Elliptic Pdes, Measures and Capacities: From the Poisson Equation to Nonlinear Thomas-fermi Problems 3037191406, 978-3-03719-140-8

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Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet, one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PDEs and GMT in a simple and elegant way by investigating properties like existence and regularity of solutions of linear and nonlinear elliptic PDEs. Inspired by a variety of sources, from the pioneer balayage scheme of Poincaré to more recent results related to the Thomas–Fermi and the Chern–Simons models, the problems covered in this book follow an original presentation, intended to emphasize the main ideas in the proofs. Classical techniques like regularity theory, maximum principles and the method of sub and supersolutions are adapted to the setting where merely integrability or density assumptions on the data are available. The distinguished role played by capacities and precise representatives is also explained.

ISBN 978-3-03719-140-8

www.ems-ph.org

Ponce | Tracts in Mathematics 23 | Fonts Nuri /Helvetica Neue | Farben Pantone 116 / Pantone 287 | RB 40 mm

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OGR H

This book invites the reader to a trip through modern techniques in the frontier of elliptic PDEs and GMT, and is addressed to graduate students and researchers having some deep interest in analysis. Most of the chapters can be read independently, and only basic knowledge of measure theory, functional analysis and Sobolev spaces is required.

From the Poisson Equation to Nonlinear Thomas–Fermi Problems

AP

Other special features are: • the remarkable equivalence between Sobolev capacities and Hausdorff contents in terms of trace inequalities; • the strong approximation of measures in terms of capacities or densities, normally absent from GMT books; • the rescue of the strong maximum principle for the Schrödinger operator involving singular potentials.

Elliptic PDEs, Measures and Capacities

Augusto C. Ponce

MO

Elliptic PDEs, Measures and Capacities

Tr a c ts i n M a t h e m a t i c s 2 3

Elliptic PDEs, Measures and Capacities

Augusto C. Ponce

Augusto C. Ponce

Tr a c ts i n M a t h e m a t i c s 2 3

WARD

EMS Tracts in Mathematics 23

EMS Tracts in Mathematics Editorial Board: Michael Farber (Queen Mary University of London, UK) Carlos E. Kenig (The University of Chicago, USA) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. For a complete listing see our homepage at www.ems-ph.org. 8 Sergio Albeverio et al., The Statistical Mechanics of Quantum Lattice Systems 9 Gebhard Böckle and Richard Pink, Cohomological Theory of Crystals over Function Fields 10 Vladimir Turaev, Homotopy Quantum Field Theory 11 Hans Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration 12 Erich Novak and Henryk Woz´niakowski, Tractability of Multivariate Problems. Volume II: Standard Information for Functionals 13 Laurent Bessières et al., Geometrisation of 3-Manifolds 14 Steffen Börm, Efficient Numerical Methods for Non-local Operators. 2-Matrix Compression, Algorithms and Analysis 15 Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian Algebraic Topology. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids 16 Marek Janicki and Peter Pflug, Separately Analytical Functions 17 Anders Björn and Jana Björn, Nonlinear Potential Theory on Metric Spaces 18 Erich Novak and Henryk Woz´niakowski, Tractability of Multivariate Problems. Volume III: Standard Information for Operators 19 Bogdan Bojarski, Vladimir Gutlyanskii, Olli Martio and Vladimir Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane 20 Hans Triebel, Local Function Spaces, Heat and Navier–Stokes Equations 21 Kaspar Nipp and Daniel Stoffer, Invariant Manifolds in Discrete and Continuous Dynamical Systems 22 Patrick Dehornoy with François Digne, Eddy Godelle, Daan Kramer and Jean Michel, Foundations of Garside Theory 24 Hans Triebel, Hybrid Function Spaces, Heat and Navier-Stokes Equations 25 Yves Cornulier and Pierre de la Harpe, Metric Geometry of Locally Compact Groups

Augusto C. Ponce

Elliptic PDEs, Measures and Capacities From the Poisson Equation to Nonlinear Thomas–Fermi Problems

Author: Augusto C. Ponce Université catholique de Louvain Institut de Recherche en Mathématique et Physique Chemin du cyclotron 2, L7.01.02 1348 Louvain-la-Neuve Belgium E-mail: [email protected]

2010 Mathematical Subject Classification: Primary: 28-02, 31-01, 35-02, 35R06; Secondary: 26B20, 26B35, 26D10, 28A12, 28A25, 28A33, 28A78, 28C05, 31B05, 31B10, 31B15, 31B20, 31B35, 35A01, 35A02, 35A08, 35A15, 35A23, 35A35, 35B05, 35B33, 35B45, 35B50, 35B51, 35B60, 35B65, 35C15, 35D30, 35J05, 35J10, 35J15, 35J20, 35J25, 35J60, 35J61, 35J86, 35J91, 35Q40, 35Q75, 35R05, 46E27, 46E30, 46E35, 49J40, 49J45, 46N20, 49S05 Key words: Balayage method, Borel measure, Chern–Simons equation, continuous potential, diffuse measure, Dirichlet problem, elliptic PDE, Euler–Lagrange equation, extremum solution, fractional Sobolev inequality, Frostman’s lemma, Hausdorff measure, Hausdorff content, Kato’s inequality, Laplacian, Lebesgue set, Lebesgue space, Marcinkiewicz space, maximum principle, minimization problem, Morrey’s imbedding, obstacle problem, Perron’s method, Poisson equation, potential theory, precise representative, reduced measure, regularity theory, removable singularity, Riesz representation theorem, Schrödinger operator, semilinear equation, Sobolev capacity, Sobolev space, subharmonic, superharmonic, sweeping-out method, Thomas–Fermi equation, trace inequality, Weyl’s lemma

ISBN 978-3-03719-140-8 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2016 European Mathematical Society

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Typeset using the author’s TEX files: M. Zunino, Stuttgart, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface “Il doit bien se présenter des problèmes de Physique mathématique pour lesquels les causes physiques de régularité ne suffisent pas à justifier les hypothèses de régularité faites lors de la mise en équation.”1 Jean Leray

This book is devoted to classical techniques in elliptic partial differential equations (PDEs), involving solutions that are not expected to be smooth. Some of the topics that are developed are: regularity theory, maximum principles, Perron–Remak method, sub- and supersolutions, and removable singularities. They rely on tools from measure theory, functional analysis, and Sobolev spaces, see e.g. [53], [125], [134], [160], and [345]. The goal is to investigate the linear Dirichlet problem involving the Laplacian: ´ u D  in , (DP) u D 0 on @, for an arbitrary finite Borel measure ; we also consider the semilinear counterpart of problem (DP). The quantity .A/ can be interpreted as the mass or total charge contained in a subset A  . An example of solution is provided by the classical Green function with a Dirac mass  D ıa . For any smooth bounded open set , this problem has a unique solution for any measure . By a solution, we mean a summable function that verifies the equation against smooth test functions vanishing on the boundary @, see [213]. This weak formulation implicitly encodes the zero boundary condition. I have gathered several elegant proofs which are mostly available in the literature, but that are not necessarily widespread within the mathematical community. A surprising example is the simple argument leading to the fractional Sobolev embedding. I also explain the connection between trace inequalities and the strong approximation of diffuse measures. The reader should feel free to choose a topic according to his/her own interests. The chapters have been conceived to be as independent as possible. In this respect, some redundancy is expected. I have also included review sections on measure theory 1 “There

must be problems in mathematical physics for which the physical causes of regularity do not suffice to justify the regularity assumptions made when the equation is formulated.”

vi

Preface

and Sobolev spaces. The exercises provide some complementary material, and are not intended to be solved in a first reading; their solutions can be found at the end of the book. This project has originated from a set of lectures given at the Universidade Estadual de Campinas in 2005, and then from a full one-semester course in 2008. They were influenced by the enthusiasm and captivating style of H. Brezis, who had introduced me to these problems. In 2012, I deeply rewrote these notes, and the resulting monograph [288] won the Concours annuel in Mathematics of the Académie royale de Belgique. The text was then enlarged, including removable singularity principles and the Maz0 ya–Adams trace inequalities. The notion of reduced measure, introduced with Brezis and Marcus [60] and pursued in [288], has been incorporated in the formalism of the nonlinear Perron–Remak method. An updated list of corrections and misprints will be available in my personal website at uclouvain.be. The reader will find in the literature some recent advances on problems involving domains with little regularity [176], [236], [237], quasilinear operators in Euclidean spaces [165], [210], [218], or in metric spaces [29], Dirichlet problems involving trace measures on the boundary [226], connections to probability [120], [121], [200], [217], and trace inequalities [234], [239], that are not covered in this book. I am indebted to several colleagues who have helped me with their comments and support: A. Ancona, L. Boccardo, P. Bousquet, D. de Figueiredo, Th. De Pauw, B. Devyver, L. Dupaigne, A. Farina, J. Mawhin, P. Mironescu, M. Montenegro, L. Moonens, H. Nguyen Quoc, M. Pimenta, A. Presoto, J.-M. Rakotoson, P. Roselli, and J. Van Schaftingen. Some of my students, in particular J. Dekeyser and N. Wilmet, have carefully read several chapters. I especially thank H. Brezis, L. Orsina, D. Spector, L. Véron and M. Willem for their advices, Th. Hinterman for his patience while waiting for the final version of the manuscript and M. Zunino for the typesetting. Partial financial support has been provided by the Fonds de la Recherche scientifique – FNRS from Belgium. I am deeply grateful to my wife Isabelle for her encouragement. She also bravely kept Clément and Raphaël away from my computer while I was preparing the manuscript. Louvain-la-Neuve, October 2015

Augusto C. Ponce

Contents

Preface

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0

Introduction

1

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The Laplacian 1.1 Laplace and Poisson equations . . . . . . . . . . . . . . . . . . . 1.2 Mean value property . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Quantitative maximum principle . . . . . . . . . . . . . . . . . .

7 7 11 15

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Poisson equation 2.1 Finite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Distributional solutions . . . . . . . . . . . . . . . . . . . . . . . 2.3 Superharmonic functions . . . . . . . . . . . . . . . . . . . . . .

21 21 28 33

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Integrable versus measure data 3.1 Linear Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . 3.2 Nonlinear Dirichlet problem . . . . . . . . . . . . . . . . . . . .

39 39 44

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Variational approach 4.1 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Minimizers and the Euler–Lagrange equation . . . . . . . . . . . 4.3 Thomas–Fermi energy functional . . . . . . . . . . . . . . . . . .

49 49 62 70

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Linear regularity theory 5.1 Embedding in Sobolev spaces . 5.2 Weak Lebesgue functions . . . . 5.3 Critical estimates . . . . . . . . 5.4 Compactness in Sobolev spaces .

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Comparison tools 6.1 Weak maximum principle . . 6.2 Variants of Kato’s inequality 6.3 Localization properties . . . 6.4 Inverse maximum principle .

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Contents

7

111 Balayage 7.1 Weak normal derivative . . . . . . . . . . . . . . . . . . . . . . . 111 7.2 Finite charge up to the boundary . . . . . . . . . . . . . . . . . . 117

8

Precise representative 8.1 Lebesgue’s differentiation theorem 8.2 Sobolev functions . . . . . . . . . 8.3 Potentials . . . . . . . . . . . . . 8.4 Kato’s inequality revisited . . . .

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123 123 128 131 134

Maximal inequalities 9.1 Integral estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Energy estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Total charge estimate . . . . . . . . . . . . . . . . . . . . . . . .

139 139 143 150

10 Sobolev and Hausdorff capacities 10.1 Comparison properties . . . . . . . . . . . . . . . . . . . . . . . 10.2 Estimating the Sobolev capacity . . . . . . . . . . . . . . . . . . 10.3 Estimating the Hausdorff content . . . . . . . . . . . . . . . . . .

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11 Removable singularities 171 11.1 Schwarz’s principle . . . . . . . . . . . . . . . . . . . . . . . . . 171 11.2 Polar sets of second order . . . . . . . . . . . . . . . . . . . . . . 175 11.3 Carleson’s condition . . . . . . . . . . . . . . . . . . . . . . . . 176 12 Obstacle problems 12.1 Comparison between capacities . . . . . . . . . . . . . . . . . . . 12.2 Perron–Remak method . . . . . . . . . . . . . . . . . . . . . . . 12.3 Total charge and energy minimizations . . . . . . . . . . . . . . .

181 181 188 195

13 Families of solutions 205 13.1 Bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . 205 13.2 Lebesgue integrable classes . . . . . . . . . . . . . . . . . . . . . 207 13.3 Hölder-continuous solutions . . . . . . . . . . . . . . . . . . . . 209 14 Strong approximation of measures 14.1 Capacitary and density bounds . . . . . . . . . 14.2 Radon–Nikodým and Lebesgue decompositions 14.3 Perturbation of diffuse measures . . . . . . . . 14.4 Precise density bound . . . . . . . . . . . . . .

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215 215 219 225 227

Contents

ix

235 15 Traces of Sobolev functions 15.1 Existence of the trace . . . . . . . . . . . . . . . . . . . . . . . . 235 15.2 Fractional Sobolev embedding . . . . . . . . . . . . . . . . . . . 241 15.3 Range of the trace operator . . . . . . . . . . . . . . . . . . . . . 251 16 Trace inequality 257 16.1 Capacitary, geometric and pointwise interpretations . . . . . . . . 257 16.2 Hölder continuity revisited . . . . . . . . . . . . . . . . . . . . . 263 16.3 Delocalization of capacitary measures . . . . . . . . . . . . . . . 271

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18 Quasicontinuity 18.1 Continuous potentials . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Lusin property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Continuity principle . . . . . . . . . . . . . . . . . . . . . . . . .

299 299 303 306

17 Critical embedding 17.1 Bounded mean oscillation . . . . . . 17.2 Brezis–Merle inequality and beyond 17.3 Exponential Sobolev embedding . . 17.4 W 1;2 and W 2;1 capacities . . . . . .

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19 Nonlinear problems with diffuse measures 311 19.1 Unconditional existence . . . . . . . . . . . . . . . . . . . . . . . 311 19.2 Measures must be diffuse . . . . . . . . . . . . . . . . . . . . . . 315 20 Extremal solutions 321 20.1 Boundary data revisited . . . . . . . . . . . . . . . . . . . . . . . 321 20.2 Method of sub- and supersolutions . . . . . . . . . . . . . . . . . 327 20.3 Nonlinear Perron–Remak method . . . . . . . . . . . . . . . . . . 332 21 Absorption problems 21.1 Subcritical case . . . . . 21.2 Contraction and stability 21.3 Polynomial growth . . . 21.4 Exponential growth . . .

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337 337 341 344 347

22 The Schrödinger operator 351 22.1 Strong maximum principle . . . . . . . . . . . . . . . . . . . . . 351 22.2 Existence of solutions with measure data . . . . . . . . . . . . . . 356 22.3 The bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

x

Contents

Appendices . . . .

367 367 371 376 381

B Hausdorff measure B.1 Density estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Frostman’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Regularity and uniform approximation . . . . . . . . . . . . . . .

385 385 388 391

C Solutions and hints to the exercises

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Bibliography

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Index

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A Sobolev capacity A.1 Finite semi-additivity . . . . . . . . . . . A.2 Outer capacity and pointwise convergence A.3 Strong additivity . . . . . . . . . . . . . . A.4 Measures on dual Sobolev spaces . . . . .

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Chapter 0

Introduction “Sans que Gauss s’en soit douté, il a jeté des semences qui sont tombées sur une terre qui n’était pas mûre pour les recevoir. Puis, d’autres ouvriers sont venus qui ont remué ce sol ingrat, l’ont amendé, lui ont apporté les sucs nourriciers nécessaires à sa fécondité, et un jour, après un long sommeil, la graine qui n’était pas morte a germé. La plante qui en est sortie est jeune et vivante et c’est à ses fruits que l’on voit enfin la profondeur de la pensée lointaine d’où elle vient.”1 Charles de la Vallée Poussin

Classical potential theory originated in the 18th century to study the gravitational potential u generated by some density of mass , based on Newton’s gravitational theory. In the beginning of the 19th century, it was discovered that the potential satisfies the Poisson equation [283] u D ; given in terms of the Laplacian u D div .ru/ D trace .D 2 u/: During the 1830s, Gauss [145] pursued the electrostatic interpretation of the Poisson equation, where in this case  is a density of electric charges, positive or not. Even though Gauss’s argument is unsatisfactory to nowadays standards, his fundamental ideas would deeply influence PDEs and potential theory in the 20th century. Subsequent works by Dirichlet [201], Riemann [293], and Thompson [323] also relied on minimization arguments involving the energy ˆ jruj2 ; 

1 “Without

suspecting it, Gauss sowed some seeds that fell on ground that was not ready to receive them. Later, other workers came along who turned this infertile soil, enriched it and gave it the nourishing juices it needed to become fertile; and one day, after a long sleep, the seed which had not died sprouted. The plant that has emerged from it is young and vigorous, and it is by its fruits that we finally see the profundity of that distant thought whence it came.”

2

0 Introduction

but those arguments would not survive Weierstrass’s criticism, see [339] and [289]. Alternative approaches were then developed by Schwarz [301], Neumann [263], Robin [298] and others to circumvent the lack of rigor around the Dirichlet principle. Poincaré [280] and [281] gave a major contribution that enables one to solve the linear Dirichlet problem ´ u D  in , (0.1) u D 0 on @, in the classical sense, using his balayage method. This approach does not rely on variational methods, which were still unavailable by 1890. The core of his idea consists in moving electric charges from inside an open region ! b  to its boundary @!, without modifying the electrostatic potential outside !. Poincaré’s paper [281], published in the freshly founded American Journal of Mathematics, is also a landmark in the theory of PDEs, where he calls the attention to the three major models of second-order equations: elliptic (Laplace equation), parabolic (heat equation) and hyperbolic (wave equation). Hilbert tried to rescue the Dirichlet principle by using minimizing sequences of the energy. He suggested that they would have better convergence properties; for instance, convergence up to some subsequence. Hilbert’s first attempt [167] to justify the Dirichlet principle based on that idea was sketchy, and the implementation of his program turned out to be harder than expected, even in dimension two [88], [140], [153], [168], [193], [203], and [324]. Monna’s book [251] is a good historical source on the development of the Dirichlet principle in the 19th century. By introducing the concept of barrier, Lebesgue [195] clarified the fact that the Dirichlet problem should be solved in two steps: first find a solution of the Poisson equation inside the domain, and then verify that the boundary condition is indeed satisfied. Perron [277] and Remak [292] later developed independently an abstract approach that contains Poincaré’s balayage method, and implements Lebesgue’s strategy as an obstacle problem. The Perron–Remak method seems to have a different nature compared to variational tools, since we look for the smallest element in a class of superharmonic functions. It is nevertheless a disguised minimization of the total charge ˆ juj: 

We are no longer in a Hilbert space setting, but from this perspective the Perron– Remak method becomes a natural companion to variational obstacle problems (Chapter 12). Surprising counterexamples by Zaremba [347] and Lebesgue [196] showed that x due to the the Dirichlet problem need not always have a continuous solution in , possibility of having singular points on the boundary @. This was the beginning of

0 Introduction

3

modern potential theory, when the attention turned to the full characterization of the singular set of @. A major breakthrough to identify singular points and sets was made by Wiener [341] and [342], by introducing the Newtonian capacity associated to the Dirichlet energy. Generalized solutions of the Poisson equation were also introduced in the 1920s. An important one is given by the Newtonian potential generated by a finite Borel measure , see [122]: ˆ d.y/ 1 : (0.2) u.x/ D .N 2/N RN jx yjN 2

This approach was motivated by the fact that the Newtonian potential of a smooth distribution of charges satisfies the Poisson equation (Chapter 1), and that measures naturally describe densities of mass or electric charge, see p. 24 in [197] or Chapter 2 below. The function u is interpreted as the potential generated by the density , even though the Poisson equation need no longer be satisfied in a pointwise – classical – sense. F. Riesz [295] and [296] connected the integral representation (0.2) with the notions of superharmonic and subharmonic functions based on averages (Chapter 2); de la Vallée Poussin [106] developed Poincaré’s balayage method in the new setting of densities given by measures (Chapter 7). These advances were being supported by tools from functional analysis and measure theory. Frostman in his thesis [139] clarified the relation between the capacity, an analytic tool, and a more geometric object, the Hausdorff measure (Chapter 10). Their nonequivalence was further investigated by Carleson [77]. Two important notions in this direction are: .a/ Sobolev capacities capW k;q , that can be used to identify sets which are effectively detected by Sobolev functions (Chapter 8 and Appendix A); s .b/ Hausdorff contents H1 , that are better suited to investigate density properties, compared to the usual Hausdorff measures Hs (Appendix B).

