z)f(z), with d(z) given by d(z) = 1 + πs(z)π. For z ∈ O, the operator πd(z)π can be regarded as acting in the finite-dimensional space πD󸀠 (X). By an argument of [97, 1.3], this operator can be represented in the form πd(z)π = ̃ e1 (z)c(z)f 1 (z), where e 1 (z) and f 1 (z) are holomorphic functions in O taking their ̃ values in the group of invertible linear operators in πD󸀠 (X), and c(z) is of the form ̃ c(z) = ∑Nν=1 π̃ ν (z − z0 )m ν . Here, m1 ≤ . . . ≤ m N are integer numbers, π̃ 1 , . . . , π̃ N are pairwise orthogonal projections acting in the space πD󸀠 (X), and N is the rank of the projection ∑Nν=1 π̃ ν in πD󸀠 (X). It is easy to verify that N

d(z) = (π0 + e1 (z)π) (π0 + ∑ π ν (z − z0 )m ν ) (π 0 + f1 (z)π) , ν=1

where π ν = π̃ ν π, ν = 1, . . . , N. Introducing the notation b 1 (z) = f −1 (z) (π0 + f1−1 (z)π) , −1 b 2 (z) = (π 0 + e−1 1 (z)π) e (z) ,

we obtain the representation (3.26). Finally, as a(z) is invertible at points close to z0 , so is b 2 (z)a(z)b 1 (z). Hence, it follows that π0 + ∑Nν=1 π ν = 1, which completes the proof.

128 | 3 Asymptotics at Characteristic Points

Following [97], we call (3.26) a normal factorization of a(z) at the point z0 . The principal significance of such a factorization is that it allows us to highlight the structure of the inverse operator-valued function. Namely, if b 2 (z)a(z)b 1 (z) = c(z) near z0 , with c(z) given by the right-hand side of (3.26), then a−1 (z) = b 1 (z)c−1 (z)b 2 (z) in a punctured neighborhood of z0 , where N

c−1 (z) = π0 + ∑ π ν (z − z0 )−m ν . ν=1

On the other hand, if we have a normal factorization of a(z) at z0 , we can show explicitly the partial null and polar multiplicities of the singular value z0 of a(z). Namely, suppose that the numbers m ν , ν = 1, . . . , N, from (3.26) satisfy the conditions m1 m J+1 m N−I+1

≤ ≤ ≤

... ... ...

≤ ≤ ≤

mJ m N−I mN

< = >

0, 0, 0,

where 0 ≤ I ≤ N and 0 ≤ J ≤ N − I. Then, the partial null multiplicities of the singular value z0 of a(z) are equal to m N−I+1 , . . . m N , the partial polar multiplicities of the singular value z0 of a(z) are equal to m1 , . . . m J , whence m(a(z0 )) = ∑Nν=1 m ν . We also deduce that the maximum of the ranks of all eigenvectors of a(z) corresponding to a characteristic value z0 which is a normal point of a(z) is equal to the order of the pole of a−1 (z) at z0 . For a(z) ∈ Mm (Ξ), we denote by p.p. a(z) the principal part of the Laurent expansion of a(z) in a neighborhood of a singular value z0 . By definition, p.p. a(z) is a smoothing operator of finite rank on X for all z in a punctured neighborhood of z0 . Hence, the trace (denoted tr ) of p.p. a(z) is well defined. Corollary 3.2.9. Suppose a(z) ∈ Mm (Ξ) is parameter-dependent elliptic and z0 ∈ Ξ is a singular value of a(z). Then, tr p.p. a󸀠 (z)a−1 (z) =

m(a(z0 )) . z − z0

Proof. Indeed, applying (3.26) yields tr p.p. a󸀠 (z)a−1 (z) =

m(a(z0 )) 󸀠 −1 − tr p.p. (b 󸀠1 (z)b −1 1 (z) + b 2 (z)b 2 (z)) z − z0

in a neighborhood of z0 . Since both b 1 (z) and b 2 (z) are holomorphic and invertible near z0 , we conclude that p.p. b 󸀠1 (z)b −1 1 (z) = 0 , p.p. b 󸀠2 (z)b −1 2 (z) = 0 , which completes the proof. In the case when a(z) is a polynomial operator-valued function, this corollary goes back at least as far as Keldysh [128]. The general case is due to Gokhberg and Sigal [97].

3.2 Euler Theory on a Spindle

| 129

Given a(z) ∈ Mm (Ξ), we write a󸀠 (z) for the function z 󳨃→ (a(z))󸀠 , the prime meaning the transposed pseudodifferential operator. It is clear that a󸀠 (z) ∈ Mm (Ξ); moreover, a󸀠 (z) is parameter-dependent elliptic if a(z) is. Corollary 3.2.10. If a(z) ∈ Mm (Ξ) is parameter-dependent elliptic, then a(z) and a󸀠 (z) have the same singular values with the same partial null and polar multiplicities. In particular, m(a󸀠 (z0 ) = m(a(z0 )). Proof. This follows immediately from Lemmas 3.2.7 and 3.2.8 (see also Theorem 5.3 in [97]).

3.2.5 Resolvent In this section, we discuss the expansion of the principal part of the resolvent a−1 (z) from [97]. Let a(z) ∈ Mm (Ξ). We assume that a(z) is parameter-dependent elliptic. Suppose that z0 ∈ Ξ is a characteristic value of a(z) and that u(z) is a root function of a(z) at z0 . Recall that the value u(z0 ) ∈ C∞ (X) is called an eigenfunction of a(z) at z0 . Denote by r the multiplicity of u(z). The derivatives 1 (k) u (z0 ), k = 1, . . . , r − 1 , k! are said to be associated functions for the eigenfunction u(z0 ) (a priori they are in a space H s (X)). Lemma 3.2.11. For each characteristic value z0 of a(z), the associated functions of a(z) at z0 lie in a finite-dimensional subspace of C∞ (X). Proof. We argue as in the proof of Lemma 3.2.5. Pick a root function u(z) of a(z) at z0 . Write ∞

a(z) = ∑ a j (z − z0 )j , j=−μ ∞

u(z) = ∑ u k (z − z0 )k k=0

in a neighborhood of z0 . An easy verification shows that for u(z) to be of multiplicity r ≥ 1 it is necessary and sufficient that μ+ν

∑ a ν−k u k = 0

for all

ν = −μ, . . . , −1 ;

k=0 ν−1

a0 u ν + ∑ a ν−k u k = − ∑ a ν−k u k k=0

(3.27)

μ+ν

for all

ν = 0, . . . , r − 1

k=ν+1

[cf. (3.25)]. As a0 is an elliptic operator in Ψclm (X) and a−1 , . . . , a−μ are smoothing operators of finite rank on X, we deduce by induction from the second group of equalities

130 | 3 Asymptotics at Characteristic Points (3.27) that each function u k , k = 0, 1, . . . , r − 1, belongs to a subspace of C∞ (X) of finite dimension. Moreover, this subspace is completely determined by the operators a−μ , . . . , a k , which is precisely our assertion. If u 1 , . . . , u r−1 are associated functions for an eigenfunction u 0 of a(z) at z0 , then any system u 0 , u 1 , . . . , u N with N ≤ r − 1 is called a chain consisting of an eigenfunction and associated functions of a(z) at z0 . It is easy to see that a system u 0 , u 1 , . . . , u N of functions in H s (X) forms a chain if and only if there are functions u N+1 , . . . , u N+μ ∈ H s (X) such that μ+ν

∑ a ν−k u k = 0 for ν = −μ, . . . , N k=0

[cf. (3.27)]. (1) (I) Let u 0 , . . . , u 0 be a canonical system of eigenfunctions of a(z) at z0 , I being the (i) dimension of ker a(z0 ). Denote by r i the rank of u 0 . If, for each i = 1, . . . , I, the func(i) (i) (i) tions u 0 , u 1 , . . . , u r i −1 form a chain consisting of an eigenfunction and associated functions of a(z) at z0 , then the system (i)

(i)

(i)

(u 0 , u 1 , . . . , u r i −1 )

i=1,...,I

is called a canonical system of eigenfunctions and associated functions of a(z) at z0 . The following result will be needed below. It was proved by Gokhberg and Sigal [97] for meromorphic operator-valued functions. They refer to Keldysh [128] for the case of polynomials with values in operators on a Hilbert space. Lemma 3.2.12. For each characteristic value z0 of a(z), there are canonical systems (i)

(i)

(i)

(u 0 , u 1 , . . . , u r i −1 ) (i)

(i)

i=1,...,I

,

(i)

(g0 , g1 , . . . , g r i −1 )

i=1,...,I

of eigenfunctions and associated functions of a(z) and a󸀠 (z) at z0 , respectively, such that I

−1

r i +j

(i)

(i)

p.p. a−1 (z) = ∑ ∑ (z − z0 )j ∑ ⟨g k , ⋅⟩ u r i +j−k . i=1 j=−r i

(3.28)

k=0

Proof. Cf. Theorem 7.1 in [97].

3.2.6 Unitary Reduction We now turn to pseudodifferential equations on a cylinder C = R×X over a C∞ compact closed manifold X.

3.2 Euler Theory on a Spindle

| 131

Any function on C may be thought of as a function on R with values in a function space on X. In particular, we write S(C) = S(R) ⊗π C∞ (X) for the space of rapidly decreasing functions on the real axis with values in C∞ (X). If u ∈ e−γt S(C), where γ ∈ R, then the Fourier transform Fu(z) = Ft󳨃→ℜz (eℑzt u) of u is well-defined for all z lying on the horizontal line Γ γ = {z ∈ C : ℑz = γ}. Moreover, Fu(τ + iγ) is a rapidly decreasing function of τ ∈ R with respect to each seminorm in C ∞ (X). Let a(z) ∈ Mm (Ξ). Pick a γ ∈ (a, b) such that the line Γ γ is free from the poles of a(z). For each u ∈ e−γt S(C), the integral Au (t) =

1 ∫ e itz a(z)Fu(z) dz 2π

(3.29)

Γγ γt = e−γt F−1 τ󳨃→ t a(τ + iγ)F t󳨃→ τ (e u), t ∈ R ,

gives a rapidly decreasing function on R with values in C∞ (X), modulo the factor e−γt . In fact, A is a continuous mapping of e−γt S(C) → e−γt S(C), as is easy to see. Lemma 3.2.13. Suppose u ∈ e−γt S(C). Then F(Au)(z) = a(z) Fu(z) for all z ∈ Γ γ . Proof. Using the equality Fu(τ + iγ) = Ft󳨃→τ (e γt u) for τ ∈ R, we obtain F(Au)(τ + iγ) = Ft󳨃→τ (e γt Au) = a(τ + iγ)Ft󳨃→τ (e γt u) = a(τ + iγ)Fu(τ + iγ) , the second equality being due to (3.29) and the fact that a(τ + iγ)Ft󳨃→τ (e γt u) is a rapidly decreasing function of τ ∈ R with values in C∞ (X). This is the desired conclusion. Roughly speaking, Lemma 3.2.13 just amounts to saying that the Fourier transform of the temperate distribution u(t) ≡ 1 is a constant multiple of the Dirac delta function. In fact, ∫R e−iτt dt = 2π δ(τ) for each τ ∈ R. Our next goal is to extend A to a continuous mapping of weighted Sobolev spaces on the cylinder, H s,γ (C). For s ∈ {0} ∪ N and γ ∈ R, we mean by H s,γ (C) the completion of C∞ comp (C) with respect to the norm 1/2 j

‖u‖H s,γ (C) = (∫ ∑ ‖D (e R j+A≤s

γt

u)‖2H A (X) dt)

.

(3.30)

Obviously, H s,γ (C) is a Hilbert space; however, the Hilbert structure is not canonical. If s is a negative integer, we set H s,γ (C) to be the dual of H −s,−γ (C). For fractional s, the space H s,γ (C) is defined by (complex) interpolation. Lemma 3.2.14. As defined by formula (3.29), A extends to a continuous mapping H s,γ (C) → H s−m,γ (C) for each s ∈ R.

132 | 3 Asymptotics at Characteristic Points Proof. Fix a family of order reductions Λ s (τ) ∈ Ψcls (X; R), s ∈ R, on X. Then 1/2

‖u‖H s,γ (C) ∼ (∫ ‖Λ s (ℜz)Fu(z)‖2L2 (X) dz)

,

Γγ

the equivalence of two norms meaning that their ratio is bounded both above and below by positive constants independent of u. Hence, 1/2

‖Au‖H s−m,γ (C) ∼ (∫ ‖Λ

s−m

(ℜz)a(z)Fu(z)‖2L2 (X) dz)

Γγ

≤ c (sup ‖Λ s−m (τ)a(τ + iγ)Λ−s (τ)‖L(L2 (X)) ) ‖u‖H s,γ (C) , τ∈R

for all u ∈ C∞ comp (C), the constant c being independent of A and u. Since the seminorm a(τ) 󳨃→ sup ‖Λ s−m (τ)a(τ)Λ−s (τ)‖L(L2 (X)) τ∈R

is continuous on Ψclm (X; R), the proof is complete. Given f ∈ H s−m,γ (C), consider the inhomogeneous equation Au = f for an unknown function u ∈ H s,γ (C). The solvability theory of this equation is a direct consequence of the fact that the Fourier transform u(t) 󳨃→ Fu(τ + iγ) extends to a unitary isomorphism ≃ H 0,γ (C) → L2 (Γ γ , L2 (X)), up to an inessential factor 2π. This reduces the problem to a parameter-dependent equation on the base X. Lemma 3.2.15. If the weight line Γ γ lies away from the set of singular values of a(z), then the mapping A : H s,γ (C) → H s−m,γ (C) is one-to-one and onto, for every s ∈ R. Moreover, the inverse mapping is given by the formula 1 (3.31) A−1 f(t) = ∫ e itz a−1 (z)Ff(z) dz, t ∈ R . 2π Γγ

Proof. Indeed, Lemma 3.2.4 shows that a−1 (z) ∈ M−m (Ξ). Moreover, the line Γ γ is free from the poles of a−1 , which is guaranteed by the assumption. Hence, it follows by Lemma 3.2.14 that the operator A−1 given by (3.31) extends to a continuous mapping H s−m,γ (C) → H s,γ (C), for any s ∈ R. It remains to prove that A−1 A = 1 and AA−1 = 1. We restrict our attention to the first equality; the proof of the second one is similar. Obviously, it suffices to show that A−1 A = 1 on C∞ comp (C) because this subspace is dense in H s,γ (C). However, for u ∈ C∞ comp (C) we may use Lemma 3.2.13 to obtain A−1 Au (t) =

1 ∫ e itz a−1 (z)a(z)Fu(z)dz 2π Γγ

1 ∫ e itz Fu(z)dz = 2π Γγ

= u(t) ,

3.2 Euler Theory on a Spindle |

133

the latter equality being a consequence of the Fourier inversion formula. This is our claim. Lemma 3.2.15 is an underlying technical tool for studying more intricate settings of the problem Au = f .

3.2.7 Inhomogeneous Equation The factor e γt entering into (3.30) cannot control independently the behavior of functions at t = −∞ and t = ∞. To do this, we introduce yet another scale of weighted Sobolev spaces on the cylinder, which includes two weight parameters. Namely, let w = (w− , w+ ) be a pair of real numbers to inspect the growth of functions at t = ±∞. Set a cut-off function ω for the point t = −∞ on the real axis, i.e., ω is a C∞ function on R equal to 1 near t = −∞ and vanishing near t = ∞. For s ∈ R, set H s,w (C) = ωH s,−w− (C) + (1 − ω)H s,w+ (C) ,

(3.32)

the right-hand side being understood in the sense of a nondirect sum of Fréchet spaces. In particular, taking w = (−γ, γ) we get H s,w (C) = H s,γ (C) for any s, γ ∈ R. Lemma 3.2.16. As defined by (3.32), the space H s,w (C) is equivalently topologized under the norm 1/2 . ‖u‖H s,w (C) = (‖ωu‖2H s,−w− (C) + ‖(1 − ω)u‖2H s,w+ (C) ) Proof. The proof follows from the fact that the spaces H s,γ (C) are invariant under multiplication by smooth functions on R constant in the complement of a compact interval. It follows that H s,w (C) bears a Hilbert structure. Note that this structure depends on the particular choice of ω, while the space H s,w (C) itself does not. The dual of H s,w (C) is still identified with H −s,−w (C), for each s and w. Lemma 3.2.17. Let w = (w− , w+ ) satisfy −w− ≤ w+ . Then, in order that a function u belong to H s,w (C), it is necessary and sufficient that u ∈ H s,γ (C) for each −w− ≤ γ ≤ w+ . Proof. To prove the necessity, we make use of the triangle inequality to get ‖u‖2H s,γ (C) ≤ 2 (‖ωu‖2H s,γ (C) + ‖(1 − ω)u‖2H s,γ (C) ) ≤ c ‖u‖2H s,w (C) , for any u ∈ C∞ comp (C), with c a constant independent of u. The latter inequality follows from Lemma 3.2.16 and the estimates ‖ωu‖H s,γ (C) ≤ c󸀠 ‖ωu‖H s,−w− (C) , ‖(1 − ω)u‖H s,γ (C) ≤ c󸀠󸀠 ‖(1 − ω)u‖H s,w+ (C) ,

134 | 3 Asymptotics at Characteristic Points which are due to the condition −w− ≤ γ ≤ w+ . Conversely, applying Lemma 3.2.16 implies ‖u‖2H s,w (C) ≤ c (‖u‖2H s,−w− (C) + ‖u‖2H s,w+ (C) ) , for all u ∈ C∞ comp (C), the constant c depending only on ω. This proves the sufficiency. The proof above gives more, namely if u belongs to both H s,−w− (C) and H s,w+ (C), then u ∈ H s,w (C). When regarded from the point of view of the Fourier transform, the spaces H s,w (C) with −w− ≤ w+ have the following advantage. Lemma 3.2.18. Given any u ∈ H s,w (C), the Fourier transform Fu(z) is holomorphic in the strip −w− < ℑz < w+ . Moreover, if j is a nonnegative integer ≤ s, then ‖z j Fu(z)‖H s−j (X) ≤ c (

1 1 + ) ‖u‖H s,w (C) , √w− + ℑz √w+ − ℑz

(3.33)

for all z in the above strip, with c > 0 a constant independent of u and z. Proof. A familiar argument shows that it is sufficient to establish the estimate (3.33) for any function u ∈ C∞ comp (C). For this purpose, write u = ωu + (1 − ω)u. We estimate separately the Fourier transforms of ωu and (1 − ω)u. Let b 󸀠 , b 󸀠󸀠 ∈ R be such that ω(t) = 1 for t ≤ b 󸀠 and ω(t) = 0 for t ≥ b 󸀠󸀠 . Then, z j F(ωu)(z) = ∫ e−izt D j (ω(t)u(t)) dt R b 󸀠󸀠

= ∫ e−iℜzt eℑzt D j (ω(t)u(t)) dt , −∞

for all z ∈ C. Hence, it follows, by Hölder’s inequality that b 󸀠󸀠

1 2

1 2

‖z j F(ωu)(z)‖H A (X) ≤ ( ∫ e2(w− +ℑz)t dt) (∫ ‖e−w− t D j (ωu)‖2H A (X) dt) −∞

≤ c󸀠

󸀠󸀠 e(w− +ℑz)b

√w− + ℑz

R

‖ωu‖H j+A,−w− (C) ,

for each z in the half-plane ℑz > −w− . Here, the constant c󸀠 depends only on j and w− , but not on u and z. Analogously, 󸀠

‖z j F((1 − ω)u)(z)‖H A (X) ≤ c󸀠󸀠

e(−w+ +ℑz)b ‖(1 − ω)u‖H j+A,w+ (C) , √w+ − ℑz

for any z in the half-plane ℑz < w+ , the constant c󸀠󸀠 depending only on j and w+ . Combining these estimates we arrive at (3.33), as desired.

3.2 Euler Theory on a Spindle |

135

Let a(z) ∈ Mm (Ξ) be an arbitrary symbol. Our next goal is to assign an operator A : H s,w (C) → H s−m,w (C) to a(z), for weight data w = (w− , w+ ), which satisfies a < −w− ≤ w+ < b. Formula (3.29) does not fit for the definition of A as the integral on the right-hand side depends on the choice of γ in the interval [−w− , w+ ]. To cope with this difficulty, we shrink the domain of A by considering only those u ∈ H s,w (C) for which the integral in (3.29) is independent of γ ∈ [−w− , w+ ]. The following lemma highlights such functions u. Lemma 3.2.19. Let a(z) have no poles on the lines Γ−w− and Γ w+ , where a < −w− ≤ w+ < b. Then, for each u ∈ H s,w (C) with s = max(0, m), we have ∫ e itz a(z)Fu(z)dz − ∫ e itz a(z)Fu(z)dz = 2πi Γ −w−

∑

resp e itz a(z)Fu(z) .

ℑp∈(−w − ,w + )

Γ w+

Proof. Consider a closed contour l which is the boundary of the rectangle with vertices −T − iw− , T − iw− , T + iw+ and −T + iw+ (see Fig. 3.3). Choose T > 0 large enough, so that the rectangle contains all the poles of a(z) in the strip between Γ−w− and Γ w+ . From Lemma 3.2.18 we deduce that F(z) = e itz a(z)Fu(z) is a meromorphic function in the strip −w− < ℑz < w+ with values in H s−m (X) (for fixed t ∈ R). Hence, the residue formula yields T−iw −

T+iw +

∫

F(z)dz + ∫ F(z)dz −

∫

−T−iw −

T−iw −

−T+iw +

−T+iw +

T+iw +

F(z)dz −

∫

F(z)dz

−T−iw −

= 2πi

∑ ℑp∈(−w − ,w + )

and we shall have established the lemma if we prove that the integrals ±T+iw +

∫

F(z)dz

±T−iw −

ℑz ω+ −T

p

T −ω−

ℜz Fig. 3.3: Auxiliary contour l.

res p F(z),

136 | 3 Asymptotics at Characteristic Points are infinitesimal with respect to the H −s (X)-norm, when T → ∞. For this purpose, we first make use of Hölder’s inequality to obtain 1/2 󵄩󵄩 󵄩󵄩 ±T+iw+ ±T+iw + 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 ∫ F(z)dz 󵄩󵄩 ≤ c ( ∫ ‖a(z)Fu(z)‖H −s (X) |dz|) , 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 H −s (X) 󵄩󵄩󵄩±T−iw− ±T−iw − w

where c = (∫−w+ e−2tσ dσ)

1/2

−

is independent of T. Consider the family of integrals

±∞

∫ ‖a(τ + iσ)Fu(τ + iσ)‖2H −s (X) dτ

(3.34)

±T

parameterized by σ ∈ (−w− , w+ ). Since the operator-valued function a(z) is holomorphic in a half-strip larger than ±ℜz > T, −w− < ℑz < w+ , we may invoke the estimates {⟨τ⟩m ‖a(τ + iσ)‖L(H s (X),H s−t (X)) ≤ c { ⟨τ⟩m−t {

if

t ≥ 0;

if

t≤0,

for all z therein, where t ≥ m is arbitrary real number and c a constant independent of ±τ > T and σ ∈ (−w− , w+ ) (cf. Shubin [251]). Hence, it follows that ±∞

±∞

∫ ‖a(τ + iσ)Fu(τ + iσ)‖2H −s (X) dτ ≤ c2 ∫ ⟨τ⟩2s ‖Fu(τ + iσ)‖2L2 (X) dτ ±T

±T

≤ C ‖u‖2H s,σ (C) , the constant C being independent of σ ∈ (−w− , w+ ). On the other hand, the norms ‖u‖H s,σ (C) are bounded uniformly in σ ∈ (−w− , w+ ) by the norm ‖u‖H s,w (C) , which is due to Lemma 3.2.17. We thus conclude that (3.34) is a bounded function on the interval σ ∈ (−w− , w+ ). Integrating this function over σ ∈ (−w− , w+ ) and interchanging the integrals, by Fubini’s theorem, we get ±∞

w+

∫ dτ ∫ ‖a(τ + iσ)Fu(τ + iσ)‖2H −s (X) dσ < ∞ . −w −

±T

Hence, it follows that there is a sequence T ν > 0 converging to ∞, such that w+

lim ∫ ‖a(±T ν + iσ)Fu(±T ν + iσ)‖2H −s (X) dσ → 0 ,

ν→∞

−w −

which is the desired conclusion.

3.2 Euler Theory on a Spindle

| 137

The condition u ∈ H s,w (C) with s = max(0, m) might be dropped but we have not been able to do this. Pick s ∈ R with s ≥ max(0, m) and weight data w = (w− , w+ ) satisfying a < −w− ≤ w+ < b. Set Dom A = {u ∈ H s,w (C) : resp e itz a(z)Fu(z) = 0 for − w− < ℑp < w+ } , where, by abuse of notation, we suppress the dependence of Dom A on s and w. Lemma 3.2.20. If a(z) is parameter-dependent elliptic, then Dom A is a closed subspace of finite codimension in H s,w (C). In fact, codim Dom A =

∑

p(a(p)) .

−w − 0 a constant independent of data (f, ⊕u i ). Since ψ ν ≡ 1 on the support of φ ν , we obtain LP(f, ⊕u i ) = f + L1 u ν + S(f, ⊕u i ) . By construction, ‖u ν ‖H s(k+m),γ (G) ≤ C ‖(f, ⊕u i )‖Dk,γ H(G) , whence ‖L1 u ν ‖ s(k+ 2b1 ),γ ≤ C ‖(f, ⊕u i )‖Dk,γ H(G) , H (G) for the term L1 u ν contains the derivatives of u ν up to the generalized order 2bm − 1 only. Thus, LP(f, ⊕u i ) = f + S󸀠 (f, ⊕u i ), where ‖S󸀠 (f, ⊕u i )‖ s(k+ 2b1 ),γ ≤ C ‖(f, ⊕u i )‖Dk,γ H(G) . H (G) Analogously we get B j P(f, ⊕u i ) = u j + S j (f, ⊕u i ) , with ‖S j (f, ⊕u i )‖

H

s(k+m−

mj + 1 ),γ 2b 4b (S)

≤ C ‖(f, ⊕u i )‖Dk,γ H(G) .

202 | 3 Asymptotics at Characteristic Points Summarizing, we conclude that LP = I − S holds on Dk,γ H(G) with S = (S󸀠 , ⊕S i ). From Lemmas 3.4.27 and 3.4.28 it follows that S is a compact self-mapping of Dk,γ H(G). On applying the theory of compact operators we deduce that the equation (I−S)𝑣 = (f, ⊕u i ) has a solution 𝑣 ∈ Dk,γ H(G), provided that the right-hand satisfies a finite number of “orthogonality conditions” Fj (f, ⊕u i ) = 0, where Fj are continuous linear functionals on Dk,γ H(G). Then, u = P𝑣 is a solution of problem (3.68), as desired. We now study properties of solutions u ∈ H s(k+m),γ (G) to the boundary value problem Lu = (f, ⊕u i ) in G. More precisely, we are interested in the asymptotic behavior of u in a neighborhood of the characteristic point (0, 0) in G. Theorem 3.4.32. Let u ∈ H s(k+m),γ (G) and (f, ⊕u i ) ∈ Dk,δ+1/2b H(G) and let the resolvent R(σ) have no poles on the line ℑσ = o(k + m + δ + 1/2b). Then, for every integer j ≥ 0, the function u can be represented in the form ∞ μ i −1

u(x, t) = ∑ ∑ (t1/2b )−ıσ i (log t)μ c i,μ (t−1/2b x) + 𝑣(x, t) ,

(3.130)

i=1 μ=0

where σ i are poles of multiplicity μ i of the resolvent R(σ), which lie in the strip o(k + m + γ) ≤ ℑσ < o(k + m + δ + 1/2b), and c i,μ are C∞ functions of t−1/2b x. Moreover, 𝑣 satisfies ‖φ 𝑣‖

H

s(k+m),δ+ 1 2b

(G)

≤ C (‖(f, ⊕u i )‖

D

φ (L𝑣 − f) ∈ H φ (B i 𝑣 − u i ) ∈ H where φ is an arbitrary

C∞

j s(k),δ+ 2b

k,δ+ 1 2b

H(G)

+ ‖u‖H s(k+m),γ (G) ) ,

(G) ,

m s(k+m− 2bi

1 − 4b ),δ+ 2bj

(S) ,

function with support in a small neighborhood of the origin.

A function of the form t λ (log t)μ c(t−1/2b x) is said to be a special function with index ℜλ. Lemma 3.4.33. Let f be a special function with index s and each u i be a special function with index s + m − m i /2b. For any numbers k and ϵ, there exists a function w, which is the sum of special functions with indices s + m + j/2b, where 0 ≤ j ≤ o(k − s + ϵ), such that φ (Lw − f) ∈ H s(k),ϵ (G) , mi

φ (B i w − u i ) ∈ H s(k+m− 2b − 4b ),ϵ (S) , 1

where φ is an arbitrary C∞ function vanishing away from a sufficiently small neighborhood of the origin. Proof. We write each coefficient L α,j (x, t) of the differential operator L in the form L α,j = L󸀠α,j + L󸀠󸀠α,j , where L󸀠󸀠α,j is the sum of those terms of asymptotic expansion (3.69), which include the powers of t greater than ((|α| + 2bj − 2bm) + o(k − s + ϵ)) /2b .

3.4 Boundary Value Problems for Parabolic Equations |

203

Similarly, each coefficient B iα,j (x, t) of the boundary operator B i is represented in the form B iα,j = B i󸀠α,j + B i󸀠󸀠α,j , where B i 󸀠󸀠α,j is the sum of those terms of asymptotic expansion (3.70), which include the powers of t greater than ((|α| + 2bj − m i ) + o(k − s + ϵ)) /2b . These splittings of the coefficients lead to the splittings of the operators L and B i , as L = L󸀠 + L󸀠󸀠 and B i = B i 󸀠 + B i 󸀠󸀠 . Consider the auxiliary problem L󸀠 w = f i󸀠

B w=u

i

in

G0 ,

at

∂G0

(3.131)

in G0 . Show that for any integer J ≥ 0 there is a function w that is the sum of special functions with indices s + m + j/2b, where 0 ≤ j ≤ J, such that L󸀠 w − f is the sum of special functions with indices s + (J + j)/2b and every function B i󸀠 w − u i is the sum of special functions with indices s + m − m i /2b + (J + j)/2b, where j ≥ 0. We prove this fact by induction in J. Let J = 0. As w it suffices to take the solution of the problem L󸀠0 (0, 0; D x , D t )w = f in G0 , B i 󸀠0 (0, 0; D x , D t )w = u i

at

∂G0

in the domain G0 . By Theorem 3.4.19, the solution is the sum of special functions with indices s + m. Assume that J ≥ 1 is a natural number and there is a function w0 that is the sum of special functions with indices s + m + j/2b, where 0 ≤ j ≤ J − 1, such that the discrepancy L󸀠 w0 − f is the sum of special functions with indices s + (J − 1 + j)/2b and every discrepancy B i󸀠 w0 − u i is the sum of special functions with indices s + m − m i /2b +(J −1+ j)/2b, where j ≥ 0. Denote by (L󸀠 w0 − f)J the sum of the terms in L󸀠 w0 − f that have index s + J/2b, and by (B i 󸀠 w0 − u i )J the sum of the terms in B i󸀠 w0 − u i with index s + m − m i /2b + J/2b. Consider a particular solution of the problem L󸀠0 (0, 0; D x , D t )w1 = − (L󸀠 w0 − f)J B i 󸀠0 (0, 0; D x , D t )w1

i󸀠

i

= − (B w0 − u )J

in

G0 ,

at

∂G0 ,

which is the sum of special functions with index s + m + J/2b. Show that w = w0 + w1 is a desired solution of problem (3.131). We get L󸀠 w = L󸀠 w0 + L󸀠0 (0, 0; D x , D t )w1 + (L󸀠 − L󸀠0 (0, 0; D x , D t ))w1 = f + (L󸀠 w0 − f) − (L󸀠 w0 − f)J + (L󸀠 − L󸀠0 (0, 0; D x , D t ))w1 .

204 | 3 Asymptotics at Characteristic Points The term (L󸀠 w0 − f) − (L󸀠 w0 − f)J is obviously the sum of special functions with index greater than s + J/2b. Moreover, the term (L󸀠 − L󸀠0 (0, 0; D x , D t ))w1 is the sum of special functions with index s+(J+1)/2b. Similarly, one verifies that B i󸀠 w is the sum of special functions with index greater than s + m − m i /2b + J/2b, as required. We have thus proved the existence of a function w satisfying (3.131). Our next concern will be to show that the same function w fulfills all conditions of Lemma 3.4.33 if J is chosen to be large enough, so that s + J/2b > o(k + ϵ)/2b. Indeed, we have Lw = L󸀠 w + L󸀠󸀠 w = f + (L󸀠 w − f) + L󸀠󸀠 w . Multiply both sides of this equality by φ. From Lemma 3.4.29 it follows that φ(L󸀠 w − f) ∈ H s(k),ϵ (G) , φL󸀠󸀠 w ∈ H s(k),ϵ (G) . In the same way we deduce from Lemma 3.4.30 that mi

φ (B i w − u i ) ∈ H s(k+m− 2b − 4b ),ϵ (S) , 1

which completes the proof. Proof of Theorem 3.4.32. Consider the function u 1 = ψ1 u, where ψ1 is the sum of those functions φ ν of the partition of unity on G that belong to the first group. The function u 1 is a solution to Lu 1 = f1 in G0 , (3.132) B i u 1 = u 1i at ∂G0 , where f1 and u 1i vanish away from the support of ψ1 and coincide with f and u i in some neighborhood of (0, 0), respectively. The coefficients of the operator L are defined in the mere domain G. To work with the equation Lu 1 = f1 in all of G0 , we modify the coefficients in the following way. The function u 1 vanishes from the support of ψ1 . Hence, we do not violate the equality Lu 1 = f1 if we change the coefficients of L arbitrarily in the complement of the support of ψ1 and keep them unchanged on supp ψ1 . Therefore, we can continue the coefficients of the operator L in G0 in such a manner that the equality Lu 1 = f1 remains valid in G0 , and the continuation L̃ coincides with the operator L0 (0, 0; D x , D t ) for t large enough. Applying Theorem 3.4.24 to u 1 yields decomposition (3.113) for u 1 . Namely, u 1 = a + 𝑣, where ∞ μ i −1

a = ∑ ∑ (t1/2b )−ıσ i (log t)μ c i,μ (t−1/2b x) i=1 μ=0

H s(k+m),δ+1/(2b) (G

and 𝑣 ∈ 0 ). The complex numbers σ i are located in the strip o(k + m + γ) < ℑσ < o(k + m + δ + 1/2b). On substituting the decomposition u 1 = a + 𝑣 into (3.132) we get L󸀠 a = Lu 1 − L𝑣 − L󸀠󸀠 a = f1 − L𝑣 − L󸀠󸀠 a

(3.133)

3.4 Boundary Value Problems for Parabolic Equations

| 205

in G0 and B i 󸀠 a = u 1i − B i 𝑣 − B i 󸀠󸀠 a

(3.134)

at ∂G0 . Obviously, both f1 and L𝑣 belong to the space H s(k),δ+1/(2b) (G0 ), and so (3.133) shows that L󸀠 a belongs to this space if and only if so does L󸀠󸀠 a. Similarly, B i󸀠 a belongs to the space H s(k+m−m i /(2b)−1/(4b)),γ+1/(2b) (∂G0 ) if and only if so does B i 󸀠󸀠 a, which is due to (3.134). The functions La and B i a in equalities (3.133) and (3.134) are linear combinations of function of the form L α,j (x, t)D t D αx ((t1/2b )−ıσ ν (log t)μ c ν,μ (t−1/2b x)) , j

j B iα,j (x, t)D t D αx

((t1/2b )−ıσ ν (log t)μ c ν,μ (t−1/2b x)) ,

|α| + 2bj ≤ 2bm , |α| + 2bj ≤ m i ,

respectively. On differentiating the first expressions we get linear combinations of functions of the form L α,j (x, t) t λ (log t)μ c(t−1/2b x), where ℜλ > o(k + γ)/2b. Differentiating the second expressions leads to linear combinations of functions of a similar form with ℜλ > o(k + m − m i /2b + γ)/2b. We are now in a position to complete the proof. By Lemma 3.4.33 with s = o(k + δ)/2b and ϵ = δ + j/2b, there is a function w with the property that φ (Lw − L󸀠 a) ∈ H s(k),δ+j/2b (G0 ) , mi

φ (B i w − B i󸀠 a) ∈ H s(k+m− 2b − 4b ),δ+j/2b (∂G0 ) . 1

On the other hand, an easy calculation using Lemmas 3.4.29 and 3.4.30 shows that φ L󸀠󸀠 a ∈ H s(k),δ+j/2b (G0 ) , mi

φ B i 󸀠󸀠 a ∈ H s(k+m− 2b − 4b ),δ+j/2b (∂G0 ) . 1

Therefore, from equalities (3.133) and (3.134) it follows that φ (L(𝑣 + w) − f) = φ (f1 − f) + φ (Lw − L󸀠 a) − φ L󸀠󸀠 a ∈ H s(k),δ+j/2b (G0 ) and analogously mi

φ (B i (𝑣 + w) − u i ) ∈ H s(k+m− 2b − 4b ),δ+j/2b (∂G0 ) , 1

as desired. If the data (f, ⊕u i ) of the problem satisfy a stronger condition, we can sharpen the asymptotics of u.

206 | 3 Asymptotics at Characteristic Points Corollary 3.4.34. Let u ∈ H s(k+m),γ (G) be a solution of equation Lu = f in G satisfying boundary conditions (3.68), where (f, ⊕u i ) ∈ Dk,δ+j/2b H(G) with δ ≥ γ and j > 1. If R(σ) has no poles on the line ℑσ = o(k + m + δ + j/2b), then u has the form μ i −1

u(x, t) = ∑ ∑ (t1/2b )−ıσ i (log t)μ c i,μ (t−1/2b x) + 𝑣(x, t) , σ i μ=0

where σ i are poles of multiplicity μ i of the resolvent R(σ) thatch lie in the strip o(k + m + γ) ≤ ℑσ i < o(k + m + δ + j/2b), and c i,μ are C∞ functions of t−1/2b x. Moreover, 𝑣 satisfies j

φ 𝑣 ∈ H s(k+m),δ+ 2b (G) , j

φ (L𝑣 − f) ∈ H s(k),δ+ 2b (G) , mi

j

φ (B i 𝑣 − u i ) ∈ H s(k+m− 2b − 4b ),δ+ 2b (S) , 1

where φ is an arbitrary C∞ function vanishing from a sufficiently small neighborhood of the origin. Proof. The formula follows from Theorem 3.4.32 if we substitute δ + (j − 1)/2b for δ and 1 for j there.

3.4.8 Regularity of Solutions Our next concern is the regularity of the solution depending on the regularity of the data. Let u ∈ H s(k+m),γ (G) satisfy the equation Lu = f in G and boundary conditions (3.68), where (f, ⊕u i ) ∈ Dk+1/(2b),γ−1/(2b) H(G). The number 2bk is assumed to be an integer, while k may be fractional. Show that if the line ℑσ = o(k + m + γ) is free from the poles of the resolvent R(σ), then u ∈ H s(k+m+1/(2b)),γ−1/(2b) (G). To this end, we represent u in the form u = ∑ φν u , ν

where φ ν are functions of the partition of unity on G used above. Assume that φ ν belongs to the second or the third group. Applying the operators L and B i to φ ν u, we obtain L(φ ν u) = φ ν f + R ν (u) , B i (φ ν u) = φ ν u i + R iν (u) , where R ν and R iν are differential operators of generalized order less than 2bm and m i , respectively. From this it follows that φ ν u ∈ H s(k+m+1/(2b)) (G). We will not prove this fact in detail. For functions φ ν that vanish in a neighborhood of characteristic boundary points, this result is established by a standard method, see for, instance, [1].

3.4 Boundary Value Problems for Parabolic Equations

| 207

Consider a function 𝑣 = φ ν u, where φ ν belongs to the first group. By assumption, 𝑣 vanishes from a small neighborhood of characteristic points. It satisfies L𝑣 = f1 i

B𝑣=

u 1i

in

G,

at

∂G ,

where f1 = f and u 1i = u i in a neighborhood of a characteristic point. Obviously, (φf, ⊕φu i ) ∈ Dk+1/(2b),γ−1/(2b) H(G). Rewrite the problem in the form L0 𝑣 = f1 − (L − L0 )𝑣

in

G0 ,

B0i 𝑣

at

∂G0

=

u 1i

− (B − i

B0i )𝑣

(cf. (3.108) and (3.109)), where (L − L0 )𝑣 ∈ H s(k+ 2b ),γ (G0 ) , 1

mi

(B i − B0i )𝑣 ∈ H s(k+m− 2b + 4b ),γ (∂G0 ) . 1

Thus, the right-hand side of the problem belongs to Dk+1/(2b),γ−1/(2b) H(G0 ). By Lemma 3.4.23, the boundary value problem has a unique solution 𝑣 in the space H s(k+m+1/(2b)),γ−1/(2b) (G0 ) provided that the line ℑσ = o((k + 1/(2b)) + m + (γ − 1/(2b))) is free from the poles of R(σ). Hence, it follows that u ∈ H s(k+m+1/(2b)),γ−1/(2b) (G), as desired. If (f, ⊕u i ) ∈ Dk+j/(2b),γ−j/(2b) H(G), then we can argue in the same way to see that u ∈ H s(k+m+j/(2b)),γ−j/(2b) (G). We thus arrive at the following result. Theorem 3.4.35. Suppose u ∈ H s(k+m),γ (G) satisfies Lu = f in G and boundary conditions (3.68) with data in Dk+j/(2b),γ−j/(2b) H(G), where j = 1, 2, . . . . If the line ℑσ = o(k + m + γ) does not contain poles of R(σ), then u ∈ H s(k+m+j/(2b)),γ−j/(2b) (G) and ‖u‖

H

s(k+m+

j j ),γ− 2b 2b

(G)

≤ C (‖u‖H s(k+m),γ (G) + ‖(f, ⊕u i )‖

D

k+

j j ,γ− 2b 2b

H(G)

) .

We are now in a position to prove the finite dimensionality of the space of solutions of the homogeneous problem Lu = 0 in G , (3.135) B i u = 0 at ∂G . Theorem 3.4.36. The space of all solutions to the homogeneous problem (3.135) in H s(k+m),γ (G) is finite dimensional. Proof. Pick an arbitrary sequence {u ν } of solutions to problem (3.135) with the property that ‖u ν ‖H s(k+m),γ (G) = 1 for all ν. By Theorem 3.4.35, we get ≤C. ‖u ν ‖ s(k+m+ 2b1 ),γ− 2b1 H (G) Choose any δ > γ, such that in the strip o(k + m + γ) < ℑσ < o(k + m + δ) there are no poles of R(σ). By the properties of the resolvent mentioned before Theorem 3.4.18,

208 | 3 Asymptotics at Characteristic Points

such a number δ indeed exists. Applying Theorem 3.4.32 we conclude that ‖u ν ‖ s(k+m+ 2b1 ),δ− 2b1 ≤ C󸀠 , H (G) with C󸀠 a constant independent of u ν . From Lemma 3.4.27 it follows that the family {u ν } is compact, and so the space of solutions to (3.135) is finite dimensional. We have considered the properties of solutions of the equation Lu = f in G with boundary conditions (3.68) in the case f ∈ H s(k),γ (G). The space H s(k),γ (G) includes merely those functions f that decrease in a neighborhood of the origin, for ∫ t−2(k+γ) |f|2 dxdt < ∞ . To pass on to smooth functions we need to preliminarily study some properties of functions of the Slobodetskii spaces H s(k),γ (G). For our purpose, it suffices to restrict the attention to the case of integral k. Let u be a function in G0 vanishing for t ≥ T, and I(u) := ∫ (t−2γ G0

∑ |D αx u|2 + t−2γ |D kt u|2 + |u|2 ) dxdt < ∞ . |α|=2bk

Since the boundary surface has the form (3.67) in a neighborhood of the origin, there is a surface S given by t = ∑ aα xα , |α|=2b

which is tangent to the hyperplane t = 0 at the origin and which lies entirely in G for t > 0. Let G󸀠 stand for the unbounded domain in the half-space t > 0 bounded by S. One extends u to all of G󸀠 by setting u = 0 for t > T. If o(k + γ) is not a nonnegative integer, then one can apply Theorem 3.4.22 to the function u in G󸀠 to get the representation u(x, t) = u 0 (x, t) +

∑

c α,j x α t j ,

|α|≤2bk−1 j≤k−1

where ‖u 0 ‖2H s(k),γ (G󸀠 ) ≤ C I(u) with C a constant independent of u. The function u 0 ex-

tends to all of G with the aid of this representation. Show that u 0 ∈ H s(k),γ (G). To this end, we make use of the following property of functions with square integrable derivatives. If D is a domain in Rn starlike with respect to a ball B r of radius r, then d n ∫ |𝑣|2 dx ≤ C ( ) (∫ |𝑣|2 dx + (dr) ∫ |∇𝑣|2 dx , ) , r D

Br

(3.136)

D

where d is the diameter of D and C a positive constant independent of r and d. We apply this inequality to a derivative 𝑣 = D αx u 0 with |α| = 2bk − 1 on an arbitrary section Gt

3.4 Boundary Value Problems for Parabolic Equations

|

209

of the domain G by a hyperplane through (0, t) orthogonal to the t -axis. Taking the section of G󸀠 by this hyperplane as B r , so that r = t1/2b , we get ∫ |D αx u 0 |2 dx ≤ C󸀠 (∫ |D αx u 0 |2 dx + t1/b ∫ ∑ |D x u 0 |2 dx) . β

Gt

Gt |β|=2bk

G󸀠t

We divide both sides of this inequality by t2γ+1/b and integrate in t from 0 to T, thus obtaining ∫ t−2(γ+ 2b ) |D αx u 0 |2 dxdt ≤ C󸀠 (‖u 0 ‖2H s(k),γ (G󸀠 ) + I(u)) ≤ C I(u) . 1

G

On substituting the other derivatives of u 0 for 𝑣 in (3.136), by induction we easily get ∫ t−2γ−

2bk−|α|−2bj ) b

j

|D t D αx u 0 |2 dxdt ≤ C I(u) ,

G

i.e., u 0 ∈ H s(k),γ (G). After a transformation of coordinates like (3.118) u takes the form k−1

u(x, t) = ∑ c j ( j=0

x t1/2b

j

) t 2b + u 0 (x, t) ,

where u 0 ∈ H s(k),γ (G). Consider the problem on the smoothness of solutions in the C s (G) -norm. For this purpose, we prove the continuity of functions in G having weak derivatives of generalized order 2bk, which are square integrable with a suitable weight exponent. Theorem 3.4.37. If I(u) < ∞ ,

(3.137)

with o(k) > 0, γ > 0 and o(k + γ) ≠ 0, 1, . . ., then the function u is continuous in G. Lemma 3.4.38. If u ∈ H s(k),γ (G0 ) and o(k + γ) > 0, then |u(x, t)| ≤ C I(u) t

o(k+γ) b

.

Proof. Write G1 for the part of G0 lying in the layer 1/2 < t < 2. This domain has no characteristic points, whence |u(x, 1)|2 ≤ C ∫

∑

j

|D t D αx u|2 dxdt ,

(3.138)

G1 |α|+2bj≤2bk

for all (x, 1) ∈ G1 , which is due to the Slobodetskii embedding theorem. To evaluate u(x, t) for all t, consider the family 𝑣s (x, t) = u(s1/2b x, st) parameterized by s > 0. It is easy to see that o(k+γ) ‖𝑣s ‖H s(k),γ (G0 ) = s b ‖u‖H s(k),γ (G0 ) .

210 | 3 Asymptotics at Characteristic Points On applying estimate (3.138) to 𝑣s we get |𝑣s (x, 1)| ≤ C s

o(k+γ) b

‖u‖H s(k),γ (G0 ) ,

where C is a constant independent of x. Since 𝑣s (x, 1) = u(s1/2b x, s), it follows that |u(s1/2b x, s)| ≤ C s

o(k+γ) b

‖u‖H s(k),γ (G0 ) ,

and so substituting s−1/2b x for x establishes the desired estimate. Lemma 3.4.39. Suppose u is a function in G0 satisfying condition (3.137). Then |u(x, t) − u(0, 0)| ≤ C I(u) t

o(k+γ) b

.

Proof. Consider the function 𝑣 = ϑ(t)u, where ϑ is a cut-off function equal to one for t < 1/2 and vanishing for t > 1. The expression I(𝑣) is finite and I(𝑣) ≤ C I(u), where C is a constant independent of u. By Theorem 3.4.22, the function 𝑣 can be written in the form 𝑣 = p + 𝑣0 , where 𝑣0 satisfies the hypotheses of Lemma 3.4.38. By this lemma, |𝑣0 (x, t)| ≤ C I(u) t

o(k+γ) b

.

Using the representation 𝑣 = p + 𝑣0 and the explicit formula for the polynomial p, we deduce immediately that |𝑣(x, t) − 𝑣(0, 0)| ≤ C I(u) t

o(k+γ) b

.

Since u coincides with 𝑣 in a neighborhood of the origin, the last estimate is also true for u, as desired. Proof of Theorem 3.4.37. Had G coincided with G0 , the theorem would follow from Lemma 3.4.38 directly. To complete the proof, it remains to settle a question on the perturbation of the equation of ∂G near the origin by higher-order terms o(|x|2b ). For this purpose, we invoke the construction described at the beginning of Section 3.4.7.

3.4.9 Some Particular Cases Assume that the equation Lu = f has the form u 󸀠t = A(x, t, D x )u + f

(3.139)

and is strongly parabolic. We look for a solution of this equation with zero Dirichlet data at the boundary of G. It is well known [195] that this problem always possesses a solution in H̊ b,1/2 (G). Show that if f ∈ H s(k),γ (G), then u ∈ H s(k+1),γ (G) for each k. This follows from Theorem 3.4.35 if we prove that R(σ) has no poles in the upper half-plane ℑσ ≥ −n/2.

3.4 Boundary Value Problems for Parabolic Equations

|

211

Assume, on the contrary, that the resolvent R(σ) has a pole σ 0 , such that ℑσ 0 > −n/2. The differential equation of (3.90) is strongly elliptic for each σ. Hence, if σ = σ 0 is a point of the spectrum of problem (3.90), then there is a nonzero solution V ∈ H b (B) of the homogeneous equation (3.90) with zero Dirichlet data. By the familiar regularity theorem for solutions of elliptic equations this solution V(ω) is actually infinitely differentiable in the closure of B. Therefore, U(ω, s) = exp(ıσ 0 s)V(ω) satisfies equation (3.86) and zero Dirichlet conditions at the boundary of B. We can now return to the variables (x, t) to see that the function −ıσ 0

u(x, t) = (t1/2b )

V (t−1/2b x)

is a solution of the equation u 󸀠t = − ∑ A α (0, 0)D αx u

(3.140)

|α|=2b

in the domain G0 and it satisfies the conditions u = (∂/∂ν)u = . . . = (∂/∂ν)b−1 u = 0 at the boundary of G0 , where ν is a vector field at the surface ∂G0 with values in vectors lying in a hyperplane t = t0 and normal to the section of ∂G0 by such a hyperplane. Consider the derivative ∂ t |u|2 . By (3.140), it is equal to ∂ t (u u)̄ = −ū ∑ A α (0, 0)D αx u − u ∑ A α (0, 0)D αx ū . |α|=2b

|α|=2b

We now integrate this equality over the part of the domain G0 lying in the layer t0 ≤ t ≤ T, where 0 ≤ t0 < T are arbitrary numbers. After integration by parts in the right-hand side of the equality and using the strong ellipticity of A, we obtain ∫ |u|2 dx − ∫ |u|2 dx ≤ −c ∫ t=T

∑ |D αx u|2 dx ,

t 0 ≤t≤T |α|=b

t=t 0

with c a positive constant independent of t0 , T and u. Let t0 → 0. From the formula u(x, t) = (t1/2b )−ıσ0 V(t−1/2b x) it follows that |u(x, t)| ≤ C t− 4b +ε , n

where ε > 0. Therefore, −

n

∫ |u|2 dx ≤ C ∫ t− 2b +2ε dx = t0 2b n

t=t 0

as t0 → 0.

t=t 0

+2ε

−

n

O (t0 2b ) → 0

(3.141)

212 | 3 Asymptotics at Characteristic Points

The right-hand side of (3.141) is always negative, which contradicts the last relation. Hence, σ 0 fails to be a pole of R(σ). It follows that the solution belongs to H s(k+1),γ (G) for each k, as desired. The assertion on the differentiability of solutions fails to hold if f does not vanish near the characteristic point. Example 3.4.40. Consider the equation u 󸀠t = −u 󸀠󸀠󸀠󸀠 xxxx + f

(3.142)

in the domain G0 := {(x, t) ∈ R2 : t > x4 }. Suppose f is equal to one in a neighborhood of the origin and u is a solution to (3.142), such that u = u 󸀠x = 0 at ∂G0 for sufficiently small t. The boundary value problem possesses a solution u, such that ∫ |u 󸀠󸀠xx |2 dxdt < ∞ . G0

If the solution u is infinitely differentiable, one can represent it in the form u(x, t) = ∑ c α,j x α t j + o(x4 + t) α≤4 j≤1

in a neighborhood of (0, 0). Substituting this formula into (3.142) and equating the values of both sides at the origin yields c0,1 + 4! c4,0 = 1 .

(3.143)

Since u(x, x4 ) = 0 for x in a small neighborhood of 0, the coefficients should also satisfy c0,0 = 0 ,

c0,1 + c4,0 = 0 ,

c1,0 = 0 ,

c2,0 = 0 ,

c3,0 = 0 .

Finally, the second boundary condition u 󸀠x (x, x4 ) = 0 implies 4c4,0 = 0, and so both c0,1 and c4,0 vanish, which contradicts (3.143). Therefore, the solution u fails to be smooth. On the other hand, the boundary value problem has a solution of the form u 0 (x, t) = t5 c(t−1/4 x), where c(ω) is C∞ , which is due to Theorem 3.4.20, and the difference u − u 0 is infinitely differentiable by the above. As we have seen, the assertion on the differentiability of solutions of inhomogeneous parabolic equations in a neighborhood of the point (0, 0) is false. However, it holds true if the order of the equation is equal to two. Consider the behavior of the solution of the Dirichlet problem for the 1D heat equation u 󸀠t = u 󸀠󸀠xx + f (3.144) in a bounded plane domain G given in a neighborhood of the origin by t > a2 x2 , where a2 > 0. We look for a solution u ∈ H̊ 1,1/2 (G) of (3.144) vanishing at the boundary of G.

3.4 Boundary Value Problems for Parabolic Equations

| 213

For equation (3.144), the homogeneous problem corresponding to (3.90) reduces to {V 󸀠󸀠 (ω) + 12 ωV 󸀠 (ω) + 12 (ıσ)V(ω) = 0 { V(ω) = 0 {

for for

ω ∈ (− a12 , ω=

± a12

.

1 a2 )

,

(3.145)

The eigenvalues of problem (3.145) just amount to the poles of the resolvent R(σ). From the Sturm–Liouville theory it follows that for every N = 1, 2, . . . there is a positive number R depending on N, such that if a2 > R, then the strip |ℑσ| < R is free of the eigenvalues of problem (3.145). Therefore, if f ∈ H 2k,k,γ (G) and a2 is large enough, then under a finite number of “orthogonality” conditions on f there is a solution u ∈ H 2(k+1),k+1,γ (G) of (3.144) vanishing at ∂G. Moreover, the space of solutions of the homogeneous problem is finite dimensional and it consists of smooth functions in G.

4 Asymptotic Expansions of Singular Perturbation Theory 4.1 Small Random Perturbations of Dynamical Systems The section is devoted to asymptotic analysis of the Dirichlet problem for a secondorder partial differential equation containing a small parameter multiplying the highest-order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain. We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node.

4.1.1 White Noise Perturbation of Dynamical Systems Consider the first boundary value problem for a second-order elliptic equation with small parameter 0 < ε ≪ 1, ℓε u := ε∆u + a1 (x, y)∂ x u + a2 (x, y)∂ y u = f(x, y),

for

(x, y) ∈ Ω ,

u = g(x, y),

for

(x, y) ∈ ∂Ω .