It is remarkable that the two concepts can be interchanged in the formalism of Maz0 ya–Adams trace inequalities (Chapter 16). These trace inequalities also provide the equivalence between the W 1;1 capacity and the HıN 1 Hausdorff capacities that was discovered by Meyers and Ziemer [245], and first suggested by Fleming’s pioneer contribution [133]. Frostman was particularly interested in the existence of minimizers of Gauss’s energy functional. The solution of Gauss’s problem would play an important role in the development of the French school on axiomatic potential theory by Brelot, Cartan, Choquet, Deny, and others, see [46], [47], [48], and [49]. The systematic use of Schwartz’s theory of distributions [303] and [304] after World War II provided some solid foundation to weak formulations of linear PDEs,

4

0 Introduction

applied until then as an ad hoc strategy to solve different specific problems [216]. A notable exception comes from the work of Sobolev, who introduced the concept of distribution of finite order in 1935, which he called fonctionnelle, see [311] and [312]; a few years later, he also defined weak derivatives as we use nowadays [313]. Modern PDE methods based on functional analysis, a priori estimates, Sobolev spaces, fixed point, and variational techniques [57] were later applied by Littman, Stampacchia, and Weinberger [213] to tackle the linear Dirichlet problem (0.1) involving measure data. They established in particular the existence and uniqueness of a solution for every finite measure  (Chapter 3). Concerning the regularity of such a solution, Stampacchia’s truncation method provides zeroth and first-order estimates in the setting of weak Lebesgue (MarcinkieN wicz) spaces. Indeed, solutions are weak L N 2 , and their gradients exist and are N weak L N 1 (Chapter 5). We have in particular the existence of a force field ru associated to any finite measure. These embeddings should be compared to the classical Sobolev–Gagliardo– Nirenberg inequality, which gives the chain of inclusions: N

W 2;1 .RN /  W 1; N

N

1

.R N /  L N

2

.RN /:

The picture is completed by the Calderón–Zygmund theory on singular integrals, which provides a weak L1 estimate for the second-order derivative D 2 u, see [72] and [149]. This estimate lies at the heart of the more familiar (strong) Lp regularity theory for densities  2 Lp ./ such that 1 < p < C1, which is obtained by interpolation. The companion nonlinear Dirichlet problem ´ u C g.u/ D  in , (0.3) u D 0 on @, has a two-fold motivation. It arises as an L1 accretivity condition in the Crandall– Liggett theory applied to the porous medium equation, see Section 10.3 in [334], and is also associated to the Thomas–Fermi theory for densities  given in terms of finite sums of Dirac masses, each one representing an electric pointwise charge, see [208], [206], and [24]. We assume here that the nonlinearity gW R ! R is merely a continuous function satisfying the sign condition: for every t 2 R, g.t /t > 0: This is called an absorption problem, and the nonlinear term in the equation satisfies the contraction property: ˆ 

jg.u/j 6 jj./:

0 Introduction

5

When the density  belongs to the Lebesgue space L2 ./, this nonlinear Dirichlet problem is variational: the energy functional is bounded from below in the Sobolev space W01;2 ./, minimizers always exist regardless of the growth of g, and they satisfy the Euler–Lagrange equation (Chapter 4). Solutions also exist when  merely belongs to L1 ./, see [69] and [144], although the problem is no longer variational. This requires a different argument based on the contraction property and the strong L1 approximation of the datum (Chapter 3). The study of the nonlinear Dirichlet problem with measures is more subtle. Bénilan and Brezis [24], see also [50] and [51], discovered in 1975 that for nonlinearities of the form g.t / D jt jp 1 t with any exponent p > NN 2 , the nonlinear Dirichlet problem has no solution when  is a Dirac mass, while for p < NN 2 , solutions exist for every finite measure . Bénilan and Brezis’s paper [24] on the Thomas–Fermi problem had to wait almost thirty years to be published in its final form, although parts of the manuscript started to circulate by the end of the 1970s, and had a great influence at the time. Since then, the mathematical landscape concerning elliptic PDEs with L1 and measure data has drastically evolved and new fields have been flourishing: – equations involving quasilinear or fractional operators; – nonlinear potential and Calderón–Zygmund theories in Euclidean and metric spaces; – obstacle problems and regularity of free boundaries; – removable singularity principles for linear and nonlinear equations; – measure valued solutions; – boundary traces and probabilistic interpretation of nonlinear problems. Concerning the nonlinear Dirichlet Problem (0.3), we are interested in the full characterization of measures for which a solution exists. The answer depends on g, and some techniques involving power ([20] and [24]) and exponential ([333] and [22]) nonlinearities are (Chapter 21): .a/ maximum principles, adapted to the linear Dirichlet problem, and Kato’s inequality, to get comparison principles for the nonlinear Dirichlet problem (Chapter 6); .b/ removable singularity principles, to deduce necessary conditions for the existence of solutions (Chapter 11); .c/ strong approximation of measures, to handle measures that are merely diffuse with respect to Sobolev capacities or Hausdorff measures (Chapter 14); .d / trace inequalities, to have a priori estimates (Chapter 16); .e/ Perron–Remak methods, based on sub- and supersolutions, to establish the existence of extremal solutions (Chapter 20).

6

0 Introduction

Kato introduced his inequality in a seminal paper [175] on Schrödinger operators  C V to deal with potentials V 2 Lp , for any exponent p > 1. His goal was to clarify and avoid unnecessary assumptions usually required by variational tools. Such a program is far from being completed, since many properties of the solutions of the Poisson equation associated to the Schrödinger operator depend on the strength of the singularity of V , see [310]. The Hardy potential V .x/ D c=jxj2 , for example, is critical in many problems, including the strong maximum principle. We explain in Chapter 22 how the above tools can be successfully adapted to investigate properties of solutions involving singular potentials.

Convention. We systematically denote by  an open subset of RN in dimension N > 1. Further assumptions on  will be explicitly stated when needed. For example, by a smooth open set  we mean that there exists a smooth (C 1 ) function x and r ¤ 0 everywhere on W RN ! R such that  < 0 in ,  > 0 in RN n , @. The outer normal vector n is then defined on @ by nD

r : jrj

x ! R is smooth if there exists an infinitely differentiable We say that a function uW  N x function U W R ! R such that U D u in .

Chapter 1

The Laplacian

“Je parviens à une expression en séries, générale et simple, des attractions des sphéroïdes quelconques très peu différents de la sphère. Il est assez remarquable que cette expression soit donnée sans aucune intégration et par la seule différentiation de fonctions.”1 Pierre-Simon de Laplace

We present classical properties of harmonic and superharmonic functions, in connection with monotonicity formulas and maximum principles.

1.1 Laplace and Poisson equations The Laplacian  appeared in the 18th and 19th centuries in the studies of gravitational and electrostatic potentials. Assume that uW  ! R denotes a smooth function describing one of these physical quantities in a region where there is no mass or electric charges. Then, the total flux of the field ru through the boundary of any open ball B.xI r/ b  strictly contained in  must vanish: ˆ . ru/
n d D 0: @B.xIr /

The integral above is taken with respect to the surface measure  , which coincides with the .N 1/-dimensional Hausdorff measure Hn 1 . By the divergence theorem (see Theorem 9.2.4 in [345]), we deduce that, on every ball B.xI r/ b , we have ˆ div .ru/ D 0; B.xIr /

and so u satisfies the Laplace equation: u D 0 in , and is called a harmonic function. 1 “I obtain a general and simple expression, in terms of a series, for the attraction of arbitrary spheroids that are not very different from the sphere. It is quite remarkable that this expression is given without any integration and only through the differentiation of functions.”

8

1. The Laplacian

Example 1.1. For every a 2 RN in dimension N > 3, the function uW RN n¹aº ! R defined for x 2 RN n ¹aº by u.x/ D

jx

1 ajN

2

is harmonic, which can be checked by differentiating twice this function. An example of radially symmetric harmonic function in dimension N D 2 is u.x/ D log

1

jx

aj

:

Exercise 1.1 (radial harmonic functions). Prove that all radial harmonic functions uW RN n ¹0º ! R in dimension N > 3 are of the form ˛ u.x/ D C ˇ; jxjN 2

for some ˛; ˇ 2 R. What is the counterpart in dimension N D 2?

We may also consider the case where u denotes the electrostatic potential generated by some distribution of electric charges of density W  ! R. In this case, by Gauss’s law the total flux through the boundary of the ball B.xI r/ b  equals the total charge inside this ball: ˆ ˆ . ru/
n d D : @B.xIr /

B.xIr /

By the divergence theorem, we deduce that, for every ball B.xI r/ b , we have ˆ ˆ div .ru/ D ; B.xIr /

B.xIr /

and so the function u satisfies the Poisson equation: u D  in . A large class of solutions of the Poisson equation is given by the Newtonian potential generated by : Proposition 1.2. Let N > 3. If W RN ! R belongs to the space Cc1 .RN / of smooth functions with compact support in RN , then the Newtonian potential uW RN ! R defined for x 2 RN by ˆ 1 .y/ u.x/ D dy; .N 2/N RN jx yjN 2

where N denotes the measure of the .N 1/-dimensional unit sphere @B.0I 1/ in RN , is a bounded smooth function satisfying the Poisson equation u D  in RN .

1.1. Laplace and Poisson equations

9

The function F W RN n ¹0º ! R defined by 1 1 N .N 2/N jzj

F .z/ D

2

is called the fundamental solution of the Laplacian in dimension N > 3. Using this notation, we write the Newtonian potential as a convolution between F and : u.x/ D

ˆ

y/.y/ dy D .F  /.x/:

F .x RN

Proof of Proposition 1.2. Since the density  has compact support and F is locally integrable, u is well defined. By the change of variables z D x y, we have u.x/ D

ˆ

F .z/.x

z/ dz;

RN

whence the function u is smooth. Differentiating under the integral sign, and then undoing the previous change of variables, we get u.x/ D

ˆ

F .x

y/ .y/ dy:

(1.1)

RN

We now identify the integrand as the divergence of a smooth vector field in RN n ¹xº. For this purpose, denote Fx .y/ D F .x y/ D F .y x/. We have div .Fx r/ D rFx
r C Fx ; div . rFx / D r
rFx C  Fx : Since the function Fx is harmonic in RN n ¹xº (Example 1.1), subtracting the second identity from the first one we deduce that div .Fx r

 rFx / D Fx :

The function Fx r  rFx has compact support in RN and is smooth on N R n ¹xº. We are thus allowed to apply the divergence theorem to open sets of the form RN n BŒxI r. For every r > 0, we then get ˆ

RN nBŒxIr 

Fx .y/ .y/ dy D

where n.z/ D .z

ˆ

@B.xIr /

.Fx r

 rFx /
. n/ d;

x/=r is the outward normal vector with respect to B.xI r/.

(1.2)

10

1. The Laplacian

For every z 2 @B.xI r/, we now compute Fx .z/ D

.N

and rFx .z/
n.z/ D

1 2/N r N 1 N r N

2

1

:

Rewriting the right-hand side of identity (1.2) in terms of average integrals over spheres, we get ˆ r Fx .y/ .y/ dy D r
n d  d: N 2 @B.xIr / RN nBŒxIr  @B.xIr / Since the right-hand side converges to .x/ as r ! 0, by identity (1.1) and the dominated convergence theorem we deduce that ˆ u.x/ D lim Fx .y/ .y/ dy D .x/:  r !0

RN nBŒxIr 

According to the proposition above, for every function  2 Cc1 .RN / we have .F  / D  in RN : By an affine change of variable and by differentiation under the integral sign, we obtain the following representation formula involving the Laplacian: for every x 2 RN , ˆ .x/ D

F .x

y/ .y/ dy:

RN

Exercise 1.2 (representation formulas in dimensions 1 and 2). Prove that .a/ given  2 Cc1 .R/, for every x 2 R we have ˆ 1 .x/ D jx yj 00 .y/ dyI 2 R .b/ given  2 Cc1 .R2 /, for every x 2 R2 we have ˆ 1 log jx yj .y/ dy: .x/ D 2 R2 The Newtonian potential satisfies the Poisson equation when the density  is merely a Hölder-continuous function of exponent ˛ for some 0 < ˛ < 1, but the proof of the counterpart of Proposition 1.2 has to be substantially modified, see Lemma 4.2 in [146]. In this case, the Newtonian potential is a C 2;˛ function.

1.2. Mean value property

11

1.2 Mean value property The Laplacian also arises as the infinitesimal difference between the values of a smooth function and the averages of this function on balls: Proposition 1.3. Let uW  ! R be a smooth function. Then, for every x 2 , we have ² ³ 1 1 lim 2 u.x/: u u.x/ D r !0 r 2.N C 2/ B.xIr / Proof. By the second-order Taylor expansion at the point x, for every y 2  we have u.y/ D u.x/ C Du.x/Œy

1 x C D 2 u.x/Œy 2

x; y

x C Rx2 u.y/;

(1.3)

where the remainder function Rx2 u satisfies lim

y!x

Rx2 u.y/ D 0: jy xj2

By a linear change of variables, we get ˆ ˆ Du.x/Œy x dy D B.xIr /

B.0Ir /

Du.x/Œz dz D 0;

since the linear transformation Du.x/ is an odd function. Integrating both sides of identity (1.3) with respect to y over a ball B.xI r/ b , we then have ˆ ˆ 1 u D !N r N u.x/ C D 2 u.x/Œy x; y x dy C o.r N C2 /; 2 B.xIr / B.xIr / where !N denotes the volume of the unit ball B.0I 1/ in RN , and o.r N C2 / denotes a quantity such that o.r N C2 /=r N C2 ! 0 as r ! 0. To compute the integral in the right-hand side, we observe that by a linear change of variables we also have ˆ ˆ D 2 u.x/Œy x; y x dy D D 2 u.x/Œz; z dz B.xIr /

B.0Ir /

D

N N X X

i D1 j D1

@2 u .x/ @xi @xj

For every i; j 2 ¹1; : : : ; N º such that i ¤ j , we have ˆ zi zj dz D 0: B.0Ir /

ˆ

B.0Ir /

zi zj dz:

12

1. The Laplacian

Moreover, since the integral of the function z 7! zi2 on B.0I r/ does not depend on the coordinate i , we have ˆ ˆ r N C2 1 2 : jzj2 dz D !N zi dz D N B.0Ir / N C2 B.0Ir / We then get

ˆ

D 2 u.x/Œy

x; y

B.xIr /

x dy D !N

Thus,

r N C2 u.x/: N C2

1 r N C2 u D !N r N u.x/ C !N u.x/ C o.r N C2 /: 2 N C2 B.xIr / Dividing both sides by the volume of the ball B.xI r/, the conclusion follows. ˆ



A smooth function uW  ! R is superharmonic if u > 0 in . In this case, for every x 2  the previous proposition gives ³ ² 1 lim u u.x/ 6 0: r !0 r 2 B.xIr /

According to the mean value property of superharmonic functions, a stronger conclusion is true:

Proposition 1.4. Let uW  ! R be a smooth function. If u is superharmonic, then, for every x 2 , the function .0; d.x; @// 3 r 7 !

u B.xIr /

is non-increasing, and, for every 0 < r < d.x; @/, we have u 6 u.x/: B.xIr /

The monotonicity formula for superharmonic functions relies on the monotonicity of averages on spheres. We introduce an additional parameter  that will be used in the proof of Proposition 1.4: Lemma 1.5. Let uW  ! R be a smooth function. If u is superharmonic, then, for every x 2  and every 0 <  6 1, the function ˆ 1 u d .0; d.x; @// 3 r 7 ! N 1 r @B.xIr  / is non-increasing.

1.2. Mean value property

13

Proof of Lemma 1.5. By the change of variable formulas, ˆ ˆ N 1 u d D r u.x C rz/ d .z/: @B.xIr  /

@B.0I /

Dividing both sides by r N 1 and differentiating under the integral sign, it follows that   ˆ ˆ 1 d u d D ru.x C rz/
z d .z/ dr r N 1 @B.xIr  / @B.0I / ˆ 1 ru
n d: D N 1 r @B.xIr  / Applying the divergence theorem, we get   ˆ d 1  u d D N N 1 dr r r @B.xIr  /

1

ˆ

u:

B.xIr  /

Since by assumption u is superharmonic, the quantity in the right-hand side is nonpositive, and the conclusion follows.  Proof of Proposition 1.4. By the integration formula in polar coordinates (see Theorem 2.49 in [134] or Lemma 2.4.7 in [345]), for every 0 < r < d.x; @/ we have  ˆ ˆ r ˆ uD u d ds: B.xIr /

0

@B.xIs/

Given 0 <  < r, we make the change of variables s D rt = to get a real integral computed over the interval .0; /:  ˆ ˆ ˆ r  u d dt: uD  0 B.xIr / @B.xIr t /

By the monotonicity of the average integrals over spheres (Lemma 1.5) with parameter  D t =, for every 0 < t 6  we have ˆ ˆ ˆ rN 1 rN 1 u d: u d 6 N 1 u d D N 1   @B.xI t / @B.xIt/ @B.xIr t / Applying this estimate and the integration formula in polar coordinates, we get  ˆ ˆ ˆ ˆ rN  rN u6 N u; u d dt D N   B.xIr / 0 B.xI/ @B.xIt/

and this gives the monotonicity property of the average integral on balls: u6 B.xIr /

u: B.xI/

To conclude it suffices to let  ! 0; the average integral in the right-hand side converges to u.x/. 

14

1. The Laplacian

If u is harmonic, then the functions u and u are superharmonic, and we deduce the mean value identity for harmonic functions: for every 0 < r < d.x; @/,

B.xIr /

u D u.x/:

(1.4)

Exercise 1.3 (Liouville’s theorems). Let uW RN ! R be a harmonic function. Prove that .a/ if u 2 L1 .RN /, then u D 0;

.b/ if u is bounded, then u is constant.

The C 2 regularity of the function u is enough in the argument leading to the conclusions of Proposition 1.4 and its corollary (1.4). As an application of the mean value property, we now deduce that every C 2 harmonic function is infinitely differentiable. This is a consequence of the following property proved independently by Bôcher [36] and Koebe [183]: Proposition 1.6. Let u 2 L1 ./. If, for every x 2  and every 0 < r < d.x; @/, we have u.x/ D u; B.xIr /

then u 2 C 1 ./. The original lemma by Bôcher and Koebe also asserts that u is harmonic, but the most subtle step is to show that u is smooth, since we may then apply Proposition 1.3 to deduce that u D 0. Observe that the integral in the right-hand side is a convolution of u with respect to the characteristic function B.0Ir / of the ball B.0I r/. The proof of Proposition 1.6 consists in establishing a similar identity for the convolution with smooth radial functions. For this purpose, let us recall Cavalieri’s principle, see Proposition 6.24 in [134] or Corollary 2.2.34 in [345]: Proposition 1.7. Let .XI / be a measure space. For every nonnegative measurable function f W X ! R, we have ˆ ˆ 1 f d D .¹f > t º/ dt: X

0

We systematically consider measure spaces equipped with positive measures or finite signed measures (Section 2.1). The sets ¹f > t º may also be replaced in this identity by ¹f > t º, without modifying the conclusion. Cavalieri’s principle is sometimes considered as a consequence of Fubini’s theorem, but actually has a

1.3. Quantitative maximum principle

15

more elementary proof based on the definition of the integral. Indeed, the identity above is satisfied by step functions on measurable sets, and the general case follows by approximation of f by such functions using the monotone convergence theorem. By a change of real variable, we also have the following extension of Cavalieri’s principle for every exponent p > 0: ˆ ˆ 1 p f d D p t p 1 .¹f > t º/ dt: (1.5) X

0

Proof of Proposition 1.6. Given a nonnegative function  2 Cc1 .RN /, for every x 2  it follows from Cavalieri’s principle applied with the Lebesgue measure with density u that (cf. Example 2.2)  ˆ ˆ 1 ˆ   u.x/ D .x y/u.y/ dy D u.y/ dy dt: 

0

¹y2W.x y/>tº

Given an open set ! b , let R D d.!; @/ > 0. Choosing as  a non-increasing radial function in RN supported in the ball B.0I R/, then, for every t > 0 and every x 2 !, we have ¹y 2 W .x

y/ > t º D ¹y 2 RN W .x

y/ > t º:

Hence, this set is an open ball centered at x, strictly contained in . Thus, by the assumption on u, we have ˆ u.y/ dy D u.x/ j¹y 2 RN W .x y/ > t ºj: ¹y2W.x y/>tº

Applying again Cavalieri’s principle, we deduce that ˆ 1 ˆ N   u.x/ D u.x/ j¹y 2 R W .x y/ > t ºj dt D u.x/ 0

: RN

Thus, u coincides in ! with a convolution function with smooth kernel. In particular, u is smooth in !. 

1.3 Quantitative maximum principle In this section we prove the following quantitative version of the strong maximum principle inspired from an unpublished work of Morel and Oswald, see e.g. Lemma 3.2 in [58]:

16

1. The Laplacian

Proposition 1.8. Let uW  ! R be a smooth function, and let K   be a connected compact set. If u is superharmonic, then, for every x 2 K, we have u.x/ > inf u C ckukL1 .K/ ; 

for some constant c > 0 depending on  and K. This estimate tells us by how much the electrostatic potential increases inside  due to the total charge kukL1 .K/ , regardless of how the electric charges are spread inside the domain. The proof requires both the weak and the strong maximum principles. We begin with a variant of the weak maximum principle: Lemma 1.9. Let uW  ! R be a smooth function. If u is superharmonic, then, for every compact set K   and every x 2 K, we have u.x/ > inf u: nK

Proof of Lemma 1.9. We first suppose that we have the strict inequality u < 0 in K,

(1.6)

and let a 2 K be the minimum point of the restriction ujK . Assuming by contradiction that u.a/ < inf u, then a is a minimum point of u in , whence u.a/ > 0 nK

which contradicts the strict inequality in (1.6). We deduce in this case that, for every x 2 K, we have u.x/ > u.a/ > inf u: nK

We now assume that u 6 0 in K. Given  > 0, let u W  ! R be the function defined by u .x/ D u.x/ C '.x/;

where 'W  ! R is any smooth bounded function such that ' < 0 in K. The specific expression of ' is not important, and we could take '.x/ D jxj2 in some neighborhood of K. By the superharmonicity of u and the choice of ', we now have u < 0 in K. For every x 2 K, we deduce from the first part of the proof that u .x/ > inf u ; nK

whence u.x/ C '.x/ > inf u > inf u nK

As  ! 0, the conclusion follows.

nK

k'kL1 ./ : 

1.3. Quantitative maximum principle

17

We deduce a more familiar statement of the weak maximum principle: x ! R be a Corollary 1.10. Let   RN be a bounded open set, and let uW  continuous function. If u is smooth and superharmonic in , then, for every x 2 , we have u.x/ > inf u: @

Proof of Corollary 1.10. Assume by contradiction that there exists a 2  such that u.a/ < inf u, and take  > 0 such that @

u.a/ < inf u

(1.7)

:

@

By the boundedness of , the set K D ¹x 2 W u.x/ 6 inf u

º

@

is compact. We also have inf u > inf u nK

principle above we deduce that

@

. Since a 2 K, by the weak maximum

u.a/ > inf u > inf u nK

;

@

which contradicts (1.7).



Using the linearity of the Laplacian, we deduce from Corollary 1.10 that classical solutions of the Dirichlet problem for the Poisson equation are unique in bounded domains. More precisely, given W  ! R, the Dirichlet problem ´ v D  in , (1.8) v D 0 on @, x ! R such that v is smooth in  and continuous on . x has at most one solution vW  We next consider the strong maximum principle:

Lemma 1.11. Let uW  ! R be a smooth function. If u is superharmonic, and if there exists a 2  such that u.a/ D inf u; 

then u is constant in the connected component of  that contains a. Proof of Lemma 1.11. Given a connected set G   such that a 2 G, consider the subset A D ¹x 2 GW u.x/ D inf uº 

18

1. The Laplacian

where u achieves its infimum. Since u is continuous, A is relatively closed in G. Given x 2 A, the mean value property for superharmonic functions (Proposition 1.4) shows that, for every 0 < r < d.x; @/, u D inf u in B.xI r/. 

Thus, B.xI r/ \ G  A, and we deduce that A is relatively open in G. Since by assumption the set A is non-empty, by the connectedness of G we must have A D G.  We now prove the quantitative maximum principle, taking for granted the existence of solutions of the Dirichlet Problem (1.8) on smooth bounded open sets, see Chapter 2 in [114] or Chapter 5 in [160]. x Proof of Proposition 1.8. We may assume that u admits a smooth extension to , for otherwise we prove the inequality in a strictly contained smooth open set ! b  such that K  !. We also assume that inf u D 0. Taking an open ball B.aI 3r/ b , 

for every x 2 B.aI r/ we have

B.aI r/  B.xI 2r/  B.aI 3r/ b : Since the function u is superharmonic and nonnegative, for every x 2 B.aI r/ it follows from the mean value property (Proposition 1.4) that ˆ 1 u.x/ > u> u: !N .2r/N B.aIr / B.xI2r / Taking a nonzero function ha;r 2 Cc1 .B.aI r// such that 0 6 ha;r 6

1 !N .2r/N

we then have u.x/ >

ˆ

in B.aI r/;

uha;r : 

Let va;r be the solution of the Dirichlet problem ´ va;r D ha;r in ,

on @.

va;r D 0

It follows from the divergence theorem that ˆ ˆ ˆ u.x/ > uha;r D u va;r D va;r u 





ˆ

@

u rva;r
n d:

1.3. Quantitative maximum principle

19

By the weak maximum principle (Corollary 1.10), the function va;r is nonnegative. Since va;r vanishes on @, we have rva;r
n 6 0 on this set. Thus, for every x 2 B.aI r/ we get ˆ u.x/ >

va;r . u/:



By the nonnegativity of the integrand, we then have ˆ ˆ u.x/ > va;r . u/ > .inf va;r / . u/; K

K

K

for every x 2 B.aI r/. We now cover the set K   with finitely many such balls B.a1 I r1 /;

:::;

B.ak I rk /

that intersect K. Since K is compact and connected, by the strong maximum principle (Lemma 1.11) each one of the functions vai ;ri is positive in K. For every k S B.ai I ri /, it then follows that x2 i D1

u.x/ > . min

inf vai ;ri /

i 2¹1;:::;kº K

where c is a positive constant.

ˆ

K

. u/ D ckukL1 .K/ ; 

Chapter 2

Poisson equation

“Les fonctions de domaine ont un sens physique très clair : ce sont les nombres qui mesurent des grandeurs. À cet égard, ces nombres s’introduisent en physique plus primitivement même que les fonctions de point, lesquelles ne servent le plus souvent qu’à étalonner des qualités.”1 Henri Lebesgue

We now consider solutions of the Poisson equation u D  in the sense of distributions. As a consequence of the Riesz representation theorem, every weakly superhamonic function satisfies the Poisson equation for some nonnegative Borel measure .

2.1 Finite measures We briefly recall in this section the definition and some properties of finite Borel measures. Definition 2.1. Given a measure space X equipped with a  -algebra †, a finite measure  is a set function W † ! R such that, for every sequence .An /n2N of disjoint subsets belonging to †, we have 

1 [

kD0



Ak D

1 X

.Ak /:

kD0

In particular, we have .;/ D 0. Measures are natural objects in gravitational and electrostatic problems, since .A/ can be physically interpreted as the mass or electric charge contained in the set A. 1 “Set functions have a clear physical meaning: these are the numbers that measure quantities. In this respect, these numbers are introduced in physics more primitively than point functions, which are most often used to calibrate properties.”

22

2. Poisson equation

Exercise 2.1 (monotone set lemma). Let  be a finite measure on a measure space X. Prove that .a/ if .An /n2N is a nondecreasing sequence of sets in †, then 

1 [

kD0

 Ak D lim .An /I n!1

.b/ if .An /n2N is a non-increasing sequence of sets in †, then 

1 \

kD0

 Ak D lim .An /: n!1

In analogy with Definition 2.1, one also considers nonnegative measures , defined in the  -algebra †, taking values in the interval Œ0; C1. An important example is the Lebesgue measure on RN . Exercise 2.2. Let  be a nonnegative measure on a measure space X. Prove that .a/  is monotone: if A; B 2 † and A  B, then .A/ 6 .B/;

.b/  is subadditive: if .An /n2N is a sequence of elements in †, not necessarily disjoint, then 1 1 [
X  Ak 6 .Ak /: kD0

kD0

If X is a locally compact topological space, an important example of  -algebra is the class of Borel subsets of X: this is the smallest  -algebra containing all compact – or equivalently all open – subsets of X. The typical example of locally compact space X we consider is given by an open subset   RN endowed with the Euclidean topology. We then denote by M./ the vector space of finite Borel measures on , and we equip this space with the total variation norm defined by kkM./ D sup ¹.A/

.B/W A; B 2 B./º:

Exercise 2.3. Prove that k
kM./ is a norm in M./. One of the reasons for choosing such a norm is to recover the usual L1 norm when dealing with densities of the Lebesgue measure: Example 2.2. For every summable function f W  ! R, denote by f the measure given in terms of the Lebesgue measure with density f : for every Borel set A  , ˆ f .A/ D f: A

2.1. Finite measures

23

By definition of the integral, this measure f satisfies ˆ ˆ df D f 



for every measurable step function W  ! R, and so for every bounded measurable function . The supremum in the definition of the total variation norm is achieved using the sets A D ¹f > 0º and B D ¹f < 0º, and we have ˆ ˆ kf kM./ D f f D kf kL1 ./ : ¹f >0º

¹f 0,

(ii) for every measurable set A  X n E, we have .A/ 6 0. Denoting by E   any Borel set given by the Jordan decomposition theorem, the positive part of  is the nonnegative measure C defined for every Borel set A   by contraction of  on E as C .A/ D bE .A/ D .A \ E/; and the negative part of  is the nonnegative measure  defined for every Borel set A   by contraction of  on  n E as  .A/ D

bnE .A/ D

.A n E/:

The notations max ¹; 0º to denote the measure C and min ¹; 0º for  are also used. By the additivity of , we have  D C

 D max ¹; 0º C min ¹; 0º;

and the total variation measure jj is defined as jj D C C  :

24

2. Poisson equation

Exercise 2.4. Prove identity (2.1). The vector space of finite Borel measures M./ equipped with the total variation norm is a Banach space, but is not separable since, for every distinct points a; b 2 , the Dirac mass gives kıa

ıb kM./ D ıa ./ C ıb ./ D 2:

Here, the Dirac mass ıa is the measure defined for every Borel set A  RN by ´ 1 if a 2 A, ıa .A/ D 0 if a 62 A. One may try to approximate a given finite measure  in  using some sequence of measures .n /n2N having better properties like density or capacitary upper bounds (cf. Chapter 14). In general, this is very difficult to achieve – or simply impossible – using the strong convergence with respect to the total variation norm: lim kn

n!1

kM./ D 0:

For instance, the Dirac mass ıa with a 2  cannot be strongly approximated by summable functions since, for every f 2 L1 ./, we have kf

ıa kM./ D kf kL1 ./ C 1:

Behind this obstruction lies a more general fact: singular measures cannot be strongly approximated by absolutely continuous measures. Exercise 2.5. Prove that if .fn /n2N is a sequence in L1 ./ converging strongly to some measure  in M./, then  D f for some summable function f . In many situations it suffices to have convergence in the weak sense, sometimes also called vague convergence: Definition 2.4. Let  2 M./. A sequence .n /n2N in M./ converges weakly to  in the sense of measures if, for every  2 Cc0 ./, we have ˆ ˆ lim  dn D  d; n!1





where Cc0 ./ denotes the set of continuous functions with compact support in .

2.1. Finite measures

25

Another characterization of the total variation norm in connection with this notion of convergence is the following: kkM./ D sup

²ˆ



 dW  2

Cc0 ./

³ and jj 6 1 in  :

(2.2)

This identity is based on the following remarkable regularity property of measures, discovered by Lebesgue in RN , see Section 7.2 in [134]: Proposition 2.5. Let X be a locally compact space, and let  be a finite Borel measure on X. Then, for every Borel set A   and every  > 0, we have (i) inner regularity: there exists a compact set K  A such that j.A n K/j 6 , (ii) outer regularity: there exists an open set U  A such that j.U n A/j 6 . According to this proposition, a finite measure is determined by its values on all compact or all open subsets. Exercise 2.6. Let  2 M./. .a/ Prove that

kkM./ D sup ¹.K/

.L/W K; L   are compact and disjointº:

.b/ Deduce identity (2.2). The total variation norm is lower semicontinuous with respect to the weak convergence: Proposition 2.6. If .n /n2N is a sequence in M./ converging weakly to  in the sense of measures on , then kkM./ 6 lim inf kn kM./ : n!1

Proof. For every  2 Cc0 ./ such that jj 6 1 in , we have ˇ ˇˆ ˇ ˇ ˇ  dn ˇ 6 kn kM./ : ˇ ˇ 

Taking the limit as n ! 1, the weak convergence of the sequence .n /n2N yields that ˆ  d 6 lim inf kn kM./ : 

n!1

The estimate follows from the characterization (2.2) by taking the supremum of the left-hand side with respect to . 

26

2. Poisson equation

We now prove that every finite Borel measure can be approximated by smooth functions in the weak sense: Proposition 2.7. For every  2 M./, there exists a sequence of summable funcx converging weakly to  in the sense of measures on , and tions .fn /n2N in C 1 ./ such that lim kfn kL1 ./ D kkM./ : n!1

The argument is based on a convolution of  using a sequence of mollifiers .n /n2N : for every n 2 N, n 2 Cc1 .RN / is a nonnegative function such that ˆ n D 1 RN

and, for every  > 0, lim

n!1

ˆ

RN nB.0I/

n D 0:

A convenient choice of mollifiers consists in taking n .x/ D

1 x  nN n

for some fixed function  2 Cc1 .RN / and some sequence of positive numbers .n /n2N converging to zero. Proof of Proposition 2.7. Given a sequence of mollifiers .n /n2N , for every n 2 N let n  W RN ! R be the convolution defined for x 2 RN as ˆ n  .x/ D n .x y/ d.y/: 

x and we also have n   2 L1 .RN /. If in addition In particular, n   2 C 1 ./, n is an even function, then, for every  2 Cc0 ./, it follows from Fubini’s theorem that  ˆ ˆ ˆ ˆ  n   D n .x y/.x/ dx d.y/ D n   d: 







x from Since  2 Cc0 ./, the sequence .n  /n2N converges uniformly to  in , which we deduce the weak convergence of .n /n2N to  in the sense of measures. To conclude with fn D n  , by the lower semicontinuity of the norm under weak convergence (Proposition 2.6) it suffices to check that, for every n 2 N, we have kn  kL1 ./ 6 kkM./ :

2.1. Finite measures

27

This is a consequence of Fubini’s theorem. Indeed, for every x 2 , ˆ jn  .x/j 6 n .x y/ djj.y/: 

Thus, by Fubini’s theorem, we get ˆ ˆ kn  kL1 ./ 6 n .x 



 ˆ y/ dx djj.y/ 6



and the conclusion follows.

djj.y/ D kkM./ ; 

We close this section with the weak compactness property of bounded sequences of measures due to Radon, see p. 1337 in [290]: Proposition 2.8. If .n /n2N is a bounded sequence in M./, then there exists a subsequence .nk /k2N converging weakly to some  2 M./ in the sense of measures on . The main ingredient in the proof of Proposition 2.8 is the Riesz representation theorem that was originally stated by F. Riesz [294] as a way of identifying a continuous linear functional in the space of continuous functions, see Section 7.1 in [134]: Proposition 2.9. Let X be a locally compact metric space. If T W Cc0 .X/ ! R is a linear functional such that, for every  2 Cc0 .X/, jT ./j 6 C sup jj; X

then there exists a unique finite Borel measure  on X such that, for every  2 Cc0 .X/, ˆ T ./ D  d: X

Proof of Proposition 2.8. Let D be a countable subset of Cc0 ./. Using a diagonalization argument, there exists a subsequence .nk /k2N such that, for every  2 D, the limit ˆ  dnk lim k!1



exists. Choosing D to be a dense subset of Cc0 ./ with respect to the sup norm, it follows that such a limit exists for every  2 Cc0 ./, and we define ˆ T ./ D lim  dnk : k!1

Since



ˇˆ ˇ ˇ ˇ ˇ  dn ˇ 6 kn kM./ sup jj; kˇ k ˇ 



28

2. Poisson equation

for every k 2 N, the functional T W Cc0 ./ ! R satisfies jT ./j 6 .lim inf knk kM./ / sup jj: k!1



Hence, by the Riesz representation theorem, there exists a finite measure  on  such that, for every  2 Cc0 ./, ˆ ˆ  dnk D T ./ D lim  d: k!1





This gives the conclusion.



2.2 Distributional solutions We are interested in solutions of the Poisson equation for some measure data: Definition 2.10. Let  2 M./. We say that u is a solution of the Poisson equation u D  in the sense of distributions in  if u 2 L1loc ./ and if u satisfies, for every ' 2 Cc1 ./, the integral identity ˆ ˆ u ' D ' d: 



If uW  ! R is a smooth function, then div.ur'

'ru/ D u '

' u:

Hence, by the divergence theorem we have that ˆ ˆ u ' D '. u/; 



for every ' 2 Cc1 ./. We now consider two examples inducing legitimate measures, that rely on the representation formula (Proposition 1.2): for every ' 2 Cc1 .RN /, 'D

.F  '/ D

F  .'/;

where F is the fundamental solution of the Laplacian.

(2.3)

2.2. Distributional solutions

29

Example 2.11. For every a 2 RN , the function

Fa W RN ! R

defined for y 2 RN n ¹aº by satisfies the Poisson equation

Fa .y/ D F .y

a/

Fa D ıa

N

in the sense of distributions in R . Indeed, by the definition of the convolution product and the representation formula (2.3), for every ' 2 Cc1 .RN / we have ˆ ˆ Fa ' D .F  '/.a/ D '.a/ D ' dıa : RN

RN

Example 2.12. Let N > 3. The Newtonian potential generated by a nonnegative measure  2 M.RN / is the function NW RN ! Œ0; C1

defined by N.x/ D

ˆ

F .x RN

y/ d.y/ D

1 .N 2/N

ˆ

RN

We note that N belongs to L1loc .RN / and satisfies

jx

d.y/ yjN

2

:

N D 

in the sense of distributions in RN . Indeed, by Fubini’s theorem and by the representation formula (2.3), for every ' 2 Cc1 .RN / we have ˆ ˆ ˆ N ' D .F  '/ d D ' d: RN

RN

RN

The original approach in the 1920s and 1930s to investigate the Poisson equation with measure data using the Newtonian potential was later superseded by the formulation in the sense of distributions in the 1940s thanks to the flexibility and greater generality of the latter. We now verify that every solution of the Poisson equation with nonnegative measure datum equals the Newtonian potential modulo a harmonic function: Proposition 2.13. Let N > 3, and let  2 M./ be a nonnegative measure. If u 2 L1loc ./ is a solution of the Poisson equation with density , then there exists a harmonic function hW  ! R such that u D N C h almost everywhere in .

30

2. Poisson equation

Such a decomposition goes back to F. Riesz’s characterization of superharmonic functions [296], and is a consequence of the following regularity result proved independently by Wiener (p. 13 in [343]) and Weyl (Lemma 2 in [340]): Proposition 2.14. Let u 2 L1loc ./. If, for every ' 2 Cc1 ./, we have ˆ u ' D 0; 

then there exists a harmonic function uW N  ! R such that uN D u almost everywhere in . A function u 2 L1loc ./ satisfying the integral identity above is called weakly harmonic. According to Weyl’s lemma above, the notions of weakly harmonic functions and harmonic functions coincide up to negligible sets. The two main ingredients are the mean value identity (1.4) for harmonic functions and the Bôcher–Koebe lemma (Proposition 1.6). Proof of Proposition 2.14. To simplify the proof, we assume that  D RN (cf. proof of Proposition 2.18). For any  2 Cc1 .RN /, we have   u 2 C 1 .RN / and ˆ .  u/.x/ D ./  u.x/ D .x y/u.y/ dy; RN

for every x 2 RN . Since the function RN 3 y 7! .x by the assumption on u we deduce that

y/ belongs to Cc1 .RN /,

.  u/.x/ D 0: Thus,   u is harmonic in RN . In particular,   u enjoys the mean value property for harmonic functions. We now apply this identity to a sequence of mollifiers .n /n2N , and write, for every ball B.xI r/ and every n 2 N, n  u.x/ D

B.xIr /

n  u:

Since the sequence .n  u/n2N converges to u in L1 .B.xI r//, by this mean value identity the sequence .n  u/n2N converges pointwise to the function uN r W RN ! R defined for x 2 RN by

uN r .x/ D

u: B.xIr /

In particular, uN r is continuous. By the partial converse of the dominated convergence theorem (Proposition 4.9), passing if necessary to a subsequence, .n  u/n2N also

2.2. Distributional solutions

31

converges almost everywhere to u. Thus, u D uN r almost everywhere in RN , and then, for every x 2 RN , we deduce that uN r .x/ D

B.xIr /

uN r :

For every r; s > 0, we have uN r D uN s in RN . Indeed, both functions are continuous in RN and coincide almost everywhere with u in RN . Denoting by uN this common function, we now have, for every x 2 RN and every r > 0, u.x/ N D

B.xIr /

u: N

By the Bôcher–Koebe lemma, uN 2 C 1 .RN /. Finally, by the characterization of the Laplacian as an infinitesimal difference of averages (Proposition 1.3), uN is harmonic. Alternatively, for every ' 2 Cc1 .RN /, we have ˆ ˆ ˆ uN ' D uN ' D u ' D 0; RN

RN

RN

whence uN D 0 in RN , and the conclusion follows.



Proof of Proposition 2.13. Since u and N both satisfy the Poisson equation with density , for every ' 2 Cc1 ./ we have ˆ .u N/ ' D 0: 

By Weyl’s lemma, the function u N is equal almost everywhere to some harmonic function in .  A simple sufficient criterion to decide whether the Newtonian potential is finite at some given point is the following: Proposition 2.15. Let N > 3, and let  2 M.RN / be a nonnegative measure. If x 2 RN is such that, for every r > 0, .B.xI r// 6 C r s for some constant C > 0 and some exponent s > N

2, then N.x/ < C1.

The proof is a consequence of the following representation of the Newtonian potential in terms of the total charge of balls, see §12 bis in [38] and §46 in [139], based on Cavalieri’s principle, see Proposition 1.7:

32

2. Poisson equation

Lemma 2.16. Let N > 3, and let  2 M.RN / be a nonnegative measure. For every x 2 RN , we have ˆ 1 .B.xI r// dr 1 : N.x/ D N 0 rN 2 r Proof of Lemma 2.16. Applying Cavalieri’s principle to the function y 7! we have ˆ ˆ 1 ° ± d.y/ 1 N D > t dt:  y 2 R W yjN 2 jx yjN 2 RN jx 0

1 jx yjN

2

,

Using the change of variables t D 1=r N 2 , we identify the set where  is integrated as the open ball B.xI r/. We then get ˆ 1 ˆ d.y/ 1 D .N 2/ .¹y 2 RN W jx yj < rº/ N 1 dr N 2 yj r 0 RN jx ˆ 1 .B.xI r// dr; D .N 2/ rN 1 0 and this gives the conclusion.



Proof of Proposition 2.15. Using Cavalieri’s representation of N.x/, we apply the density estimate for small balls and the finiteness of the measure for large balls as follows: ˆ 1 ˆ 1 .B.xI r// .B.xI r// N N.x/ D dr C dr N 1 r rN 1 0 1 ˆ 1 ˆ 1 dr s .N 1/ N : 6C r dr C .R / N 1 0 1 r Since s > N

2 and N > 2, the right-hand side is finite and the conclusion follows. 

Exercise 2.7 (Cavalieri’s representation of M. Riesz’s potentials). Let  2 M.RN / be a nonnegative measure. For every 0 < ˛ < N and every x 2 RN , prove that ˆ ˆ 1 d.y/ .B.xI r// dr : D .N ˛/ N ˛ yj rN ˛ r RN jx 0 The counterpart of the Newtonian potential in dimension two for nonnegative measures  2 M.R2 / with compact support is ˆ 1 1 N.x/ D d.y/: log 2 R2 jx yj

2.3. Superharmonic functions

33

Because of the signed kernel, we need to be more careful to get a representation in the spirit of Lemma 2.16. A trick to fall back to the case of positive kernels consists in writing the Newtonian potential for some d > 0 as 1 N.x/ D 2

ˆ

d

log

jx

R2

yj

log d .R2 /: 2

d.y/

For every d > 0 such that supp   B.xI d /, we have 1 N.x/ D 2

ˆ

log

B.xId /

d jx

yj

d.y/

log d .R2 /: 2

By Cavalieri’s principle, we deduce that 1 N.x/ D 2

ˆ

d

.B.xI r// 0

dr r

log d .R2 /: 2

(2.4)

Proceeding as in the previous proof, we conclude that N.x/ < C1 at points satisfying the density estimate .B.xI r// 6 C r s for some s > 0.