(4.1)

Here, Ω = {(x, y) ∈ R2 : x2 + y2 < R2 } is the disk of radius R with center at the origin, ∆ is the Laplace operator in the plane, the coefficients a1 and a2 are assumed to be smooth functions in a neighborhood of the closure of Ω, and f , g are smooth functions in the closure of Ω and at the boundary of Ω, respectively. From a priori estimates of the Schauder type it follows that for every fixed ε > 0 problem (4.1) has a unique solution u = u(x, y; ε), see, for instance, [84, Ch. 3]. We are interested in studying the asymptotic behavior of the solution u as ε → 0. Assume that the dynamical system ẋ = a1 (x, y) , ẏ = a2 (x, y) ,

(4.2)

t ≥ 0, has a unique stationary solution x = 0, y = 0 in Ω. We focus on the case where this solution is asymptotically stable, i.e., the trajectories of the system point towards the domain Ω and tend to the origin. It should be noted that boundary value problem (4.1) contains a small parameter multiplying the highest-order derivatives. It is known [113, 284] that the behavior of the solution to (4.1) depends on the characteristics of the limit equation ℓ0 u = f . The well-known perturbation method for constructing an asymptotic solution of problem (4.1) starts with a “good” solution of the limit problem. If such a solution is available, it can be glued together with a boundary layer constructed by transition to stretched https://doi.org/10.1515/9783110534979-004

4.1 Small Random Perturbations of Dynamical Systems

| 215

coordinates x󸀠 = √εx and y󸀠 = √εy. This method falls short of providing an asymptotic solution of problem (4.1), for under the presence of singular point of the vector field a = (a1 , a2 ) the limit problem fails to possess any “good” solution. Moreover, the approximation obtained in this way grows exponentially as ε → 0, which makes the use of boundary layer inefficient. Hence, the study of (4.1) requires more advanced techniques. Problem (4.1) appears in the study of white noise effect in the stability theory of fixed points of dynamical system (4.2). For this purpose, we consider the perturbed equations in the form of stochastic differential equations dX t = a1 (X t , Y t )dt + √2ε dW t1 , dY t = a2 (X t , Y t )dt + √2ε dW t2

(4.3)

under the initial condition X0 = x, Y 0 = y. Here, W t1 (ω) and W t2 (ω) are independent one-dimensional Winer processes defined on a probability space (X, A, P), where X is an arbitrary nonempty set, A a sigma-algebra, and P a probability measure. The solution X t (ω), Y t (ω) of this system is a stochastic process that depends on the parameter ε > 0. It is well known [280] that the trajectories of (4.3) leave any bounded domain in R2 with probability one. Hence, there is no stability of the fixed point x = 0, y = 0 under white noise perturbations. Denote by τ Ω (ω) = inf{t ≥ 0 : (X t (ω), Y t (ω)) ∉ Ω} the first exit time from the domain Ω. It is of interest to compute the mean exit time Eτ Ω of stochastic trajectories (X t (ω), Y t (ω)), when the noise intensity is sufficiently small, i.e., 0 < ε ≪ 1. It is worth pointing out that the solution of the elliptic equation (4.1) is associated with certain probabilistic parameters of stochastic trajectories. For instance, if f ≡ −1 and g = 0, then u(x, y; ε) ≡ τ Ω (see [240], p. 110). Hence, it follows that asymptotic analysis of solution of boundary value problem (4.1) as ε → 0 is of great importance in the research of dynamical systems (4.2) under white noise perturbation. As a prime example let us consider boundary value problem (4.1) in dimension one, i.e., n = 1, εu 󸀠󸀠xx − xu 󸀠x = −1, for x ∈ (−1, 1) , u(−1) = 0, u(1) = 0, where 0 < ε ≪ 1. The solution is given by x

s

−1

0

1 u(x; ε) = − ∫ exp(s2 /2ε) ∫ exp(−z2 /2ε)dzds . ε Using Laplace’s method one finds an asymptotic expansion for the solution as ε → 0. Namely, επ (1 + ε + 3ε2 + O(ε3 )) , u(x; ε) = e 1/2ε √ 2

216 | 4 Asymptotic Expansions of Singular Perturbation Theory as ε → 0, the expansion being uniform in x in each interval |x| ≤ 1 − δ with δ > 0. It should be noted that the solution has a growing exponential in its asymptotic expansion, which does not depend on x in the main term. Apparently, such intriguing effects can also appear in the two-dimensional case to be studied below. Such problems have been investigated using probabilistic methods. In particular, an exponential estimate for the solution of (4.1) was found in [280] for the case of f ≡ −1 and g = 0. The work [186] presents a method for obtaining the leading term of asymptotic expansion of the solution u. In [120], one finds a proof of this formula for the case of potential vector fields a, and in [216] a proof for arbitrary vector fields. However, the construction of full asymptotic expansion for solution (4.1) still remains an open problem. The present section is devoted to asymptotic analysis of boundary value problem (4.1) in the spatial case, where the characteristics of the limit equation bear radial symmetry. In other words, the stationary solution of system (4.2) is actually a proper node. Then equation (4.1) can be written in the usual polar coordinates (r, φ) as ε(∂2r + r−1 ∂ r + r−2 ∂2φ )u + b(r)∂ r u = f(r, φ), u = g(φ),

if

r 0, the solution u ∈ C2 (Ω) of boundary value problem (4.4) has the form of (4.9) and u(r, φ; ε) = G0 + o(1), as ε → 0, uniformly in r ∈ [ε1/2−δ ; R − δ] and φ ∈ (0, 2π], for any δ > 0.

4.1.3 The Case of the Homogeneous Boundary Condition As in Section 4.1.2, the solution is constructed in the form of (4.5). Then its coefficients fulfill the boundary value problem l k U k (r, ε) = F k (r),

for

r 0 and k ≠ 0. Using the maximum principle we immediately obtain max |U k (r; ε)| ≤

r∈[0;R]

R2 max |F k (r)| , k 2 ε r∈[0;R]

(4.11)

4.1 Small Random Perturbations of Dynamical Systems

| 221

for all ε > 0. It follows that the Fourier series (4.5) converges together with the first and second derivatives. Our next concern will be the asymptotics of the constructed solution u(r, φ; ε) as ε → 0. Let f ∈ C2 (Ω). Then, for any k > 0, the Fourier coefficient F k (r) possesses the asymptotics F k (r) = r|k| (F k,0 + rF k,1 + O(r2 )) , as r → 0. Note that if b(r) = −r, θ(r) = r2 /2 and F0 (r) does not vanish identically, and then U0 (r; ε) has exponential growth as ε → 0. To wit, U0 (r; ε) = −c(ε) e R

2

/2ε

∞

∑ j=1

ω2j ε j , R2j

as ε → 0, uniformly in r ∈ [ε1/2−δ , R − δ], for any δ > 0, where c(ε) = F0,0 + ε1/2 √2πF0,1 + O(ε) , ω2j = (2j − 2)!! . The remaining coefficients U k (r; ε) with k ≠ 0 fulfill inequalities (4.11). Furthermore, there is an available estimate for the remainder of the Fourier series U(r, φ; ε) − U0 (r; ε) = O(ε−1 )

(4.12)

as ε → 0, uniformly in (r, φ) ∈ Ω. We denote by ⟨f(r, φ)⟩ the average value of the function f(r, φ) on the interval [0, 2π], i.e., 2π

1 ⟨f(r, φ)⟩ = ∫ f(r, φ)dφ = F0 (r) . 2π 0

Theorem 4.1.2. Suppose that b ∈ C1 [0, R] and f ∈ C1 (Ω), g ≡ 0. Then, for any ε > 0, the solution u ∈ C2 (Ω) of the boundary value problem (4.4) has the form (4.5). If, moreover, ⟨f(r, φ)⟩ is different from zero, then the solution has the asymptotics u(r, φ; ε) = e θ(R)/ε

ε2 F0,0 (1 + O(ε1/2 )) , Rb(R)

as ε → 0, uniformly in r ∈ [ε1/2−δ , R − δ] and φ ∈ (0, 2π], for any δ > 0. If ⟨f(r, φ)⟩ = 0, then U0 (r; ε) ≡ 0, and it is necessary to analyze the behavior of U k (r; ε) as ε → 0 to construct the asymptotics of the solution. A rough estimate O(ε−1 ) follows from the maximum principle. However, this estimate can be specified in the particular case b(r) = −r by evaluating the asymptotics of the Fourier coefficients U k (r; ε), as ε → 0. To do this, one ought to investigate certain Laplace-type integrals, see [68].

222 | 4 Asymptotic Expansions of Singular Perturbation Theory

4.1.4 The Case of a Right-Hand Side of Zero Average Value Using variation of constants, one easily obtains a particular solution V k (r; ε) to inhomogeneous differential equation (4.10). That is, r

s

0

0

2 2 1 V0 (r; ε) = ∫ s−1 e s /2ε ∫ ze−z /2ε F0 (z)dz ds = 0 , ε

for F0 (r) = 0. If k ≠ 0, then r

V k (r; ε) = Φ k (r; ε) ∫ ze

r −z2 /2ε

F k (z)Ψ k (z; ε)dz − Ψ k (r; ε) ∫ ze−z

2

/2ε

F k (z)Φ k (z; ε)dz .

0

R

The solution of (4.10) with b(r) = −r is constructed in the form U k (r; ε) = V k (r; ε) + C k (ε)Φ k (r; ε) + D k (ε)Ψ k (r; ε) ,

(4.13)

for k ∈ Z. Once again, we set D k (ε) ≡ 0 to exclude singularities at zero. Then C0 (ε) ≡ 0 and R

C k (ε) =

2 Ψ k (R; ε) ∫ ze−z /2ε F k (z)Φ k (z; ε) dz , Φ k (R; ε)

0

for k ≠ 0. Formula (4.13) allows us to derive the asymptotics of the Fourier coefficients U k (r; ε), as ε → 0 (see Section 4.1.6). More precisely, we get ∞

∞

j=2

j=2

U k (r; ε) = ln ε ∑ τ k,j (r)r−j ε j/2 + ∑ σ k,j (r)r−j ε j/2 ,

(4.14)

as ε → 0, uniformly in r ∈ [ε1/2−δ ; R − δ], for any δ > 0. The coefficients σ k,j (r) and τ k,j (r) are bounded functions of r ∈ [0, R]. Theorem 4.1.3. Let b(r) = −r. Suppose f ∈ C1 (Ω) has zero average value on the interval [0, 2π] and g ≡ 0. Then the Fourier coefficients of the solution u ∈ C2 (Ω) to the boundary value problem (4.4) have asymptotics (4.14) uniformly in r ∈ [ε1/2−δ , R − δ], for all δ > 0.

4.1.5 Conclusion We construct an explicit formal solution of the Dirichlet problem (4.1) and establish its asymptotic character, as ε → 0. If ⟨f(r, φ)⟩ ≠ 0, then the solution grows exponentially, as ε tends to zero. If ⟨f(r, φ)⟩ = 0, then the solution has power-logarithmic asymptotics.

4.1 Small Random Perturbations of Dynamical Systems |

223

4.1.6 Appendix Here we compute asymptotic estimates for the solution U k (r; ε) of the boundary value problem (4.10), as ε → 0. All asymptotic series written here are uniform with respect to the parameter r ∈ [ε1/2−δ , R − δ], where δ > 0 is an arbitrary small number. We rewrite U k (r; ε) as U k (r; ε) = Φ k (r; ε) (J 2k (r; ε) + C k (ε)) + Ψ k (r; ε) (J 1k (0; ε) − J 1k (r; ε)) , where

r

J 1k (r; ε) := ∫ ze−z

2

/2ε

F k (z)Φ k (z; ε)dz ,

2

/2ε

F k (z)Ψ k (z; ε)dz .

R r

J 2k (r; ε) := ∫ ze−z R

The functions J 1k (r; ε) and J 2k (r; ε) are Laplace-type integrals bearing asymptotic estimates ∞

J 1k (r; ε) = ∑ α k,l (r)ξ −2l , l=1

J 2k (r; ε) = e−ξ

2

/2

∞

∑ β k,l (r)ξ −2l , l=1

as ξ → ∞. The coefficients β k,l (r) are linear combinations of F k (r) and its derivatives, in particular, β k,1 (r) = −r2 F k (r) , β k,2 (r) = −b k,2 r2 F k (r) − r3 F 󸀠k (r) , etc., while r

α k,l (r) = a k,2l r2l ∫ z1−2l F k (z)dz . R

The functions α k,l (r) and β k,l (r) are bounded. The construction of the asymptotic expansion of J 1k (0; ε) is slightly more complicated. Let k = 2m with a nonnegative integer m. Then, using an explicit representation for the integrand, we find that m

1 1 ̃ (ε)ε m+1 , J 2m (0; ε) = ∑ Ã 2m,2j ε j + J 2m j=1

where R

à 2m,2j = −a2m,2j ∫ z1−2j F2m (z) dz , 0 j

R/√ε

m 1+2(|m|−j+l) 2 ̃1 (ε) = ∑ a2m,2j ∑ ∫ z J 2m F̃ 2m (√εz)e−z /2 dz = O(1) , l l! 2 j=1 l=0 0

224 | 4 Asymptotic Expansions of Singular Perturbation Theory as ε → 0. Here, F̃ 2m (r) = r−2|m| F2m (r) = O(1), as r → 0. If k = 2m + 1, then n

F k (z) = z|k| ( ∑ F k,l z l + F̃ k,n+1 (z)) , l=0

where F̃ k,n+1 (z) =

O(z n+1 ) as

z → 0. Furthermore, we get

Φ k (z; ε) = e ξ

2

/2

N

( ∑ a k,j ξ −2j + Φ̃ k,N+1 (ξ)) , j=1

where Φ̃ k,N+1 (ξ) = O(ξ −2N−2 ), as ξ → ∞. It follows that 1 J2m+1 (0; ε) = I 1k (ε) + I 2k (ε) + I 3k (ε) + I 4k (ε) ,

where I 1k (ε)

n

R

l=0

0

n

R

l=0

0

Nl

:= − ∑ F k,l ∫ z|k|+1+l ∑ j=1

a k,2j ε j dz , z2j

I 2k (ε) := − ∑ F k,l ∫ z|k|+1+l Φ̃ k,N l +1 ( N

R

l=1

0

z )dz , √ε

I 3k (ε) := − ∑ a k,2l ε l ∫ z|k|+1−2j F̃ k,n+1 (z)dz , R

z I 4k (ε) := − ∫ z|k|+1 F̃ k,n+1 (z)Φ̃ k,N+1 ( )dz , √ε 0

where N > 1 and n > 0. For each l ≥ 0 we choose N l = [(l + k + 1)/2] in I 1k (ε). Then Nn

I 1k (ε) = ∑ λ̃ k,2j ε j , j=1

where λ̃ k,2j are constants. We consider the terms of I 2k (ε) with l = 2p and l = 2p + 1, to wit, R

I 2k,2p

z )dz = ∫ z2|m|+2p+2 Φ̃ k,|m|+p+2 ( √ε 0

∞

=ε

|m|+p+3/2

∞

( ∫ − ∫ )ξ 2|m|+2p+2 Φ̃ k,|m|+p+2(ξ)dξ 0

R/√ε

= ν k,2p ε|m|+p+3/2 +

∞

∑ j=|m|+p+2

λ k,j ε j

4.2 Formal Asymptotic Solutions

|

225

and R

I 2k,2p+1 = ∫ z2|m|+2p+3 Φ̃ k,|m|+p+2( 0 1

z )dz √ε

R/√ε

= ε|m|+p+2 ( ∫ + ∫ )ξ 2|m|+2p+3 (a k,2|m|+2p+4 ξ −2|m|−2p−4 + Φ̃ k,|m|+p+3 (ξ))dξ 0

1

= (ν k,2p+1 + μ k,2p+1 ln ε) ε|m|+p+2 +

∞

λ k,j ε j .

∑ j=|m|+p+3

The coefficients λ k,j , ν k,j , μ k,2p+1 can be computed explicitly and they do not depend on ε. This gives an asymptotic estimate for the sum ∞

∞

j=2

j=2

I 1k (ε) + I 2k (ε) = ln ε ∑ μ̃ k,j ε j/2 + ∑ λ̃ k,j ε j/2 , as ε → 0. In

I 3k (ε)

and

I 4k (ε)

we choose N = N n−1 for n > 1. Then N n−1

I 3k (ε) + I 4k (ε) = ∑ λ̃ k,2j ε j + O(ε N n−1 +1 ) , j=1 1 as ε → 0, n > 1. Thus, we arrive at an asymptotic estimate for J 2m+1 (0; ε) ∞

∞

j=2

j=2

1 (0; ε) = ln ε ∑ A2m+1,j ε j/2 + ∑ B2m+1,j ε j/2 , J 2m+1

with some constants A2m+1,j and B2m+1,j independent of ε. From this, the desired asymptotic expansion for U k (r; ε) follows.

4.2 Formal Asymptotic Solutions We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameters multiplying the derivative in t. The behavior of the solution at characteristic points of the boundary is of special interest. The behavior is well understood if a characteristic line is tangent to the boundary with a contact degree of at least 2. We allow the boundary to not only have a contact degree of less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.

4.2.1 Asymptotic Phenomena Discontinuities and quick transitions occur in various branches of physics. The mathematical questions involved are also rather classical. However, they are quite alive

226 | 4 Asymptotic Expansions of Singular Perturbation Theory

today and they will remain so for some time, cf. [85]. Quick transitions frequently befall in situations in which one perhaps would not speak of a discontinuity. A case in point is Prandtl’s ingenious concept of the boundary layer, which he presented at the 1904 Leipzig Mathematical Congress, see [224]. This is a narrow layer along the surface of a body, traveling in a fluid, across which the flow velocity changes quickly. The paper began the study of fluid dynamical boundary layers by analyzing viscous incompressible flow past an object as the Reynolds number becomes infinite. Friedrichs called asymptotic all those phenomena that show discontinuities, quick transitions, nonuniformities, or their incongruities resulting from the approximate description. In the mathematical treatment of such phenomena, physicists have developed systematic mathematical procedures. In such an approach one may introduce an appropriate quantity with respect to powers of a parameter, ε. This expansion is to be set up in such a way that the quantity is continuous for ε > 0 but discontinuous for ε = 0. Naturally, a series expansion with this character must have peculiar properties. In general, these series do not converge. The use of a series that does not necessarily converge is a typical instance of a “formal procedure.” The idea of giving validity to these formal series goes back at least as far as Poincaré [221]. He proved that these formal series represent asymptotic expansions of actual solutions. Thus it became clear in which way formal series solutions may be regarded as “valid.” Let us explain asymptotic phenomena in connection with singular perturbation problems. In a singular perturbation problem one is concerned with a differential equation of the form A(ε)u ε = f ε with initial or boundary conditions B(ε)u ε = g ε , where ε is a small parameter. The distinguishing feature of this problem is that the orders of A(ε) and B(ε) for ε ≠ 0 are higher than the orders of A(0) and B(0), respectively. The differential problem in question is referred to as a perturbed problem when ε ≠ 0 and a degenerate problem when ε = 0. We are not interested in solutions of this problem for each fixed value of the parameter ε, but rather in the dependence of such solutions on this parameter, in particular, in a neighborhood of ε = 0. A discussion of the role of singular perturbation phenomena in mathematical physics can be found in [132]. Some difficulties are inherent in singular perturbation problems. Solutions of the degenerate problem will not, in general, be as smooth as solutions of the perturbed problem. Moreover, solutions of the degenerate problem usually will not satisfy as many initial or boundary conditions as solutions of the perturbed problem do. Hence, if solutions of the perturbed problem are to converge to solutions of the degenerate problem, the notion of convergence will probably have to be rather weak. Due to the “loss” of initial or boundary data it may also happen that solutions of the perturbed problem converge in a stronger sense in the interior of the underlying domain, than in the vicinity of the boundary. This is precisely the boundary layer phenomenon observed by Prandtl. By now there is a vast amount of literature on singular perturbation problems for ordinary differential equations, both linear and nonlinear. An extensive bibliography of this literature is contained in [292]. There is also a considerable

4.2 Formal Asymptotic Solutions

|

227

amount of literature on singular perturbation problems for partial differential equations. A comprehensive theory of such problems was initiated by the remarkable paper of Vishik and Lyusternik [284]. They obtained asymptotic expressions for solutions of the perturbed problem for linear equations using boundary layer techniques. In this paper, the main condition on the dependence of A(ε) on a small parameter was formulated and the asymptotics as ε → 0 of the solution of the Dirichlet problem was constructed. It also contains a sizable bibliography. Huet [110] published several theorems on convergence in singular perturbation problems for linear elliptic and parabolic partial differential equations. One particular feature distinguishes this paper from those previously mentioned. This is that convergence theorems are first proven in a Hilbert space setting and then applied to the differential problems as opposed to starting directly with the differential equations. In the elliptic case, theorems on local convergence and convergence of tangential derivatives at the boundary are also proven. The work [110] is fundamental to the considerations in [101], aimed at obtaining rate of convergence estimates for solutions of singular perturbations of linear elliptic boundary value problems. The problem can be described as follows. Let X be a compact smooth manifold and let ε be a positive real parameter. Consider two elliptic boundary value problems on X, (εA1 )+ A0 )u ε = f and A0 u = f , where the order of A1 is greater than the order of A0 . The problem is to determine in what sense u ε converges to u on X as ε → 0 and to estimate the rate of convergence. In the 1970s pseudodifferential problems with small parameters were studied in [54] and [63]. For boundary value problems of the general type the theory of singular perturbations was developed in the 1980s by Frank, see [81]. In [204] the Vishik–Lyusternik method is developed for general elliptic boundary value problems in domains with conical points. However, this paper falls short of providing explicit Shapiro–Lopatinskii type condition of ellipticity with a small parameter; this latter is replaced by a priori estimates for corresponding problems for ordinary differential equations on the half-axis. In [288], Volevich completed the theory of differential boundary value problems with small parameter by formulating the Shapiro–Lopatinskii type ellipticity condition and proving that it is equivalent to a priori estimates uniform in the parameter. It should be noted that paper [288] restricts itself to operators with constant coefficients in the half-space. Asymptotic analysis includes two basic steps. The first is the actual construction of asymptotics. One has to choose the form in which the formal asymptotic expansion of a solution is to be sought and specify how to construct this expansion. The second step includes the justification of asymptotics, i.e., a proof that the formal asymptotic expansion is an asymptotic solution indeed. This is achieved by estimating the discrepancy. Matching of asymptotic expansions of solutions of boundary value problems is presented in the book [113]. The purpose of our work is to describe the boundary layer near a characteristic point of the boundary. We restrict the discussion to the Dirichlet problem for the heat equation in a bounded plane domain G, which contains a small parameter multiplying the time derivative. The boundary points at

228 | 4 Asymptotic Expansions of Singular Perturbation Theory which the tangent is orthogonal to the time axis are characteristic. The boundary of G is, moreover, allowed to have singularities at characteristic points. We construct an explicit asymptotic solution of the problem in a neighborhood of a characteristic point. It has the form of a Puiseux series in fractional powers of t/ε up to an exponential factor. Our asymptotic formula demonstrates rather strikingly that the boundary layer degenerates at a characteristic point unless the contact degree of the boundary and a characteristic line is sufficiently large (at least 2).

4.2.2 Blow-Up Techniques Consider the first boundary value problem for the heat equation in a domain G ⊂ R2 of the type of Fig. 3.2 with f(x) = x p . The boundary of G is assumed to be C∞ except for a finite number of characteristic points. At points like P1 and P2 the boundary curve possesses a tangent that is horizontal, hence ∂G is characteristic for the heat equation at such points. The characteristic touches the boundary with the degree eq2, which is included in the treatise [143]. At points like P2 , the boundary curve is not smooth, but it smoothly touches a characteristic from below and above. Such points are, therefore, cuspidal singularities of the boundary; explicit treatable cases have been studied in [14]. In this section, we restrict our discussion to characteristic points like P3 and P5 . These are cuspidal singularities of the boundary curve, which touches smoothly a vertical line at P3 and P5 . Thus, the boundary meets a characteristic at P3 and P5 at contact degree < 2. The study of the regularity of such points for solutions of the first boundary value problem for the heat equation goes back at least as far as [92]. The classical approach of [92] rests on the potential theory. A modern approach to studying boundary value problems in domains with singular points is based on so-called blow-up techniques, cf. [228]. In [13] it was applied to the first boundary value problem for the heat equation in domains with boundary points like P3 and P5 to get both a regularity theorem and the Fredholm property in weighted Sobolev spaces. The first boundary value problem for the heat equation in G is formulated as follows: Write Σ for the set of all characteristic points P1 , P2 , . . . on the boundary of G. Given functions f in G and u 0 on ∂G \ Σ, find a function u on G \ Σ, which satisfies εu 󸀠t − u 󸀠󸀠x,x = f u = u0

in

G,

at

∂G \ Σ ,

(4.15)

where ε ∈ (0, ε0 ] is a small parameter. By the local principle of Simonenko [254], the Fredholm property of problem (4.15) in suitable function spaces is equivalent to the local invertibility of this problem at each point of the closure of G. Here, we focus upon the points like P3 .

4.2 Formal Asymptotic Solutions

| 229

Suppose the domain G is described in a neighborhood of the point P3 = (x0 , t0 ) by the inequality t − t0 > |x − x0 |p , (4.16) where p is a positive real number. There is no loss of generality in assuming that P3 is the origin and |x − x0 | ≤ 1. We now blow up the domain G at P3 by introducing new coordinates (ω, r) with the aid of x = t1/p ω , (4.17) t = εr , where |ω| < 1 and r ∈ (0, 1/ε). It is clear that the new coordinates are singular at r = 0, for the entire segment [−1, 1] on the ω -axis is blown down into the origin by (4.17). The rectangle (−1, 1) × (0, 1/ε) transforms under the change of coordinates (4.17) into the part of the domain G near P3 lying below the line t = 1. Note that for ε → 0 the rectangle (−1, 1) × (0, 1/ε) stretches to the whole half-strip (−1, 1) × (0, ∞). In the domain of coordinates (ω, r) problem (4.15) reduces to an ordinary differential equation with respect to the variable r with operator-valued coefficients. More precisely, under the transformation (4.17) the derivatives in t and x change by the formulas ε

∂u ∂u 1 ω ∂u = − , ∂t ∂r r p ∂ω ∂u 1 ∂u = , ∂x (εr)1/p ∂ω

and so (4.15) transforms into r Q U r󸀠 −

ω 󸀠 1 󸀠󸀠 U ω,ω − r Q−1 U ω = rQ F Q p ε U = U0

in

(−1, 1) × (0, 1/ε) ,

at

{±1} × (0, 1/ε) ,

(4.18)

where U(ω, r) and F(ω, r) are pullbacks of u(x, t) and f(x, t) under transformation (4.17), respectively, and 2 Q= . p We are interested in the local solvability of problem (4.18) near the edge r = 0 in the rectangle (−1, 1) × (0, 1/ε). Note that the ordinary differential equation degenerates at r = 0, since the coefficient r2/p of the higher-order derivative in r vanishes at r = 0. For the parameter values ε > 0, the exponent Q is of crucial importance for specifying the ordinary differential equation. If p = 2, then it is a Fuchs type equation, these are also called regular singular equations. The Fuchs type equations fit well into an algebra of pseudodifferential operators based on the Mellin transform. If p > 2, then the singularity of the equation at r = 0 is weak and so the regular theory of finite smoothness applies. In the case p < 2, the degeneracy at r = 0 is strong, and the equation cannot be treated except by the theory of slowly varying coefficients [228].

230 | 4 Asymptotic Expansions of Singular Perturbation Theory

4.2.3 Formal Asymptotic Solution To determine appropriate function spaces in which a solution of problem (4.18) is sought, one constructs formal asymptotic solutions of the corresponding homogeneous problem. That is, ω 󸀠 1 󸀠󸀠 − r Q−1 U ω = 0 in (−1, 1) × (0, ∞) , r Q U r󸀠 − Q U ω,ω p ε (4.19) U(±1, r) = 0

(0, ∞) .

on

We first consider the case p ≠ 2. We look for a formal solution to (4.19) of the form U(ω, r) = e S(r) V(ω, r) ,

(4.20)

where S is a differentiable function of r > 0 and V expands as a formal Puiseux series with nontrivial principal part V(ω, r) =

1 reN

∞

∑ V j−N (ω) rej ; j=0

the complex exponent N and the real exponent e must be determined. Perhaps the factor r−eN might be included into the definition of exp S as exp(−eN ln r); however, we prefer to highlight the key role of Puiseux series. Substituting (4.20) into (4.19) yields ω 1 󸀠󸀠 r Q (S󸀠 V + V r󸀠 ) − Q V ω,ω − r Q−1 V ω󸀠 = 0 in (−1, 1) × (0, ∞) , p ε V(±1, r) = 0

on

(0, ∞) .

In order to reduce this boundary value problem to an eigenvalue problem we require the function S to satisfy the eikonal equation r Q S󸀠 = λ with a complex constant λ. This implies r1−Q S(r) = λ 1−Q up to an inessential constant to be included into a factor of exp S. In this manner, the problem reduces to ω 1 󸀠󸀠 r Q V r󸀠 − Q V ω,ω − r Q−1 V ω󸀠 = −λV in (−1, 1) × (0, ∞) , p ε (4.21) V(±1, r) = 0 If e =

Q−1 k

on

(0, ∞) .

for some natural number k, then ∞

r Q V r󸀠 = ∑ e(j − N − k) V j−N−k re(j−N) , j=k ∞

󸀠󸀠 󸀠󸀠 V ω,ω = ∑ V j−N re(j−N) , j=0 ∞

󸀠 re(j−N) , r Q−1 V ω󸀠 = ∑ V j−N−k j=k

4.2 Formal Asymptotic Solutions

| 231

as is easy to check. On substituting these equalities into (4.21) and equating the coefficients of the same powers of r we get two collections of Sturm–Liouville problems −

1 󸀠󸀠 V + λ V j−N = 0 ε Q j−N V j−N = 0

in

(−1, 1) ,

at

∓1,

(4.22)

for j = 0, 1, . . . , k − 1, and −

1 󸀠󸀠 ω 󸀠 V + λ V j−N = V j−N−k − e(j − N − k) V j−N−k p ε Q j−N V j−N = 0

in

(−1, 1) ,

at

∓1,

(4.23)

for j = mk, mk + 1, . . . , mk + (k − 1), where m takes on all natural values. Given any j = 0, 1, . . . , k − 1, the Sturm–Liouville problem (4.22) obviously has simple eigenvalues 1 π 2 λ n = − Q ( n) , 2 ε for n = 1, 2, . . ., a nonzero eigenfunction corresponding to λ n being sin 2π n(ω + 1). It follows that π V j−N (ω) = c j−N sin n(ω + 1) , (4.24) 2 for j = 0, 1, . . . , k − 1, where c j−N are constant. Without restriction of generality we can assume that the first coefficient V−N in the Puiseux expansion of V is different from zero. Hence, V j−N = c j−N V−N for j = 1, . . . , k − 1. For simplicity of notation, we drop the index n. On having determined the functions V−N , . . . , V k−1−N , we turn our attention to problems (4.23) with j = k, . . . , 2k − 1. Set f j−N =

ω 󸀠 V − e(j − N − k) V j−N−k , p j−N−k

then for the inhomogeneous problem (4.23) to possess a nonzero solution V j−N it is necessary and sufficient that the right-hand side f j−N be orthogonal to all solutions of the corresponding homogeneous problem, to wit V−N . The orthogonality refers to the scalar product in L2 (−1, 1). Let us evaluate the scalar product (f j−N , V−N ). We get (f j−N , V−N ) = c j−N−k (

1 󸀠 , V−N ) − e(j − N − k) (V−N , V−N )) (ωV−N p

and 󵄨 1 󸀠 󸀠 (ωV−N , V−N ) = ω |V−N |2 󵄨󵄨󵄨󵄨−1 − (V−N , V−N ) − (V−N , ωV−N ) 󸀠 = −(V−N , V−N ) − (ωV−N , V−N ) ,

the latter equality being due to the fact that V−N is real-valued and vanishes at ±1. Hence, 1 󸀠 , V−N ) = − (V−N , V−N ) (ωV−N 2

232 | 4 Asymptotic Expansions of Singular Perturbation Theory

and (f j−N , V−N ) = −c j−N−k (

1 + e(j − N − k)) (V−N , V−N ) , 2p

(4.25)

for j = k, . . . , 2k − 1. Since V−N ≠ 0, the condition (f j−N , V−N ) = 0 fulfills for j = k if and only if eN =

1 . 2p

(4.26)

Under this condition, problem (4.23) with j = k is solvable and its general solution has the form V k−N = V k−N,0 + c k−N V−N , where V k−N,0 is a particular solution of (4.23) and c k−N an arbitrary constant. Moreover, for (f j−N , V−N ) = 0 to fulfill for j = k + 1, . . . , 2k − 1, it is necessary and sufficient that c1−N = . . . = c k−1−N = 0, i.e., all of V1−N , . . . , V k−1−N vanish. This, in turn, implies that f k+1−N = . . . = f2k−1−N = 0, whence V j−N = c j−N V−N , for all j = k + 1, . . . , 2k − 1, where c j−N are arbitrary constants. We choose the constants c k−N , . . . , c2k−1 in such a way that the solvability conditions of the next k problems are fulfilled. More precisely, we consider the problem (4.23) for j = 2k, the right-hand side being ω 󸀠 ω 󸀠 − e(k − N) V k−N,0 ) + c k−N ( V−N − e(k − N) V−N ) f2k−N = ( V k−N,0 p p ω 󸀠 − e(k − N) V k−N,0 ) + c k−N (f k−N − ek V−N ) . = ( V k−N,0 p Combining (4.25) and (4.26) we conclude that (f k−N − ek V−N , V−N ) = −ek (V−N , V−N ) = (1 − Q) (V−N , V−N ) is different from zero. Hence, the constant c k−N can be uniquely defined in such a way that (f2k−N , V−N ) = 0. Moreover, the functions f2k+1−N , . . . , f3k−1−N are orthogonal to V−N if and only if c k+1−N = . . . = c2k−1−N = 0. It follows that V j−N vanishes for each j = k + 1, . . . , 2k − 1. Continuing in this fashion we construct a sequence of functions V j−N (ω, ε), for j = 0, 1, . . ., satisfying equations (4.22) and (4.23). The functions V j−N (ω, ε) are defined uniquely up to a common constant factor c−N . They depend smoothly on the parameter ε p . Moreover, V j−N vanishes identically unless j = mk with m = 0, 1, . . . . Therefore, V(ω, r, ε) = =

1 reN 1 r Q/4

∞

∑ V mk−N (ω, ε) remk m=0 ∞

∑ Ṽ m (ω, ε) r(Q−1)m

m=0

is a unique (up to a constant factor) formal asymptotic solution of problem (4.21) corresponding to λ = λ n .

4.2 Formal Asymptotic Solutions

| 233

Theorem 4.2.1. Let p ≠ 2. Then an arbitrary formal asymptotic solution of homogeneous problem (4.19) has the form U(ω, r, ε) =

c r Q/4

exp (λ

∞ r1−Q Ṽ m (ω, ε) ) ∑ (1−Q)m , 1 − Q m=0 r

where λ is one of eigenvalues λ n = −1/ε Q ( π2 n)2 . Proof. The theorem follows readily from (4.20). In the original coordinates (x, t) close to the point P3 in G the formal asymptotic solution looks like ε Q/4 λ x ε (1−Q)m t 1−Q ∞ u(x, t, ε) = c ( ) exp ( , ( ) ) ∑ Ṽ m ( 1/p , ε) ( ) t 1−Q ε t t m=0

(4.27)

for ε > 0. If 1 − Q > 0, i.e., p > 2, expansion (4.27) behaves in much the same way as boundary layer expansion in singular perturbation problems, since the eigenvalues are all negative. The threshold value p = 2 is a turning contact order under which the boundary layer degenerates.

4.2.4 The Exceptional Case p = 2 In this section we consider the case p = 2 in detail. For p = 2, problem (4.19) takes the form 1 󸀠󸀠 ω 󸀠 r U r󸀠 − U ω,ω − Uω = 0 in (−1, 1) × (0, ∞) , ε 2 (4.28) U(±1, r) = 0 on (0, ∞) . The problem is specified as a Fuchs type equation on the half-axis with coefficients in boundary value problems on the interval [−1, 1]. Such equations have been well understood, see [64] and elsewhere. If one searches for a formal solution to (4.28) of the form U(ω, r) = e S(r) V(ω, r) , then the eikonal equation rS󸀠 = λ gives S(r) = λ ln r, and so e S(r) = r λ , where λ is a complex number. It, therefore, makes no sense to look for V(ω, r), it being a formal Puiseux series in fractional powers of r. The choice e = (Q − 1)/k no longer works, and so a good substitute for a fractional power of r is the function 1/ ln r. Thus, ∞

V(ω, r) = ∑ V j−N (ω) ( j=0

1 j−N ) ln r

has to be a formal asymptotic solution of r V r󸀠 −

1 󸀠󸀠 ω − V 󸀠 = −λV V ε ω,ω 2 ω V(±1, r) = 0

in

(−1, 1) × (0, ∞) ,

on

(0, ∞) ,

234 | 4 Asymptotic Expansions of Singular Perturbation Theory

N being a nonnegative integer. Substituting the series for V(ω, r) into these equations and equating the coefficients of the same powers of ln r yields two collections of Sturm–Liouville problems, −

ω 1 󸀠󸀠 V − V 󸀠 + λ V−N = 0 ε −N 2 −N V−N = 0

in

(−1, 1) ,

at

∓1,

(4.29)

for j = 0, and −

ω 1 󸀠󸀠 − V 󸀠 + λ V j−N = (j − N − 1)V j−N−1 V ε j−N 2 j−N V j−N = 0

in

(−1, 1) ,

at

∓1,

(4.30)

for jeq1. Problem (4.29) has a nonzero solution V−N if and only if λ is an eigenvalue of the operator 1 ω 𝑣 󳨃→ 𝑣󸀠󸀠 + 𝑣󸀠 , ε 2 2 whose domain consists of all functions 𝑣 ∈ H (−1, 1) vanishing at ∓1. Then, equalities (4.30) for j = 1, . . . , N mean that V−N+1 , . . . , V0 are actually root functions of the operator corresponding to the eigenvalue λ. In other words, V−N , . . . , V0 is a Jordan chain of length N +1 corresponding to the eigenvalue λ. Note that for j = N +1 the righthand side of (4.30) vanishes, and so V1 , V2 , . . . is also a Jordan chain corresponding to the eigenvalue λ. This suggests that the series breaks beginning at j = N + 1. Moreover, a familiar argument shows that problem (4.29) has eigenvalues 1 π 2 1 λ n = − ( n) + o ( ) , ε 2 ε for n = 1, 2, . . ., which are simple if ε is small enough. Hence, it follows that N = 0 and π V0 (ω, ε) = c0 sin n(ω + 1) + o(1) , (4.31) 2 for ε → 0. Theorem 4.2.2. Suppose p = 2. Then an arbitrary formal asymptotic solution of the homogeneous problem (4.19) has the form U(ω, r, ε) = r λ V0 (ω, ε), where λ is one of the eigenvalues λ n . Proof. The theorem follows immediately from the above discussion. In the original coordinates (x, t) near the point P3 in G the formal asymptotic solution proves to be x ε −λ u(x, t, ε) = c ( ) V0 ( 1/2 , ε) , t t for ε > 0. This expansion behaves similarly to the boundary layer expansion in singular perturbation problems, since the eigenvalues are negative provided that ε is sufficiently small.

4.2 Formal Asymptotic Solutions

| 235

4.2.5 Degenerate Problem If ε = 0, then the homogeneous problem corresponding to the local problem (4.18) degenerates to 󸀠󸀠 U ω,ω =0

in

(−1, 1) × (0, ∞) ,

U=0

at

{±1} × (0, ∞) .

(4.32)

Substituting the general solution U(ω, r) = U1 (r)ω + U0 (r) of the differential equation into the boundary conditions readily implies U ≡ 0 in the half-strip, i.e., (4.32) has only zero solution. Corollary 4.2.3. If p ≤ 2, then the formal asymptotic solution of (4.19) converges to zero uniformly in t > 0 bounded away from zero, as ε → 0. Moreover, for p > 2 it vanishes exponentially. Proof. This follows immediately from Theorems 4.2.1 and 4.2.2. On the contrary, if p < 2, then the formal solution of problem (4.19) is hardly of asymptotic character, as ε → 0. Indeed, in this case the difference 1 − Q is negative, and so we get ε (1−Q)m ( ) →∞, t as ε → 0, while the exponent exp(λ/(1 − Q) (t/ε)1−Q ) tends to 1, see (4.27).

4.2.6 Generalization to Higher Dimensions The explicit formulas obtained above easily generalize to the evolution equation related to the b-th power of the Laplace operator in Rn , where b is a natural number. Consider the first boundary value problem for the operator ε∂ t + (−∆)b in a bounded domain G ⊂ Rn+1 . Note that the choice of sign (−1)b is explained exceptionally well by our wish to deal with parabolic (not backward parabolic) equations. By ε > 0 a small parameter is meant. The boundary of G is assumed to be C∞ except for a finite number of characteristic points. These are those points of ∂G at which the boundary touches with a hyperplane in Rn+1 orthogonal to the t -axis. As above, we restrict our attention to the analysis of the Dirichlet problem near a characteristic point like P3 or P5 in Fig. 3.2. The first boundary value problem for the evolution equation in G is formulated as follows. Let Σ be the set of all characteristic points of the boundary of G. Given any functions f in G → R u 0 , u 1 , . . . , u b−1 on ∂G \ Σ, find a function u on G \ Σ satisfying εu 󸀠t + (−∆)b u = f j ∂ν

u = uj

in

G,

at

∂G \ Σ ,

(4.33)

236 | 4 Asymptotic Expansions of Singular Perturbation Theory for j = 0, 1, . . . , b − 1, where ∂ ν is the derivative along the outward unit normal vector of the boundary. We focus upon a characteristic point P3 of the boundary which is assumed to be the origin in Rn+1 . Suppose the domain G is described in a neighborhood of the origin by the inequality t > f(x) , (4.34) where f is a smooth function of x ∈ Rn \ 0 homogeneous of degree p > 0. We blow up the domain G at P3 by introducing the new coordinates (ω, r) ∈ D × (0, 1/ε) with the aid of x = t1/p ω , (4.35) t = εr , where D is the domain in Rn consisting of those ω ∈ Rn that satisfy f(ω) < 1. Under this change of variables the domain G near P3 transforms into the half-cylinder D × (0, ∞), the cross-section D × {0} blowing down into the origin by (4.35). Note that for ε → 0 the cylinder D × (0, 1/ε) stretches into the whole half-cylinder D × (0, ∞). In the domain of coordinates (ω, r) problem (4.33) reduces to an ordinary differential equation with respect to the variable r with operator-valued coefficients. It is easy to see that under transformation (4.35) the derivatives in t and x change by the formulas 11 (ω, u 󸀠ω ) , pr

ε u 󸀠t = u 󸀠r − u 󸀠x k =

1 u󸀠 , (εr)1/p ω k

for k = 1, . . . , n, where (ω, u 󸀠ω ) = ∑nk=1 ω k (∂u)/(∂ω k ) stands for the Euler derivative. Thus, (4.33) transforms into r Q U r󸀠 +

1 1 󸀠 (−∆ ω )b U − r Q−1 (ω, U ω ) = rQ F p εQ j

∂ν U = Uj

in

D × (0, 1/ε) ,

at

∂D × (0, 1/ε)

(4.36)

for j = 0, 1, . . . , b − 1, where U(ω, r) and F(ω, r) are pullbacks of u(x, t) and f(x, t) under the transformation (4.35), respectively, and Q=

2b . p

We are interested in the local solvability of problem (4.36) near the base r = 0 in the cylinder D × (0, 1/ε). Note that the ordinary differential equation degenerates at r = 0, since the coefficient r Q of the higher-order derivative in r vanishes at r = 0. The theory of [228] still applies to characterize those problems (4.36) that are locally invertible. To describe function spaces that give the best fit for solutions of problem (4.36), we construct formal asymptotic solutions of the corresponding homogeneous problem.

4.2 Formal Asymptotic Solutions

| 237

That is, r Q U r󸀠 +

1 1 󸀠 (−∆ ω )b U − r Q−1 (ω, U ω )=0 Q p ε

in

D × (0, ∞) ,

∂ αω U = 0

on

∂D × (0, ∞) ,

(4.37)

for all |α| ≤ b − 1. We assume that p ≠ 2b. Similar arguments apply to the case p = 2b, the only difference being in the choice of the Ansatz (English: ansatz), see Section 4.2.4. We look for a formal solution to (4.37) of the form U(ω, r) = e S(r) V(ω, r), where S is a differentiable function of r > 0, and V expands as a formal Puiseux series with the nontrivial principal part V(ω, r) =

1 reN

∞

∑ V j−N (ω) rej , j=0

where N is a complex number and e a real exponent to be determined. On substituting U(ω, r) into (4.19) we extract the eikonal equation r Q S󸀠 = λ for the function S(r), where λ is a (possibly complex) constant to be defined. For Q ≠ 1 this implies r1−Q S(r) = λ , 1−Q up to an inessential constant factor. This way the problem reduces to r Q V r󸀠 +

1 1 (−∆ ω )b V − r Q−1 (ω, V ω󸀠 ) = −λV p εQ ∂ αω V = 0

in

D × (0, ∞) ,

on

∂D × (0, ∞) ,

(4.38)

for all |α| ≤ b − 1. Analysis similar to that in Section 4.2.3 shows that the right choice of e is e = (Q − 1)/k for some natural number k. On substituting the formal series for V(ω, r) into (4.38) and equating the coefficients of the same powers of r we get two collections of problems 1 (−∆)b V j−N + λ V j−N = 0 in D , εQ (4.39) ∂ α V j−N = 0 at ∂D , for all |α| ≤ b − 1, where j = 0, 1, . . . , k − 1, and 1 1 󸀠 (−∆)b V j−N + λ V j−N = (ω, V j−N−k ) − e(j − N − k) V j−N−k Q p ε ∂ α V j−N = 0

in

D,

at

∂D ,

(4.40)

for all |α| ≤ b − 1, where j = k, k + 1, . . . , 2k − 1, and so on. Given any j = 0, 1, . . . , k − 1, problem (4.39) is essentially an eigenvalue problem for the strongly nonnegative operator (−∆)b in L2 (D), whose domain consists of all

238 | 4 Asymptotic Expansions of Singular Perturbation Theory functions of H 2b (D) vanishing up to the order b − 1 at ∂D. The eigenvalues of the latter operator are known to be all positive and form a nondecreasing sequence λ󸀠1 , λ󸀠2 , . . ., which converges to ∞. Hence, (4.39) admits nonzero solutions only for λn = −

1 󸀠 λ , εQ n

where n = 1, 2, . . . . In general, the eigenvalues {λ󸀠n } fail to be simple. The generic simplicity of the eigenvalues of the Dirichlet problem for self-adjoint elliptic operators with respect to variations of the boundary have been investigated by several authors, see [215] and the references given there. We focus on an eigenvalue λ󸀠n of multiplicity 1, in which case the formal asymptotic solution is especially simple. By the above, this condition is not particularly restrictive. If λ = λ n , there is a nonzero solution e n (ω) of this problem, which is determined uniquely up to a constant factor. This yields V j−N (ω) = c j−N e n (ω) ,

(4.41)

for j = 0, 1, . . . , k − 1, where c j−N are constant. Without restriction of generality we can assume that the first coefficient V−N in the Puiseux expansion of V is different from zero. Hence, V j−N = c j−N V−N for j = 1, . . . , k − 1. For simplicity of notation, we drop the index n. On taking the functions V−N , . . . , V k−1−N for granted, we now turn to problems (4.23) with j = k, . . . , 2k − 1. Set f j−N =

1 󸀠 ) − e(j − N − k) V j−N−k , (ω, V j−N−k p

then for the inhomogeneous problem (4.40) to admit a nonzero solution V j−N it is necessary and sufficient that the right-hand side f j−N be orthogonal to all solutions of the corresponding homogeneous problem, to wit V−N . The orthogonality refers to the scalar product in L2 (D). Let us evaluate the scalar product (f j−N , V−N ). We get (f j−N , V−N ) = c j−N−k (

1 󸀠 ((ω, V−N ), V−N ) − e(j − N − k) (V−N , V−N )) p

and, by Stokes’ formula, n

󸀠 ), V−N ) = ∫ |V−N |2 (ω, ν) ds − ∑ ∫ V−N ((ω, V−N ∂D

k=1 D

∂ (ω k V−N ) dω ∂ω k

󸀠 = −n‖V−N ‖2 − ((ω, V−N ), V−N ) ,

the latter equality being due to the fact that V−N is real-valued and vanishes at ∂D. Hence, n 󸀠 ((ω, V−N ), V−N ) = − ‖V−N ‖2 2

4.2 Formal Asymptotic Solutions

and (f j−N , V−N ) = −c j−N−k (

n + e(j − N − k)) ‖V−N ‖2 , 2p

239

|

(4.42)

for j = k, . . . , 2k − 1. Since V−N ≠ 0, the condition (f j−N , V−N ) = 0 is fulfilled for j = k if and only if eN =

n . 2p

(4.43)

Under this condition, problem (4.40) with j = k is solvable, and its general solution has the form V k−N = V k−N,0 + c k−N V−N , where V k−N,0 is a particular solution of (4.40) and c k−N an arbitrary constant. Moreover, for (f j−N , V−N ) = 0 to be fulfilled for j = k + 1, . . . , 2k − 1 it is necessary and sufficient that c1−N = . . . = c k−1−N = 0, i.e., all of V1−N , . . . , V k−1−N vanish. This, in turn, implies that f k+1−N = . . . = f2k−1−N = 0, whence V j−N = c j−N V−N for all j = k + 1, . . . , 2k − 1, where c j−N are arbitrary constants. We choose the constants c k−N , . . . , c2k−1 in such a way that the solvability conditions of the next k problems are fulfilled. More precisely, we consider the problem (4.40) for j = 2k, the right-hand side being 1 1 󸀠 󸀠 ) − e(k − N)V k−N,0 ) + c k−N ( (ω, V−N ) − e(k − N)V−N ) f2k−N = ( (ω, V k−N,0 p p 1 󸀠 ) − e(k − N)V k−N,0 ) + c k−N (f k−N − ekV−N ) . = ( (ω, V k−N,0 p Combining (4.42) and (4.43) we conclude that (f k−N − ek V−N , V−N ) = −ek (V−N , V−N ) = (1 − Q) (V−N , V−N ) is different from zero. Hence, the constant c k−N can be uniquely defined in such a way that (f2k−N , V−N ) = 0. Moreover, the functions f2k+1−N , . . . , f3k−1−N are orthogonal to V−N if and only if c k+1−N = . . . = c2k−1−N = 0. It follows that V j−N vanishes for each j = k + 1, . . . , 2k − 1. Proceeding in this manner we construct a sequence of functions V j−N (ω, ε), for j = 0, 1, . . ., satisfying equations (4.39) and (4.40). The functions V j−N (ω, ε) are defined uniquely up to a common constant factor c−N . They depend smoothly on the parameter ε p . Moreover, V j−N vanishes identically unless j = mk with m = 0, 1, . . . . Therefore, V(ω, r, ε) = =

1 reN

∞

∑ V mk−N (ω, ε) remk m=0 ∞

1 ∑ Ṽ m (ω, ε) r(Q−1)m r n/2p m=0

240 | 4 Asymptotic Expansions of Singular Perturbation Theory

is a unique (up to a constant factor) formal asymptotic solution of problem (4.38) corresponding to λ = λ n . Summarizing, we arrive at the following generalization of Theorem 4.2.1. Theorem 4.2.4. Let p ≠ 2b. Then an arbitrary formal asymptotic solution of the homogeneous problem (4.37) has the form U(ω, r, ε) =

c r n/2p

exp (λ

∞ Ṽ m (ω, ε) r1−Q ) ∑ (1−Q)m , 1 − Q m=0 r

where λ is one of the eigenvalues λ n = −1/ε Q λ󸀠n . Thus, the construction of the formal asymptotic solution U of the general problem (4.33) follows by the same method as in Section 4.2.3. In the original coordinates (x, t) close to the point P3 in G the formal asymptotic solution looks like ∞ λ󸀠 t1−Q x ε (1−Q)m ε n/2p exp (− , ) ∑ Ṽ m ( 1/p , ε) ( ) u(x, t, ε) = c ( ) t ε 1 − Q m=0 t t

(4.44)

for ε > 0. If 1 − Q > 0, i.e., p > 2b, the expansion (4.44) behaves in much the same way as the boundary layer expansion in singular perturbation problems, since the eigenvalues are all negative. The threshold value p = 2b is a turning contact order under which the boundary layer degenerates. The computations of this section obviously extend both to eigenvalues λ n of higher multiplicity and arbitrary self-adjoint elliptic operators A(x, D) in the place of (−∆)b . When solving the nonhomogeneous equations (4.40), one chooses the only solution that is orthogonal to all solutions of the corresponding homogeneous problem (4.39). This special solution actually determines what is known as the Green operator. However, formula (4.44) becomes less transparent. Thus, we omit the details.

4.2.7 Parameter-Dependent Norms For p < 2b, the expansion (4.44) fails to be asymptotic in small ε > 0, even if (x, t) is bounded away from the boundary of D. An asymptotic character of this series can only be revealed using parameter-dependent norms. Indeed, if ε → 0, then the summands on the right-hand side of (4.44) increase, unless the quotient t/ε does not exceed 1. Hence, ε is allowed to tend to zero only under the condition that t/ε < 1. Then expansion (4.44) still reveals a certain asymptotic character. Within the framework of analysis on manifolds with singularities one exploits the weighted norms 1 t t −2μ (∫ exp (2γ Q ) ( ) |u(x, t, ε)|2 dxdt) ε t ε

1/2

D

on functions defined near the singular point, where γ and μ are real numbers, cf. [13].

4.3 The Shapiro–Lopatinskii Condition

| 241

4.3 The Shapiro–Lopatinskii Condition This section is concerned with elliptic problems, including a small parameter multiplying higher-order derivatives. We found algebraic conditions on the operator and boundary conditions that guarantee the Fredholm property, and prove an a priori estimate for the solution with a constant independent of the small parameter. These results are known for elliptic boundary value problems with small a parameter in the half-space R+n . We extend them to the case of bounded domains with smooth boundary. The small parameter coercive conditions are formulated and a two-sided estimate is proved.

4.3.1 Boundary Value Problems with Small Parameter This section studies linear elliptic differential equations with a small parameter at the highest derivative when boundary conditions also contain the same parameter. We consider operators acting on functions in some domain D and depending on a small parameter ε ≥ 0 in a special way. More precisely, ε2m−2μ A2m (x, D)u + ε2m−2μ−1 A2m−1 (x, D)u + . . . + A2μ (x, D)u = f , supplemented by the boundary conditions ε b j −β j B j,b j (x󸀠 , D)u + ε b j −β j −1 B j,b j−1 (x󸀠 , D)u + . . . + B j,β j (x󸀠 , D)u = u j , for x󸀠 ∈ ∂D, with j = 1, . . . , m, where A2m−i (x, D) and B j,b j−i (x󸀠 , D) are differential operators of order 2m − i and b j−i , respectively, with variable coefficients. This family of operators is assumed to be elliptic for each ε ≥ 0. The subject of our interest is the behavior of solutions u ε (x) when ε tends to zero. It is known (see, for example, [113]) that solutions u ε defined on a manifold with boundary may have the so-called boundary layer. In this case, the solution u ε (x) converges to u 0 (x), when ε → 0, uniformly in each strictly inner bounded subdomain D̃ ⊂ D, but need not converge at the boundary points x󸀠 ∈ ∂D. Essentially, this concept was introduced by Prandtl in 1904 (for an accurate historical background see, e.g., [12]). He studied fluid flow with small viscosity over a surface and explained how insignificant friction forces influence the main perfect fluid flow. His idea is based on splitting a solution into two parts, namely a solution near the boundary and a solution far away from the boundary, and stretching the coordinates in the normal direction of the boundary. In the paper [284] Lyusternik and Vishik extended this idea to differential equations that depend on a small parameter polynomially. Some applications of the Lyusternik–Vishik method for partial differential equations (PDE) are discussed in [275]. This method suggests to look for a solution as the sum of a regular part,

242 | 4 Asymptotic Expansions of Singular Perturbation Theory

which depends uniformly on ε, and an additional function, which grows rapidly when ε → 0. The regular part is found by using the ordinary method of small parameter; the boundary layer is supposed to be a solution of some ordinary differential equation (ODE) in the direction normal to ∂D. This ODE is obtained using coordinate stretching in the direction normal to the boundary, as was proposed by Prandtl. In [284] the problem was considered in the case of Dirichlet boundary conditions B j and strong uniform ellipticity of the operator A. Then the method was adapted to domains with conical points by Nazarov in [204]. In [81] it was extended to pseudodifferential operators and general elliptic boundary problems. In all these works uniform estimates in norms depending on ε for solutions were found under some generalized coercivity condition, and the estimates justify the formal asymptotic series obtained by the Lyusternik–Vishik method. However, the comprehensive theory of elliptic equations with small parameter was constructed by Volevich in [288]. For the problem (A, B) in the half-space R+n := {(x󸀠 , x n ) ∈ Rn : x n > 0} he introduced a Shapiro– Lopatinskii condition with small parameter and proved its necessity and sufficiency for the existence of two-sided uniform estimates for (A, B). Volevich used the norms proposed by Demidov in [54]. The paper by Volevich falls short of providing complete arguments in the case of arbitrary smooth bounded domains, and the aim of the present study is to extend the results of Volevich to the case of bounded domains D with smooth boundary ∂D using the local principle of elliptic theory (see, for example, [1]).