2.3 Superharmonic functions In the formalism of distributions, we may weaken the notion of superhamonic functions to allow merely summable or locally summable functions: Definition 2.17. A function u 2 L1loc ./ is superharmonic if, for every nonnegative function ' 2 Cc1 ./, we have ˆ

u ' > 0:



Exercise 2.8 (L1 stability). Prove that if .un /n2N is a sequence of superharmonic functions in  converging to u in L1 ./, then u is also superharmonic in . The mean value property (Proposition 1.4) is inherited by functions that are superharmonic in the sense of distributions. The argument has some similarities with Weyl’s lemma (Proposition 2.14), and in this case we provide the proof in full generality.

34

2. Poisson equation

Proposition 2.18. Let u 2 L1loc ./. If for every nonnegative function ' 2 Cc1 ./ ˆ u ' > 0; 

then, for every x 2 , the function .0; d.x; @// 3 r 7 !

u B.xIr /

is non-increasing. Proof. Given an open set ! b , take a nonnegative function  2 Cc1 .RN / such that ! supp  b . We first prove that the smooth function   u is superharmonic in !. Indeed, for every x 2 ! we have ˆ .  u/.x/ D ./  u.x/ D .x y/u.y/ dy: 

The function y 2 RN 7! .x y/ is smooth and supported in x supp , which is a compact subset of  for every x 2 !. By assumption on u, we deduce that .  u/.x/ 6 0; whence   u is superharmonic in !. From the mean value property for smooth superharmonic functions (Proposition 1.4), it follows that, for every x 2 ! and every 0 < s 6 r < d.x; @!/, we have u6   u: B.xIr /

B.xIs/

We now apply this inequality to a sequence of mollifiers .n /n2N , and let n ! 1 to deduce that u6 u: B.xIr /

B.xIs/

Since ! b  is arbitrary, the conclusion follows.



We now prove that every superharmonic function is the potential associated to some density of nonnegative charges: Proposition 2.19. Let u 2 L1loc ./. If u is superharmonic, then there exists a nonnegative measure  2 Mloc ./ such that u D  in the sense of distributions in .

2.3. Superharmonic functions

35

This property is a consequence of the more general Schwartz’s characterization of nonnegative distributions [303], whose main ingredient is the Riesz representation theorem (Proposition 2.9): Proposition 2.20. If F W Cc1 ./ ! R is a linear functional such that F > 0 in the sense of distributions in , then there exists a nonnegative measure  2 Mloc ./ such that, for every ' 2 Cc1 ./, we have ˆ F .'/ D ' d: 

2 Cc1 ./, for every Proof of Proposition 2.20. Given a nonnegative function 1 bounded function ' 2 C ./ the function .k'kL1 ./ '/ is admissible in the inequality F > 0. By the linearity of F , we then have F .' / 6 F . /k'kL1 ./ : Since this property holds for every bounded function ' 2 C 1 ./, we have jF .' /j 6 F . /k'kL1 ./ : Given an open subset ! b , we choose 2 Cc1 ./ such that D 1 in !. For every ' 2 Cc1 .!/, we then have ' D ' and, in addition, k'kL1 ./ D k'kL1 .!/ . Therefore, jF .'/j 6 F . /k'kL1 .!/ :

The linear functional F thus has a unique extension as a continuous linear functional on Cc0 .!/. By the Riesz representation theorem (Proposition 2.9), there exists a unique finite measure ! on ! such that, for every ' 2 Cc1 .!/, we have ˆ F .'/ D ' d! : !

The measure ! is nonnegative as a consequence of the inner regularity of finite measures (Proposition 2.5). Indeed, given a compact set K  !, let .'n /n2N be a bounded nonnegative sequence in Cc1 .!/ converging pointwise to the characteristic function K . For every n 2 N, we have ˆ 'n d! D F .'n / > 0: !

Letting n ! 1, the dominated convergence theorem yields ˆ ! .K/ D K d! > 0: !

By the inner regularity of ! , we deduce that ! > 0 on all Borel subsets of !.

36

2. Poisson equation

Taking an increasing sequence of open sets .!n /n2N strictly contained in  such S !n , we define the Borel set function  on every Borel set A   by that  D n2N

.A/ D lim !n .A \ !n /: n!1

The above limit is finite or diverges to infinity due to the monotonicity of the sequence. Moreover, for every set A strictly contained in , the limit above becomes stationary, whence .A/ D !n .A/ for some n 2 N sufficiently large. In particular,  is a locally finite measure on , and represents the functional F in the sense of distributions in .  Proof of Proposition 2.19. The functional F defined for ' 2 Cc1 ./ by ˆ F .'/ D u ' 

is nonnegative by the superharmonicity of u. The conclusion follows from Schwartz’s characterization of nonnegative distributions.  We now prove the following counterpart of the quantitative maximum principle (Proposition 1.8) for solutions of the Poisson equation with nonnegative measure data: Proposition 2.21. Let  2 Mloc ./, and let K   be a connected compact set. If u 2 L1loc ./ satisfies the Poisson equation with density , and if  is nonnegative, then, for almost every x 2 K, we have u.x/ > ess inf u C c.K/; 

for some constant c > 0 depending on  and K. We can approximate the solution u by convolution without interfering with the equation, as long as we stay away from the boundary: Lemma 2.22. Let  2 Mloc ./. If u 2 L1loc ./ satisfies the Poisson equation with density , then, for every open set ! b  and every function  2 Cc1 .RN / such that ! supp  b , we have .  u/ D    in !. Proof of Lemma 2.22. The function   u is smooth in ! and, for every x 2 !, we have ˆ .  u/.x/ D ./  u.x/ D .x y/u.y/ dy: 

2.3. Superharmonic functions

37

The function RN 3 y 7! .x y/ is well defined and smooth, and is supported in the compact subset x supp  of . Since u satisfies the Poisson equation with density , we deduce that ˆ .  u/.x/ D .x y/ d.y/ D   .x/: 

This concludes the proof of the lemma.



Proof of Proposition 2.21. Let ! b  be an open subset containing K, and let .n /n2N be a sequence of mollifiers such that ! supp n b . By the previous lemma and the nonnegativity of , the function n  u is smooth and superharmonic in !. By the quantitative maximum principle (Proposition 1.8), for every connected compact subset L  ! there exists c > 0 such that, for every x 2 L and every n 2 N, ˆ n  u.x/ > inf .n  u/ C c n  : !

L

Note that inf .n  u/ > ess inf u: !



Cc0 .!/

Moreover, for every function  2 such that 0 6  6 1 in !, we have ˆ ˆ n   > .n  / : supp 

!

Taking  with connected support and L D supp , we then get ˆ n  u.x/ > ess inf u C c .n  / : 

!

The sequence .n  /n2N converges weakly to  in the sense of measures on  (cf. proof of Proposition 2.7). Passing to a subsequence if necessary, we may assume that the sequence .n u/n2N converges almost everywhere to u in ! (Proposition 4.9). Thus, for almost every x 2 supp , we have ˆ u.x/ > ess inf u C c  d: 

!

Taking  D 1 on K, by nonnegativity of the measure  the conclusion follows.



Chapter 3

Integrable versus measure data

“I have nostalgic memories from the long days we spent together working on the big tables outside the Memorial Union, facing the inspiring view of Lake Mendota, when we discovered, sitting at our table next to the lake, that u C u3 D ıa has no solution in R3 .” Haïm Brezis

We introduce the linear Dirichlet problem ´ u D  in ; uD0

on @;

for finite Borel measures , and then explain a major difference concerning the existence of solutions between linear and nonlinear Dirichlet problems with L1 or measure data.

3.1 Linear Dirichlet problem Test functions with compact support are unable to detect boundary values of solutions of the Poisson equation. To study solutions of the linear Dirichlet problem ´ u D  in , uD0

on @,

with zero boundary condition, we adopt the notion of weak solution introduced by Littman, Stampacchia, and Weinberger, see Definition 5.1 in [213] or Définition 9.1 in [316]. For this purpose, we enlarge the class of admissible test functions, replacing Cc1 ./ by x D ¹ 2 C 1 ./W x  D 0 on @º: C01 ./

40

3. Integrable versus measure data

Definition 3.1. Let  be a bounded open set, and let  2 M./. A function uW  ! R is a solution of the linear Dirichlet problem with density  if (i) u 2 L1 ./,

x (ii) for every  2 C01 ./,

ˆ



u  D

ˆ

 d: 

To motivate such a choice of test functions, observe that if  is a smooth bounded x satisfies, for every  2 C01 ./, x the integral identity open set and if u 2 C 1 ./ ˆ ˆ u  D  u; 



then by the divergence theorem we have ˆ @ u d D 0: @n @ x such that @ D , and we deduce For every  2 C 1 .@/, there exists  2 C01 ./ @n that u D 0 on @. The boundary datum is thus implicitly encoded in this weak formulation, since our test functions are allowed to have a nontrivial normal derivative on the boundary. The Sobolev spaces W01;q ./ are introduced in the next chapter for every exponent 1 6 q < C1, and provide a natural setting to give a meaning to boundary data in the sense of traces (cf. Section 15.1). An alternative approach to define a solution of the Dirichlet problem is thus to require that (i0 ) u 2 W01;1 ./, (ii0 ) for every ' 2 Cc1 ./, ˆ



ru
r' D

ˆ

' d:



The equivalence between the two notions will be discussed later (Proposition 6.3). Another strategy to detect the zero boundary data based on averages of the solution near the boundary, avoiding the use of Sobolev spaces, is studied in Section 20.1. The definition based on the Sobolev space W01;1 ./ has the advantage of making more transparent the meaning of the boundary condition in the weak formulation, and might look more appealing to those who are familiar with variational methods in the setting of the Hilbert space W01;2 ./. A disadvantage is that one has to make sure each time that ru 2 L1 ./ and that u 2 W01;1 ./. Questions concerning the regularity of solutions of the Dirichlet problem with measure data are investigated in Chapter 5.

3.1. Linear Dirichlet problem

41

We first establish the existence of solutions for any measure data: Proposition 3.2. Let  be a bounded open set. For every  2 M./, the linear Dirichlet problem with density  has a solution u such that kukL1 ./ 6 C kkM./ ; for some constant C > 0 depending on . The proof relies on the Riesz representation theorem in the setting of continuous linear functionals in Lp spaces, see Theorem 4.11 in [53], Theorem 6.15 in [134], or Theorem 5.4.3 in [345]: Proposition 3.3. Let .XI / be a measure space, and let 1 6 p < C1. If the linear functional T W Lp .XI / ! R is such that jT .f /j 6 C kf kLp .XI/ ; 0

for every f 2 Lp .XI /, then there exists a unique function g 2 Lp .XI / such that ˆ T .f / D fg d; X

for every f 2 Lp .XI /. In addition, we have kgkLp0 .XI/ D kT k.Lp .XI//0 : The conjugate exponent p 0 is given for p > 1 by p0 D

p p

1

;

and satisfies the identity p1 C p10 D 1: When p D 1, we take p 0 D C1. The estimate involving the norm of the linear functional T has an independent proof, and the argument can be used to improve the integrability of a summable function satisfying some functional inequality: Exercise 3.1 (Lp boundedness). Let 1 6 p < C1. Prove that if f 2 L1 .XI / is such that, for every g 2 L1 .XI /, ˇˆ ˇ ˇ ˇ ˇ fg d ˇ 6 C kgk p0 L .XI/ ; ˇ ˇ X

then we have f 2 Lp .XI / and kf kLp .XI/ 6 C .

42

3. Integrable versus measure data

Another fundamental existence tool from functional analysis is the Hahn–Banach theorem, see Corollary 1.2 in [53]. We state it for arbitrary normed vector spaces, although it will be applied below in the setting of the separable Lp Lebesgue spaces for some exponent p < C1. Proposition 3.4. Let E be a normed vector space, and let V  E be a vector subspace. For every continuous linear functional LW V ! R, there exists a continuous x E ! R such that linear extension LW x E 0 D sup ¹L.v/W v 2 V and kvk 6 1º: kLk We also need the following estimate (Lemma 5.2) due to Stampacchia in the spirit of the De Giorgi and Nash continuity property of solutions of elliptic PDEs: given x we have N < r < C1, then, for every  2 C01 ./, kkL1 ./ 6 C kkLr ./ :

(3.1)

We postpone the proof of this inequality to Section 5.1. This estimate should be compared to the easier critical case r D C1: kkL1 ./ 6 C 0 kkL1 ./ ; which follows from the weak maximum principle (Corollary 1.10). The latter is not enough on establishing the existence of an L1 solution since the counterpart of the Riesz representation theorem for p D C1 is false, see p. 102 in [53]. Proof of Proposition 3.2. Given N < r < C1, denote by V the following vector subspace of Lr ./: x V D ¹W  2 C01 ./º; x by and consider the linear functional T W V ! R defined for every  2 C01 ./ ˆ T ./ D  d: 

By the weak maximum principle (Corollary 1.10), T is well defined. By Stampacx we have chia’s estimate (3.1), for every  2 C01 ./ jT ./j 6 kkM./ kkL1 ./ 6 C kkM./ kkLr ./ :

(3.2)

Applying the Hahn–Banach theorem, we deduce that T has a continuous linear extension Tx W Lr ./ ! R. By the Riesz representation theorem (Proposition 3.3), 0 there exists u 2 Lr ./ such that, for every f 2 Lr ./, we have ˆ Tx .f / D uf: 

3.1. Linear Dirichlet problem

43

Restricting this identity to elements in the subspace V , we deduce that, for every x we have  2 C01 ./, ˆ ˆ x  d D T ./ D u : 



Therefore, u satisfies the linear Dirichlet problem with density . By the Riesz representation theorem, we also have kTx k.Lr .//0 D kukLr 0 ./ . Thus, the solution u we have found satisfies kukLr 0 ./ 6 C kkM./ ; whence by the Hölder inequality we have the L1 estimate of u.



We now prove uniqueness of solutions of the Dirichlet problem for smooth bounded domains: Proposition 3.5. Let  be a smooth bounded open set, and let  2 M./. If u1 and u2 satisfy the linear Dirichlet problem with density , then u1 D u2 almost everywhere in . The proof relies on the classical existence and regularity theory of smooth solutions of the Dirichlet problem for the Poisson equation, see Appendix B in [118] or Chapter 6 in [146]. More precisely, if  is a smooth bounded open set, then, for every x the Dirichlet problem h 2 C 1 ./, ´ v D h in , vD0

on @,

x has a solution v 2 C01 ./. Proof of Proposition 3.5. By the linearity of the Laplacian, the function u1 x satisfies, for every  2 C01 ./, ˆ .u1 u2 /  D 0:

u2



x we have Thus, for every h 2 C 1 ./, ˆ .u1 

u2 /h D 0:

Applying this identity to a bounded sequence of functions .hn /n2N converging pointwise to sgn .u1 u2 /, it follows from the dominated convergence theorem that ˆ ju1 u2 j D 0; 

and we have the conclusion.



44

3. Integrable versus measure data

It is also possible to establish the existence of solutions of the Dirichlet problem with measure data based on the integral representation in terms of the Green function on domains with some smoothness properties, see Chapter 1 in [226]. In this case, the solution with density  can be written as ˆ u.x/ D G.x; y/ d.y/; 

where, for every x 2 , the Green function G.x;
/ is the solution of the linear Dirichlet problem ´ G.x;
/ D ıx in , G.x;
/ D 0

on @.

3.2 Nonlinear Dirichlet problem The nonlinear Dirichlet problem ´ u C g.u/ D  uD0

in , on @,

behaves differently according to whether  is an L1 function or a finite measure. The meaning of solution is an adaptation of the linear case: Definition 3.6. Let gW R ! R be a continuous function, let  be a bounded open set, and let  2 M./. A function uW  ! R is a solution of the nonlinear Dirichlet problem with density  if (i) u 2 L1 ./ and g.u/ 2 L1 ./, x (ii) for every  2 C 1 ./, 0

ˆ



u  C

ˆ



g.u/ D

ˆ

 d: 

To give a flavor of what happens in this case, let us momentarily consider the case where the nonlinearity g is nondecreasing, treated by Brezis and Strauss, see Theorem 1 in [69]: Proposition 3.7. Let  be a smooth bounded open set, and let gW R ! R be a nondecreasing continuous function. For every  2 L1 ./, the nonlinear Dirichlet problem with nonlinearity g and density  has a solution.

3.2. Nonlinear Dirichlet problem

45

The proof is based on the existence of variational solutions of the nonlinear Dirichlet problem from the next chapter. More precisely, combining Propositions 4.21, 4.23, and 4.24 below, we deduce that, for every  2 L2 ./, there exists u 2 W01;2 ./ such that g.u/ 2 L2 ./, and, for every v 2 W01;2 ./, we have ˆ



ru
rv C

ˆ



g.u/v D

ˆ

v: 

In particular, u is a solution of the nonlinear Dirichlet problem in the sense of x  W 1;2 ./ and, for every  2 C 1 ./, x Definition 3.6 since C01 ./ 0 0 ˆ



ru
r D

ˆ

u :



The second ingredient concerns an estimate for variational solutions of the linear Dirichlet problem (Proposition 4.6), which is related to the contraction property of solutions of the nonlinear problem with absorption (cf. Proposition 21.5): Lemma 3.8. Let  be a smooth bounded open set. If u 2 W01;2 ./ satisfies the linear Dirichlet problem with density  2 L2 ./, then ˆ  sgn u > 0: 

Proof of Lemma 3.8. For every v 2 W01;2 ./ we have ˆ



v D

ˆ



ru
rv:

On the other hand, given a smooth function H W R ! R such that H.0/ D 0 and H 0 is bounded, then H.u/ 2 W01;2 ./ (Exercise 4.6). We may then use H.u/ as test function, and get ˆ ˆ H.u/ D H 0 .u/jruj2 : 



Thus, for a nondecreasing function H , the left-hand side is nonnegative: ˆ H.u/ > 0: 

To conclude, it suffices to take a bounded sequence of functions .Hn /n2N as above converging pointwise to the sign function in R. 

46

3. Integrable versus measure data

Proof of Proposition 3.7. Let .n /n2N be a sequence of functions in L2 ./ converging strongly to  in L1 ./, and let un 2 W01;2 ./ be a solution of the nonlinear Dirichlet problem with density n . For every m; n 2 N, we subtract the equation satisfied by un from the equation satisfied by um , and we get un / D .m

.um

By Lemma 3.8, we then have ˆ Œg.um / g.un / sgn .um

n /

un / 6



ˆ

Œg.um /

g.un /:

n / sgn .um

.m

un /:



Using the monotonicity of g, we deduce the contraction estimate: kg.um /

g.un /kL1 ./ 6 km

(3.3)

n kL1 ./ :

Since .n /n2N is a Cauchy sequence in L1 ./, this implies that .g.un //n2N is also a Cauchy sequence in L1 ./. Thus, .g.un //n2N converges in L1 ./. We now deduce the convergence of the sequence .un /n2N in L1 ./ as a consequence of the linear estimate (Proposition 3.2). Indeed, for every m; n 2 N, we have kum

un kL1 ./ 6 C kum 6 C kg.um /

un kL1 ./ g.un /kL1 ./ C C km

n kL1 ./ :

Thus, kum

un kL1 ./ 6 2C km

n kL1 ./ :

In particular, .un /n2N is also a Cauchy sequence in L1 ./, and so it converges in L1 ./ to some function u, and .g.un //n2N converges in L1 ./ to g.u/. Since, for x we have every n 2 N and every  2 C01 ./, ˆ ˆ ˆ un  C g.un / D n ; 





as n ! 1 we deduce that u solves the nonlinear Dirichlet problem with density  2 L1 ./.  The contraction estimate (3.3) relies on the monotonicity of the nonlinearity g. There is an alternative argument, due to Gallouët and Morel [144], to prove the L0 convergence of .g.un //n2N for nonlinearities merely satisfying the sign condition: for every t 2 R, g.t /t > 0:

3.2. Nonlinear Dirichlet problem

47

Their strategy consists in showing that the sequence .g.un //n2N is equi-integrable: for every  > 0, there exists ı > 0 such that, for every Borel set E   verifying jEj 6 ı, we have ˆ jg.un /j 6 : E

We pursue this approach in Section 19.1. One might hope to use the same kind of argument to solve the nonlinear Dirichlet problem for an arbitrary finite measure , but this is not possible. Indeed, the approximation of a measure by functions can be done in the weak sense of measures, but not strongly (Exercise 2.5). Moreover, any attempt to bypass this obstruction fails, since there are measures  for which the nonlinear Dirichlet problem does not have a solution. The first example of such a surprising phenomenon was discovered by Bénilan and Brezis (see Remark A.4 in [24] and Théorème 1 in [51]), and shows a major difference between the nonlinear L1 and measure theories: Proposition 3.9. Let N > 3 and a 2 . If p > u C jujp

1

N , N 2

then the nonlinear equation

u D ıa

has no solution in the sense of distributions in . Proof. We assume for simplicity that a D 0 and  is the unit ball B.0I 1/ centered at 0. Suppose by contradiction that the equation has a solution. Given ' 2 Cc1 .RN / such that supp '  B.0I 1/ and n 2 N , let 'n W B.0I 1/ ! R be the function defined for x 2 B.0I 1/ by 'n .x/ D '.nx/:

Using 'n as a test function, we have ˆ ˆ u 'n C jujp B.0I1/

B.0I1/

1

u'n D

By the dominated convergence theorem, ˆ jujp lim n!1 B.0I1/

Moreover, ˆ

B.0I1/

u 'n D

ˆ

1 B.0I n /

1

ˆ

B.0I1/

'n dı0 D '.0/:

u'n D 0:

2

u 'n D n

ˆ

1 B.0I n /

u.x/ '.nx/ dx:

(3.4)

48

3. Integrable versus measure data

By the Hölder inequality and by a change of variable, we deduce that ˇˆ ˇ ˇ ˇ

B.0I1/

ˇ ˆ ˇ u 'n ˇˇ 6 n2

p

1 / B.0I n

2

Dn

N p0

juj

ˆ

p

lim

juj

N , N 2

n!1 B.0I1/

p0

1 B.0I n /

1 B.0I n /

Note that 2 pN0 6 0 if and only if p > theorem we get ˆ

 p1  ˆ

 p1  ˆ

j'.nx/j dx

B.0I1/

p0

j'j

 10



1 p0

p

:

and thus by the dominated convergence

u 'n D 0:

As n ! 1, we deduce from (3.4) that '.0/ D 0. To get a contradiction, it thus suffices to take from the beginning a test function ' such that '.0/ ¤ 0. 

Chapter 4

Variational approach

“. . . le condizioni particolarissime in cui l’Hilbert tratta il problema di Dirichlet appaiono elemento integrante delle sue deduzioni e pare lascino ben poca speranza che con ragionamenti analoghi possa trattarsi, senza profonde modificazioni, il problema generale.”1 Beppo Levi

We prove the existence of variational solutions of the Dirichlet problem ´ u C g.u/ D  in , uD0

on @,

when the nonlinearity g satisfies the sign condition and the density  belongs to the dual Sobolev space .W01;2 .//0 .

4.1 Sobolev spaces Before starting with the variational problem, we explain the setting where the energy functional is minimized: the Sobolev space W01;2 ./. More generally, we consider Sobolev spaces associated to any exponent 1 6 q < C1 as follows: Definition 4.1. Let  be a bounded open set, and let u 2 Lq ./. We say that u belongs to the Sobolev space W01;q ./ if there exists G 2 Lq .I RN / such that, x RN /, we have for every ˆ 2 C 1 .I ˆ ˆ u div ˆ D G
ˆ: 



1 “The very special conditions in which Hilbert studies the Dirichlet problem seem to be an essential part of the argument, and leave little hope that the general problem could be handled without substantial changes.”

50

4. Variational approach

Both integrands are summable over  since the domain is assumed to be bounded. Two functions G1 ; G2 2 Lq .I RN / satisfying this identity for the same function u are equal almost everywhere in , see Corollary 4.24 in [53] or Theorem 4.3.10 in [345]: this important property is sometimes called the fundamental theorem of the calculus of variations. We systematically use the notation ru D

G

for the weak gradient. We then recover the formula ˆ ˆ u div ˆ D ru
ˆ 



x vanishing on which follows from the divergence theorem for functions u 2 C01 ./ the boundary of a smooth bounded open set . The Sobolev space W01;q ./ is well suited to study minimization problems and to give a meaning to weak formulations of Dirichlet problems involving a zero boundary condition. The reason is that it enjoys two fundamental properties: it is complete with respect to its natural norm satisfying q q kukqW 1;q ./ D kukL q ./ C krukLq ./ ;

(4.1)

and is sensitive to boundary conditions. Exercise 4.1 (zero boundary datum). Let  be a smooth bounded open set, and let x Prove that u 2 W 1;q ./ if and only if u D 0 on @. u 2 C 1 ./. 0

The completeness of the Sobolev spaces relies on a stability property satisfied by sequences of Sobolev functions. Proposition 4.2. For every bounded open set , W01;q ./ is a complete metric space. Proof. Given a Cauchy sequence .un /n2N in W01;q ./, the sequence .un /n2N is Cauchy in Lq ./ and the sequence .run /n2N is Cauchy in Lq .I RN /, and so they converge to u 2 Lq ./ and to F 2 Lq .I RN /, respectively. To conclude the proof, we show that (4.2) u 2 W01;q ./ and ru D F 1 x N using the stability property: for every ˆ 2 C .I R / and every n 2 N, we have ˆ ˆ un div ˆ D run
ˆ; 

whence, as n ! 1, we get

ˆ





u div ˆ D

ˆ



F
ˆ:

Thus, assertion (4.2) holds, and the conclusion follows.



4.1. Sobolev spaces

51

The stability property provides one with examples of Sobolev functions which need not be differentiable: Exercise 4.2. Prove that the function uW B.0I 1/ ! R defined for x 2 B.0I 1/ by u.x/ D 1

jxj

belongs to W01;q .B.0I 1// for every exponent 1 6 q < C1, and ru.x/ D almost everywhere in B.0I 1/.

x=jxj

Exercise 4.3. Let ˛ > 0. Prove that the function uW B.0I 1/ ! R defined for x ¤ 0 by u.x/ D

1 jxj˛

1

belongs to W01;q .B.0I 1// for every exponent q such that q.˛ C 1/ < N . The convolution product gives a convenient way to study properties of Sobolev functions in W01;q ./ via approximation by smooth functions with compact support in RN : Proposition 4.3. Let  be a bounded open set. If u 2 W01;q ./, then, for every  2 Cc1 .RN /, we have   u 2 Cc1 .RN / and r.  u/ D   ru in RN . Proof. For every x 2 RN , we have   u.x/ D

ˆ

.x

y/u.y/ dy:



Thus,   u 2 Cc1 .RN /, and by differentiation under the integral sign we get ˆ ˆ r.  u/.x/ D rx .x y/u.y/ dy D ry .x y/u.y/ dy: 



For every e 2 RN , we have e
r D div .e/. Thus, by the linearity of the integral and by the definition of the weak gradient of u we get ˆ e
r.  u/.x/ D divy .e/.x y/ u.y/ dy 

D

ˆ

.x 

y/ e
ru.y/ dy

D e
.  ru/.x/: Since this identity holds for every e 2 RN , we have the conclusion.



52

4. Variational approach

Exercise 4.4 (constant Sobolev functions). Prove that if u 2 W01;q ./ is such that ru D 0, then u D 0 almost everywhere in . Exercise 4.5 (Leibniz rule). Prove that, for every u; v 2 W01;q ./ \ L1 ./, we have uv 2 W01;q ./ and r.uv/ D ru v C u rv: Exercise 4.6 (chain rule). Let H 2 C 1 .R/. Prove that if H.0/ D 0 and H 0 is bounded, then, for every u 2 W01;q ./, we have H.u/ 2 W01;q ./ and rH.u/ D H 0 .u/ru: In smooth domains , the approximation of Sobolev functions can be performed using functions compactly supported in : Proposition 4.4. For every smooth bounded open set , the set Cc1 ./ is dense in W01;q ./. We recover in this case the usual approach to define the space W01;q ./ as the completion of Cc1 ./ with respect to the Sobolev norm (4.1). We take this statement for granted. Proposition 4.4 can be proved combining the characterization of the kernel of the trace operator (see Theorem 6.6.4 in [187]) with Exercise 15.1 below. Note however that if  is a ball, then Proposition 4.4 has a straighforward proof based on scaling: Exercise 4.7 (approximation with compact support). For every u 2 W01;q .B.0I 1//, prove that there exists a sequence .un /n2N in the set Cc1 .B.0I 1// which converges to u in W01;q .B.0I 1//. The fact that nonzero constant functions cannot belong to W01;q ./ is quantified by the Poincaré inequality, see Corollary 9.19 in [53] or Theorem 6.4.7 in [345]: Proposition 4.5. Let  be a bounded open set. Then, for every u 2 W01;q ./, we have kukLq ./ 6 C krukLq ./ ; for some constant C > 0 depending on the diameter diam . Proof. Let Q.aI r/ be a cube such that  b Q.aI r/. Using the fundamental theorem of calculus and the Hölder inequality, one shows that, for every ' 2 Cc1 .Q.aI r//, k'kLq .Q.aIr // 6 rkr'kLq .Q.aIr // :

4.1. Sobolev spaces

53

Given a sequence of mollifiers .n /n2N in Cc1 .RN / such that Csupp n b Q.aI r/, we have n  u 2 Cc1 .Q.aI r//. By the inequality above for smooth functions and by Proposition 4.3, we then have kn  ukLq .Q.aIr // 6 rkn  rukLq .Q.aIr // : Letting n ! 1, the conclusion follows.



Taking advantage of the Hilbert space structure of W01;2 ./, we now establish the existence of solutions of the linear Dirichlet problem for the Poisson equation for every density  in the dual space .W01;2 .//0 : ´

u D  uD0

in , on @.

The main ingredient is the Fréchet–Riesz representation theorem in Hilbert spaces, see Theorem 5.5 in [53] or Theorem 5.3.1 in [345]. Proposition 4.6. Let  be a bounded open set. Then, for every  2 .W01;2 .//0 , there exists a unique function u 2 W01;2 ./ such that, for every z 2 W01;2 ./, we have ˆ 

ru
rz D Œz:

Proof. By the Poincaré inequality (Proposition 4.5), the bilinear form ˆ W01;2 ./  W01;2 ./ 3 .u; z/ 7 ! ru
rz 

is an inner product in W01;2 ./ and induces a norm that is equivalent to the W 1;2 norm. By the Fréchet–Riesz representation theorem in Hilbert spaces, there exists a unique function u 2 W01;2 ./ such that, for every z 2 W01;2 ./, we have ˆ Œz D ru
rz:  

A function u 2 W01;2 ./ satisfying the conclusion of Proposition 4.6 is called a (variational) solution of the Dirichlet problem in W01;2 ./ with density . If u and  are smooth functions in , then u satisfies pointwise the Poisson equation with density . Assuming in addition that u has a smooth extension to the boundary of a smooth domain , then from the integral formulation above we find that such an extension vanishes identically on @ (cf. Exercise 4.1), whence u satisfies the Dirichlet problem in the classical sense.

54

4. Variational approach

This formalism includes the case of L2 data: Example 4.7. Every function  2 L2 ./ can be interpreted as an element of the dual space .W01;2 .//0 by acting on every z 2 W01;2 ./ as ˆ Œz D z: 

Indeed, by the Hölder inequality we have z 2 L1 ./ and ˇˆ ˇ ˇ ˇ ˇ jŒzj D ˇ z ˇˇ 6 kkL2 ./ kzkL2 ./ 6 kkL2 ./ kzkW 1;2 ./ : 

Thus, the solution of the Dirichlet problem in W01;2 ./ with density  exists and x satisfies, for every  2 C01 ./, ˆ ˆ ˆ u  D ru
r D : 





Exercise 4.8 (weak maximum principle). Prove that if  2 L2 ./ is a nonnegative function, then the solution of the Dirichlet problem in W01;2 ./ with density  is also nonnegative almost everywhere. The Fréchet–Riesz representation theorem is well suited to solve linear problems. In the study of nonlinear problems, the Rellich–Kondrashov compactness theorem is an important tool to deal with bounded sequences of Sobolev functions, see Theorem 9.16 in [53], Theorem 5.7.1 in [125], or Theorem 6.4.6 in [345]: Proposition 4.8. Let  be a bounded open set. Then, for every bounded sequence .un /n2N in W01;q ./, there exists a subsequence .unk /k2N converging strongly in Lq ./. The proof relies on M. Riesz’s compactness criterion of equi-integrable sequences of functions, see Theorem 4.26 in [53] or Theorem 4.4.2 in [345], which is the counterpart in Lebesgue spaces of the classical Ascoli–Arzelà compactness theorem for continuous functions. The subsequence given by Proposition 4.8 can be assumed to converge almost everywhere. To be more precise, we may extract a further subsequence from .unk /k2N which converges almost everywhere in . This is a general property of sequences of functions converging strongly in Lebesgue spaces, and relies on the following partial converse of the dominated convergence theorem, see Theorem 4.9 in [53] or Proposition 4.2.10 in [345]:

4.1. Sobolev spaces

55

Proposition 4.9. Let .XI / be a measure space. If .fn /n2N is a sequence converging strongly to f in Lq .XI /, then there exist a subsequence .fni /i 2N and h 2 Lq .XI / such that (i) the subsequence .fni /i 2N converges to f almost everywhere in X,

(ii) for every i 2 N, we have jfni j 6 h almost everywhere in X.

It suffices to take any subsequence .fni /i 2N satisfying the condition 1 X kfni

fni

1

i D1

kLq .XI/ < C1I

by Fatou’s lemma, the function h D jfn0 j C finite almost everywhere.

1 P

i D1

jfni

fni

1

j; is well defined and

Exercise 4.9 (Lq convergence under composition). Prove that if .fn /n2N is a sequence converging strongly to f in Lq .XI /, then, for every continuous function H W R ! R such that jH.t /j 6 C jt jq ; the sequence .H.fn //n2N converges strongly to H.f / in L1 .XI /.

For every exponent q > 1, we now prove that the strong limit of the subsequence .unk /k2N given by the Rellich–Kondrashov compactness theorem also belongs to W01;q ./, see Theorem 6.1.7 in [345]. This is an elegant application of the Riesz representation theorem (Proposition 3.3): Proposition 4.10. Let  be a bounded open set. If q > 1, and if .un /n2N is a bounded sequence in W01;q ./ converging weakly to u in Lq ./, then u 2 W01;q ./ and we have krukLq ./ 6 lim inf krun kLq ./ : n!1

x RN /, by the definition of the weak Proof. For every n 2 N and every ˆ 2 C 1 .I gradient and by the Hölder inequality, we have ˇ ˇ ˇˆ ˇˆ ˇ ˇ ˇ ˇ ˇ un div ˆˇ D ˇ run
ˆˇ 6 krun kLq ./ kˆk q0 : L ./ ˇ ˇ ˇ ˇ 



Since the sequence .un /n2N converges weakly in Lq ./, as n ! 1 in the estimate above we get ˇ ˇˆ ˇ ˇ ˇ u div ˆˇ 6 .lim inf krun kLq ./ /kˆk q0 : (4.3) L ./ ˇ ˇ 

n!1

56

4. Variational approach

x RN / in Lq0 .I RN /, we deduce that the linear functional By density of C 1 .I ˆ x RN / 3 ˆ 7 ! C 1 .I u div ˆ (4.4) 

0

has a unique extension as a continuous linear functional in Lq .I RN /. Thus, by the Riesz representation theorem there exists G 2 Lq .I RN / such that, for every x RN /, we have ˆ 2 C 1 .I ˆ ˆ u div ˆ D G
ˆ: 



Therefore, u 2 W01;q ./ and ru D G. Since the norm of the linear functional (4.4) equals kGkLq ./ , by the functional estimate (4.3) we also have krukLq ./ D kGkLq ./ 6 lim inf krun kLq ./ : n!1

The proof is complete.



Exercise 4.10 (failure of the closure property). Prove that there exists a sequence .un /n2N that is bounded in the Sobolev space W01;1 .B.0I 1// and converges pointwise to 1 in B.0I 1/. Explain why such a sequence cannot be bounded in W01;q .B.0I 1// for any 1 < q < C1. The Sobolev–Gagliardo–Nirenberg inequality is another fundamental property in applications involving Sobolev spaces, see Theorem 9.9 in [53] or Lemma 6.4.2 in [345]: Proposition 4.11. If 1 6 q < N , then for every ' 2 Cc1 .RN / k'k

Nq q

LN

.RN /

6 C kr'kLq .RN / ;

for some constant C > 0 depending on N and q. The heart of the matter consists in first establishing the estimate for the exponent q D 1, which was carried out independently by Gagliardo [143] and Nirenberg [265]: k'k

N

LN

1 .RN /

6 C kr'kL1 .RN / :

(4.5)

For every ' 2 Cc1 .RN / and every ˛ > 1, we still have j'j˛ 2 Cc1 .RN /. Applying , we then deduce the case q > 1 inequality (4.5) to j'j˛ and choosing ˛ D .NN 1/q q due to Sobolev [313].

4.1. Sobolev spaces

The Sobolev exponent q  D

Nq N q

57

satisfies the identity

1 1 D  q q

1 : N

In view of the approximation property of Sobolev functions (Proposition 4.3), we have: Corollary 4.12. Let  be a bounded open set. If 1 6 q < N , then we have  W01;q ./  Lq ./ and, for every u 2 W01;q ./, kukLq ./ 6 C krukLq ./ ; for the same constant C > 0 as in Proposition 4.11. Proof. Let u 2 W01;q ./. Given a sequence of mollifiers .n /n2N in Cc1 .RN /, the Sobolev–Gagliardo–Nirenberg inequality above and Proposition 4.3 show that kn  ukLq .RN / 6 C kn  rukLq .RN / : 

As n ! 1, we deduce from Fatou’s lemma that u 2 Lq ./, and the estimate holds.  The range of Lebesgue functions  that can be interpreted as elements in the dual space .W01;2 .//0 is larger than L2 ./ (cf. Example 4.7): 2N

Example 4.13. In dimension N > 3, every function  2 L N C2 ./ acts continuously on W01;2 ./. Indeed, by the Sobolev embedding, for every z 2 W01;2 ./ we 2N have z 2 L N 2 ./. Thus, by the Hölder inequality we have that z 2 L1 ./ and ˇ ˇˆ ˇ ˇ ˇ kzk 2N 6 C kk 2N krzkL2 ./ : jŒzj D ˇ z ˇˇ 6 kk 2N L N C2 ./



LN

2

./

L N C2 ./

If u satisfies the Dirichlet problem in W01;2 ./ with density , then using the solution u itself as test function we get ˆ jruj2 D Œu 6 C kk 2N krukL2 ./ ; L N C2 ./



and this implies the estimate krukL2 ./ 6 C kk

2N

L N C2 ./

:

58

4. Variational approach

Sobolev functions with exponent q D N need not be bounded in dimension N > 2: Exercise 4.11 (unbounded W 1;N function). Let N > 2 and ˛ > 0. Prove that the function uW B.0I 1/ ! R defined for x ¤ 0 by  1 ˛ u.x/ D log jxj

belongs to W01;N .B.0I 1// for every ˛
N , Sobolev functions are bounded, and this can be obtained from the following representation formula based on the divergence theorem: Proposition 4.14. For every ' 2 Cc1 .RN / and every x 2 RN , we have ˆ 1 x y '.x/ D dy: r'.y/
N RN jx yjN Proof. Given x 2 RN , consider the smooth vector field ˆW RN n ¹xº ! RN defined for y 2 RN n ¹xº by x y : ˆ.y/ D '.y/ jx yjN Since the divergence of the vector field z 7! z=jzjN vanishes in RN n ¹0º, we have div ˆ.y/ D r'.y/


x y : jx yjN

For every  > 0, applying the divergence theorem to ˆ on the domain RN n BŒxI  we get ˆ ˆ x y x y r'.y/
dy D d .y/ ˆ.y/
N N jx yj jx yj R nBŒxI @B.xI/ ˆ 1 D N 1 ' d:  @B.xI/ Letting  ! 0, we deduce the representation formula from the dominated convergence theorem and the continuity of ' at x. 

4.1. Sobolev spaces

59

The representation formula above was used by Sobolev to establish his inequality for exponents 1 < q < N . Denoting I1 .z/ D

1 jzjN 1

;

we thus have the pointwise estimate j'j 6

1 I1  jr'j: N

(4.6)

Proposition 4.15. If q > N , then, for every ' 2 Cc1 .RN /, we have k'kL1 .RN / 6 C k'kW 1;q .RN / ; for some constant C > 0 depending on N and q. Proof. Assume that ' 2 Cc1 .B.aI r//, for some a 2 RN and r > 0. By the pointwise estimate (4.6) and the Hölder inequality, we have that 1 kI1 kLq0 .B.0I2r // kr'kLq .B.aIr // : N

k'kL1 .RN / D k'kL1 .B.aIr // 6 Since q 0 .N

0

1/ < N , we have I1 2 Lqloc .RN /. We then deduce that k'kL1 .RN / 6 C r 1

N q

kr'kLq .B.aIr // :

(4.7)

Using the approximation property (Proposition 4.3), we may apply this estimate to deduce the corollary, but the constant in this case depends on the diameter of . We need a localization argument to remove the dependence on the domain. For this purpose, we cover RN with balls .B.an I 1//n2N that are regularly distributed in RN . Using such a covering, take a partition of the unity . n /n2N such that n 2 Cc1 .B.an I 1//. We now apply estimate (4.7) to the function ' n with r D 1: k'

Choose the functions We then have

n kL1 .RN /

n

k'

6 C kr.'

such that kr n kL1 .RN /

n /kLq .B.an I1// :

n kL1 .RN /

6 C1 , independently of n.

6 C k'kW 1;q .B.an I1// :

60

4. Variational approach

By assuming that the balls are regularly distributed in RN , we have that each point 1 P x 2 RN belongs to at most N balls. Since n D 1, it follows that there exists j 2 N such that

j .x/

> 1=N . Thus,

j'.x/j 6 N j'.x/

j .x/j

nD0

6 N C k'kW 1;q .B.aj I1// ;

whence for every ' 2 of '.

k'kL1 .RN / 6 N C k'kW 1;q .RN / ;

Cc1 .RN /.

The constant N C is now independent of the support 

Using the approximation property (Proposition 4.3), we deduce the boundedness of Sobolev functions in the supercritical range: Corollary 4.16. Let  be a bounded open set. If q > N , then W01;q ./  L1 ./ and, for every u 2 W01;q ./, we have kukL1 ./ 6 C kukW 1;q ./ ; for the same constant C > 0 as in Proposition 4.15. Proof. Take a sequence of mollifiers .n /n2N in Cc1 .RN /. For every n 2 N, we have n  u 2 Cc1 .RN / and kn  ukL1 .RN / 6 C kn  ukW 1;q .RN / 6 C kukW 1;q ./ : As n ! 1, we deduce that u 2 L1 ./, and the estimate holds.



Exercise 4.13 (continuity of W01;q functions for q > N ). Let  be a bounded open set. Prove that if q > N , then, for every u 2 W01;q ./, there exists a continuous x function vW RN ! R such that v D u almost everywhere in  and v D 0 in RN n . We also consider later on the Sobolev spaces W 1;q ./ and their higher order counterpart W k;q ./ for k 2 N , whose definition follows by induction on k. Definition 4.17. A function u 2 Lq ./ belongs to the Sobolev space W 1;q ./ if there exists a weak gradient ru 2 Lq .I RN / such that, for every ˆ 2 Cc1 .I RN /, ˆ ˆ u div ˆ D ru
ˆ: 



Since this definition involves fewer test functions, the Sobolev space W 1;q ./ is typically larger than W01;q ./. Examples of Sobolev functions are provided by functions which are smooth except for some small singular set. The size of the singular set can be geometrically quantified by the Hausdorff measure HN 1 , whose definition is given in Appendix B.