4.3.2 Asymptotic Expansion Now we apply the Vishik–Lyusternik method to the problem (A, B) to find an asymptotic expansion of solution u. The domain D is required to be bounded and have smooth boundary ∂D. Let us introduce the new coordinates (y1 , y2 , . . ., y n−1 , z) in D, such that y ∈ ∂D is a variable on the surface ∂D and z is the distance to ∂D. By A󸀠 (y, z, D y , D z , ε), B󸀠 (y, D y , D z , ε) we denote operators A and B in the new variables. We are looking for a solution of (A, B) in the form u(x, ε) = U(x, ε) + V(y, z/ε, ε), where U is the regular part of u and V is the boundary layer. Suppose that the function V(y, z/ε) satisfies the following three conditions: 1) V(y, z/ε, ε) is a sufficiently smooth solution of the homogeneous equation AV = 0; 2) V(y, z/ε, ε) depends on the “fast” variable t = z/ε; 3) V(y, z/ε, ε) differs from zero only in a small strip near the boundary ∂D. The regular part and the boundary layer are looked for as formal asymptotic series

∞

∞

U(x, ε) = ∑ ε k u k (x) ,

V(y, z/ε, ε) = ∑ ε k 𝑣k (y, z/ε) .

k=0

k=0

4.3 The Shapiro–Lopatinskii Condition

| 243

The first series is called the outer expansion and the second one is called the inner expansion. The outer expansion is obtained using the standard procedure of the small parameter method. We substitute the series for U(x, ε) into the equation A(x, D, ε)u = f and collect the terms with the same power of ε. This gives us the system A2μ u 0 = f ,

(4.45) k

A2μ u k = − ∑ A2μ+i u k−i ,

(4.46)

i=1

for unknowns u k . To determine the coefficients 𝑣k (y, z/ε) of the inner expansion we apply the operator A󸀠 (y, z, D y , D z , ε) to V(y, z/ε, ε). Condition 1) implies ∞

∑ ε k A󸀠 (y, z, D y , D z , ε)(𝑣k (y, z/ε)) = 0 . k=0

Let us rewrite the operator A󸀠 (y, z, D y , D z , ε) in the variables (y, z/ε = t). For the homogeneous part A󸀠k of degree k we have A󸀠k (y, z, D y , D z , ε) = ε−2μ+k A󸀠k (y, εt, εD y , D t ) . On expanding A󸀠k (y, εt, εD y , D t ) as a Taylor series about the point (y, 0, 0, D t ) we obtain ∞

A󸀠k (y, z, D y , D z , ε) = ε−2μ+k (A󸀠󸀠k (k, 0, 0, D t ) + ∑ ε l A k,l (y, t, D y , D t )) , l=1

where the operators A k,l have smooth coefficients. Therefore, ∞

A󸀠 (y, z, D y , D z , ε) = ε−2μ (A󸀠󸀠 (y, 0, 0, D t ) + ∑ ε l A l (y, t, D y , D t )) , l=1

A l depends linearly on A k,l . So it defines equations for 𝑣k k

A󸀠󸀠 (ξ, 0, 0, D t )𝑣k (y, t) = − ∑ A l (y, t, D y , D t )𝑣k−l . l=1

Now we substitute the partial sums n

U n = ∑ ε k u k (x) , k=0 n

V n = ∑ ε k 𝑣k (y, t) k=0

244 | 4 Asymptotic Expansions of Singular Perturbation Theory

into the original equation and boundary conditions and find the discrepancy. For A(x, D, ε), it looks like A(x, D, ε)(u(x, ε) − U n − V n ) = f − A2μ (x, D)u 0 − (A(x, D, ε)U n − A2μ (x, D)u 0 + A󸀠 (y, z, D y , D z , ε)V n ) = O(ε n+1 ) . Hence, if we are able to find appropriate Banach spaces ‖u‖󸀠 , ‖u‖󸀠∂D , such that the operator (A(x, D, ε), B(x󸀠 , D, ε)) is bounded uniformly in ε, then the difference u(x, ε) − U n − V n is small, and so the formal series indeed approximates the solution u(x, ε). This problem was solved by Volevich [288] for the case where D is the half-space R+n = {(x󸀠 , x n ) ∈ Rn : x n > 0}. He used the norms for functions in D and their traces that were introduced in [54]. They are of the form ‖u; H r,s (Rn )‖ = ‖(1 + |ξ|2 )s/2 (1 + ε2 |ξ|2 )(r−s)/2 u‖̂ L2 , ‖u; H ρ,σ (Rn−1 )‖ = ‖u‖L2 (Rn−1 ) + ‖|η|σ (1 + ε2 |η|2 )(ρ−σ)/2 u‖L2 (Rn−1 ) , where σ ≥ 0. In these norms there is an estimate m

‖u; H r,s (R+n )‖ ≤ C (‖A(x, D)u; H r,s (R+n )‖ + ∑ ‖B j (D󸀠 , ε)u; H r−b j −1/2,s−β j −1/2 (Rn−1 )‖ j=1

+ ‖u; L

2

(R+n ))

,

(4.47)

where C does not depend on ε. A trace theorem for the norms ‖u; H r,s (Rn )‖ and ‖u; H ρ,σ (Rn−1 )‖ is also proved in [288]. Theorem 4.3.1. For r > l + 1/2 and s ≥ 0, s ≠ l + 1/2, we have ‖D ln u(⋅, 0); H r−l−1/2,s−l−1/2(Rn−1 )‖ ≤ c ‖u; H r,s (R+n )‖ , with c a constant independent of ε. Volevich [288] also proves that estimate (4.47) holds true if the operator (A, B) satisfies the small parameter ellipticity condition, the Shapiro–Lopatinskii condition with small parameter, and the system of equations (4.45) and (4.46) is correctly solvable. The problem (A, B) is called an elliptic problem with a parameter if it satisfies all these conditions. The first two conditions are, of course, of greater interest than the last one. They read as follows: The small parameter ellipticity condition: The operator A(x, D, ε) is said to be small parameter elliptic at some point x0 if its principal polynomial A0 (x0 , ξ, ε) admits an estimate |A0 (x0 , ξ, ε)| ≥ c x0 |ξ|2μ (1 + ε|ξ|)2m−2μ from below. The Shapiro–Lopatinskii condition with small parameter: The problem (A, B) satisfies the Shapiro–Lopatinskii condition for every ε ≥ 0.

4.3 The Shapiro–Lopatinskii Condition |

245

As was mentioned, this work is aimed at extending the result of [288] to the case of bounded domains D with smooth boundary ∂D. To this end, we develop the local principle that underlies elliptic theory (see, for example, [1]) in the case of problems with parameter.

4.3.3 The Main Spaces Our first task is to introduce the main spaces. Hereinafter D stands for a bounded domain with smooth boundary in Rn . The spaces H r,s (Rn ) and H r,s (R+n ) are exactly the same as those used in the work of Demidov (see, for instance, [54]). Namely, H r,s (Rn ) consists of all functions u ∈ H r (Rn ), which have the finite norm ‖u‖r,s , and H r,s (R+n ) is the factor space H r,s (Rn )/H−r,s (Rn ), where H−r,s (Rn ) is the subspace of H r,s (Rn ) consisting of all functions with support in {x ∈ Rn : x n ≤ 0}. As usual, the factor space is endowed with the canonical norm ‖[u]; H r,s (R+n )‖ = inf ‖u‖r,s . u∈[u]

When it does not cause any misunderstanding we denote this norm simply by ‖u‖r,s . Analogously, we introduce the spaces of functions defined in some domain D. To wit, r,s

H r,s (D) := H r,s (Rn )/HRn \D (Rn ) , r,s

where functions of HRn \D (Rn ) are supported outside the domain D. This space is also given the canonical norm ‖u; H r,s (D)‖, which we sometimes denote by ‖u‖r,s for short. Lemma 4.3.2. Let f be a smooth function in Rn , such that f(x) = 1 for x ∈ D. Then ‖u; H r,s (D)‖ = ‖fu; H r,s (D)‖. Lemma 4.3.3. If u ∈ H r,s (Rn ) and supp u ⊂ D, then ‖u; H r,s (D)‖ = ‖u; H r,s (Rn )‖. For positive integer numbers s and r ≥ s the space H r,s (D) proves to be the completion of C∞ (D)̄ with respect to the norm ‖u; H r,s (D)‖r,s . The elliptic technique used in this work includes the “rectification” of the boundary. Therefore, the invariance of ‖ ⋅ ‖r,s with respect to a change of variables is one of the key points. For every fixed ϵ ≥ 0, the norms ‖ ⋅ ‖r,s are the ordinary Sobolev norms, and the main question is what kind of coordinate transformations save the form of the dependence of ‖ ⋅ ‖r,s on ε. The following statement displays how ε enters into the norms ‖ ⋅ ‖r,s . Lemma 4.3.4. For natural r and s satisfying r ≥ s, the norm ‖u; H r,s (D)‖2 has a representation r

∑ i=0 i is even

r

a r,s,i (ε)‖∆ i/2 u‖2L2 (D) + ∑ a r,s,i (ε)‖∇i u‖2L2 (D) , i=1 i is odd

where a r,s,i (ε) are polynomials of degree 2i and a r,s,0 (ε) ≠ 0.

246 | 4 Asymptotic Expansions of Singular Perturbation Theory

Proof. Applying the binomial formula we get s

s

(1 + |ξ|2 ) = ∑ C is |ξ|2i

(1 + ε2 |ξ|2 )

and

r−s

r−s

= ∑ C ir−s ε2i |ξ|2i .

i=0

i=0

Hence, on multiplying the left-hand sides of these equalities we obtain s

(1 + |ξ|2 ) (1 + ε2 |ξ|2 )

r−s

r

= ∑ a r,s,i (ε)|ξ|2i , i=0

where i

i−j j

a r,s,i (ε) = ∑ C s C r−s ε2j .

(4.48)

j=0

Here, we assume C kr = 0 when k > r. If ε = 0 or r = s, then a r,s,i = C is . Therefore, a r,s,i (ε) ≠ 0 for all ε and 0 ≤ i ≤ r. As a consequence, we get r

‖u‖2r,s = ∑ a r,s,i (ε)‖|ξ|i u‖̂ 2L2 . i=0

Furthermore, ‖|ξ|

i

u‖̂ 2L2

󵄩2 󵄩 {󵄩󵄩󵄩∆ i/2 u 󵄩󵄩󵄩 2 , 󵄩L 󵄩 = {󵄩 󵄩󵄩∇i u 󵄩󵄩󵄩2 , 󵄩 󵄩 󵄩L2 {󵄩

if i is even, if i is odd,

which establishes the lemma. Now everything is prepared for proving the invariance of the norm ‖ ⋅ ‖r,s with respect to local changes of variables x = T(y). Lemma 4.3.5. Let r, s ∈ Z≥0 satisfy r ≥ s. The norm ‖u; H r,s (D)‖ is invariant with respect to any local changes of variables in D of the form x = T(y), such that 1) T : U → U 󸀠 is a C r -diffeomorphism of domains U and U 󸀠 in Rn , both U and U 󸀠 intersecting D; 2) T(U ∩ D) = U 󸀠 ∩ D; 3) T(U ∩ ∂D) = U 󸀠 ∩ ∂D. Our task is to prove that there is a constant C > 0 independent of ε, with the property that ‖T ∗ u; H r,s (D)‖ ≤ C ‖u; H r,s (D)‖ ,

(4.49)

for all smooth functions u in the closure of D supported in some compact set K ⊂ ̄ Here, by T ∗ u(y) := u(T(y)) the pullback of u by the diffeomorphism T is meant. U 󸀠 ∩ D). If u is supported in K, then T ∗ u is supported in T −1 (K), which is a compact subset of U 󸀠 ∩ D̄ by the properties of T. Since this applies to the inverse T −1 : U 󸀠 → U, it follows from (4.49) that the space H r,s (D) survives under the local C r -diffeomorphisms of D.̄

4.3 The Shapiro–Lopatinskii Condition

| 247

Proof. For the proof we make use of another norm in H r,s (D), which is obviously equivalent to ‖u; H r,s (D)‖ and more convenient here. To wit, ‖u; H r,s (D)‖ ≅ ∑ a r,s,|α|(ε)‖∂ α u; L2 (D)‖

(4.50)

|α|≤r

(or r

‖u; H r,s (D)‖ ≅ ∑ a r,s,i (ε) ‖u; H i (D)‖ , i=0

as is easy to verify), where a r,s,i (ε) are the polynomials of Lemma 4.3.4. Choose a compact set K in U 󸀠 ∩ D.̄ As has been mentioned, if u is a smooth function in D with support in K, then T ∗ u is a smooth function in D with support in T −1 (K) ⊂ U ∩ D. Obviously, ‖T ∗ u; H r,s (D)‖ = ‖u ∘ T; H r,s (U ∩ D)‖ = ∑ a r,s,|α|(ε)‖∂ α (u ∘ T); L2 (U ∩ D)‖ . |α|≤r

By the chain rule, β

∂ αy (u(T(y))) = ∑ c α,β (y) (∂ x u) (T(y)) , 0=β≤α ̸

for any multi-index α with |α| ≤ r. Here, the coefficients c α,β (y) are polynomials of degree |β| of partial derivatives of T(y) up to the order |α|−|β|+1 ≤ r. Since T : U → U 󸀠 is a diffeomorphism of class C r , all the c α,β (y) are bounded on the compact set T −1 (K) and the Jacobian det T 󸀠 (y) does not vanish on T −1 (K). This implies ‖T ∗ u; H r,s (D)‖ ≤ c ∑ a r,s,|α|(ε) ∑ ‖(∂ x u) ∘ T; L2 (T −1 (K))‖ β

|α|≤r

β≤α β

≤ c ∑ a r,s,|α|(ε) ∑ ‖∂ x u; L2 (K)‖ , |α|≤r

β≤α

where c = c(T, r, K) is a constant independent of u and different in diverse applications. Interchanging the sums in α and β yields ‖T ∗ u; H r,s (D)‖ ≤ c ∑ ( ∑ a r,s,|α|(ε)) ∑ ‖∂ β u; L2 (D)‖ , |β|≤r

|α|≤r α≥β

0=β≤α ̸

for all smooth functions u in D with support in K. Therefore, if there is a constant C > 0 such that ∑ a r,s,|α|(ε) ≤ C a r,s,|β|(ε) , |α|≤r α≥β

for each multi-index β of norm |β| ≤ r, then the lemma follows. Since r

∑ a r,s,|α|(ε) ≤ c ∑ a r,s,i (ε) , |α|≤r α≥β

i=|β|

248 | 4 Asymptotic Expansions of Singular Perturbation Theory

with c a constant dependent only on r and n, we are left with the task to show that there is a constant C > 0 independent of ε, such that r

∑ a r,s,i (ε) ≤ C a r,s,i0 (ε) , i=i 0

for all i0 = 0, 1, . . . , r. This latter estimate is, in turn, fulfilled if we show that a r,s,i (ε) ≤ C a r,s,i−1(ε) ,

(4.51)

for all i = 1, . . . , r, where C is a constant independent of ε ∈ [0, 1]. By formula (4.48), i−s−1

i−j j

a r,s,i (ε) = ∑ C s C r−s ε2j , j=0

hence, the estimate (4.51) is fulfilled for sufficiently small ε > 0 with any constant C greater than C is /C i−1 s . Since (4.51) is valid for all ε in any interval [ε0 , 1] with ε0 > 0, the proof is complete. Remark 4.3.6. The case of the inner point is not singled out in Lemma 4.3.5. Clearly, the problem is easier away from the boundary, for neither the condition 2) nor the condition 3) are required any longer. Lemma 4.3.7. The spaces H r,s (D) are invariant with respect to the local change of variables described in Lemma 4.3.5, when r, s are positive real numbers and r ≥ s. Proof. This follows from Lemma 4.3.5 by using standard interpolation techniques (see e.g., [22]). To use local techniques it is convenient to define the spaces H ρ,σ (∂D) by locally rectifying the boundary surface. Since the boundary is compact, there is a finite covering {U i }Ni=1 of ∂D consisting of sufficiently small open subsets U i of Rn . Let {ϕ i } be a partition of unity in a neighborhood of ∂D subordinate to this covering. If U i is small enough, there is a smooth diffeomorphism h i of U i onto an open set O i in Rn , such that h i (U i ∩ D) = O i ∩ R+n and h i (U i ∩ ∂D) = O i ∩ Rn−1 , where Rn−1 = {x ∈ Rn : x n = 0}. The transition mappings T i,j = h−1 i ∘ h j prove to be local diffeomorphisms of D, as explained in Lemma 4.3.5. For any smooth function u on the boundary, the norm ∗ ρ,σ (Rn−1 )‖ is obviously well defined, and we set ‖(h−1 i ) (ϕ i u); H N 󵄩 ∗ ρ,σ n−1 󵄩 ‖u; H ρ,σ (∂D)‖ := ∑ 󵄩󵄩󵄩󵄩(h−1 (R )󵄩󵄩󵄩󵄩 , i ) (ϕ i u); H

(4.52)

i=1

−1 ∗ ρ,σ (∂D) is introduced to be the where (h−1 i ) (ϕ i u) = (ϕ i u) ∘ h i . As usual, the space H ∞ completion of C (∂D) with respect to the norm (4.52).

4.3 The Shapiro–Lopatinskii Condition |

249

When combined with the trace theorem for the function spaces H r,s (R+n ) and proved in [288] and Lemma 4.3.5, a familiar trick readily shows that the Banach spaces H ρ,σ (∂D) are actually independent of the particular choice of the covering of ∂D by coordinate patches {U i } in Rn , the special coordinate system h i : U i → Rn in U i , and the partition of unity {ϕ i } in a neighborhood of ∂D subordinate to the covering {U i }. Any other choice of these data leads to an equivalent norm (4.52) in C∞ (∂D). H ρ,σ (Rn−1 )

Lemma 4.3.8. As defined above, the spaces H ρ,σ (∂D) are invariant with respect to local diffeomorphisms of the boundary surface ∂D. It is easy to give the concept of local diffeomorphisms of ∂D a sense similar to that of Lemma 4.3.5.

4.3.4 Auxiliary Results When compared to the usual local techniques of elliptic theory, the theory of elliptic boundary value problems with small parameter include only three additional estimates uniform in the parameter. To wit, 1) the invariance of the norm with respect to local changes of variables on the compact manifold D; 2) estimates of the form ε k ‖∂ α u‖r,s ≤ c ‖u‖r󸀠 ,s󸀠 with c independent of ε; 3) inequalities like ‖u‖r,s ≤ δ ‖u‖r󸀠 ,s󸀠 + C(δ) ‖u‖L2 with r󸀠 ≥ r, s󸀠 ≥ s and δ > 0 a fixed arbitrary small parameter. As usual, we write α, β, and γ for multi-indices. By β ≤ α it is meant that β i ≤ α i for all i = 1, . . . , n. We first recall several basic inequalities concerning Sobolev spaces. Directly from the multinomial theorem we obtain |ξ α | ≤

1 |ξ||α|/2 , C αn

(4.53)

where C αn is the multinomial coefficient. This inequality, if combined with the Plancherel theorem, yields ‖∂ α u‖L2 ≤

1 ‖∆|α|/2 u‖L2 , C αn

for all u ∈ H |α| := H |α| (Rn ), where ∆|α|/2 is a fractional power of the Laplace operator in Rn . Moreover, we use the following consequence of the embedding theorem for Sobolev spaces (see, e.g., [36]). Theorem 4.3.9. Suppose u is a square integrable function with compact support in Rn n and α ∈ Z≥0 fixed. If, in addition, the weak derivatives ∂ β u are square integrable for all

250 | 4 Asymptotic Expansions of Singular Perturbation Theory β ≤ α, then ‖∂ β u‖L2 ≤ C ‖∂ α u‖L2 , where C = sup{|x|2 : x ∈ supp u}. We also need some basic inequalities for the norms ‖ ⋅ ‖r,s . Lemma 4.3.10. Let u ∈ H r,s (Rn ) be a function with compact support, k ≥ 1 an integer and α a multi-index. Then: 1) We have ε‖u‖r,s ≤ c ‖u‖r+1,s , where c depends on the support of u but not on u and ε. 2) If k > |α|, then ε k ‖∂ α u‖r,s ≤ c ‖u‖r+k,s , the constant c being independent of u and ε. 3) If k ≤ |α|, then ε k ‖∂ α u‖r,s ≤ c ‖u‖r+|α|,s+|α|−k , where c is independent of u and ε. Proof. Using the expression for the norm in H r,s (Rn ) we get ε ‖∆1/2 u‖r,s = ε ‖|ξ|(1 + |ξ|2 )s/2 (1 + ε2 |ξ|2 )(r−s)/2 u‖̂ L2 ≤ ‖u‖r+1,s . As ‖u‖r,s ≤ c ‖∆1/2 u‖r,s , part 1) is true. Part 2) is proved in much the same way if one applies k−|α| times what has already been proved in part 1). To prove part 3) we split the majorizing factor as ε k |ξ||α| = (ε|ξ|)k |ξ||α|−k . The first factor contributes with order k to the terms with ε, while the second one does so with |α| − k to the others. Part 2) actually holds for all functions in H r+k,s even if u fails to be of compact support. Lemma 4.3.11. Let δ be an arbitrary small positive number. Then there is a constant C(δ), such that ‖u‖r−1,s−1 ≤ δ ‖u‖r,s + C(δ) ‖u‖L2 , for all u ∈ H r,s (Rn ). Proof. Set ⟨ξ⟩ = √1 + |ξ|2 for ξ ∈ Rn . Given any R > 0, we obtain ‖u‖2r−1,s−1 = ∫ |ξ|>R

⟨ξ⟩2s ⟨εξ⟩2(r−s) |u|̂ 2 dξ + ∫ ⟨ξ⟩2(s−1) ⟨εξ⟩2(r−s) |u|̂ 2 dξ ⟨ξ⟩2 |ξ|≤R

1 ≤ ‖u‖2r,s + (1 + R2 )s−1 (1 + ε2 R2 )r−s ‖u‖2L2 . 1 + R2 Choosing R > 0 in such a way that δ2 ≤ (1 + R2 )−1 , we establish the estimate, as is easy to check.

4.3 The Shapiro–Lopatinskii Condition |

251

4.3.5 The Main Result Now we are in a position to present the main result of this work. We impose two restrictions on the boundary value problem under study, namely, the condition of ellipticity and the Shapiro–Lopatinskii condition with small parameter. To formulate these denote by A0 the principal part of the operator A, which is understood here as A0 (x, D, ε) := ε2m−2μ A2m,0 (x, D) + . . . + ε A2μ+1,0 (x, D) + A2μ,0 (x, D) , where A j,0 (x, ξ) stands for the principal homogeneous symbol of the differential operator A j (x, D) of order j, with 2μ ≤ j ≤ 2m. Recall that the differential operator A(x, D, ε) is said to satisfy the small parameter ellipticity condition in the domain D if n > 2, and for every x ∈ D the polynomial A0 (x, ξ, ε) admits an estimate |A0 (x, ξ, ε)| ≥ c x |ξ|2μ (1 + ε|ξ|)2m−2μ , for all ξ ∈ Rn and ε ∈ [0, 1], where c x > 0 is a constant that depends only on the point x. In the case n = 2, the polynomial A0 (x, ξ 󸀠 , ξ n , ε) considered with respect to the variable ξ n is assumed to possess exactly m roots in the upper complex half-plane and m roots in the lower half-plane, for every x ∈ D, ε > 0, ξ 󸀠 ∈ Rn−1 . As is well known in elliptic theory, in the case n > 2 the ellipticity condition guarantees that the polynomial A0 (x, ξ 󸀠 , ξ n , ε) has m roots in the upper half-plane and m roots in the lower one. So, this property can be taken as the basis for the small parameter ellipticity definition. By the Shapiro–Lopatinskii condition with a small parameter it is just meant that the boundary value problem (A(x, D, ε), B(x󸀠 , D, ε)) satisfies the usual Shapiro– Lopatinskii condition for each fixed x󸀠 ∈ ∂D and ε ∈ [0, 1]. This latter condition means that the polynomials B j (x󸀠 , ξ, ε) are linearly independent modulo A(x󸀠 , ξ, ε) for each point x󸀠 ∈ ∂D and ε ≥ 0. Theorem 4.3.12. Under the above conditions, if moreover r ≥ 2m and s ≥ 2μ, then there is an estimate m

‖u‖r,s ≤ C (‖A(x, D, ε)u‖r−2m,s−2μ + ∑ ‖B j (x󸀠 , D, ε)u‖r−b j −1/2,s−β j −1/2 + ‖u‖L2 (D) ) , j=1

(4.54) with C a constant independent of u and ε. The proof exploits localization techniques. First, using a finite covering {U i } of D by sufficiently small open sets (e.g., balls) in Rn , we represent any function u ∈ H r,s (D) as the sum of functions u i ∈ H r,s (D) compactly supported in U i ∩ D, just setting u i = ϕ i u for a suitable partition of unity {ϕ i } in D subordinate to the covering {U i }. Secondly, for each summand u i we formulate its own elliptic problem and find a priori estimates for its solutions. If U i does not meet the boundary of D, then the support of u i is a compact

252 | 4 Asymptotic Expansions of Singular Perturbation Theory subset of D, and the proof of (4.54) reduces to the global analysis in Rn considered in [288]. For those U i that intersect the boundary of D we choose a change of variables n x = h−1 i (z) to rectify the boundary surface within U i . To wit, h i (U i ∩D) = O i ∩R+ , where O i is an open set in Rn , and so in the coordinates y the estimate (4.54) reduces to that in the case D = R+n treated in [288]. Thirdly, we glue together all a priori estimates for u i thus obtaining a priori estimate (4.54) for u. Perhaps the focus of local techniques is on steps 2) and 3). Taking for granted the estimates of step 2), we complete the proof of Theorem 4.3.12. Proof. For each point x0 ∈ D we choose a neighborhood U x0 in D in which the estimate of Theorem 4.3.13 holds. Moreover, for each point x0 ∈ ∂D we choose a neighborhood U x0 in Rn , such that the estimate of Theorem 4.3.14 is valid. Shrinking U x0, if necessary, one can assume that the surface U x0 ∩ ∂D can be rectified by some diffeomorphism h i : U x0 → Rn , as explained above. The family {U x0 }x0∈D is an open covering of D, hence it contains a finite family {U i } that covers D. Fix a C∞ partition of unity {ϕ i } in a neighborhood of D subordinate to the covering {U i }. Given any u ∈ H r,s (D), we get u = ∑ ui i

in D, where u i := ϕ i u belongs to H s,r (D) and supp u i ⊂ U i ∩ D. By assumption, for any function u i estimate (4.54) holds with a constant C depending on i. As the family {U i } is finite, there is no restriction of generality in assuming that C does not depend on i. Hence, ‖u‖r,s ≤ C ∑ i m

× (‖A(x, D, ε)u i ‖r−2m,s−2μ + ∑ ‖B j (x󸀠 , D, ε)u i ‖r−b j −1/2,s−β j −1/2 + ‖u i ‖L2 (D) ) . j=1

By the Leibniz formula, A(x, D, ε)u i = ϕ i A(x, D, ε)u + [A, ϕ i ] u , B j (x󸀠 , D, ε)u i = ϕ i B j (x, D, ε)u + [B j , ϕ i ] u , where [A, ϕ i ]u = A(ϕ i u)−ϕ i Au is the commutator of A and the operator of multiplication with ϕ i , and similarly for [B j , ϕ i ]. The commutators are known to be differential operators of order less than that of A and B j , respectively. From the structure of the operator A(x, D, ε) we see that the summands of [A, ϕ i ]u are of the form ε2m−2μ−k a k,β (x)∂ β u ,

(4.55)

where k = 0, 1, . . . , 2m − 2μ, |β| ≤ 2m − k − 1 and a k,β are smooth functions in the closure of D independent of u.

4.3 The Shapiro–Lopatinskii Condition

| 253

To estimate the norm of (4.55) in H r−2m,s−2μ , we apply Lemma 4.3.10 and consider separately the cases 2m − 2μ − k > |β| , 2m − 2μ − k ≤ |β| . If, e.g., |β| ≥ 2m − 2μ − k, then ε2m−2μ−k ‖a k,β ∂ β u‖r−2m,s−2μ ≤ c ε2m−2μ−k ‖a k,β ∂ β u‖r−2m+|β|,s−2m+|β|+k , where |β| − 2m + k ≤ −1. It follows that ε2m−2μ−k ‖a k,β ∂ β u‖r−2m,s−2μ ≤ c ‖u‖r−1,s−1 , with c a constant independent of u, ε. Such terms are handled by Lemma 4.3.11. Analogously, we estimate the summands (4.55) with 2m −2μ − k > |β| and the commutators [B j , ϕ i ], which establishes (4.54).

4.3.6 Local Estimates in the Interior Theorem 4.3.13. For every x0 ∈ D there exists a neighborhood U x0 in D and a constant C independent of ε, such that ‖u‖r,s ≤ C (‖A(x, D, ε)u‖r−2m,s−2μ + ‖u‖L2 ) ,

(4.56)

for all functions u ∈ H r,s (D) with compact support in U x0 , where r ≥ 2m and s ≥ 2μ are integers. This theorem is not contained in [288], which focuses on differential operators with constant coefficients in Rn . Proof. If u ∈ H r,s (D) is compactly supported in D, it can be thought of as an element of H r,s (Rn ) as well. The norm of u in H s,r (D) just amounts to the norm of u in H s,r (Rn ). Hence, the paper [288] applies if A(x, D, ε) has constant coefficients, as is the case, e.g., for A0 (x0 , D, ε), the principal part of A(x, D, ε) with coefficients frozen at x0 . According to [288], there is a constant C > 0 independent of ε, such that ‖u‖r,s ≤ C ‖A0 (x0 , D, ε)u‖r−2m,s−2μ ,

(4.57)

for all functions u ∈ H r,s (D) of compact support in D. We are thus left with the task to majorize the right-hand side of (4.57) by that of (4.56) uniformly in ε ∈ [0, 1] on functions with compact support in U x0 . To this end, we write A0 (x0 , D, ε) = A(x, D, ε) − (A(x, D, ε)−A0 (x, D, ε)) − (A0 (x, D, ε)−A0 (x0 , D, ε)) ,

254 | 4 Asymptotic Expansions of Singular Perturbation Theory

whence ‖A0 (x0 , D, ε)u‖r−2m,s−2μ ≤ ‖A(x, D, ε)u‖r−2m,s−2μ + ‖(A(x, D, ε) − A0 (x, D, ε))u‖r−2m,s−2μ + ‖(A0 (x, D, ε) − A0 (x0 , D, ε))u‖r−2m,s−2μ . (4.58) Our next concern will be to estimate the last two summands on the right-hand side of (4.58). We begin with the first of these two. By the very structure of the operator A(x, D, ε), the difference A(x, D, ε)−A0 (x, D, ε) is the sum of terms of the form ε2m−2μ−k a k,β (x)∂ β u , where k = 0, 1, . . . , 2m − 2μ, |β| ≤ 2m − k − 1 and a k,β are smooth functions in the closure of D [cf. (4.55)]. Hence, the reasoning used in the proof of Theorem 4.3.12 shows that the second summand on the right-hand side of (4.58) is dominated uniformly in ε ∈ [0, 1] by the norm ‖u‖r−1,s−1 . On applying Lemma 4.3.11 we conclude that ‖(A(x, D, ε)−A0 (x, D, ε))u‖r−2m,s−2μ ≤ δ ‖u‖r,s + C(δ) ‖u‖L2 ,

(4.59)

where δ > 0 is an arbitrarily small parameter and C(δ) depends only on δ but not on u and ε. It remains to estimate the last summand on the right-hand side of (4.58). Let us write A0 (x, D, ε) = ∑ ε|β|−2μ A0,β (x)∂ β , 2μ≤|β|≤2m

where A0,β are smooth functions on the closure of D. Then ‖(A0 (x, D, ε) − A0 (x0 , D, ε))u‖r−2m,s−2μ ≤

∑

ε|β|−2μ ‖(A0,β (x) − A0,β (x0 ))∂ β u‖r−2m,s−2μ .

2μ≤|β|≤2m

To evaluate the summands we invoke the equivalent expression for the norm in H r−2m,s−2μ (D) given by (4.50). The typical term is a r−2m,s−2μ,|α|(ε) ε|β|−2μ ‖∂ α ((A0,β (x) − A0,β (x0 ))∂ β u) ‖L2 (D) with |α| ≤ r − 2m and a r−2m,s−2μ,|α|(ε) are the polynomials defined in of (4.48). By the Leibniz formula, ∂ α ((A0,β (x)−A0,β (x0 ))∂ β u) = (A0,β (x)−A0,β (x0 ))∂ α+β u + [∂ α , A0,β ] ∂ β u , where the commutator [∂ α , A0,β ] is a differential operator of order |α| − 1 with smooth coefficients in D. Observe that |α| + |β| ≤ r. Arguing as above, we easily derive an estimate like (4.59) for the sum ∑ |α|≤r−2m

a r−2m,s−2μ,|α|(ε) ε|β|−2μ ‖[∂ α , A0,β ]u‖L2 (D) ,

4.3 The Shapiro–Lopatinskii Condition |

255

whenever u ∈ H r,s (D) is of compact support in D. It is the term a r−2m,s−2μ,|α|(ε) ε|β|−2μ ‖(A0,β (x) − A0,β (x0 ))∂ α+β u‖L2 (D) that admits a desired estimate only in the case if the support of u is small enough. (Recall that u is required to have compact support in U x0 .) Since the coefficients A0,β (x) are Lipschitz continuous in D, for any arbitrarily small δ󸀠 > 0, there is a positive ϱ = ϱ(δ󸀠 ), such that ‖(A0,β (x) − A0,β (x0 ))∂ α+β u‖L2 (D) ≤ δ󸀠 ‖∂ α+β u‖L2 (D) , for all functions u ∈ H r,s (D) with compact support in B(x0 , ϱ), the ball of radius ϱ with center x0 . Summarizing, we conclude that for each δ > 0 there is a constant C = C(δ) independent of ε, such that ‖(A0 (x, D, ε) − A0 (x0 , D, ε))u‖r−2m,s−2μ ≤ δ ‖u‖r,s + C(δ) ‖u‖L2 ,

(4.60)

for all functions u ∈ with compact support in B(x0 , ϱ), provided that ϱ = ϱ(δ) is sufficiently small. Needless to say, C(δ) need not coincide with the similar constant of inequality (4.59), however, we may assume this without loss of generality. On gathering estimates (4.58) and (4.59), (4.60), and substituting them into (4.56) we arrive at H r,s (D)

(1 − 2Cδ) ‖u‖r,s ≤ C (‖A(x, D, ε)u‖r−2m,s−2μ + 2C(δ) ‖u‖L2 ) , for all u ∈ H r,s (D) with compact support in B(x0 , ϱ). Of course, this latter inequality does not yield any estimate for ‖u‖r,s unless 1 − 2Cδ > 0. Thus, choosing δ < 1/2C we get 2CC(δ) ‖u‖r,s ≤ (‖A(x, D, ε)u‖r−2m,s−2μ + ‖u‖L2 ) , 1−2Cδ if C(δ) ≥ 1/2.

4.3.7 The Case of Boundary Points Localization at a boundary point x0 ∈ ∂D requires not only small parameter ellipticity of the operator A(x, D, ε), but also the Shapiro–Lopatinskii condition with a small parameter. Theorem 4.3.14. For every point x0 ∈ ∂D there is a neighborhood U x0 in Rn , such that m

‖u‖r,s ≤ C (‖A(x, D, ε)u‖r−2m,s−2μ + ∑ ‖B j (x󸀠 , D, ε)u‖r−b j −1/2,s−β j −1/2 + ‖u‖L2 (D) ) , j=1

for all functions u ∈ with compact support in U x0 independent of both u and ε ∈ [0, 1]. H r,s (D)

(4.61) ∩ D, where C is a constant

256 | 4 Asymptotic Expansions of Singular Perturbation Theory Proof. Choose a neighborhood U of x0 in Rn and a diffeomorphism z = h(x) of U onto an a neighborhood O of the origin 0 = h(x0 ) in Rn with the property that h(U ∩ D) = O ∩ R+n and h(U ∩ ∂D) = {z ∈ O : z n = 0}. If u ∈ H r,s (D) is a function with compact support in U ∩ D, then the pullback ũ = (h−1 )∗ u belongs to H r,s (R+n ) and has compact support in O ∩ R+n , which is due to Lemma 4.3.5. On setting A♯ := (h−1 )∗ Ah∗ , ♯

B j := (h−1 )∗ B j h∗ , for j = 1, . . . , m, we obtain the pullbacks of the operators A and B j under the diffeo♯ morphism h : U ∩ D → O ∩ R+n . It is easy to see that A♯ and B j are differential operators with small parameter ε ∈ [0, 1] on O ∩ R+n in the sense explained above. We write ♯ Ã := A♯ and B̃ j := B j for short. Since the spaces H r,s (D) and H ρ,σ (∂D) are invariant under local diffeomorphisms of D, it follows that the estimate (4.61) is equivalent to m

̃ ‖u‖̃ r,s ≤ C (‖A(z, D, ε)u‖̃ r−2m,s−2μ + ∑ ‖B̃ j (z󸀠 , D, ε)u‖̃ r−b j −1/2,s−β j −1/2 + ‖u‖̃ L2 (D) ) , j=1

(4.62) for all functions ũ ∈ H r,s (R+n ) with compact support in O ∩ R+n , where C is a constant independent of ũ and ε. From the transformation formula for principal symbols of differential operators it follows that the problem {Ã 0 (0, D, ε)ũ = f ̃ for z n > 0 , { ̃ B (0, D, ε)ũ = ũ j for z n = 0 , { j,0 where j = 1, . . . , m, satisfies both the ellipticity condition and the Shapiro–Lopatinskii condition with a small parameter in the half-space. We now apply the main result of [288], which says that there is a constant C > 0 independent of ε, such that the inequality m

‖u‖̃ r,s ≤ C (‖Ã 0 (0, D, ε)u‖̃ r−2m,s−2μ + ∑ ‖B̃ j,0 (0, D, ε)u‖̃ r−b j −1/2,s−β j −1/2 + ‖u‖̃ L2 (R+n ) ) j=1

holds true for all functions ũ ∈ The estimate (4.62) follows from the latter estimate in much the same way as the estimate (4.3.13) does from (4.57), see the proof of Lemma 4.3.13. The only difference consists in evaluating the boundary terms. However, estimates on the boundary are readily reduced to those in the half-space if one exploits the embedding theorem, see Theorem 4.3.1. Namely, H r,s (R+n ) with compact support in the closed half-space.

󵄩󵄩 ̃ 󸀠 󵄩 󵄩󵄩(B j (z , D, ε) − B̃ j,0 (0, D, ε))ũ ; H r−b j −1/2,s−β j −1/2 (Rn−1 )󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ c 󵄩󵄩󵄩󵄩(B̃ j (z󸀠 , D, ε) − B̃ j,0 (0, D, ε))ũ ; H r−b j ,s−β j (R+n ) , 󵄩󵄩󵄩󵄩 with c a constant independent of ũ and ε.

4.4 Pseudodifferential Calculus with a Small Parameter

| 257

4.3.8 Conclusion Theorem 4.3.12 answers the question about the Fredholm property of the elliptic problem (A, B) in the case where D is a bounded domain with smooth boundary. As is shown in Section 4.3.2, this allows one to apply the boundary layer method. However, the smoothness of the boundary is a strong restriction; all arguments of this study go through if the boundary is of mere class C r . Still, the solvability and regularity in smooth bounded domains lay the foundation for further research and are a necessary step in constructing the theory of small parameter ellipticity in domains with singular points.

4.4 Pseudodifferential Calculus with a Small Parameter We develop a new approach to the analysis of pseudodifferential operators with the small parameter ε ∈ (0, 1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on ε. Instead, we consider the cylinder [0, 1] × X over X and study pseudodifferential operators on the cylinders that act, by their very nature, on functions depending on ε as well. The action in ε reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of ε.

4.4.1 Singular Problems with a Small Parameter An excellent introduction into asymptotic phenomena in mathematical physics is the survey [85], which remains as being of current importance. Most differential equations of physics possess solutions that involve quick transitions, and it is an interesting task to study the features of these equations that make such quick transitions possible. A case in point is Prandtl’s ingenious conception of the boundary layer. This is a narrow layer along the surface of a body, traveling in a fluid, across which the flow velocity changes quickly. Prandtl’s observation of this quick transition was the starting point for his theory of fluid resistance, see [224]. A large class of discontinuity phenomena in mathematical physics may be interpreted as boundary layer phenomena. There was never any doubt that the boundary layer theory gave a proper account of physical reality, but its mathematical aspects remained a puzzle for some time. Only when this theory is fitted into the framework of asymptotic analysis, does its mathematical structure become transparent. In such a systematic approach one may develop an appropriate quantity in powers of a parameter ε. This expansion is to be set up in such a way that the quantity is continuous for ε > 0 but discontinuous for ε = 0. A series expansion with this char-

258 | 4 Asymptotic Expansions of Singular Perturbation Theory

acter must have peculiar properties. In general, these series do not converge. The idea of giving validity to these formal series is classical and it goes back to Poincaré [221]. The boundary layer in linear differential equations has been studied in detail in [284]. On using this method it is possible to describe the iteration processes that formally yield an asymptotic representation of the solution for small ε. To prove this asymptotic representation one needs a priori estimates for solutions of boundary value problems in function spaces with weight norms. The well-known method of construction of such estimates (see [143, S. 4] makes it possible to obtain them from uniform estimates of boundary value problems with a small parameter in the higher derivatives. The asymptotic phenomena of ordinary differential equations have also been studied in connection with nonlinear equations. An interesting problem concerns periodic solutions of a differential equation of the form εu 󸀠󸀠 = f(u, u 󸀠 ). The question is what happens with these periodic solutions as ε → 0, in particular if the limit equation f(u, u 󸀠 ) = 0 has no periodic solution. Of course, there could be no boundary layer effect in the strict sense since there is no boundary. What happens is that the limit function, if it exists, satisfies the equation f(u, u 󸀠 ) = 0, except at certain points where the derivative u 󸀠 has a jump discontinuity. Strong results on asymptotic periodic solutions have been obtained by Levinson since 1942, see [169].

4.4.2 Loss of Initial Data In this section we demonstrate the behavior of solutions of the initial problem to a firstorder ordinary differential equation as the parameter ε tends to zero. This question is extremely elementary, but nevertheless leads in a natural way to the boundary layer phenomenon. To wit, {ε u 󸀠 (x) + q(x)u(x) = f(x) for x ∈ (a, b) , (4.63) { u(a) = u 0 , { where q and f are continuous functions on the interval [a, b) and ε a small positive parameter. We prescribe an initial value u 0 for the solution of our differential equation at the point a and ask how the solution of this initial value problem behaves as ε → 0. Note that for ε = 0 the differential equation reduces to the equation of order zero qu = f in (a, b). One may, therefore, wonder whether the solution of problem (4.63) approaches the solution f(x) q(x) of the equation of order zero. Now the solution of the zero-order equation is already determined and one cannot expect that the initial condition will be satisfied in the limit. This question and related questions can easily be answered with the aid of explicit formulas.

4.4 Pseudodifferential Calculus with a Small Parameter |

259

An elementary calculation shows that x

x

x

a

a

x󸀠

1 1 1 u(x) = exp (− ∫ q(ϑ)dϑ) u 0 + ∫ exp (− ∫ q(ϑ)dϑ) f(x󸀠 )dx󸀠 , ε ε ε for x ∈ [a, b). The first term on the right-hand side satisfies the homogeneous differential equation ε u 󸀠 +qu = 0 in (a, b) and the initial condition u(a) = u 0 . If q is positive in (a, b), then this term converges to zero uniformly away from a, as ε → 0. The second term on the right-hand side is a solution of the inhomogeneous solution ε u 󸀠 + qu = f in (a, b) and satisfies the homogeneous initial condition u(a) = 0. If the solution of the zero-order equation is continuously differentiable in [a, b), then one can transform the formula for the solution u of problem (4.63) to elucidate the character of convergence of u for ε → 0. Namely, on integrating by parts one obtains x

f(x) f(a) 1 u(x) = + exp (− ∫ q(ϑ)dϑ) (u 0 − ) q(x) ε q(a) a

x

x

a

x󸀠

󸀠

1 f(x󸀠 ) ) dx󸀠 , − ∫ exp (− ∫ q(ϑ)dϑ) ( ε q(x󸀠 )

(4.64)

for all x ∈ [a, b). Assume that q is positive in the interval (a, b). Then the second term on the righthand side of (4.64) converges to zero uniformly in x ∈ (a, b) bounded from a, when ε → 0. Moreover, this term vanishes for all x ∈ [a, b), if the solution of the zero-order equation takes on the value u 0 at a. The last term on the right-hand side converges to zero for each x ∈ [a, b), as ε → 0, which is due to Lebesgue’s dominated convergence theorem. From what has been said it follows that under appropriate conditions the solution of the initial problem indeed converges to the solution of the zero-order equation in (a, b). This solution fails to assume the initial value. The process of losing an initial value takes place through nonuniform convergence. If the parameter ε is small enough, the solution will run near the limit solution except in a small segment at the initial point a, where it changes quickly in order, as it were, to retrieve the initial value about to be lost. Thus a “quick transition” is found to occur. It must occur since an initial condition is about to be lost, and this loss, in turn, is necessary since the order of the differential equation is about to drop, cf. [85]. The leading symbol that controls the asymptotic behavior of the solution of the initial problem (4.63) for ε → 0 proves to be σ 0 (x, ξ, ε) := ıεξ + q(x) for (x, ξ) ∈ T ∗ [a, b) and ε ∈ [0, 1). We get 1/2

|σ 0 (x, ξ, ε)| = (ε2 |ξ|2 + |q(x)|2 ) ≥ c ⟨εξ⟩ ,

(4.65)

260 | 4 Asymptotic Expansions of Singular Perturbation Theory where c is the smaller of the numbers 1 and inf |q(x)|, the infimum being over all x ∈ [a, b). From (4.65) it follows that σ 0 (x, ξ, ε) is different from zero for all (x, ξ) ∈ T ∗ [a, b) and ε ∈ [0, 1), provided that inf |q(x)| > 0. Moreover, vice versa, if (4.65) is fulfilled with some constant c > 0 independent of (x, ξ) ∈ T ∗ [a, b) and ε ∈ [0, 1), then inf |q(x)| > 0. The first-order ordinary differential equations satisfying condition (4.65) are called small parameter elliptic. This condition just amounts to saying that the equation is elliptic of order 1 for each fixed ε ∈ (0, 1), and it degenerates to a zero-order elliptic equation when ε → 0. The results discussed in connection with the simple equation of the first order are rather typical and they may frequently serve as a guide in understanding other asymptotic phenomena.

4.4.3 A Passive Approach to Operator-Valued Symbols Pseudodifferential operators with a small parameter are most obviously introduced within the framework of operator-valued symbols. We describe here the so-called “passive” approach to operator-valued symbols, which was used in [73] for edge and corner theory. In this case, it proves to be equivalent to the usual theory based on edge and corner Sobolev spaces with group action κ λ . However, it is more convenient to deal with. The passive approach allows one to reduce pseudodifferential operators with operator-valued symbols to the case of integral operators in L2 -spaces, so that the calculus of operator-valued symbols becomes quite similar to that of scalar-valued symbols. To the best of our knowledge it goes back at least as far as [124]. The term “passive” comes from analogy with transformation theory. Recall that a geometrical transformation y = f(x) may be treated either from an “active” or a “passive” point of view. According to the “active” approach the transformation moves geometrical points x 󳨃→ y = f(x), while in the “passive” approach the points are fixed, and we change only the coordinate system. For example, a linear change y i = a ij x j (we use the Einstein summation notation) may be thought of as a linear transformation of the space Rn or as a change of a basis in this space. Of course, both descriptions are equivalent. We demonstrate this approach by calculus of pseudodifferential operators on a product manifold. Consider M = X × Y, where X, Y are smooth compact, closed manifolds with dim X = n and dim Y = m. Suppose we work with the usual symbol classes Sμ on M and corresponding classes of pseudodifferential operators Lμ acting in Sobolev spaces H s (M). We aim to describe these objects using a fibering structure. That is, we would like to introduce appropriate classes of operator-valued symbols on X with values in pseudodifferential operators on Y to recover the classes Sμ on M. Moreover, we would like to represent H s (M) as L2 -spaces L2 (X, H s (Y), ‖⋅‖ξ ) to recover the action of pseudodifferential operators from L μ in the spaces H s (M).

4.4 Pseudodifferential Calculus with a Small Parameter |

261

A symbol a(x, y, ξ, η) on M is treated as a symbol on the fiber Y with estimates depending on the base covariable ξ . To any appearances the estimates might include a group action κ λ in function spaces on Y. Our present approach is based on a “passive” treatment of the group action κ λ . The κ λ does not act on functions, instead we introduce a special family of norms in H s (Y). In more detail, consider the Sobolev space H s (Y) with a family of norms ‖ ⋅ ‖ξ depending on a parameter ξ ∈ Rn , 󵄨 󵄨󵄨2 ̂ ‖u(y)‖2ξ = ∑ ∫ 󵄨󵄨󵄨󵄨⟨ξ, η⟩s ψ j u(η)󵄨󵄨󵄨 dη .

(4.66)

j Rm

Here, {V j } is a coordinate covering of Y, {ψ j } a subordinate partition of unity, ⟨υ⟩ is a smoothed norm function, i.e., ⟨υ⟩ := f(|υ|), where f is a C∞ function satisfying f(|υ|) ≥ 1, f(|υ|) ≡ |υ|

for

|υ| ≥ 1 ,

and ⟨ξ, η⟩ = f(√|ξ|2 + |η|2 ). Of course the norm (4.66) depends on s, but we drop it in the notation. Next, consider a function u(x) on X with values in H s (Y) equipped with the family of norms ‖ ⋅ ‖ξ given by (4.66). Definition 4.4.1. By L2 (X, H s (Y), ‖ ⋅ ‖ξ ) it is meant the completion of the space C∞ (X, H s (Y)) with respect to the norm 2 ̂ ‖u(x)‖2 = ∑ ∫ ‖φ i u(ξ)‖ξ dξ .

(4.67)

i Rn

Once again, {φ j } is a partition of unity on X subordinate to a coordinate covering {U j } of this manifold. Roughly speaking, (4.67) is an L2 -norm of the scalar-valued function ̂ ‖φ i u(ξ)‖ξ . We now are in a position to define the desired symbol classes Σ m on X with values in pseudodifferential operators on Y. Definition 4.4.2. A smooth function a(x, ξ) on Rn × Rn whose values are pseudodifn ferential operators on Y is said to belong to Σ μ if, for any α, β ∈ Z≥0 , the operators β

∂ αx D ξ a(x, ξ) : H s (Y) → H s−μ+β (Y) are bounded uniformly in ξ with respect to the norms ‖ ⋅ ‖ξ in both spaces H s (Y) and H s−μ+β (Y). That is, there are constants C α,β independent of ξ , such that β

‖∂ αx D ξ a(x, ξ)‖ξ ≤ C α,β .

(4.68)

Any symbol a(x, y, ξ, η) ∈ Sμ defines a symbol a(x, ξ) ∈ Σ μ on X with values in pseudodifferential operators on Y. We can actually stop at this point. All of what follows is a simple consequence of the generalization of these definitions. As has been mentioned, in a more general con-

262 | 4 Asymptotic Expansions of Singular Perturbation Theory

text of pseudodifferential operators with operator-valued symbols these techniques were elaborated in [124]. It is easy to see that the norms ‖ ⋅ ‖ξ in H s (Y) are equivalent for different values ξ ∈ Rn , but this equivalence is not uniform in ξ . More precisely, on applying Peetre’s inequality one sees that the norms vary slowly in ξ . Lemma 4.4.3. There are constants C and q such that ‖u‖ξ1 ≤ C ⟨ξ1 − ξ2 ⟩q , ‖u‖ξ2

(4.69)

for all ξ1 , ξ2 ∈ Rn and smooth functions u on Y. (In fact, we get C = 2|s| and q = |s|.) On the other hand, the norm ‖ ⋅ ‖ξ is independent of the coordinate covering and partition of unity up to uniform equivalence. Lemma 4.4.4. The embedding ι : H s2 (Y) → H s1 (Y) for s1 ≤ s2 admits the following norm estimate ‖ι‖ξ ≤ C ⟨ξ⟩s1 −s2 .

(4.70)

Proof. Since ‖u j ‖2H s1 (Y),ξ = ∫ |⟨ξ, η⟩s1 ̂ u j (η)|2 dη Rm

= ∫ |⟨ξ, η⟩s2 ̂ u j (η)|2 ⟨ξ, η⟩2(s1 −s2 ) dη , Rm

the estimate (4.70) follows readily from the fact that ⟨ξ, η⟩s1 −s2 ∼ (1 + |ξ|2 + |η|2 )(s1 −s2 )/2 ≤ (1 + |ξ|2 )(s1 −s2 )/2 ∼ ⟨ξ⟩s1 −s2 , for s1 − s2 ≤ 0. Theorem 4.4.5. Let a(x, ξ) ∈ Σ μ . If μ < 0, then a(x, ξ) : H s (Y) → H s (Y) is a bounded operator and its norm satisfies an estimate ‖a(x, ξ)‖ξ ≤ C ⟨ξ⟩μ . Proof. By definition, the mapping a(x, ξ) : H s (Y) → H s−μ (Y) is bounded uniformly in ξ . On applying Lemma 4.4.4 we conclude, moreover, that H s−μ (Y) is embedded into H s (Y) with estimate ‖ι‖ξ ≤ C ⟨ξ⟩μ . This gives the desired result. This result plays an important role in the parameter-dependent theory of pseudodifferential operators.

4.4 Pseudodifferential Calculus with a Small Parameter

| 263

Lemma 4.4.6. For each s ∈ R, it follows that L2 (X, H s (Y), ‖ ⋅ ‖ξ ) ≅ H s (X × Y) . As usual, the norm in H s (X × Y) is defined by 2 ̂ ‖u(x, y)‖2 = ∑ ∫ ∫ |⟨ξ, η⟩s ϕ i ψ j u(ξ, η)| dξdη . i,j Rn ×Rm

For symbols a(x, ξ) ∈ Σ m , we introduce a quantization map a 󳨃→ A = Q(a) by setting Q(a) = ∑ φ i (x) Op (a(x, ξ)) φ̃ i (x) . i

Theorem 4.4.7. For a(x, ξ) ∈ Σ m , the operator A = Q(a) extends to a bounded mapping A : L2 (X, H s (Y), ‖ ⋅ ‖ξ ) → L2 (X, H s−μ (Y), ‖ ⋅ ‖ξ ) . Proof. In Fourier representation f = Au gives ̂ − ξ 󸀠 , ξ 󸀠 )u(ξ ̂ 󸀠 )dξ 󸀠 , f ̂(ξ) = ∫ a(ξ Rn

whence ̂ − ξ 󸀠 , ξ 󸀠 )u(ξ ̂ 󸀠 )‖H s−μ (Y),ξ dξ 󸀠 ‖f ̂(ξ)‖H s−μ (Y),ξ ≤ ∫ ‖a(ξ Rn

̂ − ξ 󸀠 , ξ 󸀠 )u(ξ ̂ 󸀠 )‖H s−μ (Y),ξ 󸀠 dξ 󸀠 ≤ C ∫ ⟨ξ − ξ 󸀠 ⟩q ‖a(ξ Rn

̂ − ξ 󸀠 , ξ 󸀠 )‖L(H s (Y),H s−μ (Y)),ξ 󸀠 ‖u(ξ ̂ 󸀠 )‖H s (Y),ξ 󸀠 dξ 󸀠 ≤ C ∫ ⟨ξ − ξ 󸀠 ⟩q ‖a(ξ Rn

̂ 󸀠 )‖H s (Y),ξ 󸀠 dξ 󸀠 . = C ∫ O(⟨ξ − ξ 󸀠 ⟩−∞ )‖u(ξ So, we have reduced the problem to the boundedness of integral operators in L2 with kernels O(⟨ξ − ξ 󸀠 ⟩−∞ ). This is evident. Obviously, the results of this section make sense in a much more general context, where the spaces H s (Y) and H s−μ (Y) on the fibers of X × Y over X are replaced by abstract Hilbert spaces V and W endowed with slowly varying families of norms parameterized by ξ ∈ Rn . This way, we obtain a rough class of pseudodifferential operators on X whose symbols take their values in L(V, W) with uniformly bounded norms and which map L2 (X, V, ‖ ⋅ ‖ξ ) continuously to L2 (X, W, ‖ ⋅ ‖ξ ). In Section 4.4.6 we develop this construction for another well-motivated choice of Hilbert spaces V and W.