4.1. Sobolev spaces

61

Proposition 4.18. Let S   be a compact set, and let u 2 C 1 . n S /. If we have HN 1 .S / D 0, and if u 2 Lq ./ and ru 2 Lq .I RN /, then u 2 W 1;q ./. The singular set S is negligible with respect to the Lebesgue measure, and the gradient ru is understood in the classical sense in  n S . We prove this proposition taking for granted the fact that sets with zero Hausdorff measure HN 1 also have zero Sobolev capacity capW 1;1 (Proposition 10.1). Lemma 4.19. If S  RN is a compact set such that HN 1 .S / D 0, then there exists a uniformly bounded sequence .'n /n2N in Cc1 .RN / such that 'n D 1 in a neighborhood of S and lim .k'n kL1 .RN / C kr'n kL1 .RN / / D 0:

n!1

Proof of Lemma 4.19. From the assumption on S , we have capW 1;1 .S / D 0. Thus, there exists a sequence .'n /n2N in Cc1 .RN / satisfying the limit above, and such that 'n > 1 in a neighborhood of S . To conclude, take a smooth bounded function H W R ! R such that H 0 is bounded, H.0/ D 0, and H.t / D 1 for t > 1. The sequence .H.'n //n2N has the required properties.  Proof of Proposition 4.18. Let .'n /n2N be a sequence in Cc1 .RN / which satisfies the conclusion of the lemma. The function uˆ.1 'n / is smooth and has compact support in  n S , for every ˆ 2 Cc1 .I RN /. Applying the divergence theorem, we get ˆ ˆ ˆ ru
ˆ.1 'n / D u div ˆ .1 'n / C u ˆ
r'n : (4.8) 





Assuming in addition that u is bounded, the last integral converges to zero as n ! 1. Since the sequence .'n /n2N is uniformly bounded, we get ˆ ˆ ru
ˆ D u div ˆ: (4.9) 



If u is not bounded, we then take a smooth bounded function T W R ! R such that T 0 is bounded, and apply identity (4.9) with u replaced by nT .u=n/ for n 2 N . Hence, ˆ ˆ 

T 0 .u=n/ru
ˆ D

nT .u=n/ div ˆ:



Assume that T 0 .0/ D 1 and T .0/ D 0. Letting n ! 1 in the identity above,  and applying the dominated convergence theorem, we deduce (4.9).

62

4. Variational approach

When  is a smooth bounded open set, it is possible to associate to every Sobolev function in W 1;q ./ a notion of boundary value (Section 15.1) in such a way that W01;q ./ coincides with the vector subspace of all functions vanishing on the boundary (Exercise 15.1). We refer the reader to the extensive literature on Sobolev spaces, e.g., [53], [125], [202], and [345], for other properties concerning approximation, embedding and extension of Sobolev functions.

4.2 Minimizers and the Euler–Lagrange equation We search for solutions of the nonlinear Dirichlet problem ´ u C g.u/ D  in , uD0

on @,

with density  2 .W01;2 .//0 by looking for a minimum point of the energy functional ˆ ˆ 1 2 E.v/ D jrvj C G.v/ Œv 2   in the Sobolev space W01;2 ./. Here, GW R ! R is the primitive function of g, defined for t 2 R by ˆ t G.t / D g.s/ ds: 0

The nonlinear term g is assumed to satisfy the following sign condition: Definition 4.20. A function gW R ! R satisfies the sign condition if, for every t 2 R, we have g.t /t > 0: In other words, g.t / and t have the same sign, whence the primitive function G is nonnegative. The existence of a minimizer of E follows from a minimization technique suggested by Hilbert [167] and [168]; see also [88]: Proposition 4.21. Let  be a bounded open set, and let gW R ! R be a continuous function satisfying the sign condition. If  2 .W01;2 .//0 , then there exists u 2 W01;2 ./ such that, for every v 2 W01;2 ./, we have E.u/ 6 E.v/:

4.2. Minimizers and the Euler–Lagrange equation

63

The direct method of the calculus of variations (see [89] and [90]) consists in taking a minimizing sequence .un /n2N – in our case a sequence in W01;2 ./ satisfying lim E.un / D

n!1

inf

v2W01;2 ./

E.v/;

and then in showing that some subsequence .unk /k2N converges to a minimizer of the functional E. To implement this strategy, we first need an estimate that guarantees that E is bounded from below in W01;2 ./ and that minimizing sequences are bounded in W01;2 ./: Lemma 4.22. Let  be a bounded open set, and let gW R ! R be a continuous function satisfying the sign condition. If  2 .W01;2 .//0 , then, for every v 2 W01;2 ./, we have 6 C.E.v/ C kk2 1;2 kvk2 1;2 /; 0 W0

.W0

./

.//

for some constant C > 0 depending on N and . Proof of Lemma 4.22. Let v 2 W01;2 ./. Since g satisfies the sign condition, we have G.v/ > 0 almost everywhere in , and ˆ 1 jrvj2 6 E.v/ C Œv: 2  By the continuity of the linear functional , we have Œv 6 AkvkW 1;2 ./ ; where A D kk.W 1;2 .//0 . By the Poincaré inequality (Proposition 4.5), we also have ˆ kvk2W 1;2 ./ 6 C1 jrvj2 : 

Combining these estimates we get 1 kvk2W 1;2 ./ 6 E.v/ C AkvkW 1;2 ./ : 2C1 For every  > 0, we have (Exercise 4.14) kvkW 1;2 ./ 6 kvk2W 1;2 ./ C

1 ; 4

and we deduce that  1 2C1

 A A kvk2W 1;2 ./ 6 E.v/ C : 4

Choosing  such that A D 1=4C1 , the conclusion follows.



64

4. Variational approach

Exercise 4.14. For every a; b 2 R and every  > 0, prove that ab 6 a2 C b 2 =4: Proof of Proposition 4.21. Let .un /n2N be a minimizing sequence in W01;2 ./. By the previous lemma, the functional E is bounded from below and the sequence .un /n2N is bounded in W01;2 ./. By the Rellich–Kondrashov compactness theorem (Proposition 4.8), there exists a subsequence .unk /k2N converging in L2 ./ to some function u. From the closure lemma (Proposition 4.10), it follows that u 2 W01;2 ./ and ˆ ˆ 2 jruj 6 lim inf jrunk j2 : k!1



By the weak continuity of  2

.W01;2 .//0 ,



we also have

Œu D lim Œunk : k!1

Applying the converse of the dominated convergence theorem (Proposition 4.9), we may assume that the subsequence .unk /k2N converges almost everywhere to u. Thus, by nonnegativity of G and by Fatou’s lemma, we have ˆ ˆ G.unk /: G.u/ 6 lim inf k!1





We conclude that E.u/ 6 lim inf E.unk / D k!1

inf

v2W01;2 ./

E.v/:

Since u 2 W01;2 ./, equality must hold, so u is a minimizer of E.



The next goal is to show that any minimizer u of the functional E satisfies the Euler–Lagrange equation in the weak sense. More precisely, we prove that g.u/ 2 L1 ./ and, for every ' 2 Cc1 ./, we have ˆ ˆ ru
r' C g.u/' D Œ': 



In the variational setting, it is convenient to deal with test functions in W01;2 ./, not necessarily smooth. Proposition 4.23. Let  be a bounded open set, and let gW R ! R be a continuous function satisfying the sign condition. If  2 .W01;2 .//0 and if u 2 W01;2 ./ is a minimizer of the functional E, then we have g.u/ 2 L1 ./ and, for every v 2 W01;2 ./ \ L1 ./, ˆ ˆ ru
rv C g.u/v D Œv: 



4.2. Minimizers and the Euler–Lagrange equation

65

The existence of solutions of the Euler–Lagrange equation has been established by Brezis and Browder [54] using an approximation argument on the density . We prove directly that minimizers of the energy functional satisfy the Euler–Lagrange equation. The proof of this proposition can be seen as a 1-dimensional calculus problem: if u is a minimizer of E, then, for every v 2 W01;2 ./, the function R 3 t 7 ! E.u C t v/

achieves its minimum at t D 0. However, we cannot directly deduce that ˇ d ˇ E.u C t v/ˇ D 0; tD0 dt

(4.10)

because we do not have enough information on the growth rate of g: the quantity E.u C t v/ could equal infinity for t ¤ 0. Proof of Proposition 4.23. We begin by showing that (4.10) holds if the test function v vanishes in sets where juj is large: Claim. The Euler–Lagrange equation is satisfied agaist every test function v 2 W01;2 ./ \ L1 ./ such that v D 0 in the set ¹juj > º for some  > 0. Proof of the claim. Let v 2 W01;2 ./ \ L1 ./. For every t 2 R , we write the differential quotient as ˆ ˆ ˆ t G.u C t v/ G.u/ E.u C t v/ E.u/ D Œv: ru
rv C jrvj2 C t 2  t   By the continuity of g, we have that G 0 D g and then, for every x 2 , lim

t!0

G.u.x/ C t v.x// t

G.u.x//

D g.u.x//v.x/:

Assuming that v D 0 in ¹juj > º for some  > 0, it follows from the mean value theorem that ˇ ˇ ˇ G.u C t v/ G.u/ ˇ ˇ ˇ 6 sup ¹jg./jW jj 6  C jt jkvkL1./ ºkvkL1./ : ˇ ˇ t

In particular, the differential quotient in the left-hand side is uniformly bounded as t ! 0. By the dominated convergence theorem, we deduce that ˆ ˆ G.u C t v/ G.u/ lim D g.u/v: t!0  t  Therefore, the function R 3 t 7 ! E.u C t v/ is differentiable at 0. Since the minimum is attained at this point, u satisfies the Euler–Lagrange equation with test function v. 4

66

4. Variational approach

Given v 2 W01;2 ./ \ L1 ./, we now approximate v by a sequence .vk /k2N satisfying the assumptions of the claim. For this purpose, take a sequence of positive numbers .˛k /k2N and a function H 2 Cc1 .R/ such that supp H  Œ 1; 1. For each k 2 N, the function vk D H.˛k u/v belongs to W01;2 ./ \ L1 ./ (cf. Exercises 4.5 and 4.6). Since vk D 0 in the set ¹juj > 1=˛k º, by the claim we get ˆ ˆ ru
rvk C g.u/vk D Œvk : 



If H.0/ D 1 and if the sequence .˛k /k2N converges to zero, then .vk /k2N converges to v in W01;2 ./. Hence, from the previous identity, it follows that ˆ ˆ lim g.u/vk D ru
rv C Œv: (4.11) k!1





Although the minimizer u itself need not belong to L1 ./, we can also apply formula (4.11) with v D u since, for every k 2 N, the function H.˛k u/u satisfies the hypotheses of the claim. To prove that g.u/ 2 L1 ./, we thus assume that v D u, and pick a nonnegative cut-off function H . By the sign condition, the sequence .g.u/vk /k2N is nonnegative. Thus, by Fatou’s lemma and identity (4.11), we have g.u/u 2 L1 ./ and ˆ ˆ g.u/u 6 jruj2 C Œu 6 Œu: 



Since jg.u/j 6 sup jg.t /j¹juj61º C g.u/u; jtj61

we deduce that g.u/ 2 L1 ./. Returning to an arbitrary function v 2 W01;2 ./ \ L1 ./, we have jg.u/vk j 6 jg.u/jkH kL1 .R/ kvkL1 ./ : Since g.u/ 2 L1 ./, the dominated convergence theorem shows that ˆ ˆ g.u/vk D g.u/v; lim k!1

whence

ˆ





g.u/v D



ˆ



ru
rv C Œv:

This establishes the Euler–Lagrange equation.



4.2. Minimizers and the Euler–Lagrange equation

67

The previous proof was inspired by discussions with P. Bousquet. Observe that we have shown something stronger than g.u/ 2 L1 ./, namely g.u/u 2 L1 ./: When  2 L2 ./, we gain more integrability over g.u/: Proposition 4.24. Let  be a bounded open set, and let gW R ! R be a continuous function satisfying the sign condition. If g is odd and nondecreasing, if  2 L2 ./, and if u 2 W01;2 ./ satisfies the Euler–Lagrange equation associated to E, then g.u/ 2 L2 ./ and kg.u/kL2 ./ 6 kkL2 ./ : By an approximation argument on the test functions, we deduce in this case that the Euler–Lagrange equation holds for every test function v 2 W01;2 ./, not necessarily bounded. The main ingredient in the proof of the L2 boundedness of g.u/ is an L1 estimate on level sets of u: for every  > 0, ˆ ˆ jg.u/j 6 jj: (4.12) ¹juj>º

¹juj>º

The connection between the integral in the left-hand side and the norm kg.u/kL2 ./ is obtained using Cavalieri’s principle (Proposition 1.7) to integrate the measurable function jg.u/j with respect to the Lebesgue measure with density jg.u/j over :  ˆ ˆ 1ˆ 2 jg.u/j D jg.u/j dt: 

0

¹jg.u/j>tº

When g is odd and nondecreasing, every set ¹jg.u/j > t º is of the form ¹juj > º for some  > 0, whence we may apply the L1 estimate (4.12). Proof of Proposition 4.24. We first establish the key estimate: Claim. Inequality (4.12) holds for every  > 0. Proof of the claim. Let .Hn /n2N be a sequence of smooth bounded functions Hn W R ! R such that Hn .0/ D 0, having bounded derivatives Hn0 . We have Hn .u/ 2 W01;2 ./ \ L1 ./ by the chain rule for Sobolev functions (Exercise 4.6). Applying the Euler–Lagrange equation with test function v D Hn .u/, we get ˆ ˆ ˆ 0 2 Hn .u/jruj C g.u/Hn .u/ D ŒHn .u/ D Hn .u/: 





68

4. Variational approach

If in addition Hn is nondecreasing, then Hn0 .u/ > 0, and we have ˆ ˆ g.u/Hn .u/ 6 Hn .u/: 

(4.13)



Given  > 0, we choose the sequence .Hn /n2N converging pointwise to the function sgn W R ! R defined by 8 ˆ , if  < t < , 1 if t 6

,

and such that, for every n 2 N, jHn j 6 1. Letting n ! 1 in estimate (4.13), the dominated convergence theorem shows that ˆ ˆ g.u/ sgn .u/ 6 sgn .u/: 



Since g satisfies the sign condition, we get ˆ ˆ ˆ ˆ jg.u/j D g.u/ sgn .u/ 6 sgn .u/ 6 ¹juj>º





¹juj>º

This proves the claim.

jj: 4

We have not yet proved that g.u/ 2 L2 ./. For this purpose, we first establish an estimate for the norm kg.u/kL2 .¹jg.u/j 0. By Cavalieri’s principle with measurable function jg.u/j and Lebesgue measure with density jg.u/j¹jg.u/jtº

Applying once again Cavalieri’s principle, this time with measurable function jg.u/j and Lebesgue measure with density jj¹jg.u/j 3, and let  be a bounded open set. If r > N2N and if C2 0 v 2 Lr ./, then there exists N 2 Lr ./ such that, for every  2 Lr ./, we have E TF ./ N 6 E TF ./: Moreover, N satisfies the Euler–Lagrange equation associated to the Thomas–Fermi energy functional E TF in the sense that jj Nr

2

N C uN D v

almost everywhere in . The proof of Proposition 4.25 is also based on the direct method of the calculus of variations. We first show that the Thomas–Fermi functional E TF is bounded from below, and minimizing sequences are bounded in Lr ./: 2N N C2

and if

Combining the two estimates and choosing  6 1=2r , the conclusion follows.



Lemma 4.26. Let N > 3, and let  be a bounded open set. If r > 0 v 2 Lr ./, then, for every  2 Lr ./, we have 0

r TF r kkL ./ C kvkL r ./ 6 C.E r 0 ./ /;

for some constant C > 0 depending on r. Proof of Lemma 4.26. For every  2 Lr ./, we have ˆ ˆ u  D jru j2 > 0: 



Thus,

ˆ ˆ 1 jjr 6 E TF ./ C v: r   By Young’s inequality, for every  > 0 there exists C1 > 0 such that ˆ ˆ ˆ r0 v 6 C1 jvj C  jjr : 





72

4. Variational approach

We now establish the lower semicontinuity of the Thomas–Fermi functional E TF with respect to sequences converging weakly in Lr ./: and if Lemma 4.27. Let N > 3, and let  be a bounded open set. If r > N2N C2 r0 r v 2 L ./, then, for every bounded sequence .n /n2N in L ./ converging weakly to  in Lr ./, we have E TF ./ 6 lim inf E TF .n /: n!1

Proof of Lemma 4.27. By the lower semicontinuity of the norm with respect to the weak convergence (see Proposition 3.5 in [53] or Theorem 5.4.6 in [345]), we have ˆ ˆ (4.16) jjr 6 lim inf jn jr : 

n!1



By the definition of weak convergence, we also have ˆ ˆ v D lim inf vn : 

n!1

(4.17)



To prove that ˆ

u  6 lim inf 

n!1

ˆ

(4.18)

un n ; 

we first observe that, for every w 2 W01;2 ./, by the Sobolev embedding (Corol0 lary 4.12) we have w 2 Lr ./ for r > N2N . By the weak convergence of the C2 r sequence .n /n2N in L ./, we then have ˆ ˆ ˆ ˆ run
rw: ru
rw D wn D lim w D lim 



n!1



n!1



Taking w D u and applying the Hölder inequality, we then get ˆ



jru j2 D lim

n!1

ˆ



run
ru 6 lim inf krun kL2 ./ kru kL2 ./ : n!1

Simplifying both sides, we deduce that ˆ ˆ ˆ ˆ 2 2 u  D jru j 6 lim inf jrun j D lim inf un n : 



n!1



n!1



Combining properties (4.16)–(4.18), we get the lower semicontinuity of the Thomas– Fermi functional E TF . 

4.3. Thomas–Fermi energy functional

73

Proof of Proposition 4.25. Let .n /n2N be a minimizing sequence of the functional E TF in Lr ./. By Lemma 4.26, the sequence .n /n2N is bounded in Lr ./, whence there exists a subsequence .nk /k2N converging weakly in Lr ./ to some function . N It thus follows from the lower semicontinuity of the Thomas–Fermi functional E TF (Lemma 4.27) that E TF ./ N 6 lim inf E TF .nk / D k!1

inf

2Lr ./

E TF ./:

Hence, N minimizes the functional E TF . To deduce the Euler–Lagrange equation satisfied by , N we observe that, for every r  2 L ./, we have ˆ ˆ ˆ ˇ d TF ˇ r 2 E .N C t  /ˇ D jj N  N C ruN
ru v : (4.19) tD0 dt    The first integral above is obtained using the dominated convergence theorem. Concerning the second integral, we use the linearity of u with respect to the parameter  to write ˆ ˆ uCt .N C t  / D jruCt j2 N N 



D

ˆ

D

ˆ



jruN C t ru j2



jruN j2 C 2t

ˆ



ruN
ru C t 2

ˆ



jru j2 :

Since u satisfies the Dirichlet problem in W01;2 ./ with density  , we have ˆ ˆ ruN
ru D uN ; 



and we deduce from (4.19) that the minimizer N satisfies ˆ .jj N r 2 N C uN v/ D 0: 

Since this property holds for every  2 Lr ./, the pointwise identity satisfied by N follows.  Exercise 4.16. Let   R2 be a bounded open set. Prove that if r > 1 and 0 v 2 Lr ./, then the Thomas–Fermi energy functional is well defined in Lr ./ and achieves its minimum at some point N 2 Lr ./. What is the Euler–Lagrange equation satisfied by ? N

74

4. Variational approach

Other Thomas–Fermi type models are discussed in [76] and [206]. We now explain the connection between the Euler–Lagrange equation above and the nonlinear PDE from the previous section. For this purpose, assume that the potential v satisfies the linear Dirichlet problem with density , so that v D u . In this case, we can assume that  is merely an L1 function or a finite Borel measure on  (Proposition 3.2), and the solution is understood in the sense of Definition 3.1. The Euler–Lagrange equation involving such a potential becomes jj Nr

2

N C uN D u :

Corollary 4.28. Let N > 3, let  be a bounded open set, and let  2 M./. For every 0 < p < N 2 2 , the nonlinear Dirichlet problem ´ w C jwjp 1 w D  in , wD0

on @,

has a solution in the sense of Definition 3.6. Proof. By the Sobolev embedding of solutions of the linear Dirichlet problem (Proposition 5.1), the potential u belongs to the Sobolev space W01;q ./ for every exponent 1 6 q < NN 1 . This implies that u 2 Ls ./ for every 1 6 s < NN 2 (Corollary 4.12). The assumption on the exponent r in Proposition 4.25 is thus satisfied for every r > N2 . In this range, we have the existence of a function N 2 Lr ./ such that jj N r 2 N C uN D u : Denoting w D jj Nr

2

; N

we now verify that w satisfies the Dirichlet problem above with p D x we have for every  2 C01 ./ ˆ ˆ u  D  d; 

1 . r 1

Indeed,



with a similar identity for the potential uN . Therefore, multiplying the Euler–Lagrange equation by , and integrating over , we get ˆ ˆ ˆ jj N r 2 N  C  N D  d: 



To conclude, it suffices to observe that N D jwj r to 0 < r 1 1 < N 2 2 .



1 1

1

w, and that r >

N 2

is equivalent 

4.3. Thomas–Fermi energy functional

75

In the minimization problem from the previous section, we are unable to minimize the energy functional in W01;2 ./ for any given  2 L1 ./. In the setting of the Thomas–Fermi energy functional any such density is implicitly allowed, but under the growth restriction p < N 2 2 . This condition on the exponent p is not what one should expect for the nonlinear Dirichlet problem. First of all, Brezis and Strauss [69] proved the existence of solutions with L1 data regardless of the growth of the nonlinear term (Proposition 3.7). Moreover, Bénilan and Brezis [24] established the existence of solutions involving measure data for any p < NN 2 (Proposition 21.1), and we have seen that the equation cannot have a solution when the measure  contains a Dirac mass and p > NN 2 (Proposition 3.9). The argument that provides the existence of solutions involving measure data for any exponent p < NN 2 is presented in Section 21.1.

Chapter 5

Linear regularity theory

“Si la force F est proportionnelle à 1 , il suffira de trouver la valeur de 2 1 

et de la différencier par les méthodes ordinaires.”1 Joseph-Louis Lagrange

We investigate the Sobolev regularity of solutions of the linear Dirichlet problem when the density is merely an L1 function and, more generally, a finite Borel measure.

5.1 Embedding in Sobolev spaces We follow Littman, Stampacchia and Weinberger’s duality approach to prove the Sobolev regularity of solutions of the linear Dirichlet problem ´ u D  on , uD0

on @,

involving measure data, see Theorem 5.1 in [213] or Théorème 9.1 in [316]: Proposition 5.1. Let  be a smooth bounded open set, and let  2 M./. If u is the solution of the linear Dirichlet problem with density , then, for every 1 6 q < NN 1 , we have u 2 W01;q ./, and the estimate kukW 1;q ./ 6 C kkM./ holds for some constant C > 0 depending on q, N , and . By the Sobolev embedding (see Corollary 4.12), the solution u thus belongs to the Lebesgue space Lp ./ for every exponent 1 6 p < NN 2 , and satisfies the estimate kukLp ./ 6 C 0 kkM./ : 1 “Assuming

the force F is proportional to it by ordinary methods.”

1 , 2

it suffices to find the value of

1 

and to differentiate

78

5. Linear regularity theory 1;

N

On the other hand, solutions need not belong to W0 N 1 ./, and this is related to the failure of the natural counterpart of the Calderón–Zygmund regularity theory for L1 data that we explain at the end of this section. The introduction of the potential function u by Lagrange [189] – later pursued by Laplace [190], [191] and Poisson [283], [28] – was originally motivated by the study of the force field G D ru. In this respect, the scalar quantity u is supposedly easier to compute, but the force G itself is the major physical notion. The embedding of solutions of the Dirichlet problem into Sobolev spaces thus ensures the existence of the force field G, with an estimate in terms of the total mass or the total electric charge kkM./ D jj./. Proposition 5.1 relies on the following estimate due to Stampacchia (Proposizione 5.1 in [315]), in the spirit of the celebrated works of De Giorgi [103] and Nash [260] providing Hölder continuity of solutions of elliptic PDEs. The strategy of the proof below by Hartman and Stampacchia (Lemma 7.3 in [161]) is based on Stampacchia’s truncation method. Lemma 5.2. Let  be a bounded open set. If v 2 W01;2 ./ satisfies the linear Dirichlet problem ´ v D f C div F on ; vD0

on @;

for some f 2 Lr .I R/ and some F 2 Lr .I RN / with r > N , then v 2 L1 ./, and the estimate kvkL1 ./ 6 C.kf kLr ./ C kF kLr ./ /; holds for some constant C > 0 depending on r, N and . Proof of Lemma 5.2. We assume in the proof that N > 3; the case of dimension N D 2 requires some small modification concerning the Sobolev inequality. Given  > 0, let S W R ! R be the function defined for t 2 R by 8 ˆ . Note that S .v/ 2 W01;2 ./ and (cf. Exercise 5.3)

jrS .v/j2 D rv
rS .v/: Taking S .v/ as a test function of the Dirichlet problem, we get ˆ ˆ ˆ jrS .v/j2 D f S .v/ F
rS .v/: 





5.1. Embedding in Sobolev spaces

79

Since S .v/ D 0 in ¹jvj 6 º, the Hölder inequality yields ˆ jrS .v/j2 6 .kf kL2 .¹jvj>º/ C kF kL2 .¹jvj>º/ /kS .v/kW 1;2 ./ : 

Using the Poincaré inequality (Proposition 4.5), we deduce that krS .v/kL2 ./ 6 C1 .kf kL2 .¹jvj>º/ C kF kL2 .¹jvj>º/ /: On the other hand, by the Hölder and the Sobolev inequalities (Corollary 4.12), we have kS .v/kL1./ 6 kS .v/k

1

2N 2 ./

LN

1

j¹jvj > ºj 2 C N 1

1

6 C2 krS .v/kL2 ./ j¹jvj > ºj 2 C N : Combining the two estimates we deduce that, for every  > 0, 1

1

kS .v/kL1./ 6 C3 .kf kL2 .¹jvj>º/ C kF kL2 .¹jvj>º/ / j¹jvj > ºj 2 C N : By the Hölder inequality, for every r > 2 we have 1

kf kL2 .¹jvj>º/ CkF kL2 .¹jvj>º/ 6 .kf kLr .¹jvj>º/ CkF kLr .¹jvj>º/ / j¹jvj > ºj 2

1 r

:

Therefore, 1

kS .v/kL1./ 6 C3 .kf kLr ./ C kF kLr ./ / j¹jvj > ºj1C N

1 r

:

Claim. If there exist ˛ > 1 and A > 0 such that, for every  > 0, kS .v/kL1./ 6 A j¹jvj > ºj˛ ; then v 2 L1 ./ and

1

1

1

˛ kvkL1./ 6 C 0 A ˛ kvkL1./ :

Assuming the claim, we can conclude the proof of the lemma. Indeed, since r > N and kvkL1./ 6 jjkvkL1 ./ , we deduce from the claim that kvkL1 ./ 6 C4 .kf kLr ./ C kF kLr ./ /; which is the estimate we wanted to establish.

80

5. Linear regularity theory

Proof of the claim. By Cavalieri’s principle (Proposition 1.7), ˆ 1 ˆ 1 kS .v/kL1./ D j¹jS .v/j > sºj ds D j¹jvj > sºj ds: 0



Therefore, we may rewrite the assumption on v as ˆ 1 j¹jvj > sºj ds 6 A j¹jvj > ºj˛ : 

Let H W Œ0; C1/ ! R be the function defined for t > 0 by ˆ 1 H.t / D j¹jvj > sºj ds: t

The function Œ0; C1/ 3 s 7! j¹jvj > sºj is nonincreasing, and whence continuous except for countably many points. Thus, for almost every t > 0, we have H 0 .t / D

j¹jvj > t ºj :

In view of the estimate, for almost every  > 0 we then have 

H./ H ./ D j¹jvj > ºj > A 0

 ˛1

:

Integrating this inequality (Exercise 5.1), we conclude that if ˛ > 1, then H.0 / D 0 for some 0 > 0 such that 1

0 6 C5 A ˛ H.0/1

1 ˛

:

Since kvkL1./ 6 0 and H.0/ D kvkL1./ , the claim follows. The proof of the lemma is complete.

4 

Exercise 5.1 (finite time vanishing). Let ˇ < 1, and let H W Œ0; C1/ ! R be a nonnegative absolutely continuous function such that H0 6

BH ˇ

almost everywhere in Œ0; C1/, for some constant B > 0. Prove that if s > 0 is such that H.s/ > 0, then, for every 0 6 t 6 s, we have H.t / 6 .H.0/1 Deduce that H.t / D 0 for every t >

ˇ

.1

H.0/1 ˇ . .1 ˇ /B

ˇ/Bt / 1

1 ˇ

:

5.1. Embedding in Sobolev spaces

81

The existence of the weak gradient ru as an element in the Lebesgue space Lq ./ is obtained using the Riesz representation theorem (Proposition 3.3) and Stampacchia’s estimate (Lemma 5.2). x the solution of the linear Dirichlet Proof of Proposition 5.1. For every  2 C01 ./, problem with density  satisfies the inequality ˇ ˇˆ ˇ ˇ ˇ u  ˇˇ 6 kkM./ kkL1 ./ : ˇ 

x by the assumption on  we may use as test function  2 C 1 ./ x Given f 2 C 1 ./, 0 N the solution of the linear Dirichlet problem with density f . For every 1 < q < N 1 , the conjugate exponent satisfies q 0 > N , whence by Stampacchia’s estimate we have kkL1 ./ 6 C kf kLq0 ./ : Therefore,

ˇˆ ˇ ˇ ˇ ˇ uf ˇˇ 6 C kkM./ kf kLq0 ./ : ˇ 

x and then, by weak density, for every This estimate holds for every f 2 C 1 ./, 1 f 2 L ./. We deduce from the Riesz representation theorem (Exercise 3.1) that u 2 Lq ./ and kukLq ./ 6 C kkM./ : We now prove that u has a weak derivative in Lq .I RN /. For this purpose, x RN /, we use as test function  2 C 1 ./ x the solution of the given F 2 C 1 .I 0 linear Dirichlet problem with density div F . For every 1 < q < NN 1 , it follows from Stampacchia’s estimate that ˇˆ ˇ ˇ ˇ ˇ u div F ˇˇ 6 kkM./ kkL1 ./ 6 C kkM./ kF kLq0 ./ : ˇ 

Thus, the functional

x RN / 3 F 7 ! C 1 .I

ˆ 0

u div F 

admits a unique continuous linear extension in Lq .I RN /. By the Riesz representation theorem (Proposition 3.3), there exists a unique function G 2 Lq .I RN / such x RN /, that, for every F 2 C 1 .I ˆ ˆ u div F D G
F: 



82

5. Linear regularity theory

Therefore, u 2 W01;q ./ and we have kGkLq ./ 6 C kkM./ : The proof of the proposition is complete when q > 1, and by the Hölder inequality we also get the conclusion for q D 1.  In the previous proof, we assumed that a solution u exists, and this follows from Proposition 3.2, which is also based on Stampacchia’s estimate. We can in fact prove simultaneously the existence and regularity of solutions of the linear Dirichlet problem using a compactness argument. Indeed, the previous proof gives the estimate we need: for every 1 6 q < NN 1 and every v 2 W01;2 ./ satisfying the Dirichlet problem with density  2 L2 ./, kvkW 1;q ./ 6 C kkL1 ./ : Next, if .n /n2N is a sequence in L2 ./ converging in the sense of measures to  (Proposition 2.7) and if un is the solution of the Dirichlet problem in W01;2 ./ with density n (Proposition 4.6), then by the above estimate the sequence .un /n2N is bounded in W01;q ./. By the Rellich–Kondrashov compactness theorem (Proposition 4.8), we may extract a subsequence converging strongly in Lq ./ to some function u which satisfies the Dirichlet problem with density . By the closure property of Sobolev spaces (Proposition 4.10), we deduce that u 2 W01;q ./ for q > 1, and so also for q D 1. This implies that every solution of the linear Dirichlet problem has the required regularity since solutions are unique (Proposition 3.5). We now compare the Sobolev embedding of solutions of the linear Dirichlet problem ´ u D  in , uD0

on @,

in terms of L1 or measure data  (Proposition 5.1) with the classical Calderón– Zygmund Lp theory for 1 < p < C1, see Theorem 9.15 and Lemma 9.17 in [146]: Proposition 5.3. Let 1 < p < C1, and let  be a smooth bounded open set. If  2 Lp ./, then the solution u of the Dirichlet problem above belongs to W 2;p ./ \ W01;p ./, and the estimate kukW 2;p ./ 6 C kkLp ./ holds for some constant C > 0 depending on p and .

5.1. Embedding in Sobolev spaces

83

This is a remarkable contribution to the regularity theory of elliptic PDEs, and more generally to harmonic analysis through estimates on the Riesz transform. The inequality above fails for p D 1, which is the case handled by Proposition 5.1 above: Exercise 5.2 (failure of L1 theory). Let N > 3, and let ' 2 Cc1 .B.0I 1// be such that ' D 1 in some neighborhood of 0. Prove that, for every 0 < ˛ 6 1, the function uW B.0I 1/ ! R defined by 1  ˛ 1  log '; u.x/ D jxjN 2 jxj satisfies the Dirichlet problem with L1 density in B.0I 1/, but u 62 W 2;1 .B.0I 1//. One can also argue by contradiction as follows. If the estimate kukW 2;1 ./ 6 C kkL1 ./ were correct in dimension N > 3 for every solution of the linear Dirichlet problem, then by the Sobolev–Gagliardo–Nirenberg inequality (Corollary 4.12) we would have kuk

N

LN

2 ./

6 C 0 kkL1 ./ :

By an approximation argument (Proposition 2.7), this inequality would also hold with  replaced by a Dirac mass ıa , in which case kkL1 ./ should be replaced by the N

total mass kıa kM./ D 1. In particular, we would have u 2 L N 2 ./. But this is not possible since, in the unit ball B.0I 1/, the solution of the linear Dirichlet problem with density ı0 is explicitly given by   1 1 1 ; u.x/ D .N 2/N kxkN 2 N

which does not belong to L N 2 .B.0I 1//. In fact, a deep construction of Ornstein implies that even the stronger inequality N 2 X

@ u 2

kD ukL1 ./ 6

@x 2 1 L ./ i i D1

is false, see Theorem 1.3 in [182] or Theorem 1 in [268]. The connection with the Calderón–Zygmund’s singular integral theory becomes more transparent by considering the following weak type estimates that we establish N later on in this chapter (Proposition 5.7). Firstly, we have a weak L N 2 estimate for solutions of the Dirichlet problem: for every t > 0, t j¹juj > t ºj

N 2 N

6 C kukL1 ./

84

5. Linear regularity theory N

and, secondly, a weak L N 1 estimate for the first-order derivatives of solutions: for every t > 0, N 1 t j¹jruj > t ºj N 6 C 0 kukL1 ./ : These estimates are the natural companions of the following weak L1 estimate for second derivatives of solutions: for every t > 0, ˇ ˇ t ˇ¹jD 2 uj > t ºˇ 6 C 00 kukL1 ./ ;

that lies at the heart of Calderón and Zygmund’s approach, see eq. (9.30) in [146] or p. 30, eq. (9), in [317]. Once the latter weak estimate is established, the classical Lp estimates for 1 < p < C1 then follow by interpolation of continuous linear operators in Lebesgue spaces.

5.2 Weak Lebesgue functions Before establishing the weak estimates satisfied by solutions of the linear Dirichlet problem and their first-order derivatives, we recall the definition and some basic properties of weak Lebesgue functions. Definition 5.4. Let 1 6 p < C1. A Borel measurable function uW  ! R is a weak Lp function if there exists M > 0 such that, for every t > 0, 1

t j¹juj > t ºj p 6 M: For example, if u 2 Lp ./, then, for every t > 0, by the Chebyshev inequality we have 1 t j¹juj > t ºj p 6 kukLp ./ ;

whence u is a weak Lp function, but the converse is false as one can see by considering the function 1 : B.0I 1/ 3 x 7 ! N jxj p We are actually just missing the embedding into the Lebesgue space Lp ./:

Proposition 5.5. If uW  ! R is a weak Lp function and if  has finite Lebesgue measure, then, for every 1 6 r < p, we have u 2 Lr ./ and kukLr ./ 6 CM; for some constant C > 0 depending on p, r and jj.

5.2. Weak Lebesgue functions

Proof. By Cavalieri’s principle (1.5), ˆ ˆ r juj D r 

1

tr

0

1

85

j¹juj > t ºj dt:

We estimate the measure j¹juj > t ºj for t small by jj and for t large by M p =t p . Since r < p, for every s > 0 we have ˆ ˆ s ˆ 1 rM p 1 Mp r r 1 : juj 6 r t t r 1 p dt D s r jj C jj dt C r t p r sp r  0 s Thus, u 2 Lr ./. The inequality is obtained by minimizing the right-hand side with respect to s.  We now give a convenient characterization of weak Lp functions in the spirit of the Hölder inequality. The proof requires the same trick as in Proposition 5.5, and is used again in the proof of the weak elliptic estimates in Proposition 5.7. Proposition 5.6. Let uW  ! R be a Borel measurable function and 1 < p < C1. We have that u is a weak Lp function if and only if, for every Borel set A  , ˆ p 1 juj 6 M 0 jAj p ; A

0

for some constant M > 0. Proof. If u satisfies the estimate, then taking A D ¹juj > t º with t > 0 we have ˆ p 1 juj 6 M 0 j¹juj > t ºj p : ¹juj>tº

Thus, by the Chebyshev inequality, t j¹juj > t ºj 6 M 0 j¹juj > t ºj

p 1 p

:

Assuming that the sets ¹juj > t º have finite measure, we deduce that u is a weak Lp function. We can avoid such a finiteness assumption by taking instead An D ¹juj > t º \ B.0I rn /; where .rn /n2N is a sequence of positive numbers diverging to infinity. For every n 2 N, the previous argument gives 1

t jAn j p 6 M 0 : As n ! 1, we deduce that ¹juj > t º has finite measure, and satisfies the weak Lp estimate.

86

5. Linear regularity theory

Conversely, if u is a weak Lp function, then by Cavalieri’s principle (Proposition 1.7) we have ˆ ˆ 1 juj D jA \ ¹juj > t ºj dt: A

0

We estimate the measure of the set A \ ¹juj > t º by jAj or j¹juj > t ºj according to whether t is small or large: for every s > 0, ˆ ˆ s ˆ 1 Mp 1 juj 6 : jAj dt C j¹juj > t ºj dt D sjAj C p 1 sp 1 A 0 s Minimizing the right-hand side with respect to s, we have the estimate.



From the previous proof, the smallest constants M and M 0 arising in the definition of weak Lp functions and in the previous proposition are equivalent: cM 0 6 M 6 M 0 ; for some constant c > 0 depending on p. We can compare the integral condition that appears in Proposition 5.6 with the Riesz representation theorem (Exercise 3.1). Indeed, assume that we are given a nonnegative summable function uW  ! R, and we wish to prove that u 2 Lp ./. It then suffices to prove that, for every v 2 L1 ./, we have ˆ uv 6 M 00 kvk pp 1 : L



./

In Proposition 5.6, only characteristic functions v D A over Borel sets are admissible, and in this case u is merely a weak Lp function. If we further restrict the class of Borel sets we are allowed to take, for instance to open balls B.xI r/  , we fall into the class of Morrey functions satisfying ˆ p 1 u 6 M 000 r N p : B.xIr /

The latter property has a natural counterpart in the setting of measures, see e.g. Section 2 in [274]. More precisely, given 1 6 p < C1, we say that a locally finite Borel measure  in RN belongs to the Morrey space Mp .RN / if there exists C > 0 such that p 1 jj.B.xI r// 6 C r N p ; for every x 2 RN and every r > 0. We then define the Morrey norm kkMp .RN / D sup

x2RN r >0

jj.B.xI r// rN

p 1 p

:

(5.1)

5.3. Critical estimates

87

s We have for example that jj 6 ˛H1 for some 0 6 s < N and ˛ > 0 if and only N N s N s .R / (Proposition B.3), where H1 denotes the Hausdorff content if  2 M of dimension s. Embeddings in Morrey spaces, and their connection with the trace inequality, are investigated in Chapters 10, 16, and 17.

5.3 Critical estimates We now present an improvement of the Sobolev embedding of solutions of the linear Dirichlet problem in terms of weak Lebesgue estimates. Although the inequalities kuk

N

LN

2

./

6 C kkL1./

and kruk

N

LN

1 ./

6 C 0 kkL1 ./

are not satisfied, they have a true counterpart in the setting of weak Lebesgue spaces. Such weak estimates are implicitly stated in works by Zygmund (p. 247 in [349]) and Stampacchia (Lemma 7.3 in [315]) using an argument based on the Newtonian potential generated by the measure ; see also Appendix in [25]. We present a different strategy developed by Boccardo and Gallouët [30] (see also [23]), and based on Stampacchia’s truncation argument. Proposition 5.7. Let N > 3, let  be a smooth bounded open set, and let  2 M./. If u is the solution of the linear Dirichlet problem with density , then, for every t > 0, we have N 2 t j¹juj > t ºj N 6 C kkM./ and t j¹jruj > t ºj

N 1 N

6 C 0 kkM./ ;

for some constants C; C 0 > 0 depending on the dimension N . The constants involved in these critical estimates do not depend on the size of the domain. We rely on Stampacchia’s method based on the truncation function T W R ! R defined for s 2 R by

8 ˆ <  T .s/ D s ˆ : 

if s
.

88

5. Linear regularity theory

Lemma 5.8. Let  be a smooth bounded open set, and let  2 M./. If u is the solution of the linear Dirichlet problem with density , then, for every  > 0, we have T .u/ 2 W01;2 ./ and 1

1

2 : krT .u/kL2 ./ 6  2 kkM./

Proof of Lemma 5.8. We first assume that u 2 W01;2 ./ and  2 L2 ./. By Exercise 5.3 below, for every  > 0 we have T .u/ 2 W01;2 ./ and jrT .u/j2 D rT .u/
ru: Using T .u/ as a test function in the equation satisfied by u we get ˆ ˆ ˆ 2 jrT .u/j D rT .u/
ru D T .u/ 6 kkL1 ./ : 





This gives the estimate when u 2 W01;2 ./ and  2 L2 ./. Given  2 M./, we consider a sequence of functions .n /n2N in L2 ./ converging weakly to  in the sense of measures, and such that (Proposition 2.7) lim kn kL1 ./ D kkM./ :

n!1

If un 2 W01;2 ./ denotes the solution of the linear Dirichlet problem with density n , then, for every n 2 N, we have 1

1

2 krT .un /kL2 ./ 6  2 kn kM./ :

Since the sequence .un /n2N is bounded in W01;1 ./ (Proposition 5.1), it follows from the Rellich–Kondrashov compactness theorem (Proposition 4.8) and from the uniqueness of the solution of the Dirichlet problem (Proposition 3.5) that .un /n2N converges to u in L1 ./. Since .T .un //n2N is bounded in W01;2 ./, by the closure property in Sobolev spaces (Proposition 4.10) we have that T .u/ 2 W01;2 ./ and krT .u/kL2 ./ 6 lim inf krT .un /kL2 ./ : n!1

The conclusion follows.



Exercise 5.3 (truncation of Sobolev functions). Let 1 6 q < C1. Prove that, for every u 2 W01;q ./, we have T .u/ 2 W01;q ./ and ´ ru in ¹juj 6 º, rT .u/ D 0 in ¹juj > º. Deduce in particular that jrT .u/j2 D rT .u/
ru.