264 | 4 Asymptotic Expansions of Singular Perturbation Theory

4.4.4 Operators with a Small Parameter In this section we apply the “passive” approach on the product manifold X × Y, where X is a smooth compact closed manifold of dimension n and Y = {P} is a one-point manifold. Our purpose is to describe a calculus of singularly perturbed differential operators on X. Locally, they are in the form A(x, D, ε) =

∑

a α,j (x)ε j D α ,

(4.71)

|α|−j≤μ |α|≤m

where x = (x1 , . . . , x n ) are coordinates in a coordinate patch on X, D is the vector of local derivatives −ı∂ x1 , . . . , −ı∂ x n , ε ∈ (0, 1] a small parameter, and we use the standard multi-index notation for higher-order derivatives in x. Moreover, μ is an integer with 0 ≤ μ < m. If μ = 0, then (4.71) reduce to the so-called h -pseudodifferential operators, which belong to the basic techniques in semiclassical analysis, with h = ε. Singular perturbations is a maturing mathematical subject with a fairly long history and a strong promise for continued important applications throughout science, see [29, 78–81, 110, 198, 204, 224, 284, 292], etc. Volevich [288] was the first to present the small parameter theory as a part of general elliptic theory. Operators of the form (4.71) are given natural domains H r,s (X) to be mapped into r−m,s−μ H (X), where r, s are arbitrary real numbers. Seemingly, these spaces were first introduced in [54]. More precisely, H r,s (X) is the completion of C∞ (X) with respect to the norm 2 ̂ ‖u‖2r,s = ∑ ∫ ⟨ξ⟩2s |⟨εξ⟩2(r−s) φ i u| dξ , i Rn

where {φ i } is a partition of unity on X subordinate to a coordinate covering {U i }. By the very definition, H r,s (X) is a Hilbert space whose norm depends on the parameter ε. Remark 4.4.8. The space H r,s (X) is locally identified within abstract edge spaces H s (Rn , V, κ) with the group action κ on V = C given by κ λ u = λ s−r u for λ > 0, see [239]. One easily recovers the spaces H r,s (X) and H r−m,s−μ (X) as L2 (X, V, ‖⋅‖ξ ), and L2 (X, W, ‖ ⋅ ‖ξ ), respectively, where V = C and W = C are endowed with the families of norms ‖u‖ξ = |⟨εξ⟩r−s ⟨ξ⟩s u| , ‖f‖ξ = |⟨εξ⟩(r−m)−(s−μ)⟨ξ⟩s−μ f| parameterized by ξ ∈ Rn . Definition 4.4.2 applies immediately to specify the corresponding spaces Σ m,μ of operator-valued symbols a(x, ξ, ε) on T ∗ Rn depending on the small parameter ε ∈ (0, 1]. We restrict ourselves to those symbols that depend continuously on ε ∈ (0, 1]

4.4 Pseudodifferential Calculus with a Small Parameter

| 265

up to ε = 0. To wit, let Sm,μ be the space of all functions a(x, ξ, ε) of (x, ξ) ∈ T ∗ Rn and ε ∈ (0, 1], which are C∞ in (x, ξ) and continuous in ε up to ε = 0, such that β

|∂ αx D ξ a(x, ξ, ε)| ≤ C α,β ⟨εξ⟩m−μ ⟨ξ⟩μ−|β|

(4.72)

n is fulfilled for all multi-indices α, β ∈ Z≥0 , where the constants C α,β do not depend on (x, ξ) and ε. Note that in terms of the group action introduced in Remark 4.4.8 the symbol estimates (4.72) take the form α μ−|β| |κ̃ −1 , ⟨εξ⟩ ∂ x D ξ a(x, ξ, ε)κ ⟨εξ⟩ | ≤ C α,β ⟨ξ⟩ β

n for all α, β ∈ Z≥0 and ξ ∈ Rn uniformly in x ∈ Rn and ε ∈ (0, 1], cf. [239]. Given any fixed ε ∈ (0, 1], these estimates reveal the order of the operator-valued symbol a(x, ξ, ε) to be μ. Moreover, they give rise to appropriate homogeneity for symbols a(x, ξ, ε). To this end, choose α = 0, β = 0 and substitute ξ 󳨃→ λξ and ε 󳨃→ ε/λ to (4.72), obtaining |a(x, λξ, ε/λ)| ≤ C 0,0 ⟨εξ⟩m−μ ⟨λξ⟩μ ,

for all λ > 1, which reduces to |λ−μ a(x, λξ, ε/λ)| ≤ C0,0 ⟨εξ⟩m−μ (λ−2 + |ξ|2 )μ/2

(4.73)

(we tacitly use an equivalent expression √1 + |ξ|2 for ⟨ξ⟩). If λ → ∞, then the righthand side of (4.73) tends to a constant multiple of ⟨εξ⟩m−μ |ξ|μ . Lemma 4.4.9. Suppose the limit σ μ (a)(x, ξ, ε) = lim λ−μ a(x, λξ, ε/λ) , λ→∞

exists for some x ∈ Rn and all ξ ∈ Rn , ε > 0. Then σ μ (a)(x, ξ, ε) is homogeneous of degree μ in (ξ, ε−1 ). It is worth pointing out that σ μ (a)(x, ξ, ε) is actually defined on the whole semi-axis ε > 0. Proof. Let s > 0. Then σ μ (a)(x, sξ, ε/s) = lim λ−μ a(x, λsξ, ε/λs) , λ→∞

and so on setting λ󸀠 = λs we get σ μ (a)(x, sξ, ε/s) = lim s μ λ󸀠−μ a(x, λ󸀠 ξ, ε/λ󸀠 ) λ 󸀠 →∞ μ μ

= s σ (a)(x, ξ, ε) , as desired.

266 | 4 Asymptotic Expansions of Singular Perturbation Theory

In particular, Lemma 4.4.9 applies to the full symbol of the differential operator A(x, D, ε) given by (4.71). Example 4.4.10. By the very origin the full symbol a(x, ξ, ε) of (4.71) belongs to the class Sm,μ and σ μ (a)(x, ξ, ε) = lim λ−μ ( λ→∞

=

∑

∑

a α,j (x)ξ α ε j λ|α|−j )

|α|−j≤μ |α|≤m

a α,j (x)ξ α ε j

|α|−j=μ |α|≤m

is well defined. In fact, the full symbol of any differential operator A(x, D, ε) of the form (4.71) expands as a finite sum of homogeneous symbols of decreasing degree with step 1. More genm,μ erally, one specifies the subspaces Sphg in Sm,μ consisting of all polyhomogeneous symbols, i.e., those admitting asymptotic expansions in homogeneous symbols. To introduce polyhomogeneous symbols more precisely, we need a purely technical result. Lemma 4.4.11. Let a be a C∞ function of (x, ξ) ∈ T ∗ Rn \ {0} and ε > 0 satisfying β n |∂ αx D ξ a(x, ξ, 1)| ≤ C α,β ⟨ξ⟩m−|β| for |ξ| ≥ 1 and α, β ∈ Z≥0 . If a is homogeneous of degree −1 m,μ for any excision function χ = χ(ξ) for the origin in Rn . μ in (ξ, ε ), then χa ∈ S β

Proof. Since each derivative ∂ αx D ξ a is homogeneous of degree μ − |β| in (ξ, ε−1 ), it suffices to prove estimate (4.72) only for α = β = 0. We have to show that there is a constant C > 0, such that |χ(ξ)a(x, ξ, ε)| ≤ C ⟨εξ⟩m−μ ⟨ξ⟩μ , for all (x, ξ) ∈ T ∗ Rn and ε ∈ (0, 1]. Such an estimate is obvious if ξ varies in a compact subset of Rn , for χ vanishes in a neighborhood of ξ = 0. Hence, there is no restriction of generality in assuming that |ξ| ≥ R, where R > 1 is large enough, so that χ(ξ) ≡ 1 for |ξ| ≥ R. We distinguish two cases, namely ε ≤ ⟨ξ⟩−1 and ε > ⟨ξ⟩−1 . In the first case we immediately get |a(x, ξ, ε)| = ⟨ξ⟩μ |a(x, ξ/⟨ξ⟩, ε⟨ξ⟩)| ≤ C ⟨ξ⟩μ , where C is the supremum of |a(x, ξ 󸀠 , ε󸀠 )| over all x, 1/√2 ≤ |ξ 󸀠 | ≤ 1 and ε󸀠 ∈ [0, 1]. Moreover, ⟨εξ⟩m−μ is bounded from below by a positive constant independent of ξ and ε, for ε|ξ| ≤ 1. This yields |a(x, ξ, ε)| ≤ C󸀠 ⟨εξ⟩m−μ ⟨ξ⟩μ with some new constant C󸀠 , as desired.

4.4 Pseudodifferential Calculus with a Small Parameter | 267

Assume that ε > ⟨ξ⟩−1 . Then ε−1 < ⟨ξ⟩, whence |a(x, ξ, ε)| = |ε−μ a(x, εξ, 1)| ≤ C ε−μ ⟨εξ⟩m = C ⟨εξ⟩m−μ (ε−1 ⟨εξ⟩)μ , with C a constant independent of x, ξ , and ε. If μ > 0, then the factor (ε−1 ⟨εξ⟩)μ is estimated by μ/2 (ε−2 + |ξ|2 ) ≤ 2μ/2 ⟨ξ⟩μ . If μ < 0, then this estimate is obvious, even without the factor 2μ/2 . This establishes the desired estimate. The family Sm−j,μ−j with j = 0, 1, . . . is used as usual to define asymptotic sums of homogeneous symbols. A symbol a ∈ Sm,μ is said to be polyhomogeneous if there is a sequence {a μ−j }j=0,1,... of smooth functions of (x, ξ) ∈ T ∗ Rn \ {0} and ε > 0 satisfying β

n |∂ αx D ξ a μ−j (x, ξ, 1)| ≤ C α,β ⟨ξ⟩m−j−|β| for |ξ| ≥ 1 and α, β ∈ Z≥0 , such that every a μ−j is −1 homogeneous of degree μ − j in (ξ, ε ) and a expands as the asymptotic sum ∞

a(x, ξ, ε) ∼ χ(ξ) ∑ a μ−j (x, ξ, ε)

(4.74)

j=0

in the sense that a − χ ∑Nj=0 a μ−j ∈ Sm−N−1,μ−N−1, for all N = 0, 1, . . . . The appropriate concept in abstract algebra to describe expansions like (4.74) is that of filtration. To wit, m,μ

∞

m−j,μ−j

Sphg ∼ ⨁ (Sphg

m−j−1,μ−j−1

⊖ Sphg

) .

j=0 m,μ

Each symbol a ∈ Sphg possesses a well-defined principal homogeneous symbol of degree μ, namely σ μ (a) := a μ . To construct an algebra of pseudodifferential operators on X with symbolic structure one need not consider full asymptotic expansions like (4.74). It suffices to ensure that the limit σ μ (a) exists, and the difference a − χσ μ (a) belongs to Sm−1,μ−1 . For more details, we refer the reader to Section 3.3 in [81]. m,μ The class of polyhomogeneous symbols Sphg with μ < 0 gains in interest if we realize that ∞

a(x, ξ, ε) ∼ ε−μ χ(ξ) ∑ ε j a μ−j (x, εξ, 1) , j=0

where a μ−j (x, εξ, 1) are homogeneous functions of degree 0 in (ξ, ε−1 ). Thus, any symm,μ bol a ∈ Sphg with μ < 0 factors through the power ε−μ , which vanishes up to order −μ at ε = 0. We may now quantize symbols a ∈ Sm,μ as pseudodifferential operators on X in just the same way as in Section 4.4.3. The space of operators A = Q(a) with symbols a ∈ Sm,μ is denoted by Ψ m,μ (X).

268 | 4 Asymptotic Expansions of Singular Perturbation Theory Theorem 4.4.12. Let A ∈ Ψ m,μ (X). For any r, s ∈ R, the operator A extends to a bounded mapping A : H r,s (X) → H r−m,s−μ (X) , whose norm is independent of ε ∈ [0, 1]. Proof. This is a consequence of Theorem 4.4.7. m,μ

Let Ψphg (X) stand for the subspace of Ψ m,μ (X) consisting of those operators that have m,μ

polyhomogeneous symbols. For A ∈ Ψphg (X), the principal homogeneous symbol of degree μ is defined by σ μ (A) = σ μ (a), where A = Q(a). If σ μ (A) = 0, then A actually m−1,μ−1 (X). Hence, the mapping A : H r,s (X) → H r−m,s−μ (X) is compact, belongs to Ψphg for it factors through the compact embedding H r−m+1,s−μ+1(X) 󳨅→ H r−m,s−μ (X) . m,μ

n,ν

m+n,μ+ν

Theorem 4.4.13. If A ∈ Ψphg (X), B ∈ Ψphg (X), then BA ∈ Ψphg σ ν (B)σ μ (A).

(X) and σ μ+ν (BA) =

Proof. See, for instance, Proposition 3.3.3 in [81]. m,μ

As usual, an operator A ∈ Ψphg (X) is called elliptic if its symbol σ μ (A)(x, ξ, ε) is invertible for all (x, ξ) ∈ T ∗ X \ {0} and ε ∈ [0, 1]. m,μ

Theorem 4.4.14. An operator A ∈ Ψphg (X) is elliptic if and only if it possesses a −m,−μ

parametrix P ∈ Ψphg

(X), i.e., PA = I and AP = I modulo operators in Ψ −∞,−∞ (X).

Proof. The necessity of ellipticity follows immediately from Theorem 4.4.13, for the equalities PA = I and AP = I modulo Ψ −∞,−∞ (X) imply that σ −μ (P) is the inverse of σ μ (A). −m,−μ Conversely, look for a parametrix P = Q(p) for A = Q(a), where p ∈ Sphg has asymptotic expansion p ∼ p−μ + p−μ−1 + . . . . The ellipticity of A just amounts to saying that σ μ (A)(x, ξ, ε) ≥ c ⟨εξ⟩m−μ |ξ|μ , for all (x, ξ) ∈ T ∗ X \ {0} and ε ∈ [0, 1], where the constant c > 0 does not depend on x, ξ and ε. Hence, p−μ := (σ μ (A))−1 gives rise to a “soft” parametrix P(0) = Q(χp−μ ) for A. −m,−μ More precisely, P(0) ∈ Ψphg (X) satisfies P(0) A = I and AP(0) = I modulo Ψ −1,−1 (X). Now, the standard techniques of pseudodifferential calculus apply to improve the discrepancies P(0) A − I and AP(0) − I, see, for instance, [239]. To sum up the homogeneous components p−μ−j with j = 0, 1, . . ., one uses a trick of L. Hörmander for asymptotic summation of symbols, see Theorem 3.6.3 in [81].

4.4 Pseudodifferential Calculus with a Small Parameter

| 269

m,μ

Corollary 4.4.15. Assume that A ∈ Ψphg (X) is an elliptic operator on X. Then, for any r, s ∈ R and any large R > 0, there is a constant C > 0 independent of ε, such that ‖u‖r,s ≤ C (‖Au‖r−m,s−μ + ‖u‖−R,−R ) , whenever u ∈ H r,s (X). −m,−μ

Proof. Let P ∈ Ψphg tain

(X) be a parametrix of A given by Theorem 4.4.14. Then we ob‖u‖r,s = ‖P(Au) + (I − PA)u‖r,s ≤ ‖P(Au)‖r,s + ‖(I − PA)u‖r,s ,

for all u ∈ H r,s (X). To complete the proof it is now sufficient to use the mapping properties of the pseudodifferential operators P and I − PA formulated in Theorem 4.4.12.

4.4.5 Ellipticity with a Large Parameter Setting λ = 1/ε we get a “large” parameter. The theory of problems with a large parameter was motivated by the study of the resolvent of elliptic operators. Both theories are parallel to each other. Substituting ε = 1/λ to (4.71) and multiplying A by λ m−μ yields ̃ A(x, D, λ) =

∑

ã α,j (x)λ j D α

|α|+j≤m j≤m−μ

in local coordinates in X. For this operator, the ellipticity with large parameter leads to the inequality 󵄨 󵄨󵄨 󵄨󵄨 ∑ ã α,j (x)λ j ξ α 󵄨󵄨󵄨 ≥ c ⟨λ, ξ⟩m−μ |ξ|μ , 󵄨󵄨 󵄨󵄨 |α|+j=m j≤m−μ

which is a generalization of the Agmon–Agranovich–Vishik condition of ellipticity with the parameter corresponding to μ = 0, see [3, 288] and the references given there.

4.4.6 Another Approach to Parameter-Dependent Theory In this section, we develop another approach to pseudodifferential operators with a small parameter, which stems from analysis on manifolds with singularities. In this area the role of small parameter is played by the distance to singularities and it has been usually chosen as a local coordinate. Thus, the small parameter is included into the functions under study as an independent variable, and the action of operators also includes that in the small parameter. Geometrically, this approach corresponds to analysis on the cylinder C = X × [0, 1] over a compact closed manifold X of dimension n, see Fig. 4.1. With respect to the problem its base ε = 0 can be thought of as singular

270 | 4 Asymptotic Expansions of Singular Perturbation Theory ε 1 0

xn

x n−1

Fig. 4.1: A cylinder C = X × [0, 1) over X

point blown up by a singular transformation of coordinates. In this case, one restricts the study to functions that are constant on the base, taking on the values 0 or ∞. In our problem, the base is regarded as part of the boundary X × {0} of the cylinder C, and so we distinguish the values of functions on the base. The top X × {1} is actually excluded from consideration by a particular choice of function spaces on the segment Y = [0, 1], for we are interested in local analysis at ε = 0. Basically there are two possibilities to develop a calculus of pseudodifferential operators on the cylinder C. Either one thinks of them as pseudodifferential operators on X with symbols taking on their values in an operator algebra on [0, 1]. Or one treats them as pseudodifferential operators on the segment [0, 1], whose symbols are pseudodifferential operators on X. Singularly perturbed problems require the first approach with symbols taking on their values in multiplication operators in L(V, W), where V = L2 ([0, 1], ε−2γ ) , W = L2 ([0, 1], ε−2γ ) , with γ ∈ R. Any continuous function a ∈ C[0, 1] induces the multiplication operator u 󳨃→ au on L2 ([0, 1], ε−2γ ), which is obviously bounded. Moreover, the norm of this operator is equal to the supremum norm of a in C[0, 1]. Hence, C[0, 1] can be specified as a closed subspace of L(V, W). Pick real numbers μ and s. We endow the spaces V and W with the families of norms 2 −2γ ‖u‖ξ = ‖⟨ξ⟩s κ −1 ⟨ξ⟩ u‖L ([0,1],ε ) , 2 −2γ ‖f‖ξ = ‖⟨ξ⟩s−μ κ̃ −1 ⟨ξ⟩ f‖L ([0,1],ε )

parameterized by ξ ∈ Rn , where (κ λ u)(ε) = λ−γ+1/2 u(λε) , (κ̃ λ f)(ε) = λ−γ+1/2 f(λε) , for λ ≤ 1.

4.4 Pseudodifferential Calculus with a Small Parameter | 271

The space L2 (X, V, ‖ ⋅ ‖ξ ) is defined to be the completion of C∞ (X, V) with respect to the norm 2 ̂ ‖u‖2s,γ = ∑ ∫ ‖φ i u‖ξ dξ , i Rn

where {φ i } is a ing {U i }.

C∞

partition of unity on X subordinate to a finite coordinate cover-

Remark 4.4.16. The space L2 (X, V, ‖ ⋅ ‖ξ ) is locally identified within abstract edge spaces H s (Rn , V, κ) with the group action κ on V = L2 ([0, 1], ε−2γ ) defined above, see [239]. In a similar way, one introduces the space L2 (X, W, ‖ ⋅ ‖ξ ), whose norm is denoted by ‖ ⋅ ‖s−μ,γ . Set H s,γ (C) = L2 (X, V, ‖ ⋅ ‖ξ ) , H s−μ,γ (C) = L2 (X, W, ‖ ⋅ ‖ξ ) , which will cause no confusion since the right-hand sides coincide for μ = 0, as are easy to check. We are thus led to a scale of function spaces on the cylinder C, which are Hilbert. Our next objective is to describe those pseudodifferential operators on C that map H s,γ (C) continuously into H s−μ,γ (C). To this end, we specify the definition of symbol spaces, see (4.68). If a(x, ξ, ε) is a function of (x, ξ) ∈ T ∗ Rn and ε ∈ [0, 1], which is smooth in (x, ξ) and continuous in ε, then a straightforward calculation shows that 󵄩󵄩 α β 󵄨󵄨 󵄩 󵄨󵄨 β −μ 󵄩󵄩∂ x D a(x, ξ, ε)󵄩󵄩󵄩 sup 󵄨󵄨󵄨(∂ αx D ξ a)(x, ξ, ε/⟨ξ⟩)󵄨󵄨󵄨 󵄩󵄩 󵄩󵄩 L(V,W),ξ = ⟨ξ⟩ ξ 󵄨 ε∈[0,1] 󵄨 holds on all T ∗ Rn . We now denote by Sμ the space of all functions a(x, ξ, ε) of (x, ξ) ∈ T ∗ Rn and ε ∈ [0, 1], which are smooth in (x, ξ), continuous in ε, and satisfy 󵄨󵄨󵄨 α β 󵄨󵄨 (4.75) 󵄨󵄨(∂ x D ξ a)(x, ξ, ε/⟨ξ⟩)󵄨󵄨󵄨 ≤ C α,β ⟨ξ⟩μ−|β| , 󵄨 󵄨 n for all multi-indices α, β ∈ Z≥0 , where C α,β are constants independent of (x, ξ) and ε. In terms of the group action introduced in Remark 4.4.16 the symbol estimates (4.75) take the form 󵄩󵄩󵄩 ̃ −1 α β 󵄩󵄩 ≤ C α,β ⟨ξ⟩μ−|β| , 󵄩󵄩 κ ⟨ξ⟩ ∂ x D ξ a(x, ξ, ε)κ ⟨ξ⟩ 󵄩󵄩󵄩 2 󵄩 󵄩L(L ([0,1],ε−2γ )) n for all (x, ξ) ∈ T ∗ Rn and α, β ∈ Z≥0 , cf. [239]. In particular, the order of the symbol a is μ. Moreover, using group actions in fibers V and W gives a direct way to the notion of homogeneity in the calculus of operator-valued symbols on T ∗ Rn . Namely, a function a(x, ξ, ε), defined for (x, ξ) ∈ T ∗ Rn \{0} and ε > 0, is said to be homogeneous of degree μ if the equality a(x, λξ, ε) = λ μ κ̃ λ a(x, ξ, ε)κ −1 λ is fulfilled for all λ > 0. It is easy to see that a is homogeneous of degree μ with respect to the group actions κ and κ̃ if and only if a(x, λξ, ε/λ) = λ μ a(x, ξ, ε) for all λ > 0, i.e., a is homogeneous of degree μ in (ξ, ε−1 ). Thus, we recover the homogeneity of symbols invented in Section 4.4.4.

272 | 4 Asymptotic Expansions of Singular Perturbation Theory

Lemma 4.4.17. Assume that the limit σ μ (a)(x, ξ, ε) = lim λ−μ κ̃ −1 λ a(x, λξ, ε)κ λ λ→∞

exists for some x ∈ Rn and all ξ ∈ Rn and ε > 0. Then σ μ (a)(x, ξ, ε) is homogeneous of degree μ. Proof. Let s > 0 and let u = u(ε) be an arbitrary function of V. By the definition of group action, we get σ μ (a)(x, sξ, ε)u = lim λ−μ κ̃ −1 λ a(x, λsξ, ε)κ λ u λ→∞

󸀠 −1 󸀠 = s μ κ̃ s ( lim (λ󸀠 )−μ κ̃ −1 λ 󸀠 a(x, λ ξ, ε)κ λ )κ s u , λ 󸀠 →∞

the second equality being a consequence of substitution λ󸀠 = λs. Since the expression in the parentheses just amounts to σ μ (a)(x, ξ, ε), the lemma follows. The function σ μ (a) defined away from the zero section of the cotangent bundle T ∗ C is called the principal homogeneous symbol of degree μ of a. We also use this designation for the operator A = Q(a) on the cylinder which is a suitable quantization of a. Example 4.4.18. As defined above, the principal homogeneous symbol of differential operator (4.71) is σ μ (A)(x, ξ, ε) = lim λ−μ ( λ→∞

=

∑ |α|−j≤μ |α|≤m

j a α,j (x)(λξ)α (κ̃ −1 λ ε κλ) )

a α,j (x)ξ α ε j κ λ ,

∑ |α|−j=μ |α|≤m

cf. Example 4.4.10. μ

Now one introduces the subspaces Sphg in Sμ consisting of all polyhomogeneous symbols, i.e., those admitting asymptotic expansions in homogeneous symbols. To do this, we need an auxiliary result. Lemma 4.4.19. Let a be a C∞ function of (x, ξ) ∈ T ∗ Rn \ {0} and ε > 0 with a ≡ 0 for |x| ≫ 1. If a is homogeneous of degree μ, then χa ∈ Sμ for any excision function χ = χ(ξ) for the origin in Rn . β

Proof. Since each derivative ∂ αx D ξ a is homogeneous of degree μ − |β|, it suffices to prove the estimate (4.75) only for α = β = 0. We have to show that there is a constant C > 0, such that 󵄩 󵄩󵄩 −1 μ 󵄩󵄩κ̃ (χ(ξ)a(x, ξ, ε)) κ ⟨ξ⟩ 󵄩󵄩󵄩 󵄩󵄩L(L2 ([0,1],ε−2γ )) ≤ C ⟨ξ⟩ , 󵄩󵄩 ⟨ξ⟩

4.4 Pseudodifferential Calculus with a Small Parameter

| 273

for all (x, ξ) ∈ T ∗ Rn and ε ∈ [0, 1]. Such an estimate is obvious if ξ varies in a compact subset of Rn , for χ near ξ = 0. Hence, we may assume without loss of generality that |ξ| ≥ R, where R > 1 is sufficiently large, so that χ(ξ) ≡ 1 for |ξ| ≥ R. Then 󵄩󵄩 −1 󵄩 󵄩󵄩κ̃ (χ(ξ)a(x, ξ, ε)) κ ⟨ξ⟩ 󵄩󵄩󵄩 󵄩󵄩 ⟨ξ⟩ 󵄩󵄩L(L2 ([0,1],ε−2γ )) = ‖a(x, ξ, ε/⟨ξ⟩)‖L(L2 ([0,1],ε−2γ )) ≤ C ⟨ξ⟩μ , where C=

sup

(x,ξ)∈T ∗Rn

‖a(x, ξ/⟨ξ⟩, ε)‖L(L2 ([0,1],ε−2γ )) .

From the conditions imposed on a it follows that the supremum is finite, which completes the proof. In contrast to Lemma 4.4.11 no additional conditions are imposed here on a except for homogeneity. This might testify to the fact that the symbol classes Sμ give the best fit to the study of operators (4.71). The family Sμ−j with j = 0, 1, . . . is used in the usual way to define the asymptotic sums of homogeneous symbols. A symbol a ∈ Sμ is called polyhomogeneous if there is a sequence {a μ−j }j=0,1,... of smooth function of (x, ξ) ∈ T ∗ Rn \ {0} and ε > 0, such that every a μ−j is homogeneous of degree μ − j in (ξ, ε−1 ) and a expands as the asymptotic sum ∞

a(x, ξ, ε) ∼ χ(ξ) ∑ a μ−j (x, ξ, ε) ,

(4.76)

j=0

in the sense that a − χ ∑Nj=0 a μ−j ∈ Sμ−N−1 for all N = 0, 1, . . . . m Each symbol a ∈ Sphg admits a well-defined principal homogeneous symbol of degree μ, namely σ μ (a) := a μ . We quantize symbols a ∈ Sμ as pseudodifferential operators on X similarly to Section 4.4.3. Write Ψ μ (C) for the space of all operators A = Q(a) with a ∈ Sμ . Theorem 4.4.20. Let A ∈ Ψ μ (C). For any s, γ ∈ R, the operator A extends to a bounded mapping A : H s,γ (C) → H s−μ,γ (C) . Proof. This is a consequence of Theorem 4.4.7. μ

Let Ψphg (C) stand for the subspace of Ψ μ (C) consisting of all operators with polyhoμ

mogeneous symbols. For A = Q(a) of Ψphg (C), the principal homogeneous symbol of μ−1

degree μ is defined by σ μ (A) = σ μ (a). If σ μ (A) = 0, then A belongs to Ψphg (C). When combined with Theorem 4.4.21 stated below, this result allows one to describe those operators A on the cylinder that are invertible modulo operators of order −∞. μ

μ+ν

ν Theorem 4.4.21. If A ∈ Ψphg (C) and B ∈ Ψphg (C), then BA ∈ Ψphg (C) and σ μ+ν (BA) = ν μ σ (B)σ (A).

274 | 4 Asymptotic Expansions of Singular Perturbation Theory

Proof. This is a standard fact of the calculus of pseudodifferential operators with operator-valued symbols. μ

As usual, an operator A ∈ Ψphg (C) is called elliptic if σ μ (A)(x, ξ, ε) is invertible for all (x, ξ, ε) away from the zero section of the cotangent bundle T ∗ C of the cylinder. μ

Theorem 4.4.22. An operator A ∈ Ψphg (C) is elliptic if and only if there is an operator −μ

P ∈ Ψphg (C), such that both PA = I and AP = I are fulfilled modulo operators of Ψ −∞ (C). Proof. The necessity of ellipticity follows immediately from Theorem 4.4.21, for the equalities PA = I and AP = I modulo Ψ −∞ (C) imply that σ −μ (P) is the inverse of σ μ (A). Conversely, look for an inverse P = Q(p) for A = Q(a) modulo Ψ −∞ (C), where −μ p ∈ Sphg has asymptotic expansion p ∼ p−μ + p−μ−1 + . . . . The ellipticity of A just amounts to saying that σ μ (A)(x, ξ, ε) ≥ c |ξ|μ , for all (x, ξ) ∈ T ∗ X\{0} and ε ∈ [0, 1], where the constant c > 0 does not depend on x, ξ and ε. Hence, p−μ := (σ μ (A))−1 gives rise to a “soft” inverse P(0) = Q(χp−μ ) for A. More −μ precisely, P(0) ∈ Ψphg (C) satisfies P(0) A = I and AP(0) = I modulo operators of Ψ −1 (C). Now, the standard techniques of pseudodifferential calculus applies to improve the discrepancies P(0) A − I and AP(0) − I, see, for instance, [239]. We avoid the designation “parametrix” for P, since the operators of Ψ −∞ (C) need not be compact in H s,γ (C). μ

Corollary 4.4.23. Assume that A ∈ Ψphg (C) is an elliptic operator on C. Then, for any s, γ ∈ R and any large R > 0, there is a constant C > 0 independent of ε, such that ‖u‖s,γ ≤ C (‖Au‖s−μ,γ + ‖u‖−R,γ ) , whenever u ∈ H s,γ (C). −μ

Proof. Let P ∈ Ψphg (C) be the inverse of A up to operators of Ψ −∞ (C) given by Theorem 4.4.14. Then we obtain ‖u‖s,γ = ‖P(Au) + (I − PA)u‖s,γ ≤ ‖P(Au)‖s,γ + ‖(I − PA)u‖s,γ , for all u ∈ H s,γ (C). To complete the proof it is now sufficient to use the mapping properties of the pseudodifferential operators P and I − PA formulated in Theorem 4.4.12. We finish this section by evaluating the local norm in H s,γ (C) to compare this scale with the scale H r,s (X) used in Section 4.4.4. This norm is equivalent to that in L2 (Rn , V,

4.4 Pseudodifferential Calculus with a Small Parameter

| 275

‖ ⋅ ‖ξ ), which is 󵄩󵄩 󵄩2 ̂ 󵄩󵄩󵄩󵄩 2 u(ξ) dξ ‖u‖2s,γ = ∫ ⟨ξ⟩2s 󵄩󵄩󵄩κ −1 ⟨ξ⟩ 󵄩 󵄩 L ([0,1],ε−2γ ) Rn

1 2s+2γ−1

= ∫ ⟨ξ⟩ Rn

̂ ε/⟨ξ⟩)|2 dεdξ . ∫ ε−2γ |u(ξ, 0

Substituting ε󸀠 = ε/⟨ξ⟩ yields 1/⟨ξ⟩

‖u‖2s,γ

̂ ε󸀠 )|2 dε󸀠 ) dξ = ∫ ⟨ξ⟩ ( ∫ (ε󸀠 )−2γ |u(ξ, 2s

Rn

0

1

= ∫(ε󸀠 )−2(γ+∆γ) (

∫

̂ ε󸀠 )|2 dξ ) dε󸀠 , (ε󸀠 ⟨ξ⟩)2∆γ ⟨ξ⟩2(s−∆γ)|u(ξ,

⟨ξ⟩≤1/ε 󸀠

0 1

which is close to ∫0 ε−2(γ+∆γ) ‖u‖2H s,s−∆γ (X) dε with any ∆γ ∈ R. Remark 4.4.24. The modern theory of pseudodifferential operators on manifolds with singularities allows one to study the problem for compact manifolds X with boundary as well.

4.4.7 Regularization of Singularly Perturbed Problems The idea of constructive reduction of elliptic singular perturbations to regular perturbations goes back at least as far as [79]. For the complete bibliography see [81, p. 531]. The calculus of pseudodifferential operators with small parameter developed in Section 4.4.4 allows one to reduce the question of the invertibility of elliptic operators m,μ A ∈ Ψphg (X) acting from H r,s (X) into H r−m,s−μ (X) to that of the invertibility of their limit operators at ε = 0 acting in usual Sobolev spaces H s (X) → H s−μ (X). To shorten μ the notation, we write A(ε) instead of A(x, D, ε), and so A(0) ∈ Ψ phg (X) is the reduced operator. Given any f ∈ H r−m,s−μ (X), consider the inhomogeneous equation A(ε)u = f on X for an unknown function u ∈ H r,s (X). We first assume that u ∈ H r,s (X) satisfies A(ε)u = f in X. Since the symbol σ μ (A(ε))(x, ξ, ε) is invertible for all (x, ξ) ∈ T ∗ X \ {0} and ε ∈ [0, 1], it follows that A(0) is an elliptic operator of order μ. Hence, the Hodge theory −μ applies to A(0). According to this theory, there is an operator G ∈ Ψ phg (X) satisfying u = H 0 u + GA(0)u , f = H 1 f + A(0)Gf ,

(4.77)

276 | 4 Asymptotic Expansions of Singular Perturbation Theory for all distributions u and f on X, where H 0 and H 1 are L2 (X) -orthogonal projections onto the null spaces of A(0) and A(0)∗ , respectively. (Observe that the null spaces of A(0) and A(0)∗ are actually finite dimensional and consist of C∞ functions.) −μ,−μ Applying G ∈ Ψphg (X) to both sides of the equality A(0)u + (A(ε) − A(0))u = f on X we obtain u − H 0 u = Gf − G (A(ε) − A(0)) u ,

(4.78)

for each u ∈ H r,s (X). [We used the first equality of (4.77)]. This is a far-reaching generalization of formula (4.64), for the function Gf ∈ H r−(m−μ),s (X) is a solution of the unperturbed equation A(0)Gf = f , which is due to the second equality of (4.77). Let μ ≤ m. Since the “coefficients” of A(ε) are continuous up to ε = 0, it follows that (A(ε) − A(0)) u converges to zero in H r−m,s−μ (X) as ε → 0. By continuity, G (A(ε) − A(0)) u converges to zero in H r−(m−μ),s (X), and so u − H 0 u ∈ H r,s (X) converges to Gf in H r−(m−μ),s(X) as ε → 0. If μ > m, then in the same manner we can see that u − H 0 u ∈ H r,s (X) converges to Gf in H r,s (X) as ε → 0. The solution u of A(ε)u = f need not converge to the solution Gf of the reduced equation, for both solutions are not unique. Formula (4.78) describes the limit of the component u − H 0 u of u, which is orthogonal to the space of solutions of the homogeneous equation A(0)u = 0. This results gains in interest if the equation A(0)u = 0 only has zero solution, i.e., H 0 = 0. The task is now to show that from the unique solvability of the reduced equation it follows that A(ε)u = f is uniquely solvable if ε is small enough. Theorem 4.4.25. Suppose that A(ε) ∈ Ψ m,μ (X) is elliptic. If the reduced operator A(0) : H s (X) → H s−μ (X) is an isomorphism uniformly in a parameter with respect to ε ∈ [0, 1], then A(ε) : H r,s (X) → H r,s−μ (X) is also an isomorphism, for all ε ∈ [0, ε0 ] with sufficiently small ε0 . Proof. We only clarify the theoretic operator aspects of the proof. For symbol constructions we refer the reader to Corollary 3.14.10 in [81] and the comments after its proof given there. To this end, write I = GA(0) + (I − GA(0)), whence A(ε) = A(0) + (A(ε) − A(0)) GA(0) + (A(ε) − A(0))(I − GA(0)) = (I + (A(ε) − A(0))G) A(0) + (A(ε) − A(0))(I − GA(0)) , for all ε ∈ [0, 1]. As has been mentioned, the difference A(ε) − A(0) is small if ε ≤ 1 is small enough. Hence, the operator Q(ε) = I + (A(ε) − A(0))G = H 0 + A(ε)G is invertible in the scale H r,s (X), provided that ε ∈ [0, ε0 ], where ε0 ≤ 1 is sufficiently small.

4.4 Pseudodifferential Calculus with a Small Parameter | 277

If the operator A(0) ∈ Ψ μ (X) is invertible in the scale of usual Sobolev spaces on X, then the product Q(ε)A(0) is invertible for all ε ∈ [0, ε0 ]. Hence, by decreasing ε0 if necessary, we readily conclude that A(ε) is invertible for all ε ∈ [0, ε0 ], as desired. The proof above gives more, namely, A(ε) = Q(ε)A(0) + S0 (ε) , = A(0)Q(ε) + S1 (ε) ,

(4.79)

where S0 (ε) and S1 (ε) have at most the same order as A(ε) and are infinitesimally small if ε → 0. From (4.79) it follows immediately that for the operator A(ε) to be invertible for small ε, it is necessary and sufficient that A(0) be invertible. Any representation of a singularly perturbed operator A(ε) in the form (4.79) is called a regularization of A(ε).

5 Asymptotic Solution of the Schrödinger Equation 5.1 Semiclassical Approximations of Quantum Mechanics 5.1.1 Standard Approximation Diverse problems of theoretical and mathematical physics reduce to partial differential equations that contain a small parameter ℏ multiplying higher-order derivatives. To solve such equations one uses asymptotic methods for a long time. One could even say that if an equation contains no small parameter then one ought to invent such a parameter. Then the equation is written in the form A(x, ℏD)u = 0 where D is the n -tuple of local derivatives −ı∂/∂x i . We are not interested in solutions of a differential equation for each fixed value of the parameter ℏ but rather in the dependence of such solutions on this parameter, in particular, in the neighborhood of ℏ = 0. The formal series that we shall consider are not simply power series in ε, they are rather of the form eı

S(x) ℏ

∞

∑ 𝑣k (x) (−ıℏ)k ,

(5.1)

k=0

which is usually referred to as the “standard” form. Other designations for this ansatz are semiclassical approximation, canonic operator [184], etc. If the coefficients of A are constant, then the equation A(x, ℏD)u = 0 admits an exact solution of the plane wave form e ı⟨x,p⟩/ℏ . Hence, expression (5.1) can be thought of as a perturbed plane wave. One may try to find solutions of the differential equation A(x, ℏD)u = 0, which admits such a series expansion. To this end, one tries to determine the functions S and 𝑣k by inserting this series into the differential equation and setting the coefficient of every power of ε equal to zero. For the functions S and 𝑣k one then finds simple differential equations that are easily solved. On inserting the functions S and 𝑣k thus found into the series (5.1) one obtains a formal series solution of the differential equation. In general, though, this series does not converge. A formal series of the type described is said to represent the asymptotic expansion of a function u(x, ℏ) if the remainder of the terms up to the N-th order is of the order N + 1. Precisely, if e −ı

S(x) ℏ

N

u(x, ℏ) = ∑ 𝑣k (x) (−ıℏ)k + O(ℏN+1 ) k=0

uniformly in an appropriate x -interval. The problem treated by Poincaré [221] was a little different from the one just described, since he did not consider expansions with respect to powers of ε but with respect to powers of x−1 . Nevertheless, it was to be expected that the analogue of what he

https://doi.org/10.1515/9783110534979-005

5.1 Semiclassical Approximations of Quantum Mechanics

| 279

proved also holds in the case considered here. That is, each formal series of the standard type should be the asymptotic expansion of an actual solution. That this is so for equations of the second order was proved as early as in 1899 by Horn. The corresponding general theory for equations of the m-th order, developed in 1908 by Birkhoff [29], initiated an extensive literature in this and related fields. The coefficient of (−ıℏ)0 is easily evaluated to be A(x, S󸀠 (x))𝑣0 (x), where S󸀠 is the total derivative (Jacobi matrix) of S. It follows that A(x, S󸀠 (x)) = 0 ,

(5.2)

where S󸀠 is the total derivative of S. For the Schrödinger equation, (5.2) is precisely the Hamilton–Jacobi equation of classical mechanics S󸀠t +

1 ⟨S󸀠 , S󸀠 ⟩ + V(x) = 0 , 2m x x

whose solution is the classical action. In [184] equation (5.2) is called the Hamilton– Jacobi equation or the “characteristic equation” corresponding to the equation A(x, ℏD)u = 0. The characteristic equation is a first-order nonlinear partial differential equation. Such an equation may happen to have no solutions without singularities or the class of such solutions may be very poor. We thus encounter an obstacle in constructing a global asymptotic solution to the equation A(x, ℏD)u = 0. However, we may try to do this locally. To choose a unique solution of the characteristic equation, we fix a coordinate patch and consider the Cauchy problem with data S = S0 on a hypersurface x = φ(ϑ) with parameter ϑ in a domain U ⊂ Rn−1 . The solution of the Cauchy problem for a first-order nonlinear equation remarkably reduces to integration of the initial problem for the system of ordinary differential equations ẋ = A󸀠p (x, p) , ṗ = −A󸀠x (x, p) ,

(5.3)

with initial data x(0) = x0 (ϑ) , p(0) = p0 (ϑ) , where x0 (ϑ) = φ(ϑ) and p0 (ϑ) is chosen in such a way that the overdetermined Cauchy problem might be solvable. The system (5.3) is called the Hamilton system or the “bicharacteristic system” related to the partial differential equation A(x, ℏD)u = 0. The trajectories x = x(t, ϑ), p = p(t, ϑ) of (5.3) in the phase space R2n x,p are called the bicharacteristics of the equation A(x, ℏD)u = 0. As usual, the variables x are referred to as coordinates and the variables p as momenta.

280 | 5 Asymptotic Solution of the Schrödinger Equation

5.1.2 Preliminary Results A great number of papers deal with the solution of the Schrödinger equation in Rn du ı ̂ = − Hu dt ℏ in the form of asymptotic series in the powers of ℏ as ℏ → 0. For a pseudodifferential operator Ĥ the construction of asymptotic solutions (semiclassical expansion) is associated with certain difficulties, and overcoming them initiated the theory Fourier integral operators. The solutions are usually constructed in microlocal charts and then nontrivial problems of gluing them together arise. During the latest decade, a number of authors noticed that the asymptotic solutions behave well globally at any time intervals [0, T] under special initial conditions, the so-called “wave packets.” Here, the pioneer work by Cordoba and Fefferman [48] ought to be mentioned, who considered problems of representing functions in the form of wave packets superposition and the action of pseudodifferential operators and Fourier integral operators upon them. Later, these questions were studied from different points of view in the papers by Popov [222], Karasev [122], Vasil’ev [278], Fedosov [71], etc. In this chapter, we present in detail the approach of [71], which actually extends to the algebra of quantum observables. We consider the Schrödinger operator equation d Û ı = − Ĥ Û , dt ℏ ̂ U(0) = P̂ 0 , the initial operator P̂ 0 being a projector on a wave packet, and construct an asymptotic expansion of the operator Û as ℏ → 0 uniformly with respect to t ∈ [0, T]. Having in mind further applications to quantization problems for symplectic manifolds, we pay special attention to explicit invariant formulas. In particular, the separation of variables into coordinates and momenta is not used, which is very important for applications to quantization problems. Besides, we suggest a somewhat different method of constructing and proving the semiclassical expansion, which agrees with the calculus of quantum observables [69]. The result of our considerations is a formula for the trace of ı ̂  exp ( − Ht) , ℏ where  is a trace class pseudodifferential operator. In other words, we get an expreŝ in a “weak sense.” The results can be transformed to the algebra sion for exp(−(ı/ℏ)Ht) of quantum observables exactly in this form. As but one application of the trace formula we generalize the Lefschetz fixed point theorem of [16]. The contributions of fixed points are derived here by asymptotic expansions of some integrals, using a stationary phase method.

5.1 Semiclassical Approximations of Quantum Mechanics |

281

5.1.3 The Schrödinger Equation for Quadratic Hamiltonians Let R2n be the standard symplectic space with symplectic form 1 ω ij dx i ∧ dx j 2 = dp α ∧ dq α .

ω=

Here, x = (x1 , . . . , x2n ) = (p1 , q1 , . . . , p n , q n ) ∈ R2n , the Roman indices ranging from 1 to 2n, the Greek ones ranging from 1 to n, and moreover, a common agreement of summation is utilized. We make use of Weyl pseudodifferential operators with smooth symbols, i.e., asymptotic series of the form ∞

a(y) = ∑ ℏk a k (y) ,

(5.4)

k=0

with smooth coefficients a k ∈ C∞ (R2n ), and rapidly oscillating or decreasing symbols in the form ı a(y) = exp ( f(y)) , ℏ where f(y) is a smooth function with a positive imaginary part. For such symbols there is a general integral composition formula a∘b=

1 (πℏ)2n (2n)!

∫ R2n ×R2n

ı exp (2 ω(t, τ))a(y + t)b(y + τ)ω2n (dt, dτ) . ℏ

In a special case, when a or b is a polynomial in y, the composition formula reduces to the form 1 ℏ ij ∂ ∂ a ∘ b = exp ( ω (5.5) ) a(x)b(y) |x=y 2ı ∂x i ∂y j (the series breaks off, if a or b are polynomials). For smooth symbols of the form (5.4) the formula (5.5) is always valid, if its right-hand side is understood as an asymptotic series in the powers of ℏ. The operator with the Weyl symbol a will be denoted by a.̂ Consider the Schrödinger evolution equation d Û ı = − Ĥ Û , dt ℏ ̂ U(0) =1

(5.6)

corresponding to the Hamiltonian Ĥ with the real-valued quadratic Weyl symbol H(y, t) =

1 ω (A(t)y, y) + ω (a(t), y) + c(t) , 2

(5.7)

where A(t) ∈ sp(2n, R) is a real infinitesimal symplectic matrix. Let f t be a shift at time t along the integral curves of the Hamiltonian system ẏ = {H, y}, or, more exactly,

282 | 5 Asymptotic Solution of the Schrödinger Equation y = f t (x) is the solution of the initial problem ẏ i = ω ji H j (y, t) , y(0) = x , where H j = ∂H/∂y j . For the quadratic H the transformation f t is linear, i.e., f t (x) = V(t)x + 𝑣(t) ,

(5.8)

where V(t) is a symplectic matrix and 𝑣(t) a vector, satisfying V(0) = 1, 𝑣(0) = 0. The coefficients A(t) and a(t) of the Hamiltonian (5.7) are expressed through V and 𝑣 by the formulas A = V̇ V −1 , (5.9) a = 𝑣̇ − V̇ V −1 𝑣 . This reminds us of the well-known important fact that if H is a quadratic Hamiltonian, then for any pseudodifferential operator Ŝ 0 with Weyl symbol S0 (y) conjugation by the ̂ is equivalent to the shift of the symbol along the integral curves, i.e., operator U(t) ̂ = U(t) ̂ Ŝ 0 Û −1 (t) ⇔ S(t) = S0 (f −1 (y)) . S(t) t

(5.10)

This easily follows from the Heisenberg equation ı d Ŝ = − [H,̂ S]̂ , dt ℏ ̂ which is satisfied by S(t). For a quadratic Hamiltonian the commutator reduces to the Poisson bracket, and the equality (5.10) reduces to the Liouville equation dS + {H, S} = 0 dt for the symbols, which is satisfied by S0 (f t−1 (y)).

5.1.4 Exact Solution with a Rapidly Decreasing Initial Symbol After these preliminaries we introduce the basic object of this section – the solution of ̂ with the quadratic Hamiltonian (5.7) and the the Schrödinger operator equation U(t) ̂ initial value U(0), which is given by the rapidly decreasing symbol ı U(0) = exp ( Φ0 (y − x)) , ℏ

(5.11)

Φ0 = ω(y − x, B0 (y − x)) ,

(5.12)

Φ0 being a quadratic form

5.1 Semiclassical Approximations of Quantum Mechanics | 283

and B0 ∈ sp(2n, C) a complex infinitesimal symplectic matrix with positive definite imaginary part, i.e., ℑ ω(z,̄ B0 z) > 0 , (5.13) for any nonzero vector z ∈ C2n . This means that the symbol (5.11) is localized in a neighborhood of the point x ∈ R2n as much as the uncertainty principle allows. Example 5.1.1. A special case of symbol (5.11) is the symbol 1 P x (y) = 2n exp (− |y − x|2 ) ℏ of the projection operator in L2 (Rn ) onto the function 1 ı 1 1 exp (− |q x − q y |2 + p x (q x − q y )) , 4 (πℏ)n 2ℏ ℏ the so-called Gaussian wave packet, localized at the point x = (p x , q x ) ∈ R2n . ̂ can be explicitly written, its Weyl symbol having the same form as The operator U(t) (5.11) ı U(t) = exp ( Φ(t)) , (5.14) ℏ with the quadratic function Φ(t) = ω (y − f t (x), B(t)(y − f t (x))) + 2ω (y − f t (x), b(t)) + φ(t) ,

(5.15)

where B(t) still satisfies the condition of positive definiteness (5.13). An exact expression for B(t) will be given later. In the limit case B0 → 0 this expression yields the symbol of the evolution operator (5.6) in the form of rapidly oscillating exponents, but the corresponding formula is not valid for large t. The facts, in connection with the complex matrices from the Lie algebra sp (2n, C) and the Lie group Sp (2n, C), with the Cayley transformation, the “upper half-plane” H ⊂ sp (2n, C), and the “unit disc” D ⊂ Sp (2n, C) will be of great importance in subsequent constructions. This information is given in [71]. Let us look for a solution in the form U(t) = exp(ı/ℏ Φ(t)). Rewrite the Hamiltonian in the powers of y − f t (x) H(y, t) =

1 ω (A(y − f t (x)), y − f t (x)) + ω (f t (x), y − f t (x)) + H(f t (x), t) . 2

(5.16)

Substituting U(t) in the form of (5.14) with the phase function (5.15) into the Schrödinger equation for symbols dU ı =− H∘U dt ℏ gives, by (5.5), 1 1 Φ̇ = −H − ω ij H i Φ j − ω ij ω kl H ik (Φ j Φ l − ıℏΦ jl ) , 2 8

284 | 5 Asymptotic Solution of the Schrödinger Equation

where the second derivatives are denoted by Φ jl =

∂2 Φ ∂2 H , H = , ik ∂y i ∂y l ∂y i ∂y k

and a similar notation of H i , Φ j is used for the first derivatives. Substituting expressions (5.15) and (5.16) we get the equation ̇ − f)) + 2 ω(y − f, ḃ − B b)̇ − 2 ω(f ̇ , b) + ϕ̇ ω(y − f, B(y 1 = H(f, t) + ω(y − f, A(y − f)) + ω(y − f, f ̇) − ω(A(y − f) + f ̇ , B(y − f) + b) 2 1 1 + ω (B(y − f) + b, A(B(y − f) + b)) + ıℏ tr AB , (5.17) 2 4 with f = f t (x). Comparing the second degree terms we come to the Riccati matrix equation 1 Ḃ = (1 − B)A(1 + B) , (5.18) 2 which can be linearized by using Cayley transformation B = (1+ S)−1 (1+ S). This gives the equation Ṡ = −SA , for S, thus, S = S(0)V −1 (t), where V(t) is the matrix of the canonical transformation (5.8). At this step, principal difficulties usually arise, connected with the fact that the matrix 1 + S becomes degenerate and the Cayley transformation does not exist for sufficiently large t. However, under the specific initial conditions (5.11) and (5.12) one manages to avoid these difficulties due to condition (5.13). To be more exact (see the Appendix), the matrix B(t) is obtained from B0 by the right action of the real symplectic matrix V −1 (t) on the “upper half-plane” H ⊂ sp (2n, C). To wit, B(t) = R V −1 B0 = X+−1 (t)X− (t) ,

(5.19)

X± = 1 + B0 ± (1 − B0 )V −1 (t) ,

(5.20)

where which gives a matrix defined globally at any t ∈ [0, T]. The fact that (5.19) actually gives a solution of (5.18) can be checked directly by substitution, although this is rather cumbersome. The terms in (5.17), linear in y − f t (x), give the equation 1 1 ḃ = (1 − B) Ab + (1 + B)f ̇ , 2 2 which, under the initial condition b(0) = 0, has the solution b=

1 (1 + B(t))(f t (x) − x) . 2

(5.21)

5.1 Semiclassical Approximations of Quantum Mechanics | 285

Finally, the constant terms in (5.17) give ıℏ 1 ϕ̇ = ω(f t (x), b)̇ + ω(b, Ab) + tr AB − H(f t (x), t) . 2 4

(5.22)

It is possible to integrate this expression using (5.19),(5.20), and (5.21). Indeed, from (5.21) we get 1 1 ω(b, Ab) = ω (f − x, (1 − B)A(1 + B)(f − x)) 2 8 1 1 = ω(f − x, (1 + B)⋅ (f − x)) 4 2 1 d 1 1 1 = ω(f − x, (1 + B)(f − x)) − ω(f − x, (1 + B)f ̇) . 2 dt 2 2 2 Along with the first term in (5.22), after substituting (5.21), this expression takes the form 1 d 1 1 ω(f − x, (1 + B)(f − x)) + ω(f ̇ , f − x) . 2 dt 2 2 Furthermore, tr AB = tr X+−1 (X+ − 2(1 − B0 )V −1 ) A ̇ −1 = tr A − 2 tr X+−1 (1 − B0 )V −1 VV = 2 tr X+−1 Ẋ + =2

d ln det X+ , dt

for the trace of the matrix A ∈ sp (2n, R) is equal to zero and (5.9) holds. Finally, we get t

ıℏ 1 X+ (t) 1 1+B ϕ(t) = ln det + ω(f −x, (f −x)) − ∫( ω (f −x, f ̇) + H(f, t))dt . 2 2 2 2 2 0

Thus, the phase function Φ(t) is explicitly found in terms of B0 , x and the canonical transformation f t (x). Having completed the square, we will write it in the form Φ = ω(y − f +

1 + B−1 1 + B−1 (f − x), B(y − f + (f − x))) 2 2 t

+

1 ıℏ X+ (t) 1 ω (f − x, B−1 (f − x)) − ∫ ( ω (f − x, f ̇) + H(f, t))dt + ln det . 4 2 2 2 0

(5.23)

5.1.5 Symmetrized Generating Function The remaining integral in (5.23) has the meaning of a symmetrized generating function of the canonical transformation f t (x). Indeed, denoting it by S(x, t), find its total

286 | 5 Asymptotic Solution of the Schrödinger Equation

differential with respect to x. We have t

dS = ∫ (

1 1 ω (df − dx, f ̇) + ω (f − x, d f ̇) + dH(f, t)) dt 2 2

0 t

=

1 󵄨t 1 󵄨t ω (f − x, df ) 󵄨󵄨󵄨0 − ω(dx, f) 󵄨󵄨󵄨0 + ∫ (ω(df, f ̇) + dH(f, t)) dt . 2 2 0

The expression under the integral sign equals 0, for f t (x) it is a solution of the Hamiltonian system. It follows that 1 ω(f −x, df +dx) . (5.24) 2 On setting f +x = 2z we obtain the following formulas for the canonical transformation f t (x): 1 x i = z i − ω ij S j (z) , 2 1 f i = z i + ω ij S j (z) 2 (dependence on t is omitted), so S is really a generating function. Note that these formulas are similar to the Cayley transformation. dS = −

5.2 Asymptotic Solution of the Schrödinger Equation ̂ In this section we will construct an asymptotic solution U(t) to the Schrödinger operator equation with an arbitrary Hamiltonian under the initial condition (5.11). The exact solution, considered in Section 5.1.4, plays the part of a leading term.

5.2.1 Symbol Classes m,μ

Let us specify the symbol classes we will use in R2n . These are special cases of Σ ρ,σ 0,μ

classes (see [251], Appendix 2). To simplify the notation, take Σ μ = Σ1,0 . Symbols H(y, ℏ) ∈ Σ μ satisfy the estimates |∂ γ H(y, ℏ)| ≤ C γ ℏμ ⟨y⟩−|γ| and may be represented in the form N−1

H(y, ℏ) = ∑ H k (y)ℏμ+k + R N (y, ℏ) , k=0

where R N (y, ℏ) ∈ Σ μ+N . The corresponding operators Ĥ are bounded in R2n with the standard norm estimate ̂ ≤ C ℏμ , ‖H‖

5.2 Asymptotic Solution of the Schrödinger Equation

|

287

for all ℏ ∈ (0, 1], with C a constant independent of ℏ. In addition, we need the symbols of the form ı P x = exp ( Φ(y − x)) , ℏ where Φ(y − x) is a quadratic form with positive definite imaginary part, and x is fixed or belongs to a bounded set in R2n . We also consider compositions a = H ∘ P x , where H ∈ Σ μ . Such symbols a satisfy estimates |∂ y a| ≤ C N,γ h μ− 2 |γ| ⟨y⟩−N . γ

1

Finally, for any nonnegative integer m, we introduce symbol classes S m x that are of great importance for us. A symbol w(y, x, ℏ) depending on a parameter x (where x belongs to a compact set in R2n ) is of the class S m x , provided that it admits a representation in the form of a finite sum w(y, x, ℏ) =

ℏk (y − x)α w k,α (y, x, ℏ) ,

∑

(5.25)

2k+|α|=m

where w k,α ∈ Σ0 . We would like to emphasize that the exponents k may be any integer numbers, both positive and negative, a priori not bounded from below. In other words, large negative exponents k should be compensated by larger positive exponents |α|. The importance of these classes is clarified by the following lemma. ̂ Lemma 5.2.1. Let w ∈ S m x . Then the trace norm of the operator ŵ P x satisfies the estimate ‖ŵ P̂ x ‖1 ≤ C ℏ 2 , m

(5.26)

with C a constant independent of ℏ. Proof. Formula (5.25) can be rewritten in the form w=

∑

h k w̃ k,α ∘ (y − x)α ,

2k+|α|=m

with new w̃ k,α , where ∘ stands for the Weyl composition (5.5). Since the operators ŵ̃ k,α are bounded uniformly with respect to ℏ, it is sufficient to estimate the trace norm of the operator â with symbol ı a = (y − x)α ∘ exp ( Φ(y − x)) , ℏ or the operator b̂ with symbol ı b = (y − x)α exp ( Φ(y − x)) , ℏ where the usual multiplication of functions is substituted for the composition ∘. The operator â reduces to a finite sum of the b̂ type operators according to the composition

288 | 5 Asymptotic Solution of the Schrödinger Equation

law (5.5). On applying the estimate of the trace norm of [251] (Proposition A.2.4) we obtain |γ| ‖b‖̂ 1 ≤ C h−n ∑ h 2 ∫ |∂ γ b|dy . |γ|≤N − 12

The change of variables z = ℏ ‖b‖̂ 1 ≤ C h

|α| 2

(y − x) now yields γ

∑ ∫ |∂ z (z α exp(ıΦ(z))) | dz ≤ C h

|α| 2

.

|γ|≤N

The integrals converge, since ℑΦ(z) ≥ ϵ|z|2 with ϵ > 0. The last estimate implies (5.26).

5.2.2 Construction of a Formal Expansion Consider the Schrödinger evolution operator ı ̂ ̂ , U(t) = exp ( − Ht) ℏ where Ĥ has symbol H(y, h) = H0 (y) + ℏH1 (y) + . . . ∈ Σ0 , ̂ is uniformly with real leading term H0 (y). It can easily be seen that the operator U(t) bounded with respect to t ∈ [0, T] and h ∈ (0, 1]. Indeed, ı ̂ , ̂ U(t) = exp ( − Ĥ 0 t) V(t) ℏ ̂ satisfies the initial problem where exp(−ı/ℏ Ĥ 0 t) is a unitary operator and V(t) d ̂ ı ̂ , V(t) = − Ĥ̃ V(t) dt ℏ ̂ V(0) =1, with

Since

(5.27)

Ĥ − Ĥ 0 ı ı ı exp ( Ĥ 0 t) . − Ĥ̃ = −ı exp ( Ĥ 0 t) ℏ ℏ ℏ ℏ H − H0 ∈ Σ0 , ℏ

the operator −ı/ℏ Ĥ̃ has a norm bounded by a constant C uniformly with respect to ̂ ℏ ∈ (0, 1]. Then, from (5.27) it follows readily that ‖V(t)‖ ≤ exp(CT) for all t ∈ [0, T]. ̂ P̂ x , where P̂ x is We will construct an asymptotic expansion for the operator U(t) an operator with symbol ı P x (y) = exp ( Φ0 (y − x)) , ℏ

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| 289

and Φ0 is the quadratic form (5.12) with the condition (5.13). Let f t (x) be a Hamiltonian flow corresponding to the Hamiltonian H0 , and let ∂f t (x)(y − x)k (5.28) ∂x k be the linearization of f t (y) in a neighborhood of the point x. Decompose the symbol H(y, h) into two parts H (2) + H (3) , where ϕ t,x (y) = f t (x) +

∂H0 1 ∂ 2 H0 (f)(y − f)i + (f)(y − f)i (y − f)j i 2 ∂x i ∂x j ∂x is the quadratic symbol obtained by Taylor expansion of the symbol H(y, h) at the point f = f t (x), and H (3) = H − H (2) ∈ S3f t (x) is the remainder term. It is easy to see that the mapping (5.28) is a shift at time t along the integral curves of the Hamiltonian vector field with the time-dependent Hamiltonian H (2) . Let Û 0 = Û 0 (t, x) be the Schrödinger evolution operator with Hamiltonian Ĥ (2) , i.e., d Û 0 ı = − Ĥ (2) Û 0 , dt ℏ Û 0 |t=0 = 1 H (2) = H0 (f) + ℏH1 (f) +

(x is a parameter). It is clear that Û 0 and Û 0−1 are uniformly bounded with respect to ℏ, for Û 0 differs from a unitary operator by a scalar factor t

exp (−ı ∫ H1 (f τ (x))dτ) . 0

̂ Taking into account that U(t) is also bounded, we conclude that the operator Ŵ = −1 ̂ ̂ U0 U is bounded as well. Standard arguments lead to the initial problem d Ŵ ı = − Ĥ̃ (3) Ŵ , dt ℏ Ŵ |t=0 = 1 ,

(5.29)

̂ the operator Ĥ̃ (3) = Û −1 Ĥ (3) Û 0 having the symbol for W, 0 H̃ (3) (y, x, t) = H (3) (ϕ t,x (y), x, t) ∈ S3x . The solution Ŵ of (5.29) can be expressed as an asymptotic series with respect to the symbol classes S m x .

5.2.3 Asymptotic Solution We introduce the “symbol” W of Ŵ as a formal expansion ∞

W = ∑ W k (y, x, t, h) , k=0

290 | 5 Asymptotic Solution of the Schrödinger Equation where W k ∈ S kx . This expansion is obtained by iterations of the integral equation t

ı W(t) = 1 − ∫ H̃ (3) (y, x, τ) ∘ W(τ) dτ , ℏ 0

where ı/ℏH̃ (3) ∈ S1x . Theorem 5.2.2. The operator Û P̂ x admits the asymptotic decomposition N−1

Û P̂ x = Û 0 ∑ Ŵ k P̂ x + R̂ (N) ,

(5.30)

k=0

with the remainder term satisfying an estimate N ‖R̂ (N) ‖1 ≤ C h 2 .

̂ (N) = Ŵ − Ŵ (N−1) . It Proof. Denote W (N−1) = ∑N−1 k=0 W k and consider the operator X satisfies the differential equation d ̂ (N) ı ̂̃ d ı X + H(3) X̂ (N) = − ( Ŵ (N−1) + Ĥ̃ (3) Ŵ (N−1) ) dt ℏ dt ℏ and the initial condition X̂ (N) |t=0 = 0. Write Ŷ (N) for the operator on the right-hand side of the differential equation. Then t

̂ X̂ (N) (t) = ∫ W(τ) Ŷ (N) (τ) dτ , 0

which immediately yields an expression for the remainder term, namely, t

̂ R̂ (N) = Û 0 (t) ∫ W(τ) Ŷ (N) (τ) dτ P̂ x . 0

̂ Since Û 0 (t) and W(τ) are uniformly bounded with respect to the operator norm, by Lemma 5.2.1 it is sufficient to check that the symbol Y (N) (τ) = − (

d (N−1) ı ̃ (3) W + H ∘ W (N−1) ) dτ ℏ

belongs to S Nx . To check it we observe that the iterations W0 = 1 and t

ı W k (t) = − ∫ H̃ (3) (τ) ∘ W k−1 (τ) dτ ℏ 0

belong to S kx , for ı/ℏH̃ (3) (τ) ∈ S1x . Hence, it follows that Y (N) (τ) can be rewritten in the form ı Y (N) (τ) = − H̃ (3) (τ) ∘ W N−1 (τ) , ℏ and so it is clear that Y (N) indeed belongs to S Nx .

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| 291

Relation (5.30) in a shortened form can be written as ∞

Û P̂ x ∼ Û 0 ∑ Ŵ k P̂ x k=0

and transformed into some more convenient form. Rewrite it as ∞

Û P̂ x ∼ ∑ Û 0 Ŵ k Û 0−1 Û 0 P̂ x . k=0

The operator Ŵ k is given by the symbol W k (y, x, t, ℏ) ∈ S kx , so, according to formula (5.10), the operator Û 0 Ŵ k Û 0 has a symbol of the form W k (ϕ t,x (y), x, t, ℏ) ∈ S kft (x) . The symbol of Û 0 P̂ x is evaluated in Section 5.1.4. It is given by (5.14) and (5.23). Expanding the symbols W k (ϕ t,x (y), x, t, ℏ) in the powers of y − f t (x) by the Taylor formula, we see that the operator Û P̂ x is given by the symbol that can be written as asymptotic series, ı ℏl w l,α (x, t)(y − f t (x))α ∘ exp ( Φ(y, x, t)) , ℏ 2l+|α|≥0 ∑

its leading term (l = 0, α = 0) being exp(ı/ℏΦ(y, x, t)). Set W̃ (y − f t (x), f t (x), t, ℏ) =

∑

ℏl w l,α (x, t)(y − f t (x))α ,

(5.31)

2l+|α|≥0

where the series is understood as a formal one ordered by the total degree 2l+|α|. Thus, finally, we deduce that the operator Û P̂ x has a “symbol” given by the asymptotic series ı W̃ ∘ exp ( Φ) 1 ℏ

(5.32)

with respect to the trace norm.

5.2.4 One More Asymptotic Decomposition We conclude this section by obtaining an asymptotic decomposition for the operator Û P̂ x Û −1 . This formula is of great importance for further generalizations in deformation quantization. Theorem 5.2.3. The operator Û P̂ x Û −1 is given by a symbol that admits the asymptotic expansion ∂f t −1 ı W̃ ∘ exp ( Φ0 ( (5.33) ) (y − f t (x))) ∘ W̃ −1 , ℏ ∂x where W̃ is given by series (5.31) and W̃ −1 is the formal series with the property that W̃ ∘ W̃ −1 = 1.

292 | 5 Asymptotic Solution of the Schrödinger Equation Proof. First we note that the formal symbol W̃ is really invertible, for its leading term is 1. Moreover, the partial sums W̃ (N) and (W̃ −1 )(N) satisfy (N) 1 − W̃ (N) ∘ (W̃ −1 )(N) ∈ S f t (x) .

To prove (5.33), write the operator Û P̂ x Û −1 in the form Û 0 Ŵ P̂ x Ŵ −1 Û 0−1 = Û 0 Ŵ Û 0−1 (Û 0 P̂ x Û 0−1 )(Û 0 Ŵ Û 0−1 )−1 , where Ŵ is the operator defined by equation (5.29). From (5.10) we conclude that the symbol of the operator Û 0 P̂ x Û 0−1 has the form ı exp ( Φ0 (ϕ−1 t,x (y) − x) ) , ℏ and the operator Û 0 Ŵ Û 0−1 has a formal symbol W̃ similar to that in Theorem 5.2.2. It is evident that substituting the N-th partial sum for the formal symbol W̃ in (5.33) gives a remainder term with trace norm O(ℏN/2 ).

5.3 The Trace of the Schrödinger Operator By the trace formula we mean a formula for the trace of the operator exp ( −

ı ̂ Ht) â , ℏ

where the Hamiltonian H is described in (5.7) and â is a pseudodifferential operator with the Weyl symbol a(y, ℏ) ∈ Σ0 vanishing outside a compact set in R2n .

5.3.1 Anti-Wick Symbols Recall the notion of anti-Wick symbol, see, for instance, Appendix 2 in [251]. Let P̂ x be a projection in L2 (Rn ) with the Weyl symbol given in Example 5.1.1, and let aaW (x, ℏ) be a function of Σ0 . This function is called the anti-Wick symbol of an operator A if A = ∫ P̂ x aaW (x, ℏ) dx .

(5.34)

In particular, the Weyl symbol of A is equal to a(y, ℏ) = ∫ 2n exp ( −

1 |y − x|2 )aaW (x, ℏ) dx . ℏ

As is proved in [251, Appendix 2], any Weyl symbol a(y, ℏ) ∈ Σ0 can be represented in this form modulo Σ∞ , the leading terms of asymptotic expansions being the same for both the Weyl and the anti-Wick symbols.

5.3 The Trace of the Schrödinger Operator

|

293

5.3.2 The Trace Formula ̂ to (5.34) and using the asymptotic expansion Applying the operator Û = exp(−ı/ℏHt) of the operator Û P̂ x described in Theorem 5.2.2, we come to the trace formula. ̂ â is equal to Theorem 5.3.1. The trace of exp(−ı/ℏHt) N ı ℏk ∫ c k,β (x, t) (f t (x) − x)β exp ( Φ(x, t))aaW (x, h)dx + O(h 2 ) , ℏ

∑ 2k+|β|< N2

(5.35)

where c0,0 = 1 and Φ(x, t) =

1 ω (f t (x) − x, B−1 (x, t)(f t (x) − x)) 4 t

1 ıℏ X− . − ∫ ( ω (f τ (x) − x, f τ̇ (x)) + H0 (f τ (x), τ) + ℏH1 (f τ (x), τ)) dτ − ln det 2 2 2 0

Proof. The proof can be obtained by termwise integration of the asymptotic equality (5.32), using expressions (5.23) for Φ(y, x, t).

5.3.3 Trace Asymptotics for ℏ → 0 The trace formula has many applications related to eigenvalue asymptotics. We will not touch upon them in this book. Instead, we consider the trace asymptotics as ℏ → 0 and t as fixed. This way, we get a generalization of the Lefschetz fixed point formula by Atiyah and Bott [16]. Theorem 5.3.2. Let t > 0 be fixed and ℏ → 0. Then, if the mapping f t (x) has no fixed points on the support of a(y, h), we get tr exp ( −

ı ̂ Ht)â = O(ℏ∞ ) . ℏ

If the mapping f t (x) has a finite number of nondegenerate fixed points x k on the support of a(y, h), then tr exp ( −

exp ( − ℏı H0 (x k )t − ıH1 (x k )t) ı ̂ Ht) â = ∑ a0 (x k ) + O(ℏ) . ℏ xk √det(1 − f 󸀠 (x )) t

(5.36)

k

Proof. If |f t (x) − x| > ϵ on the support of a, we get ℑΦ(x, t) =

ϵ2 1 ℑω (f t (x)−x, B−1 (f t (x)−x)) + O(ℏ) ≥ C , 4 4

with a positive constant C. Hence, it follows that all the terms in (5.35) decrease exponentially.

294 | 5 Asymptotic Solution of the Schrödinger Equation

Consider now Φ(x, t) in a neighborhood of a fixed point x k (for convenience let x k = 0). Using (5.24) we write Φ(x, t) =

1 ω (f t (x) − x, B−1 (x, t)(f t (x) − x)) 4 x

ıℏ X− 1 ln det ( ) . − ∫ ω (f t (x) − x, d(f t (x) + x)) − (H0 (0) + ℏH1 (0))t − 2 2 2 0

Denoting f t󸀠 (0) by V = V(t) we get an expression for Φ(x, t) in a neighborhood of x = 0. More precisely, Φ(x, t) = −(H0 (0) + ℏH1 (0))t − +

ıℏ X− ln det ( ) 2 2

1 ω ((V − 1)x, B−1 (V − 1)x) − 14 ω ((V − 1)x, (V + 1)x) + O(|x|3 ) . 4

Now, using (5.19) and (5.20), by simple calculations we get B−1 (V − 1) − (V + 1) = X−−1 (X+ (V − 1) − X− (V + 1)) = −X−−1 B0 , which for the quadratic terms in Φ(x, t) gives the expression −ω ((V − 1)x, X−−1 B0 x) . Here, the matrix B0 for the symbol P x is equal to −ıJ, where J is the matrix of standard symplectic form in the canonical coordinates p α , q α . Calculating the Gaussian integral and taking into account that det J = 1, we arrive at (5.36).

5.3.4 A Lefschetz Fixed Point Formula Let us show how a fixed point theorem of Lefschetz type can be derived from formula (5.36), cf. [16]. Assume â is an elliptic matrix pseudodifferential operator in L2 (Rn ) with the Weyl symbol a(y, ℏ) ∈ Σ0 . By ellipticity is meant the existence of a parametrix r ̂ with a symbol r(y, ℏ) ∈ Σ0 , satisfying r ∘ a = 1 and a ∘ r = 1 outside a compact set in R2n . Moreover, let H 0 (y, ℏ) and H 1 (y, ℏ) be matrix symbols from Σ0 with the same leading term H0 (y), which is a real scalar function. Subsequent expansion terms of H 0 and H 1 , in particular, the subprincipal symbols, may differ and be matrix functions. Let the condition a ∘ H 0 = H 1 ∘ a be also fulfilled modulo Σ∞ . This condition means that the operators ı Û 0 = exp ( − Ĥ 0 t) , ℏ ı Û 1 = exp ( − Ĥ 1 t) ℏ

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem |

295

define an endomorphism of the elliptic complex â

0 󳨀→ L2 (Rn ) 󳨀→ L2 (Rn ) 󳨀→ 0 , so that the Lefschetz number is defined 󵄨󵄨 󵄨󵄨 ı ı − tr exp ( − Ĥ 0 t)󵄨󵄨󵄨󵄨 . L = tr exp ( − Ĥ 0 t)󵄨󵄨󵄨󵄨 ℏ ℏ 󵄨ker â 󵄨coker â This endomorphism is a generalization of the geometric endomorphisms considered in the paper [16]. While geometric endomorphisms are induced by mappings of the coordinate space Rn , our endomorphism corresponds to the canonical mapping f t (x) of the phase space R2n defined by a Hamiltonian flow. Theorem 5.3.3. Let the mapping f t (x) have a finite number of nondegenerate fixed points x k . Then, as ℏ → 0, we have L = ∑ exp ( − xk

tr exp (−ıH10 (x k )t) − tr exp (−ıH11 (x k )t) ı H0 (x k )t) + O(ℏ) . ℏ √det(1 − f t󸀠 (x k ))

Proof. The Lefschetz number can be evaluated by the formula L = tr Û 0 (1 − r ̂a)̂ − tr (Û 1 − â Û 0 r)̂ ,

(5.37)

where the right-hand side is actually independent of the particular choice of a parametrix, see, for instance, [70]. Choose a parametrix r ̂ such that its symbol equals zero inside a sphere containing all fixed points of f t (x). Since a ∘ H 0 = H 1 ∘ a modulo Σ∞ , the formula (5.37) may be rewritten in the form L = tr Û 0 (1 − r ̂a)̂ − tr Û 1 (1 − â r)̂ . Applying Theorem 5.3.2 to each term and taking into account that the symbols 1 − r ∘ a ̂ we establish the desired and 1 − a ∘ r are equal to 1 (according to the choice of r), equality. The simplest example of operators satisfying the hypotheses of Theorem 5.3.3 gives an elliptic operator â with matrix symbol a(y) = A i y i linear in y’s. Taking for H 0 = a∗ ∘ a = |y|2 + . . . and for H 1 = a ∘ a∗ = |y|2 + . . ., we ensure a ∘ H 0 = H 1 ∘ a modulo Σ∞ . More interesting examples arise in quantization theory for symplectic manifolds.

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem We study a nonlinear quantum chain with quartic interactions under the “narrow packet” approximation. We determine the set of times for which the evolution of decay processes is essentially specified by quantum effects.

296 | 5 Asymptotic Solution of the Schrödinger Equation

5.4.1 Wave Decay Processes In a nonlinear environment with dispersion waves may be instable under decay processes, cf., for instance, [302]. The instability is observed by effective interaction of waves with vectors k j and frequencies ω(k j ) in a neighborhood of resonances ∑ n j ω(k j ) = 0 , j

∑ nj kj = 0 ,

(5.38)

j

whence the wave amplitudes change exponentially fast in time at the initial stage, and nonlinear effects turn out to be essential for describing their dynamics. The wave decay processes are of considerable interest in problems of chemistry, the hydrodynamics of liquid and gas, plasma physics, nonlinear optics, solid physics, etc. Usually, the dynamics of wave decay processes is described in the framework of classical approach. Such an approach seems to be justified as far as the energy of interacting waves is sufficiently large and effects related to quantities of order ℏ are not about to become transparent. However, it is not always possible to neglect the influence of quantum mechanics corrections on system dynamics, even in the quasiclassical setting, particularly if the classical approximation is unstable. The study of dynamical stochasticity in classical and quantum mechanics shows that if the classical system is strongly unstable, then its quantum dynamics may essentially differ from the classical one, cf. [26, 28, 306]. The present section is devoted to quantum mechanics analysis of the dynamics of decay processes of the type (5.38), which occur in a one-dimensional nonlinear chain of connected oscillators, cf. [74]. The Hamiltonian of the system has the form N

H= ∑( n=1

p2n ϵ ν + (u n+1 − u n )2 + (u n+1 − u n )4 ) , 2m 2 4

(5.39)

where p n is the momentum of the n-th oscillator, u n the displacement of the n-th oscillator from the equilibrium position, N the number of oscillators, ϵ an elasticity constant, ν the parameter of nonlinearity, and m the mass of an oscillator. In the following, the boundary conditions are chosen to be periodic, i.e., p n+N = p n and u n+N = u n . The system (5.39) for ℏ = 0 is one of the simplest models for finding the conditions of the appearance of the stochastic properties in nonlinear systems with many degrees of freedom. It has been intensively investigated since with 1955, cf. [35, 74], etc. There is a certain connection between the instability of the decay type in question and the stochastic instability of [25]. This latter paper presents a numerical investigation of the system (5.38) in the case of initiating short waves (the “narrow packet” approximation). If the parameter ν exceeds a critical value ν c , four-wave decay processes that

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| 297

correspond to the resonances seem to appear (5.38). Under a further increase of ν diverse resonances of type (5.38) might interact with each other, which finally results in a stochastic behavior of the chain. The availability of decay processes in a classical chain seems, thus, to be a preliminary step giving rise to a stochastic instability in the system. Hence, the study of the dynamics of four-wave decay processes for system (5.39) in the quantum case seems to be well motivated.

5.4.2 Classical Limit Before discussing the decay instability in the quantum case we look more closely at some peculiarities of the dynamics of four-wave decay processes in a classical chain. To this end, in (5.39) we pass to the canonical variables a k and a∗k by ak =

1 (P k − ıω k U k∗ ) , √2mℏω k

(5.40)

where Pk =

k 1 N ∑ p n e−2π N nı , √N n=1

Uk =

k 1 N ∑ u n e2π N nı , √N n=1

ω k = 2√

ϵ k sin π . m N

In the classical case, the commutator [a j , a†k ] = 0 vanishes, and I k = ℏ |a k |2 is a classical action of the phonon with momentum k. Suppose that the initial data of system (5.39) satisfy the condition of the “narrow packet” approximation δk/k 0 ≪ 1 , (5.41) where δk = |k − k 0 | is the characteristic size of a packet of initiated modes by k, and k 0 is the characteristic wave number of the packet (k 0 ∼ N/2, the number of anti-phase oscillations). In the variables a k , a∗k the Hamiltonian (5.39) takes the form N

H = ℏ ∑ ω k a∗k a k + k=1

1 2 ℏ 2 k

where V k1 k2 k3 k4 =

∑

V k1 k2 k3 k4 a∗k1 a∗k2 a k3 a k4 δ k1 +k2 −k3 −k4 ,0 + R , (5.42)

1 ,k 2 ,k 3 ,k 4

3ν k1 k2 k3 k 4 1/2 . (sin π sin π sin π sin π ) ϵmN N N N N

In (5.42) the terms a∗k1 a∗k2 a k3 a k4 represent the resonant four-wave interaction processes of modes, which are decisive under the condition (5.41). By R are meant the nonresonance terms like a k1 a k2 a k3 a k4 , a∗k1 a k2 a k3 a k4 , etc., which can be neglected

298 | 5 Asymptotic Solution of the Schrödinger Equation

under the approximation in question, at least at the initial stage. Under the condition (5.41) one can set ω k ≈ ω k0 + c(k − k 0 ) − Ω(k − k 0 )2 , (5.43) V k1 k2 k3 k4 ≈ V0 , where c = 2√

ϵ π k0 cos π , mN N

Ω=√

ϵ π 2 k0 ( ) sin π , m N N

V0 =

3ν k0 2 (sin π ) . ϵmN N

Substituting (5.42) and (5.43) into the equations of motion ı ȧ k =

∂H , ∂a∗k

we get ı Ȧ j = −j2 Ω A j + ℏV0 ∑ A∗j2 A j3 A j4 δ j+j2 −j3 −j4 ,0 ,

(5.44)

j2 ,j3 ,j4

where A j = exp ((ω k0 + cj)t ı)a j+k0 . The equation (5.44) describes the dynamics of four-wave interactions in the chain (5.39). As is shown in [25], if ν ≪ 2π2 k 0 /3NE k0 ∼ 1/E, E being the energy of the system, then the “narrow packet” approximation survives in the course of time. It follows that equations (5.44) actually simulate the dynamics of (5.39) for all times. In the following, we think of equations (5.44) as the input ones. The equation (5.44) is equivalent to a nonlinear Schrödinger equation with periodic boundary conditions. The latter is known to be a completely integrable system both in the classical and in the quantum cases, cf. [114]. We next present a condition for the appearance of the decays. It is easy to verify that the equation (5.44) has an explicit solution of the form of the finite amplitude wave A k (t) = exp ((Ω k − ℏV0 |A k |2 )t ı) A k , (5.45) if j ≠ k , A j (t) = 0, where Ω k = k 2 Ω. Let us examine the stability of solution (5.45) with respect to the decay in neighboring modes 2k 󳨃→ (k − l) + (k + l). Suppose that the modes with j ≠ k are slightly perturbed at the initial instant, so that |A j | ≪ |A k |. By linearizing equations (5.44) in A j one easily arrives at the system ı Ȧ k = −Ω k A k + ℏV0 |A k |2 A k , ı Ȧ k−l = −Ω k−l A k−l + 2ℏV0 |A k |2 A k−l + ℏV0 A2k A∗k+l , ı Ȧ k+l = −Ω k+l A k+l + 2ℏV0 |A k |2 A k+l + ℏV0 A2k A∗k−l .

(5.46)

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem |

299

These equalities show that the dynamics of a “large” wave does not change at first approximation of perturbation theory. The amplitudes of “small” waves grow exponentially with the increment λ l = √V0 I(∆Ω) − (

2V0 I ∆Ω 2 − l2 , ) = lΩ √ 2 Ω

(5.47)

where ∆Ω = Ω k−l + Ω k+l − 2Ω k = 2l2 Ω is a destruction characteristic of resonance (5.38), and I = ℏ |A k |2 . From (5.47) we get the desired condition for the existence of decays, namely, 2V0 I/Ω > 1. In terms of the original system (5.39), this condition reads 1 π2 . (5.48) ∼ ν> 3NE k0 NE

5.4.3 Quantum Equations of Decay We now pass to the analysis of the quantum case where p n and u n in (5.39) are operators with commutativity relation [u j , p k ] = ıℏ δ jk . Changing the variables by (5.40) and (5.43), as in the classical case (see [23] for more details), we get the following system of operator equations, which describe the four-wave interactions in the quantum case ı Ȧ j = −j2 (1 + q)Ω A j + ℏV0 ∑ A†j2 A j3 A j4 δ j+j2 −j3 −j4 ,0 ,

(5.49)

j2 ,j3 ,j4

where [A j , A†k ] = δ jk , q=ℏ

π ν cot 2N , 32N √mϵ3

Ω and V0 having been defined in (5.43). The renormalization of the frequency Ω is explained by the ordering of operators. It will cause no confusion if we use the same letter Ω j to designate j2 (1 + q)Ω. To treat the system (5.49) we use the techniques of projection onto the basis of coherent states, cf. [24] and [255]. Assume that at the initial instant each mode of the bosonic field rests on a coherent state described by a number α j . We denote α j (t) = ⟨α| A j (t) |α⟩ = α j (t, α, α ∗ ) , where |α⟩ is the vector of states of the phononic field at the initial instant. From (5.49) it follows that the operator A j (t) satisfies the Heisenberg equation ıℏ Ȧ j = [A j (t), Heff ] ,

(5.50)

300 | 5 Asymptotic Solution of the Schrödinger Equation

with the effective Hamiltonian Heff = −ℏ ∑ Ω k A†k A k + k

1 2 ∑ ℏ V0 2 k ,k ,k 1

2

A†k1 A†k2 A k3 A k4 δ k1 +k2 −k3 −k4 ,0 . 3 ,k 4

Computing a matrix element of a commutator with the Hamiltonian thus yields ı α̇ j (t) = T̂ α j (t) ,

(5.51)

α j (0) = α j , where ∂ T̂ = − ∑ Ω k (α k − c.c.) ∂α k k (α ∗k1 α k2 α k3

∑

+ ℏV0

k1 ,k2 ,k3 ,k4

1 + ℏV0 ∑ 2 k ,k ,k 1

2

(α k1 α k2 3 ,k 4

∂ − c.c.)δ k1 +k2 −k3 −k4 ,0 ∂α k4

∂ ∂ − c.c.)δ k1 +k2 −k3 −k4 ,0 , ∂α k3 ∂α k4

c.c. meaning complex conjugate terms (cf. [23] for more details). The equation (5.51) is easily checked to possess a solution of the form of finite amplitude periodic wave α k (t) = exp (Ω k tı − (1 − exp(−ℏV0 tı))|α k |2 ) α k , α j (t) = 0,

if

j ≠ k .

(5.52)

Note that the solution (5.52) turns into the classical wave (5.45) when ℏ → 0, |α k | → ∞, and ℏ|α k |2 → I. We now examine the stability of solution (5.52) relatively to the decay in neighboring modes 2k 󳨃→ (k − l) + (k + l). Assume that at the initial instant the amplitudes of the modes j ≠ k are small, i.e., |α j | ≪ |α k |. In this case, one can look for a solution α k+l of (5.52) in the form of expansion in α j , α k+l (t, α, α ∗ ) = c l,0 (t, α k , α ∗k ) (1,0)

(0,1)

+ ∑ (c l,j (t, α k , α ∗k )α k+j + c l,j (t, α k , α ∗k )α ∗k+j ) j=0 ̸

+... ,

(5.53)

the dots meaning the terms containing the products α k+j1 α k+j2 , α ∗k+j1 α k+j2 , α ∗k+j1 α ∗k+j2 , etc. From the initial condition α k+l (0, α, α∗ ) = α k+l we readily deduce that c0,0 (0, α k , α ∗k ) = α k , (1,0)

c l,j (0, α k , α ∗k ) = δ lj ,

c l,0 (0, α k , α ∗k ) = 0 ;

(5.54)

(0,1)

c l,j (0, α k , α ∗k ) = 0 ,

for l ≠ 0. In (5.53) α k+j and α ∗k+j are the initial amplitudes of “small” waves, and α k the (1,0)

(0,1)

initial amplitude of a “large” wave. The coefficients c l,0 , c l,j and c l,j , etc., do not explicitly contain smallness related to the amplitudes α k+j with j ≠ 0.

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem |

(1,0)

301

(0,1)

Below, we will study the dynamics of functions c l,0 , c l,j and c l,j , for they determine the evolution of small perturbations with amplitudes α k+j . Substituting (5.53) into (5.51) and gathering the coefficients of the same powers of α k+j , we arrive at a system of equations for the coefficients that is not closed in general, i.e., the equations (1,0) (0,1) for c l,0 , c l,j and c l,j also include higher-order coefficients. However, one can show that higher-order coefficients describe the influence of small waves on each other and on the large wave. Hence, they do not essentially contribute to the dynamics of the system at the initial stage. A quasi-classical asymptotic of the contribution of higherorder coefficients is discussed in [23]. On account of the above remark, we cut off the expansion (5.53) upon the linear terms. This way, we get the following closed system of differential equations: ̂ l,0 , ı ċ l,0 = Mc (1,0) ̂ (1,0) − (Ω k+j − 2ℏV0 |α k |2 )c(1,0) + 2ℏV0 α k ∂ c(1,0) − ℏV0 α ∗ 2 c(0,1) , ı ċ l,j = Mc k l,j l,j l,−j ∂α k l,j (0,1) ̂ (0,1) − (Ω k−j − 2ℏV0 |α k |2 )c(0,1) − 2ℏV0 α k ∂ c(0,1) + ℏV0 α 2 c(1,0) , ı ċ l,−j = Mc k l,j l,−j l,−j ∂α k l,−j (5.55) cf. [23], where

∂ ∂2 1 + ℏV0 α 2k 2 − c.c. , M̂ = −(Ω k − ℏV0 |α k |2 ) α k ∂α k 2 ∂α k and where c.c. stands for complex conjugate terms. The solution of the first equation (5.55) has the form of (5.52) and describes the dynamics of a “large” wave at first approximation. The remaining system of two equations can be simplified further. For this purpose, we conclude from (5.54) and the linearity of (5.55) that only two relevant summands in (5.53) are different from zero, (1,0) (0,1) namely c l,l and c l,−l . Let us now substitute the unknown functions by (1,0)

c l,l

(0,1)

= exp ( − (Ω k−l − 2Ω k )tı)f ,

c l,−l =

αk exp ( − (Ω k−l − 2Ω k )tı)g . α ∗k

(5.56)

Under this notation the average of the operator A k+l (t) is α k+l (t) = ⟨α| A k+l (t) |α⟩ = exp ( − (Ω k−l − 2Ω k )tı)(α k+l f(t) +

αk ∗ α g(t)) . α ∗k k−l

Substituting (5.56) into (5.55) we deduce that both f and g depend only on |α k |2 and satisfy the system ı f ̇ = (2V0 I − (Ω k−l + Ω k+l − 2Ω k ))f + 2ℏV0 I ı ġ = V0 I f + ℏV0 g ,

∂f − V0 Ig , ∂I

(5.57)

302 | 5 Asymptotic Solution of the Schrödinger Equation

with initial data f(0) = 1 ,

(5.58)

g(0) = 0 ,

where I = ℏ|α k |2 stands for the classical action of the k-th mode. Equations (5.57) describe the decay instability in the quantum case. From now, on they will be referred to as equations of quantum decay. In the classical case ℏ = 0 they can be solved explicitly, which shows once again the exponential growth of “small” waves when the increment λ l (5.47) increases, provided that 2V0 I > Ω, cf. Section 5.4.2.

5.4.4 Analysis of Quantum Equations Before we move on to the analysis of equations (5.57) we make necessary simplifications. We reset f 󳨃→ exp(−ℏV0 t ı)f , g 󳨃→ exp(−ℏV0 t ı)g and Ω 󳨃→ Ω − V0 t. Moreover, we introduce the dimensionless time t 󳨃→ t/Ω and the dimensionless variable x = V0 I/Ω. For simplicity, we restrict our attention to the case l = 1. Then (5.57) takes the form ı f ̇ = 2(x − 1)f + 2εx

∂f − xg , ∂x

(5.59)

ı ġ = xf , where ε=ℏ

V0 Ω

is a quantum parameter. The system (5.59) is of mixed type with hyperbolic degeneracy on the line x = 0. The general theory yields merely that (5.59) has a real analytic solution in (t, x, ε) in some neighborhood of the plane t = 0. In Section 5.4.5 we prove that this solution actually extends analytically in (t, x, ε) to all of R3 , the extension satisfying √|f|2 + |g|2 ≤

√5 5 exp ( xt) , 2 2

(5.60)

for all t, x and ε. The last inequality shows that decays in the quantum case do not run faster than exp(γt), where γ does not depend on t. This enables us to apply the Laplace transform in the analysis of system (5.59). Since the solution of (5.59) for ε = 0 has an explicit analytic form, it is interesting to develop the quasi-classical approach for describing the dynamics of decays. Denote

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem

| 303

by fcl , gcl the solution of (5.59) for ε = 0. In Section 5.4.6 we prove that ∞

f(t, x, ε) = ∑ Ψ k fcl (t, x) ε k ,

(5.61)

k=0

Ψ being the integro-differential operator t

Ψu (t, x) = −2ıx ∫ fcl (t − s, x)

∂u (s, x) ds . ∂x

0

The series (5.61) converges uniformly on all subsets of R≥0 × R≥0 × R≥0 of the form −1

{t ≤ T} × {x ≤ X} × {ε ≤ (2Te3XT ) } . Hence, it follows that X 1 log 3X ε in the domain of quasi-classical approach x/ε ≫ 1. The time of applicability of the quasi-classical approach is, therefore, logarithmically small, i.e., T ∼ log 1/ℏ in contrast to T ∼ 1/ℏγ for classically stable dynamics. This is a consequence of the instability of the dynamics of the classical system. A similar result was earlier obtained in the paper [28], which studied the conditions of the applicability of quasi-classical approximations for describing the dynamics of nonlinear quantum systems whose classical limits have the property of stochastic instability. T∼

5.4.5 Existence of Solutions Let us formulate the problem more precisely. By (5.59), we have the following system for the approximate description of the dynamics of quantum decays f ̇ = −2ı(x − 1)f − 2ıεx

∂f + ıxg , ∂x

(5.62)

ġ = −ıxf in the half-plane (t, x) ∈ R≥0 × R under the initial conditions f(0, x) = 1 , g(0, x) = 0 .

(5.63)

In fact, the domain of x = V0 I/Ω is x > 1/2, the last condition guaranteeing the existence of decays by (5.48). The principal symbol of (5.62) is given by the matrix (

ıτ − 2εxξ 0

0 ) ıτ ,

304 | 5 Asymptotic Solution of the Schrödinger Equation with the determinant −τ(τ + 2ı εxξ). It follows that (5.62) is a mixed type system with hyperbolic degeneracy on the line x = 0. The real characteristics of this system are lines x = const., hence the Cauchy problem (5.62) and (5.63) is noncharacteristic. The system (5.62) has a normal form with respect to the time variable t, and the coefficients of the system and the Cauchy data (5.63) are entire functions of t, x, and ε. Therefore, it fulfills the conditions of the Cauchy–Kovalevskaya theorem, which implies that the problem (5.62) and (5.63) has a real analytic solution F(t, x, ε) = (

f(t, x, ε) ) g(t, x, ε)

in some neighborhood U of the hyperplane {t = 0} in R3 . The solution is unique in the class of real analytic functions. Moreover, the solution is unique in the class of continuously differentiable functions, which is due to Holmgren’s uniqueness theorem. The question arises whether the solution actually extends analytically to all of the half-space {t > 0}. To treat the problem we eliminate one unknown function of the system. Lemma 5.4.1. Given any entire function Φ0 (x), the Cauchy problem for the truncated equation { Φ̇ = −2ı(x − 1)Φ − 2ıεx ∂Φ for t > 0 , ∂x (5.64) { Φ(0) = Φ0 (x) { has a unique solution that is an entire function of (t, x, ε). Proof. Since we are working with entire functions, we can change the variables by t = ız , log x = −2ε z + w , with z, w ∈ R. For the function u(z, w) := Φ(ız, exp(−2ε z + w)) the Cauchy problem (5.64) becomes −2εz+w − 1) u { ∂u ∂z = 2 (e { u(0, w) = Φ0 (e w ). {

for

z∈R,

This latter problem has a unique entire solution, which, moreover, can be written explicitly, 1 u(z, w) = Φ0 (e w ) exp ( e w (1 − e−2εz ) − 2z) . ε Returning to the variables t and x yields x Φ(t, x, ε) = Φ0 (x) exp ( (e−2ı εt − 1) + 2ı t) , ε as desired.

(5.65)

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem |

305

Note that Φ(t, x, ε) converges to Φ0 (x) exp(−2ı t(x − 1)) for ε → 0, as is easy to see. From now on, we tacitly assume that Φ0 = 1. By abuse of notation, we use the same letter Φ to designate the solution of (5.64) with Φ0 = 1. Set t

Γ(t, x, ε) = −ıx ∫ Φ(s, x, ε) ds .

(5.66)

0

Lemma 5.4.2. Suppose P is a continuous function of (t, x, ε) in the half-space R+ × R × R. Then the solution of the Cauchy problem for the system f ̇ = −2ı(x − 1)f − 2ıεx

∂f +P, ∂x

(5.67)

ġ = −ıxf under initial conditions (5.63) is given by the formula t

f(t, x, ε) = Φ(t, x, ε) + ∫ Φ(t − s, x, ε)P(s, x, ε) ds , 0 t

g(t, x, ε) = Γ(t, x, ε) + ∫ Γ(t − s, x, ε)P(s, x, ε) ds . 0

Proof. To simplify notation, we will not indicate the dependence of f , g, etc., on x and ε. Since Φ(0) = 1 and Γ(0) = 0, both f and g satisfy (5.63). Furthermore, an easy calculation shows that t

̇ + Φ(0)P(t) + ∫ Φ(t ̇ − s)P(s) ds f ̇(t) = Φ(t) 0 t

̇ + Φ(0)P(t) − ∫ (2ı(x − 1)Φ(t − s) + 2ıεx ∂ Φ(t − s))P(s) ds = Φ(t) ∂x 0

̇ + Φ(0)P(t) − 2ı(x − 1)(f(t) − Φ(t)) − 2ıεx ∂ (f(t) − Φ(t)) = Φ(t) ∂x ∂ = −2ı(x − 1)f(t) − 2ıεx f(t) + P(t) , ∂x the last equality being a consequence of (5.64), and similarly t

̇ + Γ(0)P(t) + ∫ Γ(t ̇ − s)P(s) ds ̇ = Γ(t) g(t) 0 t

̇ + Γ(0)P(t) − ıx ∫ Φ(t − s)P(s) ds = Γ(t) 0

̇ + Γ(0)P(t) − ıx (f(t) − Φ(t)) , = Γ(t)

306 | 5 Asymptotic Solution of the Schrödinger Equation

showing the lemma. Lemma 5.4.2 allows one to reduce the Cauchy problem (5.62) and (5.63) to an integral equation of Volterra type, namely, t

f(t) = Φ(t) + ıx ∫ Φ(t − s)g(s) ds , 0 t

(5.68)

g(t) = Γ(t) + ıx ∫ Γ(t − s)g(s) ds . 0

Theorem 5.4.3. The problem (5.62) and (5.63) has a unique solution {f, g}, which is a real analytic function of (t, x, ε) on all of R × R × R, satisfying (5.60). Proof. Since both Φ and Γ are entire functions of (t, x, ε), the existence and uniqueness of a solution follow from the classical Volterra theory. This solution can actually be obtained by successive approximations. It remains to establish (5.60). To this end, we apply the successive approximation method to solve the second equation of (5.68), and then we substitute g to the first equation, thus obtaining f . For simplicity, we restrict our discussion to the case of nonnegative t, x, and ε, which involves no loss of generality. Setting g0 = Γ, we define iterations t

g k (t) = Γ(t) + ıx ∫ Γ(t − s)g k−1 (s) ds , 0

for k = 1, 2, . . . . Since t

x |g0 (t)| ≤ x ∫ exp ( (cos 2εs − 1)) ds ε 0 t

x ≤ x ∫ exp ( 2εs) ds ε 0

1 ≤ (e2tx − 1) , 2 one easily obtains by induction 1 1 1 − ) φ + tx ψ , 2 4 4 1 1 1 1 |g2 (t)| ≤ ( − + (tx)2 ) φ + tx ψ , 2 16 16 16 21 1 1 21 tx + (tx)3 ) ψ , |g3 (t)| ≤ ( − )φ+( 2 96 96 96 63 63 45 2 1 1 |g4 (t)| ≤ ( − + (tx)2 + (tx)4 ) φ + ( tx − (tx)3 ) ψ , 2 768 768 768 768 768 |g1 (t)| ≤ (

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem

where

| 307

φ = e2tx − 1 , ψ = e2tx + 1 .

Given any k = 1, 2, . . ., we get |g k (t)| ≤ (c k,0 + c k,2 (tx)2 + . . .) φ + (c k,1 tx + c k,3 (tx)3 + . . .) ψ , where c k,n = 0 for n > k. The coefficients c k,n can actually be estimated uniformly in k by 1 1 1 , (5.69) |c k,n | ≤ 2 2n n! for all n. Letting k → ∞ we deduce that the limiting function g(t) fulfills the estimate |g(t)| ≤ ≤

1 tx 2n tx 2n+1 1 ∞ 1 ∞ 1 ∑ ψ ( ) φ+ ∑ ( ) 2 n=0 (2n)! 2 2 n=0 (2n + 1)! 2 tx 2tx tx 2tx 1 1 cosh (e − 1) + sinh (e + 1) , 2 2 2 2

for all t ≥ 0. Using the definitions of functions cosh x and sinh x we readily obtain 1 (5/2) tx − e−(1/2) tx ) (e 2 1 ≤ e(5/2) tx 2

|g(t)| ≤

and t

|f(t)| ≤ e

2tx

+ x ∫ e2x(t−s)

1 (5/2) sx ds e 2

0

≤e

(5/2) tx

,

(5.70)

which implies (5.60). As has already been mentioned, the solution {f, g} of (5.62) and (5.63) is also unique in the space of continuously differentiable functions.

5.4.6 Successive Approximations Set A=(

−2ı (x − 1) −ıx

ıx ) , 0

and let λ+ = ı(1 − x) + √2x − 1 , λ− = ı(1 − x) − √2x − 1

308 | 5 Asymptotic Solution of the Schrödinger Equation stand for the eigenvalues of the matrix A. The system (5.62) for ε = 0 takes the form Ḟ cl = AFcl , with Fcl (t, x) = (

fcl (t, x) ) , gcl (t, x)

hence, the solution of the Cauchy problem (5.62) and (5.63) corresponding to ε = 0 can be written in the form ∞

Fcl = ∑ A k F0 (x) k=0

=(

tk , k!

e λ+ t −e λ− t λ + −λ − ) e −e λ− t λ + −λ −

e λ − t + λ+ −ı

λ+ t

,

(5.71)

where 1 F0 = ( ) . 0 The inequality (5.70) certainly applies to fcl , thus giving an estimate for all real t and x. In order to derive an estimate of Fcl on all of C × C, we need the following lemma. Lemma 5.4.4. As defined above, Fcl is an entire function of t and x, satisfying |Fcl (t, x)| ≤ exp ((2|1 − x| + |x|)|t|) , for all (t, x) ∈ C × C. Proof. To shorten notation, set z = 2ı (1 − x) and w = ıx. An easy calculation shows that c k z k − c k−2 z k−2 w2 + c k−4 z k−4 w4 − . . . A k F0 (x) = ( ) , −c k−1 z k−1 w + c k−3 z k−3 w3 + c k−5 z k−5 w5 + . . . where the coefficients c k , c k−1 , . . . , c0 are natural numbers determined by the table 1 1 1 1 1 1 1 1 1 1 ck

1 c k−1

1

1 1

4

6

1 2

3 4

5 6

7

1 2

3

5

7 8

1 1

1

1 1

3 6

10 15

21

1 3

6 10

15 ...

1 1 4 10 20

1 4

10

1 5

1 c1

c0 . (5.72)

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem

| 309

Using the inequality √|a|2 + |b|2 ≤ |a| + |b| and comparing (5.72) with the Pascal triangle we get |A k F0 (x)| ≤ |z|k + (|z| + |w|)k−1 |w| ≤ (|z| + |w|)k . Substituting this estimate into (5.71) yields ∞

|Fcl (t, x)| ≤ ∑ |A k F0 (x)| k=0

|t|k k!

≤ exp ((|z| + |w|)|t|) , for all (t, x) ∈ C × C, as desired.

The proof can be summarized by saying that the matrix A has an operator bound less that the Holmgren norm 2|1 − x| + |x|, from which the estimate is immediate. Theorem 5.4.3 shows immediately that F(t, x, 0) = Fcl (t, x) for all t and x, i.e., the classical solution is the pointwise limit of the quantum solution if ε → 0. Given any small ε > 0, the question arises of the range of times t for which the classical limit still satisfactorily describes the dynamics of quantum decays. To study the problem we make use of the geometric series to get an asymptotic expansion of F(t, x, ε) in powers of ε. Lemma 5.4.5. Let P be a continuous function of (t, x) in the quarter-plane R+ × R+ . Then the solution of the Cauchy problem for the system f ̇ = −2ı(x − 1)f + ıxg + P , ġ = −ıxf ,

(5.73)

under initial conditions (5.63) is given by the formula t

f(t, x) = f cl (t, x) + ∫ fcl (t − s, x)P(s, x) ds , 0 t

g(t, x) = gcl (t, x) + ∫ gcl (t − s, x)P(s, x) ds . 0

These formulas are just the well-known Duhamel formulas for an inhomogeneous linear evolution system.

310 | 5 Asymptotic Solution of the Schrödinger Equation

Proof. To shorten the notation, we write f(t) and g(t) instead of f(t, x), g(t, x), etc. Since fcl (0) = 1 and gcl (0) = 0, both f and g satisfy (5.63). Furthermore, an easy calculation shows that t

f ̇(t) = fcl̇ (t) + fcl (0)P(t) + ∫ fcl̇ (t − s)P(s) ds 0 t

= fcl̇ (t) + fcl (0)P(t) + ∫ (−2ı(x − 1)fcl (t − s) + ıxgcl (t − s)) P(s) ds 0

= fcl̇ (t) + fcl (0)P(t) − 2ı(x − 1)(f(t) − fcl (t)) + ıx(g(t) − gcl (t)) = −2ı(x − 1)f(t) + ıxg(t) + P(t) , and similarly, t

̇ = ġ cl (t) + gcl (0)P(t) + ∫ ġ cl (t − s)P(s) ds g(t) 0 t

= ġ cl (t) + gcl (0)P(t) − ıx ∫ fcl (t − s)P(s) ds 0

= ġ cl (t) + gcl (0)P(t) − ıx (f(t) − fcl (t)) , which completes the proof. Using Lemma 5.4.5 reduces the Cauchy problem (5.62) and (5.63) to an integral equation of Volterra type, namely t

f(t, x, ε) = fcl (t, x) − 2ıεx ∫ fcl (t − s, x)

∂f (s, x, ε) ds . ∂x

(5.74)

0

Equation (5.74) is an, in general, difficult integral equation to analyze, not lending itself to, for example, analysis in a Sobolev space, for the operator does not contract. Usually these linear equations are analyzed including the x∂ x as part of the unperturbed operator (via characteristics). It is important to note that the characteristics go to infinity in an infinite time, from which the global existence follows. As in Section 5.4.4, we denote by Ψ the integro-differential operator t

Ψu (t, x) = −2ıx ∫ fcl (t − s, x)

∂u (s, x) ds , ∂x

0

and then the equation (5.74) can be written in the form (I − εΨ)f = fcl ,

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem

| 311

whence f(t, x, ε) = (I − εΨ)−1 fcl (t, x) ∞

= ∑ Ψ k fcl (t, x) ε k .

(5.75)

k=0

One verifies by induction that k

k

Ψ k fcl (t, x) = x (( ∑ c−,k,j (x)t j ) e λ − t + ( ∑ c+,k,j (x)t j ) e λ + t ) , j=0

j=0

for k = 1, 2, . . ., where c±,k,j (x) are irrational functions having the only singularity at the point x = 1/2. Since fcl is an entire function, the iterations Ψ k fcl are also entire functions of t and x. Note that (5.75) is a regular asymptotic series in powers of the small parameter. No boundary layer is required, for the degeneracy at ε = 0 does not affect the nature of the Cauchy problem. Theorem 5.4.6. The series (5.75) converges uniformly in t, x, and ε on compact subsets of R × R × R of the form {|t| ≤ T} × {|x| ≤ X} × {|ε| ≤ (2Te 3XT )−1 } . Proof. From the Cauchy formula it follows that if φ(x) is an entire function of x ∈ R, then 󵄨󵄨 ∂φ 󵄨󵄨 1 󵄨 󵄨󵄨 sup 󵄨󵄨󵄨 (5.76) 󵄨󵄨󵄨 ≤ (r − r󸀠 )X sup |φ(z)| , 󵄨 ∂z 󸀠 |z|≤rX 󵄨 |z|≤r X 󵄨 for all X > 0 and 0 < r󸀠 < r. By Lemma 5.4.4 we conclude that sup |fcl (t, z)| ≤ e(3rX+2)|t| ,

|z|≤rX

for any r > 0. We next show by induction that for all k = 1, 2, . . . the estimate holds sup |z|≤X/k+1

|Ψ k fcl (t, z)| ≤ (2|t|)k e(3X+2)|t| .

For k = 1 we get, by (5.76), t

sup |Ψfcl (t, z)| ≤ sup | − 2ız| ∫ |fcl (t − s, z)| |(∂/∂z)fcl (s, z)| ds

|z|≤X/2

|z|≤X/2

0

t

≤ X ∫ exp ((3

X 2 + 2)(t − s)) exp ((3X + 2)s)ds 2 X

0 t

= 2 exp ((3

X X + 2)t) ∫ exp (3 s)ds 2 2 0

≤ 2|t| exp ((3X + 2)|t|) ,

(5.77)

312 | 5 Asymptotic Solution of the Schrödinger Equation

as desired. Having granted the inequalities (5.77) up to the number k, by (5.76) we derive sup |z|≤X/k+2

|Ψ k+1 fcl (t, z)| t

(k + 1)(k + 2) X 2X + 2) (t − s)) ≤ ∫ exp ((3 k+2 k+2 X 0

sup |z|≤X/k+1

|Ψ k fcl (s, z)|ds

t

≤ 2(k + 1) exp ((3

X k+1 + 2) t) ∫(2s)k exp (3 Xs) ds k+2 k+2 0

t

≤ 2(k + 1) exp ((3X + 2)|t|) ∫(2s)k ds 0

≤ (2|t|)

k+1

exp ((3X + 2)|t|) ,

thus completing the induction step. Since X is actually arbitrary in (5.77), we easily deduce from this inequality that k

sup |Ψ k fcl (t, z)| ≤ e(3X+2)|t| (2|t|e3X|t| ) ,

|z|≤X

for all t ∈ R. Hence, it follows that the series (5.75) converges uniformly in t, x, and ε on each compact set −1

{|t| ≤ T} × {|x| ≤ X} × {|ε| ≤ (2Te3XT ) } , for ∞

|f(t, x, ε)| ≤ exp ((3X + 2)|t|) ∑ (2|ε||t| exp(3X|t|))

k

k=0

≤

e(3X+2)|t| 1 − 2|ε||t| e3X|t|

,

showing the theorem. Theorem 5.4.6 implies that (5.71) is an asymptotic series in the powers of ε for the solution of (5.62) and (5.63) on bounded subsets of Rt × Rx , provided that ε is small enough. Let us express T as a function of ε and x from the inequality ε ≤ (2Te3XT )−1 entering into the theorem. This will enable us to evaluate the characteristic times of applicability of the classical approximation corresponding to ε = 0. Corollary 5.4.7. Let X/ε ≫ 1. Then Fcl approximates F(t, x, ε) for small ε if t ≤ T with T∼

X 1 log . 6X ε

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem |

313

Proof. Rewrite the inequality ε ≤ (2Te3XT )−1 in the form 2εT e3XT ≤ 1 .

(5.78)

Since the left-hand side is an increasing function of T ≥ 0, the set of all T satisfying (5.78) is an interval [0, T0 ], where T0 = T0 (X, ε) is the root of the equation 2εT e3XT = 1. Let us evaluate T0 . From e3XT > 1 + 3XT it follows that T < (e3XT − 1)/3X for all T ≥ 0. Hence, T1 < T0 < T2 , where T1 and T2 are the unique positive solutions of the equations e3XT1 − 1 3XT1 e =1, 3X 2ε T2 (1 + 3XT2 ) = 1 ,

2ε

respectively. The solutions of these equations can be explicitly found, more precisely, T1 =

1 X 1 log (1 + √1 + 6 ) , 3X 2 ε

T2 =

1 X (−1 + √1 + 6 ) . 6X ε

The asymptotic of T1 in the domain of quasi-classical approach x/ε ≫ 1 is actually T1 ∼

1 X log , 6X ε

as is easy to check.

5.4.7 Asymptotic under Large Time Since the time of applicability of the quasi-classical approach is logarithmically small, the question arises about the behavior of f and g for large t, cf. [23]. To study the problem we invoke the Laplace transform in the variable t > 0, which is denoted by ∞

̂ F(τ) = ∫ e−ıτt F(t) dt . 0

By Theorem 5.4.3, the solution F of (5.62) and (5.63) is of exponential growth exp(γt) in t > 0, where γ = 5/2 x. Hence, the Laplace transform of F is an analytic function of τ in the lower half-plane ℑτ < −γ. Taking into account that ̂̇ ̂ F(τ) = −F(0) + ıτ F(τ)

314 | 5 Asymptotic Solution of the Schrödinger Equation

and applying the Laplace transform to both sides of (5.62) we reduce the problem to 2εx

∂f ̂ + (τ + 2(x − 1))f ̂ − x ĝ = −ı , ∂x τ ĝ = −x f ̂ .

Eliminating ĝ from this system we arrive at the Cauchy problem for f ̂ on the half-axis x ∈ R≥0 ∂f ̂ τ2 + x2 2εx +( + 2(x − 1)) f ̂ = −ı , ∂x τ (5.79) −ı f ̂(τ, 0) = , τ−2 the initial condition being an immediate consequence of the differential equation. Note that the term in the parentheses in the differential equation is equal to (τ + ıλ+ )(τ + ıλ− ) . τ Note that (5.79) is a Fuchs type equation on the half-axis. Such equations are usually treated in weighted Sobolev spaces and no boundary conditions are posed at x = 0. In fact, we derived a condition at x = 0 from the differential equation itself. Our approach, however, is justified by the fact that we deal with the solution f ̂(τ, x), whose existence and analyticity for all x ∈ R are known a priori. The solution of the Cauchy problem (5.79) on R≥0 is given by a familiar formula a(τ, x) :=

x

f ̂(τ, x) =

y τ−2 −ı −ı − 1 ( x −y 2ε + ∫ ( ) e ε 4τ τ − 2 2ε x 2

2

+x−y) dy

y

.

(5.80)

0

Partial integration now leads now to an asymptotic expansion of f ̂ in powers of ε, which corresponds to (5.75). The asymptotic of the inverse Laplace transform of f ̂ f(t) =

1 ∫ e ıtτ f ̂(τ) dτ , 2π R−ıγ

for large t can be derived by the saddle point method. This way, we arrive at the equality x2 t , 2ε provided that εt ≫ 1, where Q is a slowly varying nonincreasing function of the variable t. f(t, x, ε) = Q(t, x, ε) exp √

5.4.8 Conclusion We have described the decay processes in a nonlinear quantum chain with the Hamiltonian (5.39). The analysis is based on the approximate equation (5.44). This approach is justified in the framework of the so-called “narrow packet” approximation.

5.4 Quantum Dynamics in the Fermi–Pasta–Ulam Problem

| 315

In the classical limit ℏ = 0, four-wave decays of the above type are possible merely in the domain (5.48), i.e., x > 1/2. In this case, the increment of instability is determined by (5.47) or A k±l (t) ∼ A k±l (0) exp (lΩ√2x − l2 t) ≤ A k±l (0) exp (Ωx t) . Quantum effects result in slowing down the development of instability for large times A k±l (t) ∼ A k±l (0) exp (Ωx √

t ) , V0 ℏ

provided that t ≫ 1/V0 ℏ. The slowing down of the development of instability was earlier observed in [40]. The second peculiarity of quantum effects in the dynamics of decays is the disappearance of the critical value ν c of the amplitude of the initial wave. In the quantum case, the periodic wave (5.52) is, therefore, always unstable. It is also worth mentioning that the result of the applicability times of the quasiclassical approach for systems (5.59) is of independent interest. Because of the exponential instability of the classical system, the time of applicability of the asymptotic series (5.75) in powers of ℏ is logarithmically small in the quasi-classical domain x/ε ≫ 1.

6 The Kelvin–Helmholtz Instability A classic example of hydrodynamic instability occurs when two fluids are separated by a free surface S across which the tangential (but not the normal) component of the fluid velocity exhibits a jump discontinuity. Such surfaces occur naturally on the downstream side of an obstacle, such as the wing of an airplane, where they separate the particles of fluid that passed over the obstacle from those that went under it. At very slow speeds, such surfaces S can hold their shapes indefinitely, as they are convected along with the fluid. However, at higher speeds they can, suddenly and without apparent warning, be crumpled like a sheet of typing paper into a thoroughly irregular shape. Such behavior, known as Kelvin–Helmholtz instability, ranks among the most extensively studied forms of hydrodynamic instability. In this chapter, we study several mathematical aspects of the following classical problem. One considers the flow of two fluids that are separated by a smooth surface St . Away from St the flow is assumed to be potential. For the purpose of analysis, on either side of St the fluid velocity, density, and stress, along with their low-order space derivatives, can be extended by continuity to St . At the initial time t = 0 a vortex distribution is given on the surface S0 . The problem consists in describing the evolution of St in the time t. We restrict our attention to the special case of this problem where the densities of the fluid on both sides of St coincide, and the flow is planar. In 1868 H. Helmholtz [107] noticed the instability of surfaces that separate domains in which a fluid flows with different velocities. W. Kelvin and J. Rayleigh [130] studied this problem in linear approximation. In particular, the stationary planar flow along the x-axis of the form {V, u(x, y) = { −V, { 𝑣(x, y) = 0

if

x>0,

if

x 0 if the curvature center belongs to X+t .) Since the flow is potential, there is the so-called Cauchy–Lagrange integral given by 1 p± (℘± )󸀠t + |∇℘± |2 + ± − ϖ = 0 , 2 ϱ where ϱ ± is the density of the fluid, which fills up X±t and ϖ is the potential of outer forces, see [241, p. 779]. On expressing the pressure p± from the Cauchy–Lagrange integral and substituting it into (6.6) we obtain (℘+ )󸀠t − (℘− )󸀠t +

ϱ+ − ϱ− 1 1 κ . ((℘− )󸀠t + |∇℘− |2 − ϖ) = − (|∇℘+ |2 − |∇℘− |2 ) + 2 ϱ+ 2 ϱ+ R

We restrict the discussion to the case where the fluids on the both sides of St have the same density, i.e., ϱ + = ϱ − , the common value being ϱ. Then the last equality takes a simpler form 1 κ (℘+ )󸀠t − (℘− )󸀠t + (|∇℘+ |2 − |∇℘− |2 ) = − . (6.7) 2 ϱ+ R We have thus arrived at the following mathematical problem: Find a curve St and harmonic functions ℘± in X±t , respectively, which satisfy equations (6.5) and (6.7). By the very hydrodynamical setting this problem certainly has infinitely many solutions. In order to achieve the uniqueness one has to pose certain initial conditions, S t = S0 ±

℘ (x, y, 0) =

for

t=0,

℘±0 (x, y).

6.1.2 Conditions on the Unknown Boundary Denote by ν and τ the unit outward normal and tangential vectors to the curve St , respectively. We first note that the conditions (6.5) imply the continuity of (∂/∂ν)℘± on St , i.e., ∂ + ∂ − (6.8) ℘ = ℘ . ∂ν ∂ν To establish this, we subtract the second equality of (6.5) from the first one, obtaining S󸀠x ((℘+ )󸀠x − (℘− )󸀠x ) + S󸀠y ((℘+ )󸀠y − (℘− )󸀠y ) = 0 .

320 | 6 The Kelvin–Helmholtz Instability Since the vector (S󸀠x , S󸀠y ) is parallel to ν, the desired assertion follows. We will be interested in equations (6.5) and (6.7) in the case where the curve St is given parametrically in the form z = x(s, t) + ıy(s, t), where s ∈ (−∞, ∞). On substituting x = x(s, t), y = y(s, t) into the equality S(t, x, y) = 0 and differentiating it in s and t we get S󸀠x x󸀠s + S󸀠y y󸀠s = 0 , S󸀠t + S󸀠x x󸀠t + S󸀠y y󸀠t = 0 , which allows us to replace the first equality of (6.5) by −y󸀠s (x󸀠t − (℘+ )󸀠x ) + x󸀠s (y󸀠t − (℘+ )󸀠y ) = 0 . One can rewrite this equality in the form ℑ (z󸀠s (z̄󸀠t − ((℘+ )󸀠x − ı(℘+ )󸀠y ))) = 0 . Since |∇℘± |2 = (

(6.9)

∂℘± 2 ∂℘± 2 ) +( ) , ∂ν ∂τ

equation (6.7) transforms, by (6.8), into (℘+ )󸀠t − (℘− )󸀠t +

∂℘− 2 ∂℘+ 2 1 κ (( ) −( ) )=− , 2 ∂τ ∂τ ϱR(z)

where R(z) =

x󸀠s y󸀠󸀠ss − x󸀠󸀠ss y󸀠s 3/2

((x󸀠s )2 + (y󸀠s )2 )

(6.10)

.

On summarizing we conclude that equations (6.5) and (6.7) can be replaced by the equivalent equations (6.8), (6.9), and (6.10).

6.1.3 Derivation of an Equation for the Curve We now use the Cauchy integral to eliminate the potentials ℘± from the relations (6.9) and (6.10). We look for a potential ℘ of the form ∞

℘(x, y, t) = ℜ P(z, t) = ℜ

1 ∫ log(z − z(s󸀠 , t)) ρ(s󸀠 )ds󸀠 , 2πı −∞

where ρ(s) is a given smooth function. Since P(z, t) is a holomorphic function of z, we get ∞

℘󸀠x

−

ı℘󸀠y

=

P󸀠z (z, t)

ρ(s󸀠 ) 1 ds󸀠 , = ∫ 2πı z − z(s󸀠 , t) −∞

(6.11)

6.1 Derivation of the Fundamental Equation

| 321

by the Cauchy–Riemann equations. Assume that the domains X±t are chosen in such a way that the vector ız󸀠s (s, t) points into X+t , so that P󸀠z (z(s, t), t) ± = lim P󸀠z (z(s, t) + ıεz󸀠s (s, t), t) . ε→0±

By the Sokhotskij–Plemelj formulas, we get ∞

P󸀠z (z(s, t), t) ± = p.v.

z 󸀠s (s, t) ρ(s󸀠 ) 1 1 󸀠 ∫ ∓ . ds ρ(s) 2πı z(s, t) − z(s󸀠 , t) 2 |z󸀠s (s, t)|2

(6.12)

−∞

Set

∞

ρ(s󸀠 ) 1 I = p.v. ds󸀠 . ∫ 2πı z(s, t) − z(s󸀠 , t) −∞

Using (6.11), one can rewrite (6.12) in the form z󸀠s ((℘± )󸀠x − ı(℘± )󸀠y ) = z󸀠s I ∓

1 ρ(s) , 2

(6.13)

whence y󸀠 x󸀠 ∂℘± = − 󸀠s (℘± )󸀠x + 󸀠s (℘± )󸀠y ∂ν |z s | |z s | 1 = − 󸀠 ℑ (z󸀠s ((℘± )󸀠x − ı(℘± )󸀠y )) |z s | 1 = − 󸀠 ℑ (z󸀠s I) , |z s | i.e., the equality (6.8) for the potential ℘ is fulfilled automatically. Furthermore, using (6.13) we can rewrite (6.9) in the form ℑ (z󸀠s (z̄󸀠t − I)) = 0 .

(6.14)

We now turn to equation (6.10). By (6.13), ∂℘± 1 = 󸀠 (x󸀠s (℘± )󸀠x + y󸀠s (℘± )󸀠y ) ∂τ |z s | 1 = 󸀠 ℜ (z󸀠s ((℘± )󸀠x − ı(℘± )󸀠y )) |z s | 1 1 ρ(s) = 󸀠 ℜ (z󸀠s I) ∓ , 2 |z󸀠s | |z s | whence

(6.15)

∂℘+ 2 ∂℘− 2 1 ρ(s) (( ) −( ) ) = − 󸀠 2 ℜ (z󸀠s I) . 2 ∂τ ∂τ |z s |

On differentiating in t under the integral sign in (6.11) and using the Sokhotskij– Plemelj formulas we obtain ∞

(℘± )󸀠t

−z󸀠t (s󸀠 , t)ρ(s󸀠 ) 󸀠 1 1 ρ(s)z̄󸀠s z󸀠t ds = ℜ p.v. ± ℜ , ∫ 2πı z(s, t) − z(s󸀠 , t) 2 |z󸀠s |2 −∞

322 | 6 The Kelvin–Helmholtz Instability

and so

∂℘+ ∂℘− ρ(s) − = 󸀠 2 ℜ (z󸀠s z̄󸀠t ) . ∂t ∂t |z s |

Substituting the last two equalities into (6.10) yields ℜ (z󸀠s (z̄󸀠t − I)) = −

κ|z󸀠s |2 . ρ(s)ϱR(z)

(6.16)

On comparing (6.14) and (6.16) we finally obtain an equation for the curve St , more precisely ∞

κ z̄󸀠s (s, t) ∂ ρ(s󸀠 ) 1 ̄ t) − p.v. z(s, ds󸀠 = − . ∫ 󸀠 ∂t 2πı z(s, t) − z(s , t) ρ(s)ϱR(z(s, t))

(6.17)

−∞

X+t

X−t

and are filled up by the same fluid, then the surface tension If the domains coefficient κ vanishes. Hence, the equation of tangential discontinuity takes the form ∞

∂ ρ(s󸀠 ) 1 ̄ t) − p.v. ∫ z(s, ds󸀠 = 0 . ∂t 2πı z(s, t) − z(s󸀠 , t) −∞

On separating the real and imaginary parts in the last equation we arrive at the system ∞

(y(s, t) − y(s󸀠 , t))ρ(s󸀠 ) 1 ∂ x(s, t) + p.v. ds󸀠 = 0 , ∫ ∂t 2π (x(s, t) − x(s󸀠 , t))2 + (y(s, t) − y(s󸀠 , t))2 −∞ ∞

(x(s, t) − x(s󸀠 , t))ρ(s󸀠 ) ∂ 1 ∫ ds󸀠 = 0 . y(s, t) − p.v. ∂t 2π (x(s, t) − x(s󸀠 , t))2 + (y(s, t) − y(s󸀠 , t))2

(6.18)

−∞

Equation (6.17) [or the system (6.18)] should, moreover, be complemented by an initial condition z(s, 0) = z0 (s) := x0 (s) + ıy0 (s) , (6.19) where z = z0 (s) is a given smooth curve. Remark 6.1.1. The function ρ(s) is called the vortex density. According to (6.15), ∂℘+ ∂℘− ρ(s) − =− 󸀠 , ∂τ ∂τ |z s | i.e., ρ(s) is determined by the jump of the tangential components of the flow velocity on the curve.

6.1.4 A Hamiltonian Form of the Equation of Tangential Discontinuity Recall that a function f(s) is called a variational derivative of a functional F(x) on functions x = x(s), if F(x + ε∆x) = F(x) + ⟨f, ∆x⟩ε + o(ε) ,

6.1 Derivation of the Fundamental Equation

for ε → 0. In this case, one writes f = Consider the functional H(x, y) =

δ δx F

| 323

or simply f = F 󸀠 (x).

1 ∫∫ log((x(s) − x(s󸀠 ))2 + (y(s) − y(s󸀠 ))2 ) ρ(s)ρ(s󸀠 )dsds󸀠 4π

on functions x = x(s) and y = y(s). An easy computation shows that ∂ 1 (x(s)−x(s󸀠 ))ρ(s󸀠 ) ds󸀠 ds , H(x + ε∆x, y) |ε=0 = ∫ρ(s)∆x(s) ∫ ∂ε 2π (x(s)−x(s󸀠 ))2 + (y(s)−y󸀠 (s))2 i.e.,

ρ(s) x(s)−x(s󸀠 ) δ H= ρ(s󸀠 )ds󸀠 . ∫ δx 2π (x(s)−x(s󸀠 ))2 + (y(s)−y󸀠 (s))2

Similarly, one verifies that ρ(s) y(s)−y(s󸀠 ) δ H= ρ(s󸀠 )ds󸀠 . ∫ δy 2π (x(s)−x(s󸀠 ))2 + (y(s)−y󸀠 (s))2 Therefore, one can rewrite equations (6.18) in the Hamiltonian form δ ∂ x(s, t) = − H(x(s, t), y(s, t)) , ∂t δy δ ∂ H(x(s, t), y(s, t)) . ρ(s) y(s, t) = ∂t δx ρ(s)

(6.20)

Thus, the system (6.18) can be thought of as an infinite-dimensional Hamiltonian system.

6.1.5 Conservation Laws On using the Hamiltonian form (6.20) one can easily construct conservation laws (i.e., the first integrals) for the system (6.18). More precisely, these are the energy conservation law δH ∂x δH ∂y ∂ H(x(s, t), y(s, t)) = ∫ ( − ) ds = 0 ∂t δx ∂t δy ∂t and two momentum conservation laws ∂ δH ∫ ρ(s)x(s, t)ds = − ∫ (x(s, t), y(s, t))ds = 0 , ∂t δy ∂ δH ∫ ρ(s)y(s, t)ds = ∫ (x(s, t), y(s, t))ds = 0 . ∂t δx

(6.21)

Note that the integrals (6.21) still survive in the presence of surface tension. Indeed, since z̄󸀠 z̄󸀠s ∂ z̄󸀠s = 󸀠s 3 (x󸀠s y󸀠󸀠ss − x󸀠󸀠ss y󸀠s ) = ı , R(z) |z s | ∂s |z󸀠s |

324 | 6 The Kelvin–Helmholtz Instability

we get, by (6.17), z̄󸀠s (s, t) ∂ κ 1 ρ(s)ρ(s󸀠 ) 󸀠 ̄ t)ds = + ∫ ρ(s)z(s, ∫∫ dsds ∫ ds ∂t 2πı z(s, t) − z(s󸀠 , t) ϱ R(z(s, t)) =0. One of the most telling experiments in the entire history of fluid dynamics was performed by Osborne Reynolds in 1883. While studying the flow of water through a horizontal glass tube, he injected a small quantity of colored water into the middle of the stream near the intake. At low velocities, Reynolds found the flow to be smooth, or laminar, with each fluid particle traveling in a straight line parallel to the axis of the tube. As the speed was gradually increased, however, the colored streak suddenly began to mix, some distance downstream, with the surrounding water, filling the entire breadth of the tube with a cloudy mass of unevenly colored liquid. When backlit by an intermittent spark (in lieu of stop action photography), the cloud resolved itself into a mass of more or less distinct curls and eddies. The sudden transition from smooth, laminar flow to turbulence as the fluid velocity is gradually increased remains one of the least adequately explained phenomena in all of classical physics. Smooth rectilinear flow through a tube of circular cross-section remains a valid solution of the Navier–Stokes equations (and relevant boundary conditions) at all flow speeds. It is not, however, a stable solution, for at high speeds unavoidable small shocks to the experimental apparatus invariably cause such flows to “morph” into more complicated ones quite suddenly. The relevant boundary value problem is known to be ill posed in the Hadamard sense, see, for instance,[270].

6.2 Small Perturbation of Tangential Discontinuity 6.2.1 Linearization of the Equation of Tangential Discontinuity Let z = z(s, t) a smooth solution to (6.18). On substituting z + εw for z into (6.18), differentiating in ε and setting ε = 0 we get the equation 1 ∂ ̄ t) + F 󸀠 (z)w = 0 , w(s, ∂t 2ı

(6.22)

for w, where ∞

w(s, t)−w(s󸀠 , t) 1 (F (z)w)(s, t) = p.v. ∫ ρ(s󸀠 )ds󸀠 . π (z(s, t)−z(s 󸀠 , t))2 󸀠

−∞

(6.23)

6.2 Small Perturbation of Tangential Discontinuity

| 325

Our next objective is to prove that F 󸀠 (z) is a pseudodifferential operator of the first ̂ order on R. Denote by w(ξ, t) the Fourier transform of w(s, t) in s, to wit, ∞

̂ w(ξ, t) = ∫ e−ıξs w(s, t)ds , −∞ ∞

w(s, t) =

1 ̂ t)dξ . ∫ e ısξ w(ξ, 2π −∞

On replacing both w(s, t) and readily

w(s󸀠 , t)

in (6.23) by their Fourier integrals we obtain

∞

1 ̂ (F (z)w)(s, t) = t)dξ ; ∫ e ısξ σ(F 󸀠 (z))(s, ξ)w(ξ, 2π 󸀠

(6.24)

−∞

the function ∞

σ(F 󸀠 (z))(s, ξ) = p.v.

󸀠

1−e ı(s −s)ξ 1 ∫ ρ(s󸀠 )ds󸀠 π (z(s, t)−z(s 󸀠 , t))2 −∞

is called the symbol of the operator (6.24). Look for asymptotics of function σ(F 󸀠 (z))(s, ξ), as |ξ| → ∞. To this end, we assume that the functions z(s, t) and ρ(s) are such that all integrals in question exist. Write ρ(s)z󸀠󸀠ss (s, t) ρ(s󸀠 ) ρ(s) ρ 󸀠 (s) 1 = + ( − ) 󸀠 +R, 󸀠 󸀠 󸀠 2 󸀠 2 2 3 s −s (z(s󸀠 , t)−z(s, t))2 (z s (s, t)) (s − s) (z s (s, t)) (z s (s, t)) where R = O(1) for s󸀠 → s. Using the equalities ∞

p.v. ∫ −∞ ∞

p.v. ∫ −∞

∞

1 − e ıϑξ 1 − cos(ϑξ) dϑ = ∫ dϑ = π|ξ| , ϑ2 ϑ2 1− ϑ

e ıϑξ

−∞ ∞

dϑ = ı ∫ −∞

sin(ϑξ) dϑ = −ıπ sgn ξ , ϑ

we rewrite the symbol of operator (6.24) in the form ρ(s)z󸀠󸀠ss (s, t) ) sgn ξ + σ −1 (F 󸀠 (z))(s, ξ) , (z󸀠s (s, t))3 (6.25) where σ −1 (F 󸀠 (z))(s, ξ) = o(1) for |ξ| → ∞ under reasonable assumptions on z(s, t) and ρ(s). The functions σ(F 󸀠 (z))(s, ξ) =

ρ(s)

(z󸀠s (s, t))2

|ξ| − ı (

ρ 󸀠 (s)

(z󸀠s (s, t))2

σ 1 (F 󸀠 (z))(s, ξ) =

−

ρ(s) |ξ| (z󸀠s (s, t))2

is said to be the principal homogeneous symbol of the pseudodifferential operator (6.24).

326 | 6 The Kelvin–Helmholtz Instability By (6.24), one can represent the operator F 󸀠 (z) in the form F 󸀠 (z)w =

ρ(s)z󸀠󸀠ss (s, t) ρ(s) ρ 󸀠 (s) Aw + − ) Hw + Sw , ( (z󸀠s (s, t))2 (z󸀠s (s, t))2 (z󸀠s (s, t))3

(6.26)

where ∞

Aw (s, t) =

1 ̂ ∫ e ısξ |ξ| w(ξ, t) dξ , 2π −∞ ∞

Hw (s, t) =

1 ̂ ∫ e ısξ (sgn ξ ) w(ξ, t) dξ , 2πı −∞

and S is a smoothing integral operator. It is easy to see that ∞

Aw (s, t) = p.v.

w(s, t) − w(s󸀠 , t) 󸀠 1 ∫ ds , π (s − s󸀠 )2 −∞ ∞

Hw (s, t) = p.v.

w(s, t) − w(s󸀠 , t) 󸀠 1 ds , ∫ π s󸀠 − s −∞

where H is the classical Hilbert operator in the variable s. The operators A and H are interrelated by ∂ . (6.27) A=H∘ ∂s

6.2.2 On the Ellipticity of the System (6.18) As usual, by the type of a nonlinear equation is meant the type of the linearized equation, provided that this latter does not depend on the particular choice of the function at which the nonlinear equation is linearized. In order to determine the type of linearized system (6.22) one has to consider its principal homogeneous part ∂ 1 ρ(s) ̄ Aw = 0 . w(s, t) + ∂t 2ı (z󸀠s (s, t))2 Set

(6.28)

1 ρ(s) = p(s, t) + ı q(s, t) . 2ı (z󸀠s (s, t))2

On separating the real and imaginary parts in (6.28) we get the system of ordinary differential equations u 󸀠t + p(s, t) Au − q(s, t) A𝑣 = 0 , (6.29) 𝑣󸀠t − q(s, t) Au − p(s, t) A𝑣 = 0 . The symbol of this system is the matrix (

ıτ + p(s, t)|ξ| − q(s, t)|ξ|

− q(s, t)|ξ| ) ıτ − p(s, t)|ξ| ,

6.2 Small Perturbation of Tangential Discontinuity

| 327

whose determinant just amounts to −τ 2 −(p2 + q2 )ξ 2 . This polynomial does not vanish for all (ξ, τ) ∈ R2 \ {0}, if (p(s, t))2 + (q(s, t))2 =

1 (ρ(s))2 ≠ 0 . 4 |z󸀠s (s, t)|4

The latter condition is always fulfilled if the curve z = z(s, t) is smooth and has no singular points. Therefore, the system (6.29) is elliptic, and so is the source system (6.18).

6.2.3 Small Perturbations of Rectilinear Tangential Discontinuity Consider the flow determined by formula (6.1). To this flow there corresponds the solution z(s, t) = s of equation (6.18) with vortex density ρ(s) ≡ −2V, cf. Remark 6.1.1. On linearizing system (6.18) at the solution z(s, t) = s with ρ(s) ≡ −2V one gets the equation ∂ V ̄ w(s, t) − Aw = 0 . ∂t ı On passing on to the real form we arrive at the system of pseudodifferential equations u 󸀠t − VA𝑣 = 0 , 𝑣󸀠t − VAu = 0 .

(6.30)

This system can be easily solved. Applying the Fourier transform yields the system of ordinary differential equations ̂ t) = 0 , û 󸀠t (ξ, t) − V|ξ|𝑣(ξ, ̂ t) = 0 . 𝑣̂󸀠t (ξ, t) − V|ξ|u(ξ, The characteristic equation of this system λ2 − V 2 ξ 2 = 0 obviously has the roots λ = ±V|ξ|, i.e., independently of the sign of V one mode proves to be exponentially growing while the other one exponentially falling. The solution of the Cauchy problem u(s, 0) = u 0 (s) , (6.31) 𝑣(s, 0) = 𝑣0 (s) , for system (6.30) can be written in the form Vt|ξ| −Vt|ξ| ̂ ̂ u(s, t) = ∫ e ısξ ( a(ξ)e + b(ξ)e ) dξ , Vt|ξ| −Vt|ξ| ̂ ̂ − b(ξ)e ) dξ , 𝑣(s, t) = ∫ e ısξ ( a(ξ)e

where

1 (û 0 (ξ) + 𝑣0̂ (ξ)) , 2 1 ̂ b(ξ) = (û 0 (ξ) − 𝑣0̂ (ξ)) . 2 ̂ a(ξ) =

328 | 6 The Kelvin–Helmholtz Instability ̂ Hence, for problem (6.30) and (6.31) to possess a solution it is necessary that a(ξ) be exponentially decaying for |ξ| → ∞, i.e., u 0 (s) + 𝑣0 (s) might be analytic in a strip. The problem has a solution for all t only in the case where u 0 (s) + 𝑣0 (s) is an entire function. If one changes the variables in (6.30) by u = P and 𝑣 = −HQ, where H is the Hilbert transform, and uses the relations (6.27) and H 2 = −I, then one arrives at the system ∂P ∂Q −V =0, ∂t ∂s ∂Q ∂P +V =0. ∂t ∂s For V = 1, this system reduces to the classical Cauchy–Riemann system of function theory.

6.2.4 The Linearization of Equation (6.17) On applying the formula for the curvature radius R(z) given at the end of Section 6.1.2 we deduce that the linearization of equation (6.17) at an arbitrary smooth function z = z(s, t) looks like κ z̄󸀠s ∂ 1 󸀠 κ ∂ w̄ + F (z)w + w̄ = 0 , (L(z)u + N(z)𝑣) + ∂t 2ı ρ(s)ϱ ρ(s)ϱR(z) ∂s

(6.32)

where L(z) and N(z) are second-order differential operators in s with coefficients depending on s. To wit, L(z) = − N(z) =

y󸀠s ∂ 2 y󸀠󸀠 3x󸀠 ∂ ( ) + ( 󸀠ss3 − 󸀠 2s ) , 󸀠 3 ∂s |z s | |z s | |z s | R ∂s

x󸀠󸀠 3y󸀠 x󸀠s ∂ 2 ∂ ( ) − ( 󸀠ss3 + 󸀠 2s ) . 󸀠 3 ∂s |z s | |z s | |z s | R ∂s

We can now return to the flow given by (6.1). If the function ρ is determined by ρ(s) = −2V, and the function z = z(s, t) reduces to z(s, t) = s (cf. Section 6.2.3), then (6.32) transforms to κ 󸀠󸀠 𝑣 =0, u 󸀠t − V A𝑣 − 2Vϱ ss (6.33) 𝑣󸀠t − V Au = 0 . The symbol of this system is the matrix (

ıτ

κ 2 ξ 2Vϱ ) ıτ ,

− V|ξ| +

− V|ξ|

with determinant ∆(ξ, τ) = −τ 2 +

κ 3 |ξ| − V 2 ξ 2 . 2ϱ

6.2 Small Perturbation of Tangential Discontinuity

|

329

Note that ∆(ξ, τ) is different from zero for ℑτ ≠ 0, i.e., the function ∆ satisfies the correctness condition of Petrovskii. It follows that the Cauchy problem for the system (6.33) is well posed in the class of functions of finite smoothness (or finite-order distributions). Let us look for a solution of the Cauchy problem for the system (6.33) with initial data (6.31). In the Fourier images it takes the form ̂ t) = â 1 (ξ)e λ(ξ)t + b̂ 1 (ξ)e−λ(ξ)t , u(ξ, ̂ t) = â 2 (ξ)e λ(ξ)t − b̂ 2 (ξ)e−λ(ξ)t , 𝑣(ξ, where λ = ıτ is defined from λ2 = −κ/2ϱ|ξ|2 + V 2 ξ 2 and 1 (û 0 (ξ) + 2 1 b̂ 1 (ξ) = (û 0 (ξ) − 2 â 1 (ξ) =

V|ξ| 𝑣0̂ (ξ)) , λ(ξ) V|ξ| 𝑣0̂ (ξ)) , λ(ξ)

1 λ(ξ) ( û 0 (ξ) + 𝑣0̂ (ξ)) , 2 V|ξ| 1 λ(ξ) û 0 (ξ) − 𝑣0̂ (ξ)) . b̂ 2 (ξ) = ( 2 V|ξ| â 2 (ξ) =

(6.34)

The formula for λ = λ(ξ) shows immediately that the amplitudes of “small” harmonics |ξ|


2ϱ 2 V κ

are bounded. The harmonics with amplitude |ξ| = (2/3)((2ϱ)/κ)V 2 is of the most rapid growth.

6.2.5 Remarks on the Linearized System Set 1 ρ(s) = p(s, t) + ı q(s, t) , 2ı (z󸀠s (s, t))2 1 1 ρ(s)z󸀠󸀠ss (s, t) ρ 󸀠 (s) − = a(s, t) + ı b(s, t) 2ı (z󸀠s (s, t))2 2ı (z󸀠s (s, t))3 κ and ε = ρ(s)ϱ . On separating the real and imaginary parts in (6.32) we rewrite the system in the form ∂ u u M11 M12 )( ) = 0 , (6.35) ( )+( ∂t 𝑣 𝑣 M21 M22

where M11 = pA + aH + εx󸀠s L(z) +

ε ∂ + S11 , R ∂s

M21 = −qA − bH + εy󸀠s L(z) + S21 ,

M12 = −qA − bH + εx󸀠s N(z) + S12 , M22 = −pA − aH + εy󸀠s N(z) +

ε ∂ + S22 , R ∂s

330 | 6 The Kelvin–Helmholtz Instability

the operators S ij being of negative order. Discarding the operators S ij and freezing the coefficients in M ij yields a system of pseudodifferential equations with constant coefficients. Denote ıτ + σ(M11 )(ξ) σ(M12 )(ξ) ∆(ξ, τ) = det ( ) , (6.36) σ(M21 )(ξ) ıτ + σ(M22 )(ξ) where σ(M 11 )(ξ) = p|ξ| − aı sgn ξ + εx󸀠s σ(L(z))(ξ) + (ε/R) ıξ , σ(M12 )(ξ) = −q|ξ| + bı sgn ξ + εx󸀠s σ(N(z))(ξ) , σ(M21 )(ξ) = −q|ξ| + bı sgn ξ + εy󸀠s σ(L(z))(ξ) , σ(M22 )(ξ) = −p|ξ| + aı sgn ξ + εy󸀠s σ(N(z))(ξ) + (ε/R) ıξ . Since σ(M11 )(ξ) + σ(M22 )(ξ) = ε (x󸀠s σ(L(z))(ξ) + y󸀠s σ(N(z))(ξ) +

2 ıξ ) R

=0, the determinant (6.36) reduces to ∆(ξ, τ) = −τ2 + σ(M11 )(ξ)σ(M22 )(ξ) − σ(M12 )(ξ)σ(M21 )(ξ) . On substituting the above explicit formulas for σ(M ij ) we get ∆(ξ, τ) = −τ 2 − (

1 (ρ(s))2 ε2 1 κ − 2 ) ξ2 + ξ3 󸀠 4 4 |z s | 2 |z󸀠s |3 ϱ R

up to a term c1 εıξ|ξ| + (c2 + ıc3 )ξ + c4 ; the explicit expressions for c j are cumbersome and will be omitted. If c1 ≠ 0 and c3 ≠ 0, then the symbol ∆(ξ, τ) does not satisfy the Petrovskii condition for the correctness of the Cauchy problem. However, we cannot deduce from this fact that the Cauchy problem for the original system (6.35) fails to be correct in the sense of Hadamard. This is explained by the following circumstances. The problem on the correctness of the Cauchy problem for pseudodifferential operators with symbols similar to ∆(ξ, τ) has not yet been studied, and so there is no reason to believe that the correctness of the Cauchy problem for such an equation is equivalent to that for the equation with frozen coefficients. The system (6.35) is “degenerated”, for the summands of ∆(ξ, τ) containing ξ 4 cancel. For such systems, the question of the correctness of the Cauchy problem does not reduce, in general, to the analogous question for the scalar equation whose symbol coincides with the determinant of the system. It is worth pointing out that on replacing the unknown function w by z̄󸀠s one obtains a “nondegenerate” system of complicated structure, which defies the study.

6.3 Analytic Continuation from a Boundary Subset |

331

Conclusion It is well known that vortex behavior plays an essential role in generating turbulence in the Navier–Stokes equations in three dimensions. The most outstanding open question is whether the three-dimensional Euler equation can develop a finite-time singularity for smooth initial data. One is thus left to wonder whether singularities in three-dimensional vortex sheets are qualitatively different from two-dimensional singularities. At the outset one must distinguish Kelvin–Helmholtz instability from true turbulence. These two phenomena are not truly separable. However, the Kelvin–Helmholtz instability is a phenomenon that can be adequately considered in two dimensions, which is due to Squire’s theorem [264]. Turbulence is an inherently three-dimensional phenomenon demanding a more extensive and sophisticated analysis. The analysis of this section shows that the character of Kelvin–Helmholtz instability is similar to that in the problem of analytic continuation. This latter can be adequately described within the framework of the so-called spaces with two-norm convergence, see [270]. On introducing the complex structure in the plane of variables (s, t) by the formula s s s (α + ıβ) ( ) := α ( ) + β J ( ) t t t, 0

with linear transformation J = (−1/V ∂ ∂ζ ̄

F+ε

V 0 ),

we rewrite the linearized system (6.33) as

H ∂ 2 ( ) ℑF = 0 , V ∂s

(6.37)

where F = u + ıH𝑣 and ζ = s − ıVt. One looks for a solution F of (6.37) in the lower half-plane ℑζ < 0 subject to the Cauchy data F = u 0 + ıH𝑣0 on the real axis t = 0. The WKB approximations of the solution for small values of ε > 0 starts with the solution of the unperturbed problem, which is precisely a problem of analytic continuation.

6.3 Analytic Continuation from a Boundary Subset The phenomenon of analytic continuation is possible because analytic functions possess a uniqueness property. More precisely, if two analytic functions in a domain D of the complex plane C coincide on a set S ⊂ D, which has at least one accumulation point in D, then these functions are identically equal. If f0 is now a function on S and there is an analytic function f in D, such that f = f0 on S, then f is called the analytic extension of f0 in D.

332 | 6 The Kelvin–Helmholtz Instability

The analytic continuation is uniquely determined through f0 ; however, it obviously does not exist for all f0 . If S has no accumulation point in D, then the uniqueness of analytic continuation fails. For example, consider the set of all nonnegative integer numbers as S and f0 (n) = n! for n = 0, 1, . . . . The function f0 possesses an analytic continuation into the half-plane ℜz > −1 given by the gamma function f(z) = Γ(z + 1). By the Bohr–Mollerup theorem, this analytic continuation is unique only under the additional condition that the continuation is logarithmically convex. Here, we are interested almost exclusively in the cases where the analytic continuation is uniquely determined through the set S and the function f0 on S. If for each point of the set S there is a neighborhood in which the function f0 expands as a power series, then these power series provide an analytic continuation of f0 to a neighborhood of S. Many manipulations of L. Euler with divergent series were first understood after B. Riemann and K. Weierstrass established function theory. It looks as if L. Euler understood well that the origin of a power series is decisive for its proper summation. He worked essentially with power series, and his summation methods represented numerous methods for analytic continuation of a power series. He thought of a series ∞

∑ an n=0

as the value of the power series ∞

f0 (z) = ∑ a n z n n=0

at the point z = 1. In order to continue f analytically at z = 1 Euler mapped the unit disk D := {z ∈ C : |z| ≤ 1} conformally onto the half-strip ℜw < 1/2 by means of the mapping w = z/(z+1). For the composition f(w) = f0 (z(w)), the point 1/2 = w(1) is the closest possible singularity, if f0 is analytic in D. To find the sum f0 (1) Euler developed f in the powers of w and substituted w = 1/2, if the power series converges at w = 1/2. If, e.g., a n = (−1)n , this gives f(w) = 1 − w and f0 (1) = 1/2. Analytic continuation is especially efficient if one has an explicit integral formula for the continuation. The first integral formula for analytic continuation was perhaps published by Carleman in his book [38]. Substantial further contributions to the area were accomplished in [76, 100, 162, 282, 307], etc. The book [5] develops considerably explicit formulas for analytic continuation in the many-dimensional complex space Cn . Let D be a simply connected domain in C bounded by a rectifiable Jourdan curve and S a nonempty arc on ∂D. If f is an analytic function in D whose weak limit values on S vanish, then f is identically zero in D. In other words, the problem of analytic continuation from S to the domain D is well posed. In this section, we study the problem by means of Legendre polynomials.

6.3 Analytic Continuation from a Boundary Subset | 333

6.3.1 The Riemann Mapping Theorem A map from the plane to the plane is called conformal at the point z0 if it preserves the angles between pairs of regular curves intersecting at z0 . If D is a domain in the complex plane C, then a function w : D → C is conformal if and only if it is holomorphic or antiholomorphic, and its derivative vanishes at no point of D. We restrict ourselves to those conformal maps that preserve orientation. As is well known, these are given by holomorphic functions, see, for instance, [99]. Each injective holomorphic function w = w(z) in a domain D maps D conformally onto a domain D󸀠 in the complex w -plane. We now wish to find a conformal map between two arbitrary domains D and D󸀠 . This is obviously not always possible because of the following. 1) The connectivity degree of D is topologically invariant, i.e., the properties of connected open sets are carried over by a holomorphic map to the target space. Example 6.3.1. The ring 1 < |z| < 2 can not be mapped conformally onto the unit disk |w| < 1. 2) The domains D and D󸀠 can be homeomorphically equivalent but not conformally equivalent. Example 6.3.2. The plane |z| < ∞ can be mapped homeomorphically but not conformally onto the unit disk |w| < 1, for by the Liouville theorem each bounded entire function is constant and so not conformal. The Riemann mapping theorem is a basic result of geometric function theory. The theorem is named after B. Riemann, who outlined a proof in his thesis (1851). A complete proof was given first in 1922 by L. Fejér and F. Riesz. Theorem 6.3.3. Let D=⊄ C be a nonempty, simply connected domain. Then for each point z0 ∈ D there is precisely one bijective conformal mapping w : D → D such that w(z0 ) = 0 , w󸀠 (z0 ) > 0 .

Proof. The proof requires considerable effort, hence we sketch only the main steps. The domain D can be mapped by an injective conformal mapping onto a domain D󸀠 lying in the unit disk. One considers the set of all functions that map D󸀠 one-to-one onto a domain D󸀠󸀠 lying in the unit disk. In this set, there is precisely one function that is extremal in an appropriate sense. Finally, one shows that this extremal function maps onto the whole unit disk D. The desired function is now obtained as composition of the mapping onto the unit disk with the extremal function.

334 | 6 The Kelvin–Helmholtz Instability Example 6.3.4. Let D = {z ∈ C : ℜz < 0} and z0 = 0, then z w(z) = , z+1 w󸀠 (0) = 1 . The Riemann mapping theorem implies that any simply connected domain D different from C can be mapped biholomorphically onto the unit disk. A mapping is called biholomorphic if it is bijective and the inverse mapping is also holomorphic.

6.3.2 Hardy Spaces We follow the classical book [99] in introducing Hardy spaces on a simply connected domain D ⊂ C. In function theory by a Hardy space H p (D) is meant a space of holomorphic functions in D. Hardy spaces are analogues of L p -spaces in functional analysis. They are named after G. F. Hardy who introduced them in 1914. Let D be the unit disk in C and p ≥ 1. The Hardy space H p (D) consists of all holomorphic functions f on D that satisfy 2π

1/p

sup ( ∫ |f(re ıφ )|p dφ)

0 0 and ζ ∈ D \ {z}, where r z,σ ∈ H ∞ (D); 2) supζ ∈C\S |C z,σ (ζ)| ≤ const.(z) σ −1 for all z ∈ D, where the constant depends on z. As but one example of the Carleman function, we mention the kernel function in the formula (6.43), i.e., 1 1 h(ζ) σ C z,σ (ζ) = ( ) 2πı ζ − z h(z) parameterized through σ > 0. The corresponding constant of 2) just amounts to (|h(z)| − 1)−1 up to an obvious factor. Assume that S is a nonempty open arc in the boundary curve C. Then there is a neighborhood U of D such that U\(C \ S) has no precompact connected components in

6.3 Analytic Continuation from a Boundary Subset

| 341

U. By the Runge theorem, each holomorphic function in a neighborhood of the compact set C \ S can be approximated uniformly on C \ S by holomorphic functions in U. For any fixed z ∈ D, the Cauchy kernel function is holomorphic in ζ in some neighborhood of C \ S. Therefore, there is a sequence (r z,n )n=1,2,... of holomorphic functions in U with the property that C z,n (ζ) :=

1 1 + r z,n (ζ) 2πı ζ − z

satisfies |C z,n (ζ)| ≤ n−1 , for all ζ ∈ C \ S and n = 1, 2, . . . . In order to get a Carleman function one interpolates the sequence (C z,n ) through a function of σ > 0, e.g., C z,σ := C z,n , if σ ∈ (n−1, n]. These are the basics of the theory of [162], which proved to be very useful in the study of the Cauchy problem for harmonic functions of several variables. Theorem 6.3.13. Let C z,σ (ζ) be a Carleman function in D. For any function f ∈ H 1 (D), it follows that f(z) = lim ∫ C z,σ (ζ)f(ζ)dζ , σ→∞

(6.47)

S

when z ∈ D. Proof. Let z ∈ D be fixed. From the Cauchy formula for H 1 -functions in D, the property 1) of the Carleman function and Corollary 6.3.6 we deduce that f(z) = ∫ C z,σ (ζ)f(ζ)dζ = ∫ C z,σ (ζ)f(ζ)dζ + ∫ C z,σ (ζ)f(ζ)dζ , C

S

C\S

for all z ∈ D. Since the angular limit values of f on C are Lebesgue integrable, we get 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ C z,σ (ζ)f(ζ)dζ 󵄨󵄨󵄨 ≤ const(z) σ −1 ‖f‖L1 (C\S) 󵄨󵄨 󵄨󵄨 󵄨 C\S 󵄨 by the property 2) of the Carleman function, where σ > 0 is arbitrary. Letting σ → ∞ yields (6.47), as desired.

6.3.8 Expansion in a Fourier Series The approximation of the Cauchy kernel function on the set C \ S is the most constructive if it occurs in the Hilbert space L2 (C \ S). The polynomials of z (i.e., holomorphic polynomials) lie densely in the space L2 (C \ S) if the complement of the compact set K := C \ S in C is connected. In order to prove this, one exploits the Runge theorem on polynomial approximations. Namely, each function f ∈ L2 (C \ S) can obviously be approximated by continuous functions on K. Since the Lebesgue measure of K in C is equal to zero, a familiar trick with the Cauchy integral formula shows that each continuous function on K can be approximated uniformly by functions that are holomorphic

342 | 6 The Kelvin–Helmholtz Instability

in a neighborhood of K. Finally, by the Runge theorem, each holomorphic function in a neighborhood of K can be approximated uniformly on K by polynomials of z, for the complement of K in C is connected. On applying the Gram–Schmidt orthogonalization we also conclude that in L2 (C \ S) there is an orthonormal basis whose elements are polynomials of z. In particular, one obtains an orthonormal basis in the space L2 (C \ S), if one applies the Gram–Schmidt orthogonalization to the system of holomorphic monomials (z n )n=0,1,.... We denote this orthonormal basis through (P n (z))n=0,1,... , so that the degree of P n (z) amounts to n. For each fixed z ∈ ̸ C \ S the Cauchy kernel function can be expanded as the Fourier series ∞ 1 1 = ∑ c n (z)P n (ζ) 2πı ζ − z n=0

(6.48)

on the set C \ S, where c n (z) = ∫ C\S

1 1 P n (ζ) |dζ| , 2πı ζ − z

for n = 0, 1, . . . . As is well known, the Fourier series converges in the L2 (C \ S) -norm and the coefficients c n (z) are holomorphic functions in the complement of C \ S. We now define n 1 1 C z,n (ζ) := (6.49) − ∑ c k (z)P k (ζ) , 2πı ζ − z k=0 for z ∈ ̸ C \ S, n = 0, 1, . . . and all ζ ∈ C different from z. When comparing (6.49) with Definition 6.3.12 of the Carleman function, one sees the only difference, namely that C z,n (ζ) is small on C \ S not in the supremum norm but merely in the L2 (C \ S) -norm for large n. As a consequence of the proof of Theorem 6.3.13 one gets the following. Theorem 6.3.14. Let S be an arc on the boundary curve C and let C \ S have a connected complement in the complex plane. For each function f ∈ H 2 (D) it follows that f(z) = lim ∫ C z,n (ζ)f(ζ)dζ , n→∞

S

if z ∈ D. Proof. It suffices to show that the integral ∫ C z,n (ζ)f(ζ)dζ C\S

6.3 Analytic Continuation from a Boundary Subset | 343

tends to zero for each fixed z ∈ D, as n → ∞. Since the angular limit values of f form an L2 -function on C, by the Hölder inequality we conclude that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ C z,n (ζ)f(ζ)dζ 󵄨󵄨󵄨 ≤ ‖C z,n ‖L2 (C\S) ‖f‖L2 (C\S) , 󵄨󵄨 󵄨󵄨 󵄨 C\S 󵄨 where the first factor is infinitesimal for large n. The exposition of Sections 6.3.7 and 6.3.8 insistently suggests that the parameter σ in the definition of the Carleman function should rather take on discrete values.

6.3.9 Approximation through Legendre Polynomials Suppose that D is a simply connected domain in C and S is a nonempty open arc on C. Choose a larger simply connected domain D󸀠 with the property that ∂D󸀠 ∩ ∂D = C \ S and the remaining boundary of D󸀠 belongs to the complement of D. Fix a point z0 ∈ ∂D󸀠 \ D in the remaining boundary of D󸀠 . By the Riemann mapping theorem for the compactified complex plane (see [99]) there is a bijective conformal mapping w = w(z) of D󸀠 onto the upper half-plane ℑw > 0, such that w(z0 ) = ∞. The image of the arc C \ S by w = w(z) is an interval [a, b] of the real axis, and the image of S is a curve in the upper half-plane that connects a with b. On applying the further mapping w 󳨃→ (b − a)−1 (2w−(b+a)) we can assume without restriction of generality that w(C \ S) = [−1, 1]. The mapping w = w(z) enables us to reduce the problem of analytic continuation from S into D to that of analytic continuation from w(S) into w(D) in the complex w -plane. If C \ S is an interval in the real axis, then one can take a system of classical orthogonal polynomials as (P n )n=0,1,... . For the interval [−1, 1] one can start with monomials (z n )n=0,1,... in L2 [−1, 1] and use the Gram–Schmidt orthogonalization to iteratively derive a system of orthogonal polynomials P n (z). This way, one arrives at the Legendre polynomials if P n (z) are required to additionally satisfy P n (1) = 1, see [293]. The socalled Rodrigues formula says that P n (z) =

1 2n n!

(

d n 2 ) (z − 1)n , dz

(6.50)

for n = 0, 1, . . . . The family (P n )n=0,1,... of Legendre polynomials forms a complete orthogonal system in L2 [−1, 1]. We get 1

∫ P m (z)P n (z)dz = −1

2 δ m,n , 2n + 1

344 | 6 The Kelvin–Helmholtz Instability

where δ m,n denotes the Kronecker delta. The Cauchy kernel expands in the Legendre polynomials on the interval [−1, 1] as in (6.48) with coefficients −1

c n (z) = ∫ 1

1 1 2n + 1 P n (ζ)dζ , 2πı ζ − z 2

for n = 0, 1, . . . . In [293, 15.4], the functions −(2πı/(2n + 1)) c n (z) are called Legendre functions of the second kind, and the expansion of the Cauchy kernel function is referred to as the Heine formula. Theorem 6.3.15. Suppose S is an open arc in the boundary curve C with the property that C \ S = [−1, 1], Moreover, let C z,n (ζ) be defined through (6.49). Then, for each f ∈ H 2 (D), one has f(z) = lim ∫ C z,n (ζ)f(ζ)dζ , n→∞

S

when z ∈ D. Proof. This is a special case of Theorem 6.3.14.

6.4 A Numerical Approach to the Riemann Hypothesis The Riemann hypothesis is that all zeros of the Riemann zeta function ζ(z) in the critical strip 0 < ℜz < 1 belong to the critical line ℜz = 1/2. This just amounts to saying that the function 1/ζ(z) extends from the interval (1/2, 1) to an analytic function in the quarter-strip 1/2 < ℜz < 1, ℑz > 0. Note that the restriction of 1/ζ(s) to (1/2, 1) is actually continuous on the closed interval. Hence, function theory allows one to rewrite the condition of analytic continuability in an elegant form that is amenable to numerical experiments. More precisely, one constructs an explicit sequence {c n } of complex n |c n | = 1 is fulfilled if and only if the Riemann hynumbers, such that the equality lim √ pothesis is true. The numbers c n are integrals of 1/ζ(s) over the interval [1/2, 1] with an explicit weight function depending on n. Computations with the newest versions of Mathematica, Maple, and Matlab performed by the author’s diploma students give certain evidence to the fact that the limit is, indeed, 1. However, standard computer n programs are not sufficient to evaluate the sequence √ |c n | with strict accuracy. The n numerical data obtained this way testify to lim √|c n | = 1 not only for 1/ζ(s) but also for other functions (e.g., 1/(s − 3/4 − ı/2)) that fail to be analytically extendable to the critical quarter-strip. Thus, the work gives rise to the problem of elaborating an n efficient program which through the behavior of the sequence √ |c n | recognizes those continuous functions on [1/2, 1] that extend to analytic functions in the quarter-strip.

6.4 A Numerical Approach to the Riemann Hypothesis

| 345

6.4.1 The Riemann Zeta Function In this section, we gather the necessary material about the Riemann zeta function. For complete proofs, the reader is referred to [47, 123, 274]. For complex numbers s = ℜs + √−1ℑs in the half-plane ℜs > 1, the Riemann function is defined by ∞ 1 ζ(s) = ∑ s , n n=1 the series converging absolutely and uniformly in each half-plane ℜs > s0 with s0 > 1. In 1737, Euler proved his product formula, which displayed a deep connection of ζ(s) with the distribution of prime numbers. Theorem 6.4.1. If ℜs > 1, then ζ(s) = ∏ (1 − p

1 −1 ) , ps

where the product runs over all prime numbers p (p = 1 is not a prime number). In order to extend ζ(s) to an analytic function on all of C, one uses the analytic extension of the gamma function constructed by Weierstraß. More precisely, ∞ 1 1 = se γs ∏ (1 + ) e−s/n Γ(s) n n=1

holds for ℜs > 0, where γ is the Euler–Mascheroni constant. The right-hand side of this equality is an entire function of s vanishing at the points s = 0, −1, −2, . . . . Lemma 6.4.2. If ℜs > 1, then ∞

∞ π s/2 2 1 ζ(s) = ( + ∫ (x s/2−1 + x−s/2−1/2 ) ∑ e−πn x dx) . Γ(s/2) s(s − 1) n=1 1

The lemma shows that the Riemann zeta function extends to a meromorphic function in the whole complex plane with the only pole at s = 1, which is simple. This function vanishes at s = −2, −4, . . .; the other zeros of ζ(s) are known to lie in the critical strip 0 < ℜs < 1. Riemann conjectured (1869) that all zeros of ζ(s) in the critical strip belong to the line ℜs = 1/2. The restriction of ζ(s) to the critical strip is symmetric with respect to both the critical line ℜs = 1/2 and the interval (0, 1) of the real axis. Moreover, it is different from zero for all s ∈ [0, 1]. Hence, the Riemann hypothesis just amounts to saying that ζ(s) has no zeros in the quarter-strip 1/2 < ℜs < 1, ℑs > 0.

346 | 6 The Kelvin–Helmholtz Instability For real x > 0, let π(x) denote the number of prime numbers p that satisfy p ≤ x. Riemann showed a formula for the difference x

π(x) − ∫

ds log s

0

in terms of x and zeros of ζ(s) lying in the critical strip. If ζ(s) has no zeros with ℜs > s0 for some 1/2 ≤ s0 < 1, then the asymptotic formula x ds + O(x s0 log x) π(x) = ∫ log s 0

holds. The Riemann hypothesis just amounts to this formula with s0 = 1/2. Some textbooks on complex analysis include the so-called prime number theorem that was proved independently by J. Hadamard and C.-J. de la Vallée–Poussin (1896). It reads as π(x) ∼ x/ log x.

6.4.2 Analytic Continuation in a Lune Denote by D := {w ∈ C : |w| < 1} the open unit disk with center at the origin in the plane of the complex variable w. Let S be a regular curve in D, whose endpoints lie on the unit circle and which does not run through 0 (i.e., 0 ∉ S). The curve S divides the disk D into two domains, and we write G for the subdomain of D that does not contain the origin 0. This way, we obtain a bounded domain with a piecewise smooth boundary, which is referred to as a lune. The boundary of G consists of two parts, one of the two is the curve S and the other an arc of the circle ∂D, see Fig. 6.2. In 1926, T. Carleman found a simple formula for analytic continuation in a corner, see [38]. For this reason, the following refined formula is named after him. This formula is well known, see [5, 100] and elsewhere. Since the proof is very simple, we give it for completeness. ℑw ı S −1

G 0

1

ℜw

−ı Fig. 6.2: A basic domain.

6.4 A Numerical Approach to the Riemann Hypothesis

|

347

Theorem 6.4.3. Suppose f is a holomorphic function in G continuous up to the boundary of G. Then w n 1 dw󸀠 , (6.51) f(w) = lim ∫ f(w󸀠 ) ( 󸀠 ) n→∞ w 2πı w󸀠 − w S

for all w ∈ G. Proof. Fix an arbitrary w ∈ G. Since 0 ∉ G and f is holomorphic in G and continuous up to the boundary, the function F(w󸀠 ) := f(w󸀠 ) (

w n ) w󸀠

is holomorphic in G and continuous on G for all n = 1, 2, . . . . By the integral formula of Cauchy one gets 1 dw󸀠 , F(w) = ∫ F(w󸀠 ) 2πı w󸀠 − w ∂G

for all w ∈ G. Substituting F yields f(w) = ∫ f(w󸀠 ) ( ∂G

w n 1 dw󸀠 ) , w󸀠 2πı w󸀠 − w

for each n = 1, 2, . . . . The integral on the right-hand side splits into two integrals, the first is over S and the second one is over ∂G \ S. So, f(w) = ∫ f(w󸀠 ) ( S

w n 1 dw󸀠 w n 1 dw󸀠 ) + ∫ f(w󸀠 ) ( 󸀠 ) . 󸀠 󸀠 w 2πı w − w w 2πı w󸀠 − w ∂G\S

On letting n → ∞ one obtains f(w) = lim ∫ f(w󸀠 ) ( n→∞

S

w n 1 dw󸀠 w n 1 dw󸀠 + lim ∫ f(w󸀠 ) ( 󸀠 ) , ) 󸀠 󸀠 w 2πı w − w n→∞ w 2πı w󸀠 − w ∂G\S

in the case when at least one of the limits exists. We now show that the second limit exists and is precisely zero. Since w ∈ G and 󸀠 w ∈ ∂G \ S, we get 󵄨󵄨 w 󵄨󵄨 |w| 󵄨󵄨󵄨 󸀠 󵄨󵄨󵄨 = 0 around the origin (take ε < dist (0, S)). Hence, C− f0 expands in this small disk as a power series whose coefficients are given by cn =

f0 (w󸀠 ) 1 ∫ 󸀠 n+1 dw󸀠 , 2πı (w ) S

cf. (6.54). The condition (6.52) forces the power series (6.54) to actually converge in the unit disk D to a holomorphic function F. By the uniqueness theorem, the integral C− f0 extends holomorphically to all of D, and this analytic continuation is F. Hence, it follows that the integral C+ f0 extends to a continuous function on G ∪ S. We now set f(w) := C+ f0 (w) − F(w) , for w ∈ G ∪ S, thus obtaining a holomorphic function in G that is continuous up to S and satisfies f(w) = f0 (w) for all w ∈ S, as desired.

6.4.3 A Carleman Formula for a Half-Disk The upper half-disk D󸀠 = {z ∈ C : |z| < 1, ℑ(z) > 0} is a canonical domain of the lune type corresponding to S = (−1, 1). Since 0 ∈ (−1, 1), the formula (6.51) is no longer applicable. To this end, one needs a transformation w = h(z) that maps D󸀠 conformally onto a lune like that in formula (6.51). We look for such a transformation in the group of fractional affine automorphisms of the unit disk D. These have the form h(z) = e ıφ

z−a , az − 1

for z ∈ D, with φ ∈ [0, 2π) and |a| < 1. We pose two additional conditions on h, namely,

350 | 6 The Kelvin–Helmholtz Instability ℑz

ℑw

w = h( z) D󸀠 S

1

S ı/2

ℜz

1

z = h −1 (

ℜw

w)

Fig. 6.3: A conformal mapping of D󸀠 onto a lune for t = 1/2.

1) 2)

h(0) = tı, where t ∈ (0, 1); h(ı) = ı.

The desired transformation is illustrated in Fig. 6.3. An easy computation shows that there is only one automorphism of the unit disk satisfying 1) and 2). This is z + tı h(z) = . (6.55) 1 − tız Corollary 6.4.5. Let f be a holomorphic function in the half-disk D󸀠 continuous up to the boundary of D󸀠 . Then 1

f(z) = lim ∫ f(z󸀠 ) ( n→∞

−1

dh(z󸀠 ) h(z) n 1 ) h(z󸀠 ) 2πı h(z󸀠 ) − h(z)

(6.56)

for all z ∈ D󸀠 . Proof. Set F(w) := f (h−1 (w)) for all w ∈ G. Since z = h−1 (w) maps the domain G conformally onto the half-disk D󸀠 and f is holomorphic in D󸀠 , the function F is holomorphic in G. Furthermore, z = h−1 (w) extends to a homeomorphism of the closure of G onto that of D󸀠 . Hence, F is continuous up to the boundary of G. By the formula (6.51), w n 1 dw󸀠 , F(w) = lim ∫ F(w󸀠 ) ( 󸀠 ) n→∞ w 2πı w󸀠 − w S

for all w ∈ G. On substituting w = h(z) and w󸀠 = h(z󸀠 ) and taking into account that S is the image of (−1, 1) by h, we arrive at 1

F(h(z)) = lim ∫ F(h(z󸀠 )) ( n→∞

−1

for all z ∈ D󸀠 , as desired.

dh(z󸀠 ) h(z) n 1 , ) 󸀠 h(z ) 2πı h(z󸀠 ) − h(z)

6.4 A Numerical Approach to the Riemann Hypothesis

| 351

Replacing the function h in (6.56) by its expression (6.55) we write the formula for analytic continuation from the interval (−1, 1) into the half-disk in explicit form. More precisely, 1

n

1 − tız 1 dz󸀠 z + tı n 1 − tız󸀠 , f(z) = lim ( ) ∫ f(z󸀠 ) ( 󸀠 ) n→∞ 1 − tız z + tı 1 − tız󸀠 2πı z󸀠 − z −1

for all z ∈ D󸀠 . Recall that t is any number in the interval (0, 1). For t = 0 we recover formula (6.51), however, this value is prohibited, because the integrand becomes singular at z󸀠 = 0. Under the conformal map (6.55) Theorem 6.4.4 is also traced back to conditions of analytic continuability of functions from the interval (−1, 1) to the upper half-disk D󸀠 . We actually rewrite the same invariant object in other holomorphic coordinates. Let f0 be a continuous function on [−1, 1]. Then F0 := f0 ∘ h−1 is a continuous function on S, the image of (−1, 1) by w = h(z). This is a regular curve in D \ {0} with endpoints ±

1 − t2 2t + ı 2 1+t 1 + t2

on the unit circle. Obviously, f0 extends to a holomorphic function f in D󸀠 , which is continuous up to (−1, 1), if and only if F0 extends to a holomorphic function F := f ∘ h−1 in G continuous up to S. By Theorem 6.4.4 F0 extends analytically to G if and only if n lim √ |c n | = 1 ,

n→∞

where cn =

F0 (w) 1 ∫ n+1 dw 2πı w S 1

=

f0 (z) 1 ∫ dh(z) 2πı (h(z))n+1 −1 1

1 − t2 1 − tız n 1 dz , = ∫ f0 (z) ( ) 2πı z + tı (z + tı)(1 − tız)

(6.57)

−1

for n = 0, 1, . . ., because dh(z) = (1 − t2 )/(1 − tız)2 dz. Corollary 6.4.6. Let f0 ∈ C[−1, 1] be a nonzero function. In order that there be a holomorphic function f in D󸀠 continuous up to (−1, 1) and satisfying f = f0 on (−1, 1), it is n necessary and sufficient that lim sup √ |c n | = 1, where c n are given by (6.57).

352 | 6 The Kelvin–Helmholtz Instability

Using the triangle inequality one readily finds an explicit estimate from above for the limit in question. Namely, 1 ) z∈(−1,1) |h(z)|

n+1

|c n | ≤ ( sup

≤(

1 ) |h(0)|

n+1

1

1 ∫ |f0 (z)| |dh(z)| 2π −1

1

1 1 − t2 ∫ |f0 (z)| dz , 2π 1 + t2 z2 −1

for all n = 0, 1, . . ., the last inequality being a consequence of the fact that the modulus of h(z) takes on its global infimum in (−1, 1) at the point z = 0, as is easy to see from Fig. 6.3. The right-hand side here is a constant multiple of |h(0)|−n , and so n |c n | ≤ lim sup √

1 1 = . |h(0)| t

(6.58)

Given any ε > 0, choose t close to 1 in the interval (0, 1), so that 1/t < 1 + ε. On n |c n | belongs to the interval [1, 1 + ε) applying Corollary 6.4.6 we see that lim sup √ for each continuous function f0 on [−1, 1], which is different from the identical zero. n Hence, the condition lim sup √ |c n | = 1 might be efficient numerically only if t is small enough.

6.4.4 Reduction of the Riemann Hypothesis Arguing as in Section 6.4.3, we look for a conformal mapping z = k(𝑣) of the critical quarter-strip H󸀠 := {𝑣 ∈ C : ℜ𝑣 ∈ (1/2, 1), ℑ𝑣 > 0} onto the half-disk D󸀠 . For this purpose, we need several lemmata. Lemma 6.4.7. The function z = tan 𝑣 maps the strip −π/4 < ℜ𝑣 < π/4 conformally onto the unit disk D. Proof. As is well known, the function z = tan 𝑣 maps the strip −π/2 < ℜ𝑣 < π/2 conformally into the complex plane C. It remains to specify the image of the strip −π/4 < ℜ𝑣 < π/4 by this mapping. Given any 𝑣 in the strip −π/4 < ℜ𝑣 < π/4, we get | tan 𝑣| = √

e2ℑ𝑣 − 2 cos(2ℜ𝑣) + e−2ℑ𝑣 . e2ℑ𝑣 + 2 cos(2ℜ𝑣) + e−2ℑ𝑣

By assumption, −π/2 < 2ℜ𝑣 < π/2, whence cos(2ℜ𝑣) > 0 and cos(2ℜ𝑣) = 0 if and only if either ℜ𝑣 = −π/4 or ℜ𝑣 = π/4. So, the quotient under the root sign is less than 1 in the open strip and it is equal to 1 if and only if ℜ𝑣 = −π/4 or ℜ𝑣 = π/4, as desired. From Lemma 6.4.7 we further deduce the following.

6.4 A Numerical Approach to the Riemann Hypothesis

ℑ𝑣

| 353

ℑz z=k (𝑣)

H󸀠 1 2

1

D󸀠

ℜ𝑣

1

ℜz

𝑣 = k −1 (z) Fig. 6.4: A conformal mapping of H󸀠 onto D󸀠 .

Lemma 6.4.8. The function z = tan 𝑣 maps the half-strip {𝑣 ∈ C : −π/4 < ℜ𝑣 < π/4 , ℑ𝑣 > 0} conformally onto the half-disk D󸀠 . The image of the interval (−π/4, π/4) by this mapping is the interval (−1, 1). Proof. Using the Euler formula one easily obtains tan 𝑣 =

2 sin(2ℜ𝑣) + ı (e2ℑ𝑣 − e−2ℑ𝑣 ) . e2ℑ𝑣 + 2 cos(2ℜ𝑣) + e−2ℑ𝑣

If ℑ𝑣 > 0, then e2ℑ𝑣 > e−2ℑ𝑣 , and so ℑ tan 𝑣 > 0, which shows the first part of the lemma. For the second part, we assume that 𝑣 ∈ (−π/4, π/4), so 𝑣 is real. Then z = tan 𝑣 is also real. Since the function tan 𝑣 is strongly monotonic, increasing on the interval (−π/2, π/2) and tan(−π/4) = −1 and tan(π/4) = 1, the assertion is clear. In order to construct a conformal mapping of the half-disk D󸀠 onto the critical quarterstrip H󸀠 , it suffices to take the composition of z = tan 𝑣 with an affine transformation of the 𝑣 -plane. That is, 3 z := k(𝑣) = tan π (𝑣 − ) , (6.59) 4 with inverse 𝑣 = k −1 (z) := 3/4 + 1/π arctan z, see Fig. 6.4. Lemma 6.4.9. If f is a holomorphic function in the critical quarter-strip H󸀠 , then the composition f ∘ k −1 is a holomorphic function in the upper half-disk D󸀠 . If, moreover, the function f is continuous up to the interval (1/2, 1), then f ∘ k −1 is continuous up to (−1, 1). Proof. The proof is obvious by the above. We formulate this lemma for the convenience of references. The function z = k(𝑣) maps the segment ℜ𝑣 = 1, ℑ𝑣 ≥ 0 homeomorphically onto the quarter-circle |z| = 1, arg z ∈ [0, π/2) and the segment ℜ𝑣 = 1/2, ℑ𝑣 ≥ 0 homeomorphically onto the quarter-circle |z| = 1, arg z ∈ (π/2, π]. The inverse mapping

354 | 6 The Kelvin–Helmholtz Instability 𝑣 = k −1 (z) extends continuously to the entire boundary of the upper half-disk D󸀠 , except for the north pole z = ı, where k −1 blows up. Thus, the mapping 𝑣 = k −1 (z) might be used to compactify the closure of H󸀠 by adding a “point at infinity” to it. A function f on the closure of H󸀠 is said to be continuous on such a one-point compactification of the closure of H󸀠 , if the composition f ∘ k −1 is continuous on the closure of D󸀠 . A Carleman-type formula with integration over the interval [1/2, 1] still holds for functions f holomorphic in H󸀠 and continuous on the compactification of H󸀠 . However, the Riemann zeta function does not belong to this class, so we shall not discuss the Carleman formula for holomorphic functions in H󸀠 . On the other hand, the criterion of analytic continuability does not require any continuity on the compactification of the closure of H󸀠 . Therefore, it extends to holomorphic functions in the critical quarterstrip in much the same way as Corollary 6.4.6. Theorem 6.4.10. Let f0 a continuous function on the interval [1/2, 1]. In order that there be a holomorphic function f in H󸀠 , such that f is continuous up to the open interval (1/2, 1) and coincides with f0 on this interval, it is necessary and sufficient that n lim sup √ |c n | = 1 ,

where 1

1 − t2 1 3 1 1 − tız n cn = ∫ f0 ( + arctan z) ( ) dz . 2πı 4 π z + tı (z + tı)(1 − tız)

(6.60)

−1

Proof. By assumption, F0 (z) := f0 (

3 1 + arctan z) 4 π

is a continuous function of z ∈ [−1, 1]. By Lemma 6.4.9, it extends to a holomorphic function in the half-disk D󸀠 continuous up to (−1, 1) if and only if f0 (𝑣) extends to a holomorphic function in the quarter-strip H󸀠 continuous up to (1/2, 1). The theorem now follows from Corollary 6.4.6. In Section 6.4.1 we made an alternative formulation of the Riemann hypothesis. We now make it more precise. Lemma 6.4.11. The Riemann hypothesis is true if and only if there exists a holomorphic function in H󸀠 that is continuous up to (1/2, 1) and equal to 1/ζ(s), for all s ∈ (1/2, 1). Proof. Since the Riemann zeta function does not vanish on the interval [1/2, 1] and has a simple pole at s = 1, the function f0 (s) := 1/ζ(s) is continuous on [1/2, 1] and has a simple zero at s = 1. If ζ(s) ≠ 0, for all s ∈ H󸀠 , then the function 1/ζ(s) is actually holomorphic in the whole critical quarter-strip. Hence, the Riemann hypothesis just amounts to the fact that f0 extends to a holomorphic function in H󸀠 continuous up to (1/2, 1).

6.4 A Numerical Approach to the Riemann Hypothesis

| 355

Corollary 6.4.12. The Riemann hypothesis holds if and only if n |c n | = 1 , lim sup √

where

1

1 1 − t2 1 1 − tız n ∫ cn = ) dz . ( 2πı (z + tı)(1 − tız) ζ ( 34 + 1π arctan z) z + tı

(6.61)

−1

Proof. This follows immediately from Lemma 6.4.11 and Theorem 6.4.10. The idea of this approach goes back at least as far as André Weil, who first proved the analogue of the Riemann hypothesis for general curves over finite fields in 1942. As but one part of his proof was to show that the logarithmic derivative of the zeta function has no poles in the “critical strip.” Weil proved crucial estimates for the radius of convergence of the power series in which the logarithmic derivative expands around the origin, see [90]. The Riemann zeta function arguments of Weil were recovered by Aizenberg et al. in [7]. The main formula of [7] contains an unfortunate misprint.

6.4.5 Numerical Experiments Formula (6.60) can be rewritten in the form cn =

1 dw , ∫ e n(− log w) f0 ((h ∘ k)−1 (w)) 2πı w S

where S ⊂ D is the image of (−1, 1) by w = h(z). Since the function − log w has no saddle points in the complex plane, the saddle point method does not apply to construct the asymptotics of c n as n → ∞. Hence, Theorem 6.4.10 seems to be of purely numerical interest. In their degree theses, A. Bühmann (2009) and M. Albinus (2010) evaluated nun merically several terms √ |c n | for the functions 1/ζ(s) and 1/(s−(3/4+ı)) on the interval [1/2, 1]. Obviously, the latter function does not extend analytically to all of H󸀠 . Hence, n |c n | for this function is larger than 1 (but ≤ 1/t). Several numerical the limit lim sup √ n values of the sequence √ |c n | corresponding to t = 1/2 are given in Fig. 6.5. Computations with the newest versions of Mathematica, Maple, and Matlab give n certain evidence to the fact that for the function 1/ζ(s) the limit of √ |c n | is, indeed, 1. However, replacing 1/ζ(z) by other functions (e.g., 1/(s − (3/4 + ı))), which fail to be analytically extendable to the critical quarter-strip computer simulations, still n suggests that lim sup √ |c n | = 1. So, standard computer programs do not allow us to specify by means of Theorem 6.4.10 those continuous functions on [1/2, 1] that extend analytically to the critical quarter-strip, cf. [7]. A severe difficulty consists in the rough evaluations of integrals depending on a parameter. For large values of the parameter, the graph of the integrand function in (6.60) fills in a rectangle. The problem

356 | 6 The Kelvin–Helmholtz Instability

n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n = 10 n = 11 n = 12 n = 13 n = 14 n = 15 n = 16

1/ζ(s)

1/(s − (3/4 + ı))

0,5361 0,7322 0,8124 0,8557 0,8827 0,9013 0,9148 0,9250 0,9330 0,9395 0,9449 0,9493 0,9531 0,9564 0,9592 0,9617

1,5046 1,1659 1,0265 0,9274 0,8374 0,7391 0,6070 0,5941 0,7636 0,8512 0,8960 0,9159 0,9195 0,9110 0,8928 0,8677

n Fig. 6.5: A computer simulation of √ |c n | for t = 1/2.

that arises is to elaborate an effective program for numerical evaluation of the limit n lim sup √ |c n |, where c n are given by the formula (6.60). This feature once again highlights the transcendental character of the Riemann problem on zeros of zeta function.

7 Nonlinear Cauchy Problems for Elliptic Equations 7.1 A Variational Approach to the Cauchy Problem We discuss the relaxation of a class of nonlinear elliptic Cauchy problems with data on a piece S of the boundary surface by means of a variational approach known in the optimal control literature as the “equation error method.” By the Cauchy problem we mean any boundary value problem for an unknown function y in a domain X with the property that the data on S, if combined with the differential equations in X, allow us to determine all derivatives of y on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy–Kovalevskaya theorem. We also admit overdetermined elliptic systems, in which case the set of those Cauchy data on S for which the Cauchy problem is solvable is very “thin.” For this reason, we discuss a variational setting of the Cauchy problem that always possesses a generalized solution.

7.1.1 Relaxations of Ill-Posed Problems This work actually began when the third author gave a seminar on the Cauchy problem for elliptic equations for scientists of the Max–Planck Institute for Gravitational Physics in Potsdam, on November 11, 2001. To highlight typical features of the problem, he restricted the discussion to the problem of analytic continuation in the complex plane, i.e., to the Cauchy problem for the Cauchy–Riemann system in the plane. If S is an open piece of the boundary surface different from the whole boundary, then the Cauchy problem fails to be solvable for all Cauchy data of compact support in S. Moreover, the problem is not stable, and the solution operator necessarily contains a limit passage even in the simplest situation. Because the Cauchy problem that the physicists were interested in was for a nonlinear elliptic differential equation with variable coefficients, the sample problem forced them to refuse a quite natural mathematical model based on the Cauchy problem for an elliptic equation. This was a good experience for the third author not to expose the theoretical obstructions but rather suggest a reasonable approach to cope with them. Since the differential equations under study are not linear, the direct use of distribution theory is limited. Instead, we invoke the calculus of variations to not only specify generalized solutions of the Cauchy problem, but also to suggest efficient methods of constructing approximate solutions. If the Cauchy problem does not possess any solutions, one has to invent them. The book [269] presents a modern treatment of the Cauchy problem for elliptic linear partial differential equations and systems. As first observed by J. Hadamard in 1917, this problem is ill posed for the Laplace operator if the Cauchy data are posed on https://doi.org/10.1515/9783110534979-007

358 | 7 Nonlinear Cauchy Problems for Elliptic Equations an open part S of the boundary surface. This actually initiated the development of the theory of ill-posed problems within which one usually studies the Cauchy problem for elliptic equations. Since ill-posed problems are not normally solvable, one usually neglects any characterization of the range and restricts oneself to constructing efficient regularization methods, thus assuming the existence of a solution. The analytical methods developed in [269] allow us to describe those Cauchy data on S for which the Cauchy problem is solvable. While they are dense in any natural function space on S, the space of such data has an infinite codimension. Hence, it follows that the range of an applied Cauchy problem cannot be recognized by any means, for the data on S are given approximately. The problem becomes more difficult if the system of differential equations is overdetermined, for in this case, the right-hand side should satisfy a local compatibility condition. This readily suggests the following generalized setting of the Cauchy problem: Given a system F(x, y, y󸀠 ) = 0 of first-order partial differential equations in a domain X ⊂ Rn , “minimize” the function F(x, y, y󸀠 ) over a space of functions y = y(x) on X satisfying the Cauchy conditions on S. Clearly, there is no restriction of generality in assuming that the system is of order one, for if not, we introduce new dependent variables in a standard way. The task of “minimizing” F(x, y, y󸀠 ) requires further explanation, for F takes its values in Cm for some m ∈ N. To this end, we consider the Euclidean norm of F(x, y, y󸀠 ), which is already a nonnegative function on X. By “minimizing” F(x, y, y󸀠 ) we mean the problem of minimizing the integral of |F(x, y, y󸀠 )|q , with q > 1, over X evaluated for those functions y = y(x) on X that fulfil the Cauchy conditions on S. This is a variational problem and it can be treated by methods of variational calculus. We have thus specified the Cauchy problem for elliptic differential equations and systems within variational problems. Even if F(x, y, y󸀠 ) is linear in y, its modulus fails to be so in general, which naturally leads to nonlinear partial differential equations. Note that if y is a solution of the original Cauchy problem, then it is also a solution of the corresponding variational problem, which justifies the variational setting. If the function |F(x, y, y󸀠 )|q is smooth enough, we are, in turn, able to reduce our variational problem to a mixed boundary value problem in the domain X. For this purpose, we write down the necessary conditions for extreme points of the functional y 󳨃→ ∫ |F(x, y, y󸀠 )|q dx X

on an affine space of functions y on X, satisfying the Cauchy conditions on S. These are the Euler equations, which constitute a system of ℓ second-order partial differential equations for ℓ unknown functions y = (y1 , . . . , yℓ ). Even if we study the Cauchy problem for an overdetermined system, the system of Euler equations is determined. It is complemented by boundary conditions of mixed type, consisting of the original Cauchy conditions on S and complementary first-order conditions on ∂X \ S.

7.1 A Variational Approach to the Cauchy Problem

| 359

We are thus led to a mixed boundary value problem in X like the Zaremba problem, cf. [305]. In many interesting cases, the mixed problem obtained in this way is a Fredholm one, see [226]. The connection between the Cauchy problem for the Laplace equation and the Zaremba problem goes back at least as far as [147]. This approach was developed for arbitrary elliptic equations in [250], where a function-theoretic construction of was refined [183]. The aim of this section is to elaborate the variational approach to the Cauchy problem for nonlinear elliptic equations and systems. When there is a “good” relaxation of the Cauchy problem, one is in a position to use iterative methods. They have found many applications in practical problems concerning the solution of diverse ill-posed problems of mathematical physics, cf. [151]. These methods have a number of advantages, which include the simplicity of computational schemes, the similarity of the schemes for problems with linear and nonlinear operators, the high accuracy of solutions, etc.

7.1.2 The Cauchy Problem Let X be a bounded domain with smooth boundary in Rn and F(x, y, p) a smooth function on the set X × Cℓ × Cℓ×n with values in Cm . We interpret y as a column with entries y1 , . . . , yℓ and p as a column matrix (p1 , . . . , p n ), where p k is a column with entries p1k , . . . , pℓk . We will substitute a function y(x) on X with values in Cℓ for y and its Jacobi matrix y󸀠 (x) = (∂ k y j (x)) for p. Pick a smooth function y = y0 (x) on X with values in Cℓ and expand F(x, y, p) in y and p about y0 (x) and p0 (x) = y󸀠0 (x) by the Taylor formula, x ∈ X being fixed. This yields n

F(x, y, y󸀠 ) = F(x, y0 , p0 ) + F 󸀠y (x, y0 , p0 )(y − y0 ) + ∑ F 󸀠p k (x, y0 , p0 )(p k − p0,k ) k=1

+ o (√|y − y0

|2

+ |p − p0

|2 )

.

Definition 7.1.1. By the symbol of the system F(x, y, y󸀠 ) = 0 along a function y = y0 (x) is meant the family of linear mappings Cℓ → Cm , which is given by the matrix n

∑ F 󸀠p k (x, y0 (x), y󸀠0 (x)) (ıξ k ) k=1

parameterized by (x, ξ) ∈ T ∗ X, where T ∗ X ≅ X × Rn stands for the cotangent bundle of X. A nonlinear system F(x, y, y󸀠 ) = 0 is said to be (overdetermined) elliptic along a function y = y0 (x) if its symbol has rank ℓ for all (x, ξ) ∈ T ∗ X. If m = ℓ, the system is called

360 | 7 Nonlinear Cauchy Problems for Elliptic Equations

elliptic in the classical sense or simply elliptic. In the case of linear partial differential equations, the symbol is independent of y0 (x), and so no y0 (x) is relevant. Suppose S is a nonempty open piece of the surface ∂X. Consider the Cauchy problem { F(x, y, y󸀠 ) = 0 in X , { y = y0 on S {

(7.1)

in the domain X with data y0 on S, y0 (x) being a smooth function on S with values in Cℓ . We call (7.1) noncharacteristic if the data F(x, y, y󸀠 ) in X and y on S actually determine all the derivatives of the function y on S by means of mere functional equations. Let us look for conditions on F and S under which this is the case. Since the problem is local, we can assume without loss of generality that S is a flat piece of the surface ∂X given by x1 = 0. Write x = (x1 , x󸀠󸀠 ), where x󸀠󸀠 = (x2 , . . . , x n ). From the equality y = y0 on S we are able to find all the derivatives ∂ αx󸀠󸀠 of y on S, which are tangential to j

S. It, therefore, remains to determine the normal derivatives ∂ 1 of y on S for j ∈ N. To this end, we invoke the differential equation F(x, y, y󸀠 ) = 0 near S. If restricted to S, it reduces to F (x, y0 (x), ∂1 y(x), ∂2 y0 (x), . . . , ∂ n y0 (x)) = 0 , for all x ∈ S. By the implicit function theorem, this system uniquely determines the normal derivative ∂1 y on S if the matrix F 󸀠p1 (x, y(x), y󸀠 (x)) has rank ℓ for all x ∈ S. This matrix is nothing else than the symbol of the system F(x, y, y󸀠 ) = 0 along y = y(x) evaluated at the outward normal vector ξ = e1 for S. If the symbol has rank ℓ on all of S, then the normal derivative ∂1 y is uniquely determined. We now apply this argument again, with F(x, y(x), y󸀠 (x)) replaced by ∂1 F(x, y(x), y󸀠 (x)), to obtain ∂21 y, etc. The hypersurface S is called noncharacteristic for the system F(x, y, y󸀠 ) = 0 if the symbol of this system along any solution y = y(x) evaluated at the unit outward normal vector ν(x) for S has rank ℓ at each point x ∈ S. Obviously, if S is noncharacteristic for F(x, y, y󸀠 ) = 0, then the Cauchy problem (7.1) is noncharacteristic. Lemma 7.1.2. If the system F(x, y, y󸀠 ) = 0 is (overdetermined) elliptic along any solution y = y(x), then the Cauchy problem (7.1) is noncharacteristic. Proof. Indeed, if the system F(x, y, y󸀠 ) = 0 is (overdetermined) elliptic along any solution y = y(x), then each hypersurface S is noncharacteristic for the system, which shows the lemma.

For nonlinear elliptic partial differential equations one can consult the monograph [267].

7.1 A Variational Approach to the Cauchy Problem

| 361

7.1.3 Variational Setting We now turn to the variational setting of the Cauchy problem (7.1). To this end, we introduce a nonnegative function L(x, y, p) on X × Cℓ × Cℓ×n , which is equal to |F|2 or, more generally, to s ∘ F, with s : Cm → R a nonnegative convex continuous function that is smooth in Cm \ {0} and takes the only local minimum at the origin. Consider the variational problem on extreme points f(y) → min for the functional f(y) = ∫ L(x, y(x), y󸀠 (x)) dx

(7.2)

X

over the set Df of all functions y ∈ C1 (X, Cℓ ) that satisfy y = y0 on S, y0 being a smooth function on S with values in Cℓ . We thus fix y only on the part S of the boundary; no restrictions are posed on y in ∂X \ S. However, such conditions on ∂X \ S along with a differential equation in X arise naturally for extreme functions, cf. Section 7.1.4. Just because the integral (7.2) is bounded below, among the competing functions, it does not follow that the greatest lower bound is taken on in the class of competing functions, cf. [202, 1.3]. If y ∈ D f furnishes a local (in the C1 norm) minimum to (7.2), it follows that ℓ

n

∑ ( ∑ L󸀠󸀠i i,j=1

k,l=1

j

p k ,p l

(x, y(x), y󸀠 (x))ξ k ξ l ) υ i ῡ j ≥ 0 ,

(7.3)

for all ξ ∈ Rn and υ ∈ Cℓ , which is known as the Legendre–Hadamard condition, see [202, 1.5]. In this case, one says that the integrand L is regular if the inequality holds in (7.3) for all ξ ≠ 0 and υ ≠ 0. For ℓ = 1, the Legendre–Hadamard condition is equivalent to the convexity of L as a function of p1 , . . . , p n for each fixed x ∈ X and y ∈ Cℓ . In many cases, (7.3) already guarantees the positiveness of the second variation of the functional f along a function y, which implies that y furnishes a local minimum to (7.2) among all y in any finite dimensional affine subspace of Df , cf. Section 1.6 in [202]. The necessary and sufficient conditions presuppose the existence and differentiability of an extremal. In the case n = 1, this is often proved using the existence theorems for ordinary differential equations. However, corresponding theorems for partial differential equations are not available, so direct methods were developed to handle this problem. Write m for the infimum of f(y) over all y ∈ Df . By definition, there is a sequence {y ν } in Df , such that f(y ν ) ↘ m. It is called minimizing. Any subsequence of a minimizing sequence is also a minimizing sequence. Were it possible to extract a subsequence {y ν ι } converging to an element y ∈ Df in the C1 (X, Cℓ ) norm, then f(y ν ι ) would converge to f(y) = m, and so y would be a desired solution of our variational problem. It is possible to require the convergence of a minimizing sequence in a weaker topology than that of C1 (X, Cℓ ). However, f should be lower semicontinuous with respect to correspondingly more general types of convergence. The new spaces of functions can be the Sobolev spaces W 1,q (X, Cℓ ) with q ≥ 1. This way, at the end of the 1930s,

362 | 7 Nonlinear Cauchy Problems for Elliptic Equations

Morrey obtained very general existence theorems. Unfortunately, the solution shown to exist was known only to be in one of these general spaces and hence was not even known to be continuous, let alone of class C2 (X, Cℓ )! In order to find a convergent subsequence of a minimizing sequence, one uses a compactness argument. Under reasonable assumptions on the function L, a minimizing sequence proves to be bounded in W 1,q (X, Cℓ ). If q > 1, then the space W 1,q (X, Cℓ ) is reflexive. Therefore, every bounded sequence in W 1,q (X, Cℓ ) has a weakly convergent subsequence. Thus, any minimizing sequence {y ν } has a subsequence {y ν ι }, which converges weakly in W 1,q (X, Cℓ ) to some function y. Since L(x, y, p) is convex in p for each x and y, on invoking Jensens’ inequality one readily deduces f(y) ≤ lim inf f(y ν ι ) ,

(7.4)

as desired. We now give a simple condition of lower semicontinuity for a wide class of integrals but use a more weakened type of convergence, which is sufficiently general for the existence theory. The hypothesis that the L󸀠p i be continuous can be removed rather k

easily. Theorem 7.1.3. Suppose L and the L󸀠p i are continuous with L(x, y, p) ≥ 0 for all (x, y, p) k

and L convex in p for each (x, y). If y and each y ν are in W 1,1 (X, Cℓ ) and y ν → y weakly in W 1,1 (U, Cℓ ) for each domain U ⋐ X, then (7.4) holds with {y ν ι } replaced by {y ν }. Proof. See Theorem 1.8.2 in [202]. If L satisfies the condition c |p|q − Q ≤ L(x, y, p) ,

(7.5)

with q > 1, c being a positive constant, an existence theorem can easily be deduced from the lower-semicontinuity Theorem 7.1.3 and embedding theorems for Sobolev spaces. First of all, we observe that it is the condition (7.5) that implies the boundedness of any minimizing sequence {y ν } in W 1,q (X, Cℓ ), for integrating (7.5) readily yields ∫ |y󸀠ν (x)|q dx ≤ X

1 (f(y ν ) + Q vol(X)) , c

where vol(X) is the n -dimensional Lebesgue measure of X. Theorem 7.1.4. Assume that 1) L and the L󸀠p i are continuous in their arguments; 2) L is k

convex in p for each (x, y); 3) L satisfies (7.5) with q > 1; 4) F is a family of functions, closed under weak convergence in W 1,q (X, Cℓ ), such that each y in F coincides on S with y0 ; and 5) f(Y) < ∞ for some Y ∈ F. Then f(y) takes on its minimum for some y in F.

7.1 A Variational Approach to the Cauchy Problem |

363

Proof. It follows from (7.5) that f is bounded below, and hence it has infimum over the family F. Pick a minimizing sequence {y ν } in F. We may certainly assume that f(y ν ) ≤ f(Y). By the above, (7.5) implies that the sequence {y󸀠ν } is bounded in L q (X, Cℓ×n ). By Lemma 1.9.2 of [202], there is a constant C depending only on X, S, q, and n, such that ‖y‖W 1,q (X,Cℓ ) ≤ C (‖y󸀠 ‖L q (X,Cℓ×n ) + ‖y‖L1 (S,Cℓ ) ) ,

(7.6)

for all y ∈ W 1,q (S, Cℓ ) with q ≥ 1. Combining (7.6) and 4), we readily deduce that the sequence {y ν } is actually bounded in W 1,q (X, Cℓ ). For q > 1, bounded sets in W 1,q (X, Cℓ ) are known to be relatively compact with respect to weak convergence in W 1,q (X, Cℓ ), see, for instance, Theorem 3.2.4 (e) in [202] and elsewhere. Hence, there is a subsequence of {y ν }, still called {y ν }, such that y ν tends weakly in W 1,q (X, Cℓ ) to some function y ∈ W 1,q (X, Cℓ ). By 4), the family F is closed under weak convergence in W 1,q (X, Cℓ ), showing that y belongs to F. On applying Theorem 3.2.4 (b) of [202] once again we conclude that y ν → y weakly in W 1,1 (U, Cℓ ) for each domain U ⊂ X. Now the theorem follows from Theorem 7.1.3. Actually, our hypotheses do not imply that L(x, y, p) ≥ 0. However, L is bounded below and there is no loss of generality (since X is bounded) in assuming that L is nonnegative.

Note that the assumption 4) implicitly includes certain conditions on the data y0 . However, they are hidden in other assumptions. In order to examine the condition 4) one often uses the Mazur theorem [299], which says that any convex closed subset of a reflexive Banach space is actually weakly closed. Clearly, the theorem applies to the subset A of W 1,q (X, Cℓ ) consisting of all y, such that y = y0 on S. We thus see that under the assumptions of Theorem 7.1.4 the Cauchy problem (7.1) has at least one solution y ∈ W 1,q (X, Cℓ ) for each data y0 ∈ W 1−1/q,q (S, Cℓ ). To derive further information on the differentiability and uniqueness of this solution, we invoke the necessary conditions for the extremals of the functional f , which are of differential character. One question that is still unanswered is whether the functional (7.2) has finitely many local minima on A. This question is hardly rich in content in the general case, i.e., for general Lagrange functions L no explicit conditions guaranteeing finiteness can be shown. If, moreover, L(x, y, p) does not depend on y, then the minimum is known to be unique. Since extremal functions of (7.2) satisfy the so-called Euler equation, the uniqueness might be derived from a finiteness theorem for solutions of the Euler equation. However, for nonlinear equations, this question is at present far from being solved.

364 | 7 Nonlinear Cauchy Problems for Elliptic Equations

7.1.4 Euler’s Equations To obtain a boundary value problem for extremals of the functional (7.2), we evaluate f at a small variation (y1 + ε1 δ1 , . . . , yℓ + εℓ δℓ ) of a function y ∈ C1 (X, Cℓ ), where δ = (δ1 , . . . , δℓ ) is an arbitrary smooth function on X with values in Cℓ vanishing on S. Such variation still remains in the domain of the functional f , hence, the resulting function ε 󳨃→ f(y1 + ε1 δ1 , . . . , yℓ + εℓ δℓ ) has an extreme value at the point ε = 0 if y is an extreme point of f . On differentiating in ε i and setting ε = 0, we get n

∫ (L󸀠y i (x, y, y󸀠 )δ i + ∑ L󸀠p i (x, y, y󸀠 ) ∂ k δ i ) dx = 0 , k=1

X

(7.7)

k

for each i = 1, . . . , ℓ. Equations (7.7) are the necessary first-order optimality conditions for (7.2). If the extreme function y is smooth enough (y ∈ C 2 (X, Cℓ ) is sufficient, while Euler had no compunctions about this), then in a familiar way we deduce from (7.7) that L󸀠y i (x, y, y󸀠 ) − ∑nk=1 ∂ k L󸀠p i (x, y, y󸀠 ) = 0 { { { k { y = y0 { { { { n ∑ L󸀠 (x, y, y󸀠 )ν k = 0 { k=1 p ik

in

X,

on

∂S ,

on

∂X \ S ,

(7.8)

for all i = 1, . . . , ℓ. Here, ν(x) = (ν1 (x), . . . , ν n (x)) stands for the unit outward normal vector for ∂X at a point x. These are precisely the classical Euler equations for extreme functions of the functional (7.2). Since the total derivative ∂ k L󸀠p i (x, y, y󸀠 ) just amounts k to ℓ

ℓ

n

L󸀠󸀠p i ,x k (x, y, y󸀠 ) + ∑ L󸀠󸀠p i ,y j (x, y, y󸀠 )∂ k y j + ∑ ∑ L󸀠󸀠i k

k

j=1

j=1 l=1

j

p k ,p l

(x, y, y󸀠 )∂ k ∂ l y j ,

(7.8) is a boundary problem for a second-order quasilinear differential operator whose principal symbol along y = y(x) is n

( ∑ L󸀠󸀠i k,l=1

j

p k ,p l

(x, y, y󸀠 )ξ k ξ l )

,

(7.9)

i=1,...,ℓ j=1,...,ℓ

for ξ = (ξ1 , . . . , ξ n ) ∈ Rn . This operator is called elliptic if the matrix (7.9) is invertible for all ξ ∈ Rn \ {0}. For a regular Lagrange function L, the Legendre–Hadamard condition (7.3) actually implies that the system (7.8) of Euler’s equations is strongly elliptic in the sense of [209]. This latter notion of ellipticity is more restrictive than the proper ellipticity, which coincides with the usual ellipticity (defined in Section 7.1.2) provided that n > 2. Moreover, the principal symbol along y = y(x) of the boundary condition of (7.8) on ∂X \ S is easily seen to be n

− ( ∑ L󸀠󸀠i k,l=1

j

p k ,p l

(x, y, y󸀠 )ν k ν l )

, i=1,...,ℓ j=1,...,ℓ

(7.10)

365

7.1 A Variational Approach to the Cauchy Problem |

which, in general, fails to be invertible over each point x ∈ ∂X \ S. Hence, we deduce that, in general, the boundary conditions in (7.8) fail to be of Lopatinskii type away from the boundary of S on ∂X. We mention that, in general, (7.8) will not have regular solutions due to the mixed boundary conditions. Such problems are nowadays handled within the framework of analysis on stratified manifolds, here with boundary and smooth edge ∂S on it. The concept of ellipticity of (7.8) should be related to sufficient second-order conditions for the variational problem. The boundary value problem (7.8) still makes sense if the extreme function y = y(x) is of class W 1,q (X, Cℓ ) with q > 1 large enough. Indeed, the restriction of y to the surface S is well understood in the sense of W 1−1/q,q (S, Cℓ ), and so the first boundary condition survives. As for the system of partial differential equations and the second boundary conditions, they appeared from the very beginning in the weak form, so that the equality (7.7) is fulfilled for all test functions δ in C∞ (X, Cℓ ) vanishing on S. As so many other great questions of twentieth century mathematics, it all started with one of Hilbert’s problems presented on the occasion of the 1900 International Congress of Mathematicians in Paris, namely the 19th problem: Are the solutions of regular problems in the calculus of variations always necessarily analytic? The question is to prove that, given the smoothness of L, the minimizer (assuming it exists) is also smooth. The minimizing function y satisfies a system of linear partial differential equations of the form n

− ∑ A k,l (x) k,l=1

∂2 y +... =0, ∂x k ∂x l

(7.11)

with coefficients A k,l (x) := (L󸀠󸀠i

j

p k ,p l

(x, y(x), y󸀠 (x))) i=1,...,ℓ , j=1,...,ℓ

the dots meaning lower-order terms. In the 1950s, regularity theory for elliptic equations was based on Schauder’s s,λ estimates which, roughly speaking, guarantee that if A k,l ∈ Cloc in X (i.e., of class s+2,λ

C s,λ in each domain U ⋐ X), then the solutions of (7.11) are of class Cloc s = 0, 1, . . . . So, if it could be shown that y ∈ 2,λ

1,λ Cloc , then

in X, for 0,λ

A k,l would belong to Cloc and

y would belong to Cloc in X. A bootstrap argument would then solve Hilbert’s 19th problem. Meanwhile, the existence theory had been developed through the use of direct methods. The minimization problem has a unique solution provided that L, apart from satisfying natural growth conditions like L(x, y, p) ≤ C (1 + |y|2 + |p|2 )q/2 with q > 1, is also coercive and uniformly convex. The notion of solution had to be conveniently extended, and the admissible set Df taken to be the set of functions that, together with their first weak derivatives, belong to L q , i.e., belong to the Sobolev space W 1,q (X, Cℓ ).

366 | 7 Nonlinear Cauchy Problems for Elliptic Equations So, the existence theory gave a minimizer y ∈ W 1,q (X, Cℓ ), and the missing step for the regularity problem to be solved was from first derivatives in L q to Hölder continuous first derivatives. In terms of the elliptic system (7.11), regularity theory worked if the leading coefficients were already somewhat regular (at least continuous), since it was based on perturbation arguments and comparison of the solutions with harmonic functions. Assuming only the measurability and the boundedness of the coefficients (together with the essential structural assumptions of ellipticity) was insufficient, nothing was known about the regularity of the solutions in this case. The problem was solved by C.B. Morrey in 1938 for the special case n = 2 and by J. Nash and E. de Giorgi independently in 1957 for n > 2 and ℓ = 1. Some time later, de Giorgi showed that the theory does not extend to the case ℓ > 1 unless n ≤ 2, cf. [202, 277]. As has been clarified, already because of the mixed boundary conditions the regularity of the minimizer cannot be expected in our case.

7.1.5 Examples The simplest Cauchy problem in a domain X ⊂ Rn with data y0 on S consists in finding a function y ∈ W 1,q (X) satisfying y󸀠 = 0 in X and y = y0 on S. For nonempty S, this Cauchy problem is solvable if and only if y0 is a constant function on S. The variational approach leads to a reasonable substitution for the solution, if y0 is no longer constant. Example 7.1.5. Let ℓ = 1 and L(x, y, p) = becomes −∆y = 0 { { { y = y0 { { { 󸀠 {y ν = 0

1/2 |p|2 , where p = (p1 , . . . , p n ). Then (7.8) in

X,

on

S,

on

∂X \ S ,

which is precisely the problem of [305]. Since the Zaremba problem has a unique solution in H 1 (X) for all Dirichlet data y0 ∈ H 1/2 (S) and Neumann data y1 ∈ H −1/2 (∂X \ S), it follows that the Cauchy problem {y 󸀠 = 0 { y = y0 {

in

X,

on

S

(7.12)

has a unique solution in H 1 (X) for each y0 ∈ H 1/2 (S), the solution being understood in the variational sense. The Neumann data of this solution on ∂X \ S are induced by the differential equation y󸀠 = 0 in X\S, provided that we are interested in solutions of finite order of growth near ∂X. This heuristic observation agrees with the Euler equation. Every solution y ∈ H 1 (X) of the Zaremba problem is known to fulfil the a priori estimate ‖y‖H 1 (X) ≤ c ‖y0 ‖H 1/2 (S) ,

7.1 A Variational Approach to the Cauchy Problem |

367

with c > 0 a constant independent of y. Hence, if the Cauchy problem (7.12) possesses a true solution, then the solution of the Zaremba problem approximates it with data “noise” going to zero. This seems to no longer hold for unstable Cauchy problems. The equation y󸀠 = 0 is also equivalent to 1/Q |y󸀠 |Q = 0 with any Q > 0. This purely formal reformulation yields a natural generalization of the Zaremba problem to the nonlinear case. Example 7.1.6. Let ℓ = 1 and L(x, y, p) = 1/Q |p|Q , where p = (p1 , . . . , p n ) and Q ≥ 1. Then (7.8) becomes −∆ Q y = 0 in X , { { { y = y on S , 0 { { { Q−2 󸀠 {|∇y| y ν = 0 on ∂X \ S , where ∆ Q y = div(|∇y|Q−2 ∇y) is the so-called Q -Laplace operator, see, for instance, [281, Ch. 5]. By the above, this mixed boundary value problem possesses a unique solution y ∈ W 1,Q (X) for each y0 ∈ W 1−1/Q,Q (S). We mention that the Q -Laplace operator is of great importance in the theory of nonlinear Bessel capacity associated with L Q -Sobolev spaces. Examples 7.1.5 and 7.1.6 show that the Cauchy problem for the gradient operator can be relaxed in various ways, yielding mixed boundary value problems for the Laplace and Q -Laplace operators, respectively. It follows that the variational relaxation is arbitrary. Moreover, the examples indicate that certain mixed boundary value problems can be regarded as first-order optimality systems for some Cauchy problems, while the corresponding Cauchy problems are not of interest in applications per se.

7.1.6 Mixed Problems The Cauchy problem for nonlinear partial differential equations can, thus, be handled within nonlinear mixed boundary value problems, which arise as Euler equations for suitable extremal problems. By construction, the condition on the complementary part of the boundary is completely determined by the initial equation, i.e., it bears zero data, cf. (7.8). When linearizing the Cauchy problem for a nonlinear first-order elliptic system, one arrives at the Cauchy problem for an inhomogeneous equation Ay = z in X. The variational approach leads to minimizing the integral of the Lagrange function L(x, y, y󸀠 ) := |Ay − z|2 over the set of functions taking the prescribed data y0 on S. Euler’s equations for this extremal problem just amount to the mixed boundary value problem for solutions of the equation A∗ Ay = A∗ z in X with data y = y0 on S and n(Ay) = n(z) on ∂X \ S, where n(z) stands for the Cauchy data of z on ∂X with respect to the formal adjoint A∗ of A. We are, thus, lead to a mixed boundary value problem of

368 | 7 Nonlinear Cauchy Problems for Elliptic Equations

Zaremba type with nonzero data. To pose nonzero data, it suffices to slightly modify the functional. More precisely, given any function y1 on ∂X \ S with values in Cℓ , we consider the functional f(y) = ∫ L(x, y(x), y󸀠 (x)) dx − ∫ ⟨y1 (x), y(x)⟩x ds , (7.13) X

∂X\S

ds being the surface measure on ∂X. We are looking for extreme points of the functional over the set A of all functions y ∈ W 1,q (X, Cℓ ) satisfying y = y0 on S, where y0 is a given function on S with values in Cℓ . The functional f is well defined on A if we require that the data y1 belong to 󸀠 L q (∂X \ S, Cℓ ), with 1/q + 1/q󸀠 = 1. The arguments of Section 7.1.3 still work, with slight changes, to show that the variational problem has at least one solution y ∈ A, cf. [247]. Lemma 7.1.7. As defined above, the extremals of the functional (7.13) satisfy the Euler equations L󸀠y i (x, y, y󸀠 ) − ∑nk=1 ∂ k L󸀠p i (x, y, y󸀠 ) = 0 { { { k { y = y0 { { { { n L󸀠 (x, y, y󸀠 )ν = y i ∑ k 1 { k=1 p ik

in

X,

on

∂S ,

on

∂X \ S ,

for all i = 1, . . . , ℓ. Proof. We apply arguments similar to those in Section 7.1.4. Assume that f takes on a local minimum at a function y ∈ A. We evaluate f at a small variation (y1 +ε1 δ1 , . . . , yℓ +εℓ δℓ ) of the function y, where δ = (δ1 , . . . , δℓ ) is an arbitrary smooth function on X with values in Cℓ vanishing on S. Such a variation still remains in the set A; hence, the resulting function ε 󳨃→ f(y1 + ε1 δ1 , . . . , yℓ + εℓ δℓ ) has an extreme value at the point ε = 0 if y is an extreme point of f . On differentiating in ε i and setting ε = 0 we get n

∫ (L󸀠y i (x, y, y󸀠 )δ i + ∑ L󸀠p i (x, y, y󸀠 ) ∂ k δ i ) dx − ∫ y1i δ i ds = 0 , k

k=1

X

∂X\S

for each i = 1, . . . , ℓ. Choosing δ of compact support in X, we readily derive the system of differential equations in X. Moving the derivatives ∂ k from δ i to L󸀠p i (x, y, y󸀠 ) by partial integration k

now yields n

∫ ( ∑ L󸀠p i (x, y, y󸀠 )ν k − y1i ) δ i ds = 0 , ∂X\S

k=1

k

for arbitrary functions δ ∈ C∞ (X, Cℓ ), which vanish on S. This is a weak form of the boundary condition on ∂X \ S, as desired.

7.1 A Variational Approach to the Cauchy Problem

| 369

By no direct means it is obvious that there exists a function y ∈ W 1,q (X, Cℓ ) satisfying the inhomogeneous mixed boundary conditions in the Euler equations, let alone the differential equations in X. It is just Theorem 7.1.4 that gives a positive answer. As described in Lemma 7.1.7, the mixed boundary problems are, therefore, of relevance for the Cauchy problem. If S = ∂X is the whole boundary, the variational setting of the Cauchy problem reduces to the Dirichlet problem for a nonlinear elliptic system of partial differential equations in X. While by the Laplacian associated with a linear operator A it is usually meant A∗ A, the nonlinear theory suggests another reasonable substitution for the Laplace operator. Namely, given any nonlinear first-order operator F(x, y, y󸀠 ), by the associated Laplacian the Euler operator is meant, n

(L󸀠y i (x, y, y󸀠 ) − ∑ ∂ k L󸀠p i (x, y, y󸀠 )) k=1

k

, i=1,...,ℓ

where L = 1/2 |F|2 . Example 7.1.5 demonstrates rather strikingly that these two (different in general) definitions can coincide, provided that the first is properly written. For linear Cauchy problems and 1 f(y) = ∫ |F(x, y, y󸀠 )|2 dx , 2 X

this is not surprising.

7.1.7 Inverse Problem Approach The aim of this section is to specify the Cauchy problem for nonlinear elliptic equations within a certain class of inverse problems. This brings new insight into the equation error method developed above. By inverse problem we mean the problem of finding a missing model parameter from the observed data. Of course, pure mathematics deals mostly with inverse problems, although such problems require further refinement, as has been historically developed. For example, one looks for an unknown coefficient of a nonlinear differential operator if some information on solutions of a boundary value problem for this operator is available. As but one example, we mention the electrical impedance tomography inverse problem, which consists in determining the electrical conductivity of a body by making voltage and current measurements at the boundary of the body. Substantial progress has been made on this problem since Calderón’s pioneer contribution [37], and it is also known as the Calderón problem, in the case where the measurement are made on the whole boundary. The problem can be reduced to studying the Dirichlet to Neumann map associated to the Helmholtz equations (−∆ + q)y = 0 in X. In [131], it is shown that measuring the Dirichlet to Neumann map on an arbitrary open subset S of the boundary, one can determine uniquely the potential q.

370 | 7 Nonlinear Cauchy Problems for Elliptic Equations

Inverse problems are typically ill posed. Of the three conditions for a well-posed problem suggested by J. Hadamard, the condition of stability is most often violated, which also affects the existence. We now turn to the Cauchy problem (7.1) of finding a function y ∈ W 1,q (X, Cℓ ) in X, such that F(x, y, y󸀠 ) = 0 in X and y = y0 on S, where S is an open piece of ∂X. Hadamard’s proof of the ill-posedness of the Cauchy problem for linear elliptic operators is based on the analytic regularity of solutions to linear elliptic boundary problems. This regularity was extended to nonlinear elliptic problems in [201], and so Hadamard’s argument also applies to general nonlinear elliptic problems. By the Cauchy–Kovalevskaya theorem, the problem is locally (in a neighborhood of S in Rn ) solvable for all real analytic data y0 . Since the real analytic functions are dense in any reasonable function space on S, the Cauchy problem is densely solvable, i.e., it possesses a solution for a dense subspace of the Cauchy data y0 on S. Under a uniqueness assumption, it is not solvable for nonzero data y0 with compact support on S. Hence, it follows that the inverse operator, if it exists, is not continuous. To achieve the existence, we observe that if the Cauchy problem models a real physical situation, then the existence is clear by the very setting. The unsolvability can only occur if there are small errors in the data. Moreover, the boundary condition y = y0 on S can always be satisfied if y ∈ W 1,q (X, Cℓ ) is no longer required to fulfil the equation F(x, y, y󸀠 ) = 0 in X. For such y we might evaluate the residual F(x, y(x), y󸀠 (x)) := F0 (x) and minimize s(F0 ) in the domain X. The residual can be thought of as unknown coefficient of the new differential equation F(x, y, y󸀠 ) − F0 (x) = 0 in X. We thus arrive at a new setting of the Cauchy problem. Namely, given any data y0 ∈ W 1−1/q,q (S, Cℓ ) on S, find F0 ∈ L1 (X, Cm ), such that the problem { F(x, y, y󸀠 ) − F0 = 0 in X , { y = y0 on S {

(7.14)

has a solution y ∈ W 1,q (X, Cℓ ). As posed in (7.14), the Cauchy problem is obviously solvable for any data y0 in W 1−1/q,q (S, Cℓ ). However, the solution is by no means unique. One possibility to achieve the uniqueness is to use the variational approach of Section 7.1.3. More precisely, find F0 with the smallest integral of s(F0 ) over X, such that (7.14) possesses a solution y ∈ W 1,q (X, Cℓ ). As has been mentioned, the solution is unique if F(x, y, p) is independent of y. To find an approximate solution to the variational problem obtained in this way one invokes both direct and indirect methods, the latter being based on the Euler equations. If F(x, y, y󸀠 ) = Ay is an (overdetermined) elliptic first-order differential operator, certain explicit conditions for solvability of (7.14) are known, see [269]. They look like ∞

∑ |c k |2 < ∞ , k=1

(7.15)

7.2 The Cauchy Problem for Chaplygin’s System

| 371

with c k = − ∫⟨w k , σ(A)(x, ν(x))y0 ⟩ ds + ∫⟨w k , F0 ⟩ dx , S

X

and (w k ) k=1,2,... a complete system of solutions to the transposed system A󸀠 w = 0 in a neighborhood of the closure of X. One may ask whether these conditions are efficient enough to show that for each Cauchy data y0 there is an F0 such that (7.15) is fulfilled. We have not been able to do this by directly evaluating the coefficients c k .

7.2 The Cauchy Problem for Chaplygin’s System In this section we discuss the Cauchy problem for the so-called Chaplygin system which often appears in gas, aerodynamics, and hydrodynamics. This system can be thought of as a nonlinear analogue of the Cauchy–Riemann system in the plane. We pose Cauchy data on a part of the boundary and apply a variational approach to construct a solution to this ill-posed problem. The problem actually gives insight to fundamental questions related to instable problems for nonlinear equations.

7.2.1 Preliminaries In [177], a variational approach to the Cauchy problem for nonlinear elliptic partial differential equations was elaborated. If the Cauchy data are given on a part of the boundary, the problem is well known to be instable in general, see [269]. The Cauchy problem still remains a challenge for mathematicians, for it requires a revision of the concept of solution to a differential equation, let alone to a nonlinear one. The aim of Section 7.2 is to indicate how these techniques may be used to study the Cauchy problem for a quasilinear elliptic system of practical interest on the plane. This is the so-called Chaplygin system, which is frequently used in the study of supersonic flows. Chaplygin’s system generalizes the classical Cauchy–Riemann system of complex analysis. This latter corresponds to the case a = −b = 1, where a and b are coefficients of Chaplygin’s system. Both a and b are real valued and depend on unknown functions u and 𝑣, which makes Chaplygin’s system nonlinear. Our standing assumption on a and b is that ab < 0 for all (u, 𝑣) ∈ R2 , which guarantees the ellipticity of the system. On applying the least squares method we formulate the Cauchy problem for Chaplygin’s system as an extremal problem and look for a solution in H 1 := W 1,2 . The Euler equations amount to a mixed boundary problem for the corresponding nonlinear Laplacian with boundary data that complement the Cauchy data by Chaplygin’s equation on the complementary part of the boundary. This is actually a nonlinear version of the classical Zaremba problem for the Laplace equation, see [305].

372 | 7 Nonlinear Cauchy Problems for Elliptic Equations

The Lagrange function of the variational problem in question fails to satisfy the coercivity condition, hence the classical calculus of variation, e.g., [202] does not apply to prove the existence of variational solutions. For this purpose, we need a Korn type inequality implying the boundedness of any minimizing sequence to use a weak compactness argument. In the case where the Cauchy data are given on the whole boundary, a Korn type inequality can be derived under certain additional assumptions on a and b. They are fulfilled for an important particular case of Chaplygin’s system, see [203]. We thus prove the existence. If the Cauchy data are defined on an open arc different from the whole boundary, then no Korn type inequality is possible. In this case, we restrict sets of competing functions by requiring them to be weakly bounded. In linear theory this idea is certainly not new, see, for instance, [231]. Under this assumption we prove the existence of variational solutions to the Cauchy problem for Chaplygin’s system in the general case.

7.2.2 Chaplygin’s System In the papers [133] and [134] the system for two unknown functions u and 𝑣 in the plane (x, y) ∈ R2 is studied, { u 󸀠x = a(u, 𝑣) 𝑣󸀠y , (7.16) { 󸀠 u = −a(u, 𝑣) 𝑣󸀠x , { y where u(x, y) is a flow function, 𝑣(x, y) a stress function, and a(u, 𝑣) is a given smooth function defined from experiments. This system was first introduced in [203]; for a ≡ 1 it coincides with the Cauchy–Riemann system in the complex plane z = x + ıy. In [135, 136] and [137], a more refined model in the study of flow was studied; it uses Chaplygin’s system and was named after Sergei Alekseevich Chaplygin. This is the nonlinear system { u 󸀠x = a(u, 𝑣) 𝑣󸀠y , (7.17) { 󸀠 u y = b(u, 𝑣) 𝑣󸀠x , { where a(u, 𝑣) and b(u, 𝑣) are certain smooth functions defined from experiments, see also [138]. Denote by A the nonlinear differential operator corresponding to Chaplygin’s system, i.e., u 󸀠 − a(u, 𝑣) 𝑣󸀠y A(u, 𝑣) = ( 󸀠x ) . u y − b(u, 𝑣) 𝑣󸀠x The principal symbol of A along the surface u = u(x, y), 𝑣 = 𝑣(x, y) is (

ıξ − a(u, 𝑣) ıη , ) ıη − b(u, 𝑣) ıξ ,

7.2 The Cauchy Problem for Chaplygin’s System |

373

which fails to be an isomorphism if b(u, 𝑣)ξ 2 = a(u, 𝑣)η2 for some (ξ, η) ∈ R2 \ (0, 0). It follows that if a(u, 𝑣)b(u, 𝑣) < 0 for all (x, y), then Chaplygin’s system is elliptic. If a(u, 𝑣)b(u, 𝑣) ≥ 0 holds at some point (x, y) ∈ R2 , then (7.17) is not elliptic at this point. Let X be a closed bounded domain in R2 with smooth boundary and S an open arc on ∂X. Given functions u 0 (x, y) and 𝑣0 (x, y) on S, we are interested in finding functions u and 𝑣 in X, satisfying the differential equations (7.17) in X and the boundary conditions {u = u 0 on S , (7.18) { 𝑣 = 𝑣0 on S . { Choose a defining function ϱ ∈ C1 (R2 ) for X, i.e., X consists of all (x, y) ∈ R2 such that ϱ(x, y) ≤ 0 and ∇ϱ(x, y) ≠ 0 for all (x, y) ∈ ∂X. Then, ν(x, y) =

∇ϱ(x, y) |∇ϱ(x, y)|

is the unit outward normal vector ν(x, y) for the boundary of X at (x, y) ∈ ∂X. The condition that the Cauchy problem (7.17) and (7.18) with data on S is noncharacteristic reads as b(u 0 , 𝑣0 )(ϱ 󸀠x )2 − a(u 0 , 𝑣0 )(ϱ 󸀠y )2 ≠ 0 , (7.19) for all (x, y) ∈ S. If system (7.17) is elliptic along the curve u = u 0 (x, y), 𝑣 = 𝑣0 (x, y) for (x, y) ∈ S, then (7.19) is obviously fulfilled. Under the hypothesis (7.19) we shall look for a solution of the Cauchy problem. To this end, we use a variational approach to set up a proper framework for the study of the problem.

7.2.3 Variational Setting We will minimize the functional 1 I(u, 𝑣) = ∫∫ |A(u, 𝑣)|2 dxdy 2

(7.20)

X

over the set A consisting of all pairs (u, 𝑣) of real-valued functions u, 𝑣 ∈ H 1 (X) that satisfy (7.18), i.e., u = u 0 and 𝑣 = 𝑣0 on S. Pick two functions δ1 , δ2 ∈ C1 (X) whose boundary values vanish on S. Then (u + ε1 δ1 , 𝑣 + ε2 δ2 ) still belongs to A for all real ε1 and ε2 , provided that (u, 𝑣) ∈ A. Hence, if the functional I(u, 𝑣) has a local minimum at (u, 𝑣) ∈ A, then the function f(ε1 , ε2 ) = I(u + ε1 δ1 , 𝑣 + ε2 δ2 ) of ε = (ε1 , ε2 ) in R2 has a local minimum at ε = 0. It follows that 0 is a critical point of f , i.e., f 󸀠 (0) = 0, which just amounts to the Euler equations.

374 | 7 Nonlinear Cauchy Problems for Elliptic Equations

An easy computation shows that f ε󸀠1 (0) = ∫∫ ((u 󸀠x − a𝑣󸀠y )(δ󸀠1,x − a󸀠u 𝑣󸀠y δ1 ) + (u 󸀠y − b𝑣󸀠x )(δ󸀠1,y − b 󸀠u 𝑣󸀠x δ1 )) dxdy , X

f ε󸀠2 (0)

= ∫∫ ((u 󸀠x − a𝑣󸀠y )(−a󸀠𝑣 𝑣󸀠y δ2 − aδ󸀠2,y ) + (u 󸀠y − b𝑣󸀠x )(−b 󸀠𝑣 𝑣󸀠x δ2 − bδ󸀠2,x )) dxdy . X

The equations f ε󸀠1 (0) = 0, f ε󸀠2 (0) = 0 for all δ1 , δ2 ∈ C1 (X) vanishing on S can already be thought of as a first relaxation of the Cauchy problem (7.17) and (7.18). It is worth pointing out that both equations are actually well defined for each pair (u, 𝑣) ∈ A. When interpreting the second derivatives of u and 𝑣 in the sense of distributions, we can move the derivatives from δ1 and δ2 to u and 𝑣 by partial integration, thus obtaining f ε󸀠1 (0) = ∫∫ (−(u 󸀠x − a𝑣󸀠y )󸀠x − (u 󸀠x − a𝑣󸀠y )a󸀠u 𝑣󸀠y − (u 󸀠y − b𝑣󸀠x )󸀠y − (u 󸀠y − b𝑣󸀠x )b 󸀠u 𝑣󸀠x ) δ1 dxdy X

+ ∮ ((u 󸀠x − a𝑣󸀠y )ν1 + (u 󸀠y − b𝑣󸀠x )ν2 ) δ1 ds , ∂X

f ε󸀠2 (0) = ∫∫ ((u 󸀠x − a𝑣󸀠y )󸀠y a + (u 󸀠x − a𝑣󸀠y )a󸀠u u 󸀠y + (u 󸀠y − b𝑣󸀠x )󸀠x b + (u 󸀠y − b𝑣󸀠x )b 󸀠u u 󸀠x ) δ2 dxdy X

+ ∮ (−(u 󸀠x − a𝑣󸀠y )aν2 − (u 󸀠y − b𝑣󸀠x )bν1 ) δ2 ds , ∂X

C1 (X) vanishing on S. Since δ

for all δ1 , δ2 ∈ we deduce that

(7.21) 1 and δ 2 are arbitrary in the interior of X,

{−(u 󸀠x − a𝑣󸀠y )󸀠x − (u 󸀠x − a𝑣󸀠y )a󸀠u 𝑣󸀠y − (u 󸀠y − b𝑣󸀠x )󸀠y − (u 󸀠y − b𝑣󸀠x )b 󸀠u 𝑣󸀠x { 󸀠 (u − a𝑣󸀠y )󸀠y a + (u 󸀠x − a𝑣󸀠y )a󸀠u u 󸀠y + (u 󸀠y − b𝑣󸀠x )󸀠x b + (u 󸀠y − b𝑣󸀠x )b 󸀠u u 󸀠x { x

=

0,

=

0

(7.22)

in X. Substituting this into (7.21) yields {(u 󸀠x − a𝑣󸀠y )ν1 + (u 󸀠y − b𝑣󸀠x )ν2 = 0 { −(u 󸀠x − a𝑣󸀠y )aν2 − (u 󸀠y − b𝑣󸀠x )bν1 = 0 {

on

∂X \ S ,

on

∂X \ S .

(7.23)

Remark 7.2.1. The differential equations (7.22) along with the boundary conditions (7.18) and (7.23) constitute a mixed boundary problem in X, which amounts to the Euler equations for I(u, 𝑣) → min. Note that (7.23) is a system of two linear equations for the unknowns u 󸀠x − a𝑣󸀠y and u 󸀠y − b𝑣󸀠x on ∂X \ S. Its determinant is equal to bν21 − aν22 . If a(u, 𝑣)b(u, 𝑣) < 0 for all (x, y) ∈ ∂X \ S, then the determinant is different from zero, which readily implies {u 󸀠x − a(u, 𝑣) 𝑣󸀠y { 󸀠 u − b(u, 𝑣) 𝑣󸀠x { y

=

0,

=

0

7.2 The Cauchy Problem for Chaplygin’s System |

375

on ∂X\S. This is the case if Chaplygin’s system is elliptic. Thus, the Euler equations for the extremal problem I(u, 𝑣) → min include, in particular, the differential equations (7.17) on ∂X \ S.

7.2.4 Existence of Solutions Write

1 ((u 󸀠x − a(u, 𝑣)𝑣󸀠y )2 + (u 󸀠y − b(u, 𝑣)𝑣󸀠x )2 ) 2 for the integrand function of (7.20). It is clear that L(x, y, u, 𝑣, p, q) is nonnegative for all (x, y) in X and u, 𝑣 ∈ R, p, q ∈ R2 , p, and q stands for the gradients of u and 𝑣. It follows that the image of A under I(u, 𝑣) is bounded below by 0, and so its infimum is a number m ≥ 0. We aim at finding those (u, 𝑣) ∈ A at which I(u, 𝑣) takes on the value m. Note that if the Cauchy problem (7.17) and (7.18) has a solution (u, 𝑣) ∈ A, then I(u, 𝑣) = 0, and so the pair (u, 𝑣) satisfies the variational problem I(u, 𝑣) → min. Conversely, if I(u, 𝑣) vanishes for some (u, 𝑣) ∈ A, then this pair is a solution to the Cauchy problem in X. L(x, y, u, 𝑣, ∇u, ∇𝑣) =

Lemma 7.2.2. For each fixed (x, y, u, 𝑣), the function L(x, y, u, 𝑣, p, q) is convex in (p, q) ∈ R2 × R2 . Proof. Write p = (p1 , p2 ) and q = (q1 , q2 ), then 2

2

2

2

k,l=1

k,l=1

k,l=1

∑ L󸀠󸀠p k ,p l w k w l + ∑ L󸀠󸀠p k ,q l w k z l + ∑ L󸀠󸀠q k ,p l z k w l + ∑ L󸀠󸀠q k ,q l z k z l k,l=1

= (w1 − az2 )2 + (w2 − bz1 )2 ≥0, for all (w1 , w2 ) ∈ R2 and (z1 , z2 ) ∈ R2 . Since L(x, y, u, 𝑣, p, q) is of class C2 in (p, q) ∈ R2 × R2 , the nonnegative definiteness of the quadratic form on the left-hand side is equivalent to the convexity of L as a function of (p, q), see Lemma 1.8.1 of [202] and elsewhere. Thus, the general hypotheses of [202, p. 91] concerning the integrand function L are satisfied. If (u, 𝑣) ∈ A furnishes a local minimum to (7.20), then necessarily ∗

(

L󸀠󸀠 ξ k ξ l υ1 ∑2 ) ( 2k,l=1 󸀠󸀠p k ,p l 2 υ ∑k,l=1 L q k ,p l ξ k ξ l

υ1 ∑2k,l=1 L󸀠󸀠p k ,q l ξ k ξ l ) ( 2) ≥ 0 2 󸀠󸀠 ∑k,l=1 L q k ,q l ξ k ξ l υ

(7.24)

is fulfilled for all (ξ1 , ξ2 ) ∈ R2 and υ ∈ C2 , the derivatives of L being evaluated at (x, y, u, 𝑣, ∇u, ∇𝑣). This classical necessary condition is known as the Legendre– Hadamard condition, see [202, 1.5]. In this case, one says that the integrand function L is regular if the inequality holds in (7.24) for all (ξ1 , ξ2 ) and υ different from zero.

376 | 7 Nonlinear Cauchy Problems for Elliptic Equations Lemma 7.2.3. The integrand function L(x, y, u, 𝑣, ∇u, ∇𝑣) is regular if and only if Chaplygin’s system is elliptic along (u, 𝑣). Proof. A trivial verification shows that the matrix in (7.24) just amounts to ξ12 + ξ22 ( −(a + b) ξ1 ξ2

−(a + b) ξ1 ξ2 ) , a2 ξ22 + b 2 ξ12

whose determinant is (bξ12 − aξ22 )2 . Clearly, for the matrix to be positive definite it is necessary and sufficient that bξ12 − aξ22 do not vanish, which establishes the lemma. By definition, there is a sequence {(u ν , 𝑣ν )} in A, such that I(u ν , 𝑣ν ) ↘ m. It is called minimizing. Any subsequence of a minimizing sequence is also a minimizing sequence. Were it possible to extract a subsequence {(u ν ι , 𝑣ν ι )} converging to an element (u, 𝑣) ∈ A in the H 1 (X, R2 ) norm, then I(u ν ι , 𝑣ν ι ) would converge to I(u, 𝑣) = m, and so (u, 𝑣) would be a desired solution of our variational problem. It is possible to require the convergence of a minimizing sequence in a weaker topology than that of H 1 (X, R2 ). However, the functional I should be lower semicontinuous with respect to correspondingly more general types of convergence. In order to find a convergent subsequence of a minimizing sequence, one uses a compactness argument. The space H 1 (X, R2 ) is reflexive. Hence, each bounded sequence in H 1 (X, R2 ) has a weakly convergent subsequence. Thus, any bounded minimizing sequence {(u ν , 𝑣ν )} has a subsequence {(u ν ι , 𝑣ν ι )}, which converges weakly in H 1 (X, R2 ) to some function (u, 𝑣). By Mazur’s theorem, see [299] and elsewhere, any convex closed subset of a reflexive Banach space is actually weakly closed. It follows that the limit function (u, 𝑣) satisfies (7.18), i.e., it belongs to A. Moreover, Theorem 3.4.4 of [202] says that the subsequence {(u ν ι , 𝑣ν ι )} also converges strongly in L2 (X, R2 ) to (u, 𝑣). Note that L(x, y, u, 𝑣, p, q) is neither normal nor strictly convex in (p, q), for it vanishes for p1 = aq2 and p2 = bq1 . However, Theorem 4.1.1 of [202] applies to the functional I(u, 𝑣), since L is actually independent of (x, y) and polynomial in (p, q). Lemma 7.2.4. If (u ν , 𝑣ν ) and (u, 𝑣) lie in H 1 (X, R2 ) and (u ν , 𝑣ν ) → (u, 𝑣) in L1 (K, R2 ) for each compact set K interior to X, then I(u, 𝑣) ≤ lim inf I(u ν , 𝑣ν ) . Proof. See Theorem 4.1.1 of [202], which refers to Theorem 12 of [242]. We have thus proved that if there is a bounded minimizing sequence {(u ν , 𝑣ν )}, and (u, 𝑣) is a weak limit point of this sequence in H 1 (X, R2 ), then (u, 𝑣) ∈ A and I(u, 𝑣) = m, i.e., (u, 𝑣) is a minimizer. It is clear that any minimizing sequence is bounded in H 1 (X, R2 ) if the functional I(u, 𝑣) majorizes the norm of (u, 𝑣) in A in the sense that ∫∫ L(x, y, u, 𝑣, ∇u, ∇𝑣) dxdy ≥ c ‖(u, 𝑣)‖2H 1 (X,R2 ) − Q , X

(7.25)

377

7.2 The Cauchy Problem for Chaplygin’s System |

for all (u, 𝑣) ∈ A, with c and Q constants independent of (u, 𝑣). This is obviously a farreaching generalization of A. Korn’s (1908) inequality for the case of nonlinear problems. It is the so-called (strong) quasiconvexity of L(x, y, u, 𝑣, p, q) in (p, q) that plays the important role in the lower semicontinuity and existence theory. Roughly speaking, it is equivalent to the lower semicontinuity of I(u, 𝑣) with respect to the Lipschitzian convergence on X, cf. Theorem 4.4.2 of [202]. Note that the Lipschitzian convergence implies weak convergence in W 1,p (X, R2 ) for each p ≥ 1, but it does not imply strong convergence in any W 1,p (X, R2 ). If L is of class C2 (which is the case in our study), then for L to be quasiconvex, it is necessary and sufficient that L satisfies the Legendre–Hadamard condition (7.24), cf. Theorem 4.4.1 [202].

7.2.5 Stable Cauchy Problems If the set S is different from the whole boundary of X, no a priori estimate (7.25) is possible. Example 7.2.5. Choose a = −b = 1, in which case the Chaplygin system reduces to ̄ = 0 for an unknown function f = u + ı𝑣 of the Cauchy–Riemann equation (∂/∂ z)f the complex variable z = x + ıy. The Cauchy problem (7.17) and (7.18) just amounts to the problem of analytic continuation of functions f0 = u 0 + ı𝑣0 given on S. The Korn inequality reads ̄ 2L2 (X) ≥ c ‖f‖2H 1 (X) − Q , ‖(∂/∂ z)f‖ for all f ∈ H 1 (X) satisfying f |S = f0 , where c and Q are constants that are independent of f . Were such an inequality fulfilled, there might exist other constants C and Q󸀠 , such that ̄ L2 (X) + ‖f‖H 1/2 (S) + Q󸀠 ) , ‖f‖H 1 (X) ≤ C (‖(∂/∂ z)f‖ for all f ∈ H 1 (X). Since the problem of analytic continuation from S is instable unless S coincides with the whole boundary, the Korn inequality holds only if S = ∂X in which case Q = ‖f0 ‖2H 1/2 (S) , as follows easily from the Cauchy–Pompey formula. We thus restrict our attention to the case S = ∂X. Recall that functions of H 1 (X) can be essentially unbounded. We assume that a and b are bounded functions of (u, 𝑣) ∈ R2 , since otherwise A(u, 𝑣) fails to map H 1 (X, R2 ) continuously to L2 (X, R2 ). It suffices to derive a Korn inequality (7.25) for the smooth functions u and 𝑣 in X, for (u, 𝑣) ∈ C∞ (X, R2 ) satisfying (7.18) lie densely in A. An easy computation shows that ‖A(u, 𝑣)‖2L2 (X,R2 ) = ∫∫ (u 󸀠x 2 + u 󸀠y 2 + b 2 𝑣󸀠x 2 + a2 𝑣󸀠y 2 − (a + b)(u 󸀠x 𝑣󸀠y + u 󸀠y 𝑣󸀠x )) dxdy X

− ∫∫(a − b)du ∧ d𝑣 , X

(7.26)

378 | 7 Nonlinear Cauchy Problems for Elliptic Equations for all u, 𝑣 ∈ C∞ (X). This equality is of key importance in our study of the Cauchy problem (7.17) and (7.18). The differential form (a−b)du∧d𝑣 is of maximal degree in the space R2 of variables (u, 𝑣). As a and b are smooth functions of (u, 𝑣), this form is actually exact, i.e., of the form d℘ with ℘ = ℘1 du + ℘2 d𝑣, where ℘1 and ℘2 are smooth functions of (u, 𝑣). By Stokes’ formula, the second integral on the right-hand side of (7.26) is equal to ∮ ℘(u, 𝑣) , ∂X

and so it depends on the restrictions of u and 𝑣 to ∂X only. Lemma 7.2.6. Suppose that 1 2 2 { {a − 4 (a + b) { { 1 2 2 {b − 4 (a + b)

>

0,

>

0

(7.27)

uniformly in (u, 𝑣) ∈ R2 . Then there is a constant c > 0 such that ∫∫ L(x, y, u, 𝑣, ∇u, ∇𝑣) dxdy ≥ c ‖(∇u, ∇𝑣)‖2L2 (X,R2 ) − ∮ ℘(u 0 , 𝑣0 ) , X

∂X

for all (u, 𝑣) ∈ A. Proof. Indeed, condition (7.27) just amounts to saying that the quadratic form p21 + p22 + b 2 q21 + a2 q22 − (a + b)(p1 q2 + p2 q1 ) is positive definite uniformly in (u, 𝑣) ∈ R2 . Hence, it follows that there is a constant c > 0, such that ∫∫(u 󸀠x 2 + u 󸀠y 2 + b 2 𝑣󸀠x 2 + a2 𝑣󸀠y 2 − (a + b)(u 󸀠x 𝑣󸀠y + u 󸀠y 𝑣󸀠x )) dxdy ≥ c ‖(∇u, ∇𝑣)‖2L2 (X,R2 ) , X

for all (u, 𝑣) ∈ C∞ (X, R2 ). Combining this estimate with (7.26) establishes the lemma.

Obviously, the condition (7.27) is fulfilled if the sum a + b is sufficiently small in all of R2 (cf. Fig. 7.1). Theorem 7.2.7. Assume that condition (7.27) is fulfilled. Then, for each data (u 0 , 𝑣0 ) ∈ H 1/2 (∂X, R2 ), the variational problem I(u, 𝑣) → min has a unique solution (u, 𝑣) ∈ A. Proof. This follows from Lemma 7.2.6 and what has been said in Section 7.2.4 (see [177] for more details).

7.2 The Cauchy Problem for Chaplygin’s System

| 379

Chaplygin with existence of solutions

b b=

− 13 a Holomorphy 1

a

Muskat–Meres

−1

b = −a b = −3a Fig. 7.1: The range of a and b.

Under the same hypotheses, for the Cauchy problem (7.17) and (7.18) to be solvable it is necessary and sufficient that the infimum of I(u, 𝑣) over A is zero. If it exists, the solution of the Cauchy problem can be constructed by the universal iterative method 1

1

∑ (−1)k ∆ k (u ν+1 , 𝑣ν+1 ) = ∑ (−1)k ∆ k (u ν , 𝑣ν ) − ε A(u ν , 𝑣ν ) , k=0

k=0

under the Dirichlet condition (u ν+1 , 𝑣ν+1 ) = (u 0 , 𝑣0 ) on ∂X, where ε > 0 is a small parameter, cf. [146].

7.2.6 Approximate Solutions We now consider the problem I(u, 𝑣) 󳨃→ min in the case where S is different from ∂X. As has been mentioned, in this case, no a priori estimate (7.25) is available. Hence, we are not able to prove the boundedness of any minimizing sequence {(u ν , 𝑣ν )}, so the arguments of Section 7.2.4 are not valid. To get over this difficulty, a general idea is to confine the set A of competing functions so that A is itself bounded. In linear analysis this idea goes back at least as far as [231]. While additional assumptions on A may depend on the concrete problem there is also an abstract prescription. Given any R > 0, we denote by AR the subset of H 1 (X, R2 ) consisting of all pairs (u, 𝑣) of real-valued functions u, 𝑣 ∈ H 1 (X), such that u = u 0 and 𝑣 = 𝑣0 on S and,

380 | 7 Nonlinear Cauchy Problems for Elliptic Equations

moreover, ‖(∇u, ∇𝑣)‖L2 (X,R2×2) ≤ R .

(7.28)

Clearly, AR ⊂ A and each (u, 𝑣) ∈ H 1 (X, R2 ) lies in some AR , i.e., the family AR exhausts A. Theorem 7.2.8. Let R > 0 and let a and b be bounded functions on R2 . Then, for each data (u 0 , 𝑣0 ) ∈ H 1/2 (S, R2 ), the functional I(u, 𝑣) takes on its infimum on AR . Proof. Pick a minimizing sequence {(u ν , 𝑣ν )} in AR . By definition, the sequence {(∇u ν , ∇𝑣ν )} is bounded in L2 (X, R2×2 ). By Lemma 1.9.2 of [202], there is a constant C that depends only on X and S, such that ‖(u, 𝑣)‖H 1 (X,R2) ≤ C (‖(∇u, ∇𝑣)‖L2 (X,R2×2) + ‖(u, 𝑣)‖L1 (S,R2 ) , )

(7.29)

for all (u, 𝑣) ∈ H 1 (X, R2 ). Combining (7.29) and (7.28) we readily deduce that the sequence {(u ν , 𝑣ν )} is actually bounded in the H 1 (X, R2 ) -norm. The bounded sets in H 1 (X, R2 ) are relatively compact with respect to weak convergence in H 1 (X, R2 ). Hence, there is a subsequence {(u ν ι , 𝑣ν ι )} of {(u ν , 𝑣ν )}, which tends weakly in H 1 (X, R2 ) to some function (u, 𝑣) ∈ H 1 (X, R2 ). Since AR is a convex closed subset of the Hilbert space H 1 (X, R2 ), it is, by Maur’s theorem cited above, weakly closed. Hence, we conclude that the limit function (u, 𝑣) belongs to the set AR . Moreover, Theorem 3.4.4 of [202] says that {(u ν ι , 𝑣ν ι )} also converges strongly in L2 (X, R2 ) to (u, 𝑣). Now the theorem follows from the lower semicontinuity of I(u, 𝑣), as stated by Lemma 7.2.4.

One question that is still unanswered is whether the minimum point of I(u, 𝑣) in AR is unique. However, this topic lies outside the scope of this work. Remark 7.2.9. The results of this section extend to the case where the coefficients a and b depend not only on u and 𝑣 but also on x and y. This is important in practice.

7.3 Hyperbolic Formulas in Elliptic Cauchy Problems We are studying the Cauchy problem for the Laplace equation in a cylindrical domain with data on a part of its boundary that is a cross-section of the cylinder. On reducing the problem to the Cauchy problem for the wave equation in a complex domain and using hyperbolic theory we obtain explicit formulas for the solution, thus developing the classical approach of Hans Lewy (1927).

7.3 Hyperbolic Formulas in Elliptic Cauchy Problems

|

381

A Short Survey of Results The question of the well-posedness of the Cauchy problem was first raised by Hadamard, who proved in [106] that it is ill posed in the case of linear second-order elliptic equations. Hadamard’s proof is based on the analytic regularity of linear boundary value problems. This regularity was extended to nonlinear elliptic equations in [201], so that Hadamard’s argument also applies to general nonlinear elliptic equations. Hadamard also pointed out in [106] that the problem occurring in wave propagation is not at all an analytic problem, but a problem with real, not necessarily analytic, data. For general linear equations it is well known that hyperbolicity is a necessary condition for the well-posedness of the noncharacteristic Cauchy problem in C∞ , that is for the existence of solutions for general C∞ data, cf. [164, 199]. Moreover, for several classes of nonhyperbolic equations, explicit conditions on the initial data necessary for the existence of solutions were given in [211]. For nonlinear equations, [289] proves that the existence of a smooth stable solution implies hyperbolicity, stability meaning that one can perturb the initial data and the source terms in the equations. The nonlinear theory yields difficult new problems, see [109, 194], etc. There are many interesting examples, for instance in multiphase fluid dynamics, where the equations are not hyperbolic everywhere. As but one occurrence of this phenomenon, we consider Euler’s equations of gas dynamics in Lagrangian coordinates, {∂ t u + ∂ x 𝑣 { ∂ p(u) + ∂ t 𝑣 { x

=

0,

=

0,

(7.30)

which are mentioned in [194]. The system is hyperbolic when p󸀠 (u) > 0 and elliptic when p󸀠 (u) < 0. For van der Waals state laws, it happens that p is decreasing on an interval [u ∗ , u ∗ ]. A mathematical example is p(u) = u(u 2 − 1). Hadamard’s argument shows that the Cauchy problem with data taking values in the elliptic region is ill posed. If u(0, x) = u 0 (x) is real analytic near x, and u 0 (x) belongs to the elliptic interval, then any local C1 solution is analytic, see, e.g., [201]. Thus, the initial data u 0 (x) must actually be analytic for the initial value problem to have a solution. It was Hans Lewy who first used hyperbolic techniques to study problems for elliptic equations, cf. [170]. The solutions of elliptic equations with real analytic coefficients prove to be real analytic, and so they extend to holomorphic functions in a complex neighborhood of their domain. For a holomorphic function obtained this way, the derivative ∂/∂x k just amounts to the derivative ∂/∂(ıy k ), where z k = x k + ıy k are complex variables with k = 1, . . . , n. One can go to a complex space in only one variable, say x n , and the change ∂/∂x n 󳨃→ −i∂/∂y n leads to a drastic modification of the characteristic variety. The Laplace equation written in the coordinates (x󸀠 , x n ) with x󸀠 = (x1 , . . . , x n−1 ) transforms to the wave equation in the coordinates (x󸀠 , y n ). This idea is especially useful in the study of the Cauchy problem for elliptic equations. This problem is overdetermined even in the case of data given on an open part

382 | 7 Nonlinear Cauchy Problems for Elliptic Equations

of the boundary, hence it does not admit any simple formulas for solutions (see, however, [296] and [248]). Since the problem is unstable, the left inverse operator fails to be continuous. On the other hand, the Cauchy problem for hyperbolic equations is of textbook character and it admits many explicit formulas for solutions like d’Alembert, Kirchhoff, Poisson, etc., formulas, cf. [106]. An outstanding contribution to the Cauchy problem for hyperbolic equations is due to Leray, who developed multidimensional residue theory in complex analysis to handle the problem, see, e.g., [166, 167]. Having granted a solution u(x󸀠 , ıy n ) of the Cauchy problem for a hyperbolic equation, how can one restore the solution u(x󸀠 , x n ) of the Cauchy problem for the original elliptic equation? The simple substitution ıy n 󳨃→ x n does not make sense in general. For this purpose, we invoke a formula of [38], which restores the values of holomorphic functions in a corner on the diagonal through their values on an arc connecting to faces of the corner. The resulting formula for the solution of an elliptic Cauchy problem includes a limit passage and agrees perfectly with the general observation that the character of instability in an elliptic Cauchy problem is similar to that in the problem of analytic continuation, cf. [269]. As has been mentioned, the idea to use hyperbolic formulas for elliptic Cauchy problems goes back at least as far as [170]. In the 1960s it was directly applied in a number of papers by Krylov, see, for instance, [152]. In [152], an integral representation for holomorphic solutions of a partial differential equation in a complex domain is constructed through the Cauchy data of solutions on an analytic surface. However, the formula does not manifest any instability of the Cauchy problem, which shows its local character. The approach that we develop in this section has the advantage of providing a large parameter to perturb the solution of the problem. This might give rise to a calculus of Cauchy problems for elliptic equations. Since these problems are unstable, no operator calculus similar to that including elliptic boundary values problems and their parametrices on compact manifolds with boundary is possible. On introducing a large parameter into operators we are able to describe their perturbations leading to solutions. Let us dwell on the contents of the section. In Section 7.3.1 we formulate the Cauchy problem for a second-order elliptic equation in a domain X in Rn . The principal part of the equation is given by the Laplace operator, while the lower-order part may include nonlinear terms. The Cauchy data are given on a nonempty open set S of the boundary. Our standing assumption is that X is a cylinder over a bounded domain B with smooth boundary in the space Rn−1 of variables x󸀠 , and S is a smooth cross-section of X. In Section 7.3.2, we reformulate the same Cauchy problem for a hyperbolic equation. Namely, we assume that the solution u(x󸀠 , x n ) is a real analytic function of x n ∈ (b(x󸀠 ), t(x󸀠 )) for each fixed x󸀠 ∈ B. Then, it extends to a function u(x󸀠 , z n ) that is holomorphic in a narrow strip −ε < y n < ε around the interval (b(x󸀠 ), t(x󸀠 )) in the plane of the complex variable z n = x n + ıy n . The Cauchy–Riemann equations force u(x󸀠 , z n ) to fulfill (∂/∂x n )u = −ı(∂/∂y n )u in the strip (b(x󸀠 ), t(x󸀠 )) × (−ε, ε). Hence, we rewrite

7.3 Hyperbolic Formulas in Elliptic Cauchy Problems |

383

the original elliptic equation as a hyperbolic equation for a new unknown function of variables (x󸀠 , y n ). Since S is the graph of some smooth function x n = t(x󸀠 ) on B, the Cauchy data transform easily for the new unknown function. In Section 7.3.3, we test our approach in the case of two variables. It is precisely the case treated in [170], and the approach of [170] does not work for n > 2. For n = 2, the geometric picture is especially descriptive because the complexification of x2 does not lead beyond R3 . On solving the Cauchy problem for a hyperbolic equation in a conical domain in the space of variables (x󸀠 , y n ), we are left with the task of continuing the solution given on the base of an isosceles triangle analytically along the bisectrix of the angle at the vertex, for each fixed x󸀠 ∈ B. To this end, we invoke the classical formula of Carleman, established precisely for this configuration, see [38]. Of course, the use of Carleman’s formula is justified only for real analytic solutions of the original elliptic Cauchy problem. In Section 7.3.4, we give a simple proof of this formula. Numerical simulations with Carleman’s formula failed to manifest its striking efficiency. However, nowadays more efficient formulas of analytic continuation are available, cf. [5]. In Section 7.3.5, we investigate the Cauchy problem for the inhomogeneous Laplace equation in the space Rn of variables (x󸀠 , x n ) with odd n. As is shown in Section 7.3.2, it reduces to the Cauchy problem for the inhomogeneous wave equation in the space of variables (x󸀠 , y n ). The case n = 1 deserves special study, for it concerns the initial problem for ordinary differential equations. If n = 3, the Cauchy problem for the wave equation possesses a very explicit solution constructed by Poisson. For odd n ≥ 5 an explicit solution formula was derived by Hadamard in [106] by his method of descent. On substituting it into Carleman’s formula and changing integrations over y n and x󸀠 , we get a formula for solutions of the Cauchy problem for harmonic functions. In Section 7.3.6, we restrict our attention to the Cauchy problem for the inhomogeneous Laplace equation in the space Rn of variables (x󸀠 , x n ) with even n. By the above, it reduces to the Cauchy problem for the inhomogeneous wave equation in the space of variables (x󸀠 , y n ). The latter Cauchy problem admits a very explicit solution formula due to d’Alembert in the case n = 2 and due to Kirchhoff in the case n = 4. For general, even n the formula seems to have been first published in [106]. We combine it with Carleman’s formula and change the integration over y n and over x󸀠 . This yields an explicit formula for solutions of the Cauchy problem for the inhomogeneous Laplace equation. To the best of our knowledge, this formula has never been published. In Section 7.3.7, we analyze whether our approach applies to Cauchy problems for elliptic equations of order different from two. Yet another question under study is whether the method of quenching functions in the Cauchy problem for the Laplace equation presented in [296] is actually a very particular case of formulas elaborated in this work.

384 | 7 Nonlinear Cauchy Problems for Elliptic Equations xn S

x n = t(x󸀠 ) x n = b(x󸀠 ) B x󸀠

Fig. 7.2: A typical domain under consideration.

7.3.1 The Cauchy Problem Let X be a bounded domain with piecewise smooth boundary in Rn . We require X to be of cylindrical form, i.e., X is a part of the cylinder B×R intercepted by two surfaces y n = b(x󸀠 ) and y n = t(x󸀠 ) over B, where B is a bounded domain with smooth boundary in the space Rn−1 of variables x󸀠 = (x1 , . . . , x n−1 ). For simplicity, we assume that t(x󸀠 ) > b(x󸀠 ) for all x󸀠 ∈ B, the case t(x󸀠 ) = b(x󸀠 ) for some or all x󸀠 ∈ ∂B is not excluded. The Cauchy data will be posed on the top surface S := {(x󸀠 , t(x󸀠 )) : x󸀠 ∈ B}, which is tacitly assumed to be real analytic, cf. Fig. 7.2. For an elliptic second-order differential operator on the closure of X, the Cauchy data on S look like {u = u 0 on S , { ∂u = u 1 on S , { ∂ν where ν is the outward unit normal vector at S. Obviously, ν = ∇ϱ/|∇ϱ| where ϱ = x n − t(x󸀠 ). Lemma 7.3.1. If u is a smooth function near S satisfying u = u 0 on S, then 1 ∂u ∂u = (−⟨∇x󸀠 t, ∇x󸀠 u 0 ⟩ + ) ∂ν √|∇x󸀠 t|2 + 1 ∂x n on S. Proof. This is an easy exercise. Consider the nonlinear second-order partial differential equation ∆u = f(x, u, ∇u) in X, where f(x, u, p) is a real analytic function on X × R × Rn . By Lemma 7.3.1, the Cauchy problem for solutions of this equation with data on S can be formulated in the following way. Given functions u 0 and u 1 on S, find a function u in X smooth up to S that satisfies ∆u = f(x, u, ∇u) in X , { { { (7.31) u = u0 on S , { { { 󸀠 on S . { u xn = u1

7.3 Hyperbolic Formulas in Elliptic Cauchy Problems |

385

Lemma 7.3.2. There is at most one real analytic function u in X ∪ S, which is a solution of (7.31). Proof. Let u 1 and u 2 be two real analytic functions in X ∪ S satisfying (7.31). Set u = u 1 − u 2 , then u is real analytic in X ∪ S and vanishes up to the order 2 on S. Hence, it follows that ∆u = f(x, u 1 , ∇u 1 ) − f(x, u 2 , ∇u 2 ) vanishes on S. Since ∆ is a second-order elliptic operator, we readily deduce that u 󸀠󸀠x n x n = 0 on S, and so u vanishes up to order 3 on S. Hence, it follows that ∆u vanishes up to order 2 on S, and so (∂/∂x n )3 u = 0 on S. Arguing in this way, we conclude that u vanishes up to the infinite order on S. Since u is real analytic in X ∪ S, we get u ≡ 0 in X, as desired.

7.3.2 Hyperbolic Reduction Assume that u is a real analytic function in X∪S that satisfies (7.1). Then, for each fixed x󸀠 ∈ B, the function u(x󸀠 , x n ) can be extended to a holomorphic function u(x󸀠 , x n +ıy n ) in some complex neighborhood of the interval (b(x󸀠 ), t(x󸀠 )]. Without loss of generality, we can assume that this neighborhood is a triangle T(x󸀠 ) in the complex plane z n = x n + ıy n with vertexes at b(x󸀠 ) and t(x󸀠 ) ∓ ıε, where ε > 0 depends on x󸀠 . We write U(x󸀠 , x n , y n ) for the extended function, so that u(x) just amounts to U(x󸀠 , x n , 0). Since u(x󸀠 , z n ) is holomorphic in a complex neighborhood of (b(x󸀠 ), t(x󸀠 )], it follows from the Cauchy–Riemann equations that (

∂ j ∂ j ) U(x󸀠 , x n , y n ) = (−ı ) U(x󸀠 , x n , y n ) , ∂x n ∂y n

for all j = 1, 2, . . . . Therefore, the Cauchy problem (7.31) for u transforms to the problem ∆ x󸀠 U − U y󸀠󸀠n y n = f(x󸀠 , z n , U, ∇x󸀠 U, −ıU y󸀠 n ), { { { U(x󸀠 , x n , 0) = u 0 (x󸀠 , z n ), { { { 󸀠 󸀠 󸀠 { U y n (x , x n , 0) = ı u 1 (x , z n ),

if

x󸀠 ∈ B , z n ∈ T(x󸀠 ) ,

if

x󸀠 ∈ B , z n = t(x󸀠 ) ,

if

x󸀠

∈ B , zn =

t(x󸀠 )

(7.32)

,

relative to the new unknown function U(x󸀠 , x n , y n ). Equation (7.32) can hardly be specified within Cauchy problems for second-order differential equations, for the number of independent variables is n + 1, while the Cauchy data are given on a surface of dimension n − 1. Since the differential equation in (7.32) does not contain the derivative U x󸀠 n , it is easy to deduce that the smooth solution to this problem is by no means unique. This no longer holds true for the holomorphic solution because of uniqueness theorems for holomorphic functions. Moreover, if U(x󸀠 , x n , y n ) is holomorphic in z n = x n + ıy n , then the differential equation in (7.32) is satisfied for all x󸀠 ∈ B and z n ∈ T(x󸀠 ), provided that it is fulfilled for all x󸀠 ∈ B and z n = t(x󸀠 ) + ıy n with |y n | < ε.

386 | 7 Nonlinear Cauchy Problems for Elliptic Equations

Thus, when one looks for a holomorphic solution to (7.32), this problem actually reduces to the Cauchy problem for a quasilinear hyperbolic equation in the space of variables (x󸀠 , y n ), whose principal part is given by the wave operator. More precisely, U y󸀠󸀠n y n { { { { { { U(x󸀠 , x , 0) { { 󸀠 󸀠 n { U y n (x , x n , 0)

x󸀠 ∈ B , } } } |y n | < ε(x󸀠 ) , } 󸀠 󸀠 } = u 0 (x , x n ), if x ∈B, } } } 󸀠 = ı u 1 (x , x n ), if x󸀠 ∈ B , } (7.33) where the variable x n is thought of as a parameter that runs over the interval (b(x󸀠 ), t(x󸀠 )). We are actually interested in the solution of this problem corresponding to the special choice x n = t(x󸀠 ) of the parameter. In other words, we study problem (7.33) on the hypersurface x n = t(x󸀠 ) in the space of variables (x, y n ), the Cauchy data being given on the intersection of the hypersurface with the hyperplane {y n = 0}. When passing to the Cauchy problem on the hypersurface x n = t(x󸀠 ) in Rn+1 , one should interpret equations (7.33) adequately in accordance with the presence of parameter x n . Namely, each equation has to be fulfilled together with all derivatives in x n on x n = t(x󸀠 ). =

∆ x󸀠 U − f(x󸀠 , x n + ıy n , U, ∇x󸀠 U, −ıU y󸀠 n ),

if

Lemma 7.3.3. There is at most one function U(x󸀠 , x n , y n ) in a neighborhood of S, which is real analytic in y n at y n = 0 and satisfies (7.33) with x n = t(x󸀠 ). Proof. Let U1 and U2 be two functions in a neighborhood of S, which are real analytic in y n at y n = 0 and satisfy (7.33) with x n = t(x󸀠 ). In the coordinates (x󸀠 , x n , y n ), the surface S is given as the intersection of two hypersurfaces x n = t(x󸀠 ), where x󸀠 ∈ B and y n = 0. Set U = U1 − U2 , then U is real analytic in y n at y n = 0. We shall have established the lemma if we prove that each derivative (∂/∂y n )j U with j = 0, 1, . . . vanishes for x n = t(x󸀠 ) and y n = 0. For j = 0, 1, this follows immediately from the conditions that U1 and U2 fulfil on S. For j ≤ 2, this follows from the differential equation in (7.33) by induction. We check it only for the initial value j = 2, for the induction step is verified in much the same way. From (7.33) we get U y󸀠󸀠n y n = ∆ x󸀠 U1 − ∆ x󸀠 U2 󸀠 󸀠 − (f(x󸀠 , x n + ıy n , U1 , ∇x󸀠 U1 , −ıU1,y ) − f(x󸀠 , x n + ıy n , U2 , ∇x󸀠 U2 , −ıU2,y )) , n n

provided that x n = t(x󸀠 ). Since (∂/∂x n )j (U1 − U2 ) = 0 for x n = t(x󸀠 ), y n = 0, and all j = 0, 1, . . ., it follows that 󸀠

󸀠 U1,x (x󸀠 , t(x󸀠 ), 0) = (U1 (x󸀠 , t(x󸀠 ), 0)) x − U1,x n (x󸀠 , t(x󸀠 ), 0) t󸀠x k (x󸀠 ) k k

󸀠

󸀠

󸀠

= (U2 (x , t(x ), 0)) x − U2,x n (x󸀠 , t(x󸀠 ), 0) t󸀠x k (x󸀠 ) k

󸀠 = U2,x (x󸀠 , t(x󸀠 ), 0) , k

for each k = 1, . . . , n − 1. Moreover, we get 󸀠

󸀠

∂ αx󸀠 U1 = ∂ αx󸀠 U2

(7.34)

7.3 Hyperbolic Formulas in Elliptic Cauchy Problems

|

387

on the surface x n = t(x󸀠 ), y n = 0 for all multi-indices α 󸀠 = (α 1 , . . . , α n−1 ). This readily yields ∆ x󸀠 U1 = ∆ x󸀠 U2 for x n = t(x󸀠 ) and y n = 0. Substituting these equalities into the formula for U y󸀠󸀠n y n , we obtain U y󸀠󸀠n y n (x󸀠 , t(x󸀠 ), 0) = 0 for all x󸀠 ∈ B, as desired. Note that equalities (7.34) generalize to ∂ αx ∂ y nn+1 U1 = ∂ αx ∂ y nn+1 U2 for x n = t(x󸀠 ), y n = 0, and all multi-indices α = (α 1 , . . . , α n ) and α n+1 = 0, 1, . . ., as is easy to check. We have thus reduced the Cauchy problem for the Laplace equation perturbed by nonlinear terms of order ≤ 1 to the Cauchy problem for the wave equation perturbed in the same way. The reduction is justified as long as the solution under study is real analytic in x n . Perhaps the reduction does not make sense in the case n = 1, for it does not leads to any simplification. α

α

7.3.3 The Planar Case To test the hyperbolic reduction of Section 7.3.2, we consider the case n = 2 in detail, assuming f to depend on x ∈ X ∪ S only. Let X be a strip domain in R2 consisting of all x = (x1 , x2 ), such that x1 ∈ (a, b) and b(x1 ) < x2 < t(x1 ), where (a, b) is a bounded interval in R and b, t are smooth functions of x1 ∈ (a, b). Write B := (a, b) and denote by S the curve {(x1 , t(x1 )) : x1 ∈ (a, b)} which is a part of ∂X. We focus on the Cauchy problem for the inhomogeneous Laplace equation given by (7.31). When looking for a solution u of this problem that extends to a holomorphic function u(x1 , z2 ) of z2 = x2 + ıy2 in a neighborhood of {(x2 , 0) : x2 ∈ (b(x1 ), t(x1 )]}, for each fixed x1 ∈ (a, b), we arrive at U y󸀠󸀠2 y2 { { { { { { U(x , x , 0) { { 󸀠 1 2 {U y2 (x1 , x2 , 0)

=

U x󸀠󸀠1 x1 − f(x1 , x2 + ıy2 ),

if

= =

u 0 (x1 , x2 ), ı u 1 (x1 , x2 ),

if if

x1 ∈ (a, b) , } } } |y2 | < ε(x1 ) , } } x1 ∈ (a, b) , } } } x1 ∈ (a, b) , }

(7.35)

which is a Cauchy problem for the inhomogeneous wave equation with parameter x2 relative to the unknown function U(x1 , x2 , y2 ) = u(x1 , x2 + ıy2 ), cf. (7.33). We are actually interested in finding a function U that satisfies (7.35) only on the surface x2 = t(x1 ), see Fig. 7.3. It is an easy exercise to verify that the function y2

x 1 +y󸀠2

0

x 1 −y󸀠2

1 (Gf)(x1 , x2 , y2 ) = − ∫ dy󸀠2 ∫ f(x󸀠1 , x2 + ı(y2 − y󸀠2 )) dx󸀠1 2 satisfies the inhomogeneous wave equation and homogeneous (i.e., corresponding to u 0 = u 1 = 0) initial conditions in (7.35). On the other hand, d’Alembert’s formula gives a function satisfying the homogeneous (i.e., corresponding to f = 0) wave equation

388 | 7 Nonlinear Cauchy Problems for Elliptic Equations

S

x2

x2 = t(x1 ) X

y2

a

x2 = b(x1 ) x1

Fig. 7.3: The case n = 2.

b

and the inhomogeneous initial conditions in (7.35), see [50, Ch. I, Section 7.1]. In fact, this is x 1 +y2

u 0 (x1 + y2 , x2 ) + u 0 (x1 − y2 , x2 ) ı ∫ u 1 (x󸀠1 , x2 )dx󸀠1 , + P(u 0 , u 1 )(x1 , x2 , y2 ) = 2 2 x 1 −y2

(7.36) where the right-hand side is well defined for all (x1 , x2 , y2 ) satisfying x1 + y2 ∈ (a, b) and x1 − y2 ∈ (a, b). The pairs (x1 , y2 ) with this property form two cones C± in the plane, C± being the set of all (x1 , y2 ), such that x1 ∈ (a, b) and ±y2 ∈ [0, ε(x1 )), where b − a 󵄨󵄨󵄨󵄨 a + b 󵄨󵄨󵄨󵄨 ε(x1 ) = − 󵄨󵄨x1 − 󵄨 . 2 2 󵄨󵄨󵄨 󵄨󵄨 Thus, given any twice differentiable function u 0 (x1 , x2 ), the differentiable function u 1 (x1 , x2 ) of x1 ∈ (a, b) and any differentiable function f(x1 , z2 ) of both variables, the formula U = Gf + P(u 0 , u 1 ) yields a solution to the Cauchy problem (7.35) for all values of parameter x2 that do not lead beyond the domains of u 0 , u 1 , and f . If we know u 0 (x1 , x2 ) and u 1 (x1 , x2 ) for all values x2 ∈ (b(x1 ), t(x1 )], then the first initial condition of (7.35) would give U(x1 , x2 , 0) = u 0 (x1 , x2 ), and so the solution to the Cauchy problem (7.31) by u(x) = u 0 (x1 , x2 ). This just recovers the reduction but is not of use to solve the original Cauchy problem. However, on substituting x2 = t(x1 ) into U(x1 , x2 , y2 ), we obtain y2

x 1 +y󸀠2

0

x 1 −y󸀠2

1 u(x1 , t(x1 ) + ıy2 ) = − ∫ dy󸀠2 ∫ f(x󸀠1 , t(x1 ) + ı(y2 − y󸀠2 )) dx󸀠1 2 +

u 0 (x1 + y2 , t(x1 )) + u 0 (x1 − y2 , t(x1 )) 2 x 1 +y2

ı + ∫ u 1 (x󸀠1 , t(x1 ))dx󸀠1 , 2

(7.37)

x 1 −y2

for all x1 ∈ (a, b) and |y2 | < ε(x1 ). Note that (x󸀠1 , t(x1 )) fails to lie on the curve S for all x󸀠1 ∈ [x1 − y2 , x1 + y2 ] unless t(x1 ) is constant. Therefore, u(x1 , t(x1 ) + ıy2 ) is

7.3 Hyperbolic Formulas in Elliptic Cauchy Problems |

389

determined by the Cauchy data of u in some neighborhood of S. This forces us once again to confine ourselves to solutions that are real analytic in the variable x2 . For fixed x1 ∈ (a, b), the formula (7.37) gives the restriction of the function u(x1 , z2 ), holomorphic in z2 in the triangle with vertexes at b(x1 ) and t(x1 ) ∓ ıε(x1 ), to the side t(x1 ) + ı[−ε(x1 ), ε(x1 )] of the triangle. This limits the application of hyperbolic theory. Our next objective is to continue the function from the side of the triangle analytically along the bisectrix of the angle at b(x1 ). This is a problem of analytic continuation.

7.3.4 The Carleman Formula Let D be a domain in the complex plane C of variable z bounded by lines BO and OA and by a smooth curve c = AB lying inside the angle BOA. Write ∠BOA = απ with 0 < α < 2. α in the complex plane Choose the univalent branch of the analytic function √w with a slit along the ray arg w = π, which takes the value 1 at w = 1. Lemma 7.3.4. If u is a holomorphic function in D continuous up to the boundary, then 1 ζ − ζ0 1/α dζ ) − 1) ∫ u(ζ) exp N (( N→∞ 2πı z − ζ0 ζ −z

u(z) = lim

c

holds for any point z ∈ D on the bisectrix of the angle BOA, where ζ0 is a complex number corresponding to the vertex O of the angle. This formula is due to Carleman [38]. To the best of our knowledge, it was the first formula of analytic continuation using the idea of a quenching function. Since then, such formulas in complex analysis and elliptic theory are called Carleman formulas, see [5, 269]. Proof. Fix any z ∈ D lying on the bisectrix of the angle BOA. For N = 1, 2, . . ., we apply the Cauchy integral formula to the function u(ζ) exp N ((

ζ − ζ0 1/α ) − 1) , z − ζ0

which is holomorphic in D and continuous in the closure of D. Since its value at ζ = z is u(z), we get u(z) =

ζ − ζ0 1/α dζ 1 ) − 1) ∫ u(ζ) exp N (( 2πı z − ζ0 ζ −z c

+

1 ζ − ζ0 1/α dζ . ) − 1) ∫ u(ζ) exp N (( 2πı z − ζ0 ζ −z ∂D\c

(7.38)

390 | 7 Nonlinear Cauchy Problems for Elliptic Equations If ζ ∈ ∂D \ c, then (

ζ − ζ0 1/α 󵄨󵄨󵄨󵄨 ζ − ζ0 󵄨󵄨󵄨󵄨1/α π = 󵄨󵄨 ) 󵄨󵄨 exp (± ı) 󵄨 󵄨 z − ζ0 2 󵄨 z − ζ0 󵄨 󵄨󵄨󵄨 ζ − ζ0 󵄨󵄨󵄨1/α 󵄨󵄨 ı , = ± 󵄨󵄨󵄨 󵄨󵄨 z − ζ0 󵄨󵄨󵄨

and so the modulus of exp N(((ζ − ζ0 )/(z − ζ0 ))1/α − 1) equals e−N . Letting N → ∞ in (7.38) establishes the lemma. Having disposed of this preliminary step, we now turn to the problem of analytic continuation that we encountered in Section 7.3.3. We apply Lemma 7.3.4 in the plane of the complex variable z2 = x2 + ıy2 . Given any fixed x1 ∈ (a, b), we take the triangle T(x1 ) with vertexes O := b(x1 ) and A := t(x1 ) − ıε(x1 ), B := t(x1 ) + ıε(x1 ) as D, cf. Fig. 7.4. yn d(x󸀠 , ∂B) b(x󸀠 )

t(x󸀠 )

−d(x󸀠 , ∂B)

xn Fig. 7.4: Recovering a holomorphic function.

In this case, α=

2 ε(x1 ) arctan ( ) π t(x1 ) − b(x1 )

depends on x1 and the bisectrix of the angle BOA coincides with the real axis. The solution u(x1 , z2 ) is given on the edge AB, and we aim to reconstruct it in the interval (b(x1 ), t(x1 )). Theorem 7.3.5. Let n = 2. For each solution u of the Cauchy problem (7.31) in X that is real analytic up to S, the formula ε(x 1 )

1

1 t(x1 ) − b(x1 ) + ıy2 α dy2 ∫ U(x1 , t(x1 ), y2 ) exp N (( ) − 1) N→∞ 2π x2 − b(x1 ) t(x1 ) − x2 + ıy2

u(x) = lim

−ε(x 1 )

holds for all x ∈ X. Proof. This follows immediately from Lemma 7.3.4 and formula (7.37), giving an explicit continuation of the solution u(x1 , x2 ) along S to the plane of the complex variable z2 = x2 + ıy2 .

7.3 Hyperbolic Formulas in Elliptic Cauchy Problems |

391

This formula is especially simple if S is a segment x2 = t0 , i.e., the graph of a constant function t(x1 ) ≡ t0 of x1 ∈ (a, b). If, moreover, f ≡ 0, then the formula (7.37) transforms to x 1 +y2

U(x1 , t0 , y2 ) =

u 0 (x1 + y2 , t0 ) + u 0 (x1 − y2 , t0 ) ı ∫ u 1 (x󸀠1 , t0 )dx󸀠1 , + 2 2 x 1 −y2

for all x1 ∈ (a, b) and |y2 | < ε(x1 ). Substituting this into the formula of Theorem 7.3.5, we get x 1 +ε(x 1 ) N→∞

u(x󸀠1 , t0 ) ℜ K N (x1 , x2 , x1 − x󸀠1 ) dx󸀠1

∫

u(x) = lim

x 1 −ε(x 1 ) x 1 +ε(x 1 )

− lim

N→∞

∫ x 1 −ε(x 1 )

ε(x 1 )

∂u 󸀠 (x , t0 ) ( ∫ ℑ K N (x1 , x2 , y2 ) dy2 ) dx󸀠1 , ∂x2 1

(7.39)

|x 󸀠1 −x 1 |

where 󸀠

K N (x󸀠 , x n , y n ) =

1 2π

󸀠

1 α

)+ıy n exp N (( t(x x)−b(x ) − 1) 󸀠 n −b(x )

t(x󸀠 ) − x n + ıy n

.

Formula (7.39) can be regarded as an elliptic analogue of the d’Alembert formula for the wave equation. Note that, nowadays, there are many explicit formulas of analytic continuation that are simpler than the original formula of [38]. We refer the reader to [5].

7.3.5 Poisson’s Formula In this section, we discuss the case n = 3 in detail, assuming the function f to depend on x ∈ X ∪ S only. The Cauchy problems for the inhomogeneous Laplace equation reduces to the Cauchy problem for the inhomogeneous wave equation. This latter reads as U y󸀠󸀠3 y3 = ∆ x󸀠 − f(x󸀠 , x3 + ıy3 ), if x󸀠 ∈ B , } { { } { { } |y3 | < ε(x󸀠 ) , } (7.40) { 󸀠 , x , 0) 󸀠 , x ), 󸀠 ∈ B, } { } = u (x if x U(x 0 3 { } { 󸀠 󸀠 3 } ı u 1 (x󸀠 , x3 ), if x󸀠 ∈ B , } {U y3 (x , x3 , 0) = with x3 being thought of as a parameter. We aim to find a function U that fulfills (7.40) on the surface x3 = t(x󸀠 ). The advantage of the reduction lies in the fact that the Cauchy problem for hyperbolic equations is well posed in the class of smooth functions. For n = 3, there is an explicit formula for its solution due to Poisson, see [50, Ch. III, Section 6.5]. More

392 | 7 Nonlinear Cauchy Problems for Elliptic Equations

precisely, y3

U(x󸀠 , x3 , y3 ) = −

1 ∫ dy󸀠3 2π 0

+ +

√y󸀠3 2

|x 󸀠󸀠 −x 󸀠 |