5.3. Critical estimates

89

It follows from the previous lemma that, for every u 2 W01;2 ./ \ L1 ./ such that u 2 L1 ./, we have 1

1

krukL2 ./ 6 kukL2 1 ./ kukL2 1 ./ :

(5.2)

This inequality is the borderline case of the Gagliardo–Nirenberg interpolation inequality, see [143] and [265]: 1

1

krukL2q .RN / 6 C kukL2 1 .RN / kD 2 ukL2 q .RN / for every 1 6 q < C1 and u 2 W 2;q .RN / \ L1 .RN /. Proof of Proposition 5.7. We begin with the first estimate. By the interpolation inequality (Lemma 5.8), for every t > 0 we have T t .u/ 2 W01;2 ./. Thus, by the Sobolev inequality, we may estimate kT t .u/k

2N 2 ./

LN

6 C1 krT t .u/kL2 ./ :

Next, by the Chebyshev inequality, t j¹juj > t ºj

N 2 2N

6 kT t .u/k

2N 2 ./

LN

while, by the interpolation inequality, we have 1

1

2 krT t .u/kL2 ./ 6 t 2 kkM./ :

We then deduce that t j¹juj > t ºj

N 2 2N

1

1

2 6 C1 t 2 kkM./ :

We now establish the second estimate. For every t > 0 and s > 0, we have ³ ² jruj > t [ ¹juj > sº: ¹jruj > t º  juj 6 s Thus, by the subadditivity of the Lebesgue measure, we get ˇ² ³ˇ ˇ jruj > t ˇ ˇ ˇ C j¹juj > sºj : j¹jruj > t ºj 6 ˇ juj 6 s ˇ

(5.3)

We already have an estimate of the second term in the right-hand side. In order to deal with the first one, we note that (Exercise 5.3) ³ ² jruj > t D ¹jrTs .u/j > t º: juj 6 s

90

5. Linear regularity theory

By the Chebyshev and the interpolation inequalities (Lemma 5.8), we obtain ˇ² ³ˇ 1 1 ˇ jruj > t ˇ 2 ˇ 6 krTs .u/kL2 ./ 6 s 12 kk 2 ˇ : tˇ ˇ M./ juj 6 s

From (5.3), we then have

j¹jruj > t ºj 6

N s C2 N 2 kk kk C : M./ N M./ t2 sN 2

Minimizing the right-hand side with respect to s, we obtain the second estimate.



The best constant C arising in the first estimate of Proposition 5.7 was computed by Cassani, Ruf, and Tarsi (Theorem 5 in [83]) using Talenti’s comparison principle [320]. The counterparts of the estimates of Proposition 5.7 in dimension N D 2 are j¹juj > t ºj 6 C jj e and 1

j¹jruj > t ºj 2 6

C t=kukM./

C0 kukM./ : t

The first one can be obtained as in the previous proof by replacing the Sobolev inequality by the Trudinger inequality [325], see also Theorem 7.15 in [146]: ˆ ˛v 2 =krvk2 2 00 L 6 C jj : e 

The second estimate is more subtle. In Lemma A.14 in [25], it relies on the integral representation of ru in terms of the Green function G, ˆ 1 ru.x/ D rx G.x; y/ d.y/; 2  and on the pointwise estimate jrx G.x; y/j 6

C 000 : jx yj

This argument relies on the linearity of the Laplacian and on the integral representation of solutions of the Poisson equation. Recent alternative proofs (Theorem 1.1 in [116], Theorem 1.2 in [178], and Theorem 1.4 in [247]) are based on BMO estimates and reverse Hölder inequalities, but they are not as elementary as in dimension N > 3.

5.4. Compactness in Sobolev spaces

91

5.4 Compactness in Sobolev spaces By the Sobolev embedding of solutions of the linear Dirichlet problem (Proposition 5.1), we have the following compactness result in Lebesgue spaces, see Theorem 5.4 in [213]: Proposition 5.9. Let  be a smooth bounded open set, let .n /n2N be a sequence in M./, and let un be the solution of the linear Dirichlet problem with density n . If .n /n2N is bounded in M./, then there exists a subsequence .unk /k2N converging strongly in Lp ./ for every 1 6 p < NN 2 . Proof. Since the sequence .un /n2N is bounded in Lr ./ for every 1 6 r < NN 2 , it suffices to find a subsequence .unk /k2N converging strongly in L1 ./. Indeed, by the Hölder inequality we have, for every 1 6 p 6 r and every i; j 2 N, kuni

 unj kL 1 ./ kuni

unj kLp ./ 6 kuni

1  unj kL r ./ ;

where 0 6  6 1 satisfies the identity p1 D 1 C 1 r  . Taking r < NN 2 , this estimate implies that .unk /k2N is a Cauchy sequence in Lp ./ for every p < r. By Proposition 5.1, the sequence .un /n2N is bounded in W01;1 ./. Applying the Rellich–Kondrashov compactness theorem (Proposition 4.8), there indeed exists a convergent subsequence in L1 ./, and the conclusion follows.  A stronger result is actually true: Proposition 5.10. Under the assumptions of Proposition 5.9, there exists a subsequence .unk /k2N converging strongly in W01;q ./ for every 1 6 q < NN 1 . This result is proved by Boccardo and Gallouët (see assertion (21) in [30], Lemma 1 in [31], or Appendix in [108]), and relies on the following weak interpola2p , which is the harmonic average between tion inequality involving the exponent pC1 p and 1, see Theorem 4 in [34]: 1 2p pC1

D

1 p

C

1 1

2

:

Lemma 5.11. Let  be a smooth bounded open set, let  2 M./, and let u be the solution of the linear Dirichlet problem with density . If 1 6 p < C1 and u 2 Lp ./, then, for every t > 0, we have t j¹jruj > t ºj

pC1 2p

1

1

2 6 C kukL2 p ./ kkM./ ;

for some constant C > 0 depending on p.

92

5. Linear regularity theory

Proof of Lemma 5.11. For every t > 0 and s > 0, we have ˇ² ³ˇ ˇ jruj > t ˇ ˇ C j¹juj > sºj : j¹jruj > t ºj 6 ˇˇ juj 6 s ˇ

By the Chebyshev and the interpolation inequalities (Lemma 5.8), we have ˇ² ³ˇ ˇ jruj > t ˇ s ˇ ˇ ˇ juj 6 s ˇ D j¹jrTs .u/j > t ºj 6 t 2 kkM./ :

By the Chebyshev inequality, we also have j¹juj > sºj 6

1 p kukL p ./ : sp

Combining the two estimates we get j¹jruj > t ºj 6

1 s p kkM./ C p kukL p ./ : t2 s

Minimizing the right-hand side with respect to s, we obtain the inequality we want.  Proof of Proposition 5.10. Passing to a subsequence if necessary, we may assume that .un /n2N is a Cauchy sequence in Lp ./ for every 1 6 p < NN 2 . By the inclusion of weak Lr functions in Lq ./ for r > q (Proposition 5.5), for every m; n 2 N we have krum

run kLq ./ 6 C1 sup t j¹jrum t>0

1

run j > t ºj r

Given 1 6 q < NN 1 , we take 1 6 p < NN 2 such that q < in the previous estimate, it follows from Lemma 5.11 that

2p . pC1

1

krum

run kLq ./ 6 C2 kum

un kL2 p ./ kum

Choosing r D

2p pC1

1 2 un kM./ :

Therefore, .run /n2N is a Cauchy sequence in Lq .I RN /, and the conclusion follows.  The vector space X./ D ¹u 2 W01;1 ./W u 2 M./º coincides with the set of solutions of the linear Dirichlet problem with measure data (Proposition 6.3). Equipped with the norm kukM./ , we have that X./ is a Banach space (Proposition 5.1), and is compactly embedded for instance in W01;1 ./ (Proposition 5.10). By the interpolation inequality in W01;2 ./ (Lemma 5.8) and

5.4. Compactness in Sobolev spaces

93

by compactness of solutions of the Dirichlet problem in Sobolev spaces (Proposition 5.10), the vector subspace X./ \ L1 ./ is compactly embedded in W01;q ./ for every 1 6 q < 2. However, the inclusion X./ \ L1 ./  W01;2 ./ is not compact in view of a counterexample by Cioranescu and Murat, see Example 2.1 in [85].

Chapter 6

Comparison tools

“Un ensemble, en chaque point duquel le potentiel atteint sa borne inferieure dans un domaine, ne peut porter de charge positive.”1 Charles de la Vallée Poussin

We investigate maximum principles adapted to the formalism of weak solutions for the Poisson equation and its companion, the Dirichlet problem.

6.1 Weak maximum principle We begin with a substitute of the classical weak maximum principle (Corollary 1.10) in the setting of weak solutions: Proposition 6.1. Let  be a smooth bounded open set and u 2 L1 ./. If x 0 , then u > 0 almost everywhere in . in the sense of .C01 .//

u > 0

x 0 we mean that, for every nonnegative By u > 0 in the sense of .C01 .// x we have function  2 C01 ./, ˆ

u  > 0:



x are sensitive to the information that u > 0 on the boundTest functions in C01 ./ ary @: Example 6.2. Every nonnegative superharmonic function uW RN ! R satisfies x 0 in a smooth bounded open set . Indeed, given u > 0 in the sense of .C01 .// a sequence of mollifiers .k /k2N , each function k  u is nonnegative and superharx monic in RN (Lemma 2.22). In addition, for every nonnegative function  2 C01 ./ @ we have @n 6 0 on @. By the divergence theorem, for every k 2 N we thus have ˆ ˆ ˆ @ .k  u/  D  .k  u/ C .k  u/ d 6 0: @n   @ 1 “A set such that in each of its points the potential attains its infimum in a domain cannot carry a positive charge.”

96

6. Comparison tools

The conclusion then follows letting k ! 1. An alternative approach that does not rely on the convolution of u is based on the composition of test functions with convex functions (Lemma 17.6). In the proof of the proposition, we replace the term  by any nonnegative x function in C 1 ./. This is possible in view of the classical weak maximum principle. x let  2 C01 ./ x be the solution of Proof of Proposition 6.1. For every f 2 C 1 ./, the linear Dirichlet problem ´  D f in , D0

on @.

x then  is superharmonic, whence by the classical weak maximum If f > 0 in , principle we have that  > 0 in . We then deduce that ˆ uf > 0: 

x such that f > 0 in , x we may take Since this inequality holds for every f 2 C 1 ./ a sequence .fn /n2N of such functions converging almost everywhere to the characteristic function ¹u 0: ¹u 0 almost everywhere in .



It is convenient to pass from an inequality in the sense of distributions to an x 0 . Stated differently, we want to find an assumpinequality in the sense of .C01 .// tion which ensures that a supersolution of the equation u D 

in ,

is a supersolution of the Dirichlet problem ´ u D 

in ,

uD0

on @.

We first clarify the meaning of the boundary condition, in terms of Sobolev functions, that is implicit in Littman–Stampacchia–Weinberger’s formulation of the Dirichlet problem (Definition 3.6):

6.1. Weak maximum principle

97

Proposition 6.3. Let  be a smooth bounded open set. Then, for every  2 M./, we have that u is a solution of the linear Dirichlet problem with density  if and only if u 2 W01;1 ./ and the equation u D  is satisfied in the sense of distributions in : for every ' 2 Cc1 ./, ˆ ˆ ru
r' D ' d: 



We begin with an elementary approximation procedure on the test functions in x C01 ./: Lemma 6.4. Let  be a bounded open set. Then, for every nonnegative function x there exists a sequence .'n /n2N of nonnegative functions in Cc1 ./  2 C01 ./, such that x (i) .'n /n2N converges uniformly to  in , x and converges pointwise to r in . (ii) .r'n /n2N is bounded in  Proof of Lemma 6.4. Given a smooth function H W R ! R vanishing in a neighborx ! R defined by hood of 0, for every n 2 N the function 'n W  'n D H.n/ belongs to Cc1 ./. The first assertion is then satisfied by choosing H such that lim H.t / D 1:

t!C1

Concerning the convergence of the sequence of gradients, we first note that r'n D H.n/r C ŒnH 0 .n/r: Since  is nonnegative, we have r D 0 in the set  \ ¹ D 0º. Hence, the second assertion is satisfied by taking H such that lim tH 0 .t / D 0:



t!C1

The previous lemma provides a similar approximation property for a signed funcx Indeed, we first write  D 1 2 as a difference of two nonnegative tion  2 C01 ./. x We then get an approximating sequence for  by applying the functions in C01 ./. lemma separately to 1 and 2 . Proof of Proposition 6.3. If u is a solution of the Dirichlet problem, then the equation is satisfied in the sense of distributions, and by the Sobolev regularity property (Proposition 5.1) we have u 2 W01;1 ./. Conversely, assume that u 2 W01;1 ./ and

98

6. Comparison tools

that the equation is satisfied in the sense of distributions in . On the one hand, since x we have u 2 W01;1 ./, for every  2 C01 ./, ˆ ˆ u  D ru
r: 



On the other hand, taking an approximating sequence .'n /n2N in Cc1 ./ satisfying properties .i/ and .ii/ from Lemma 6.4, for every n 2 N we have ˆ ˆ ru
r'n D 'n d: 



x we deduce from the dominated convergence As n ! 1, for every  2 C01 ./ theorem that ˆ ˆ ru
r D  d: 



Hence, u is a solution of the linear Dirichlet problem with density  in the sense of Definition 3.1.  Using the same argument, we obtain the equivalence between the notions of supersolutions for the Poisson equation and the Dirichlet problem for functions vanishing on the boundary in the sense of Sobolev functions: Proposition 6.5. Let  be a smooth bounded open set and let  2 M./. Take u 2 W01;1 ./. The following assertions are equivalent: (i)

x 0, u >  in the sense of .C01 .//

(ii)

u >  in the sense of distributions in .

Proof. We only need to prove the reverse implication. Since u 2 W01;1 ./, for every x we have  2 C01 ./ ˆ ˆ 

u  D



ru
r:

Taking a nonnegative function  and a sequence .'n /n2N in Cc1 ./ as in the approximation lemma above, for every n 2 N we have ˆ ˆ ru
r'n > 'n d: 



Letting n ! 1, by the dominated convergence theorem we get ˆ ˆ ˆ ˆ 'n d D  d; ru
r'n > lim ru
r D lim 

n!1

and the conclusion follows.



n!1







6.2. Variants of Kato’s inequality

99

The condition u 2 W01;1 ./ is rather strong. A natural assumption – but more subtle to implement – is to assume that u 2 W01;1 ./. The boundary information x can be also expressed as a limit of encoded in terms of test functions in C01 ./ average integrals of u near the boundary (cf. Proposition 20.2): ˆ 1 lim u D 0; !0  ¹x2Wd.x;@/ ¹u>0º u in the sense of distributions in . The original motivation of Kato was to study properties of solutions of the Schrödinger equation which need not belong to the variational W 1;2 setting. Note that a twice differentiable function uW  ! R satisfies, for every x 2 , ´ u.x/ if u.x/ > 0, C u .x/ D 0 if u.x/ < 0. If u.x/ D 0, then uC .x/ need not exist in the classical sense. Since in this case x is a minimum point for the nonnegative function uC , we could formally say that uC .x/ > 0. We thus obtain a formal pointwise statement of Kato’s inequality, namely uC .x/ > ¹u>0º .x/u.x/: Proof of Proposition 6.6. The first step of the proof of Kato’s inequality relies on the observation that when u is smooth, then, for every smooth function H W R ! R, we have H.u/ D H 0 .u/u C H 00 .u/jruj2 ;

100

6. Comparison tools

and if in addition H is convex, then H.u/ > H 0 .u/u:

(6.1)

The next step consists in approximating u 2 L1 ./ by smooth functions – for instance via convolution – in which case we may apply the inequality above. More precisely, given a nonnegative test function ' 2 Cc1 ./, and given a sequence of mollifiers .n /n2N such that supp ' supp n b , we have .n  u/ D n  u

(6.2)

pointwise in supp ' (Lemma 2.22). Thus, integration by parts and inequality (6.1) for smooth functions yield ˆ ˆ ˆ H.n  u/ ' D H.n  u/ ' > H 0 .n  u/.n  u/ ': (6.3) 





Assuming in addition that the derivative H 0 is bounded in R, as n ! 1 we deduce from the dominated convergence theorem that ˆ ˆ H.u/ ' > H 0 .u/u ': (6.4) 



In the last step, we approximate the function R 3 t 7! t C by smooth convex functions. For this purpose, we take a sequence .Hn /n2N of smooth convex functions in R such that the sequence .Hn0 /n2N is uniformly bounded, and .a/ for every t 2 R, lim Hn .t / D t C , n!1

.b/ for every t 6 0, lim Hn0 .t / D 0, n!1

.c/ for every t > 0, lim Hn0 .t / D 1. n!1

Applying the integral inequality (6.4) with H D Hn , and letting n ! 1, we get ˆ ˆ C u ' > ¹u>0º u ': 



This is the formulation of Kato’s inequality in the sense of distributions in .



The assumption u 2 L1 ./ is not invariant in the context of Kato’s inequality: Example 6.7. For every a < c < b, the function uW .a; b/ ! R defined by u.x/ D x c satisfies uC D .uC /00 D ıc in the sense of distributions in .a; b/. Hence, uC is not an L1 function.

6.2. Variants of Kato’s inequality

101

Note however that uC is always a locally finite measure, which follows from Schwartz’s characterization of nonnegative distributions (Proposition 2.20). Indeed, by Kato’s inequality, the distribution T D uC

¹u>0º u

is nonnegative, whence T is a locally finite measure on . By linearity, we deduce that uC is also a locally finite measure on . Kato’s inequality is usually applied to a solution of some equation, in which case the assumption u 2 L1 ./ is probably enough. However, when dealing with subsolutions, or when the equation itself involves measure data, such an assumption on u becomes restrictive. In order to have a counterpart of Kato’s inequality when u is a measure, one should first understand the meaning of the product ¹u>0º u, but this is a delicate issue. Indeed, if u and v are two functions which coincide almost everywhere in , then u and v coincide as distributions, but ¹u>0º u and ¹v>0º v may be different. We propose three ways of handling the product ¹u>0º u in Kato’s inequality when u need not be an L1 function. The first strategy consists in eliminating the characteristic function ¹u>0º . For example, when u is smooth, we could write ¹u>0º u > min ¹u; 0º: Proposition 6.8. If u 2 L1 ./ is such that u 2 M./, then uC > min ¹u; 0º in the sense of distributions in . Proof. In the proof of Kato’s inequality, we rewrite (6.2) in supp ' as .n  u/ D n  u > n  min ¹u; 0º: In particular, the function in the right-hand side is nonpositive. Assuming in addition that 0 6 H 0 6 1, the integral inequality (6.3) can now be replaced by ˆ ˆ H.n  u/ ' > .n  min ¹u; 0º/ ': 



Thus, as n ! 1, (6.4) is substituted by ˆ ˆ H.u/ ' > ' min ¹u; 0º: 



Using the same approximation argument with smooth convex functions, the conclusion follows. 

102

6. Comparison tools

The second strategy consists in replacing u by some summable function smaller than u. For example, when u is smooth and u > f for some L1 function f , we have ¹u>0º u > ¹u>0º f: Proposition 6.9. Let f 2 L1 ./. If u 2 L1 ./ is such that u > f in the sense of distributions in , then uC > ¹u>0º f in the sense of distributions in . Proof. In the proof of Kato’s inequality, we rewrite (6.2) in supp ' as .n  u/ D n  u > n  f: Assuming in addition that H 0 is nonnegative, the integral inequality (6.3) becomes ˆ ˆ H.n  u/' > H 0 .n  u/.n  f / ': 



Thus, as n ! 1, (6.4) is replaced by ˆ ˆ H.u/' > H 0 .u/f ': 



Using the same approximation argument with smooth convex functions, the conclusion follows.  These two statements – or some combination of them – suffice for most purposes in applications. Besides, their proofs are as simple as the proof of the original Kato’s inequality. As a drawback, we have to give up on part of the information carried by the measure u. The third strategy consists in giving a meaning to ¹u>0º u by choosing a suitable representative in some equivalence class of u. For this purpose, we adopt the precise representative uO defined in the Lebesgue set Lu of u (Definition 8.3): for every x 2 Lu , lim

r !0 B.xIr /

ju

u.x/j O D 0:

6.2. Variants of Kato’s inequality

103

Lebesgue’s differentiation theorem (Proposition 8.1) asserts that the exceptional set  n Lu is negligible for the Lebesgue measure and that uO D u almost everywhere in . In our case, we are dealing with functions satisfying the additional property u 2 M./, and this gives some better information concerning the size of the exceptional set, namely (Proposition 8.11) capW 1;2 . n Lu / D 0; where capW 1;2 denotes the Sobolev capacity associated to the W 1;2 norm, whose definition and basic properties are explained in Appendix A. Next, every measure  has a unique Lebesgue decomposition as a sum of two measures (Proposition 14.12),  D d C c ; where the measure d is diffuse with respect to the W 1;2 capacity: for any Borel set A   such that capW 1;2 .A/ D 0, we have d .A/ D 0I the measure c is concentrated in some Borel set E   such that capW 1;2 .E/ D 0: jc j.nE/ D 0: The proof of such a decomposition in terms of the W 1;2 capacity is the same as for the classical decomposition in terms of the Lebesgue measure, in which case one measure is absolutely continuous and the other measure is singular with respect to the Lebesgue measure. Using the notation above, we have the following counterpart of Kato’s inequality, see Theorem 1.1 in [65]: Proposition 6.10. If u 2 L1 ./ is such that u 2 M./, then uC 2 Mloc ./ and the diffuse part of uC with respect to the W 1;2 capacity satisfies .uC /d > ¹u>0º .u/d : O The inequality above is meant in the sense of measures: for every Borel set A  , .uC /d .A/ > .u/d .A \ ¹uO > 0º/:

Thus, for every nonnegative function ' 2 Cc1 ./, we have ˆ ˆ ' .uC /d > ' .u/d : 

¹u>0º O

104

6. Comparison tools

We can apply Kato’s inequality to deduce a sign property of the measure .u/d on the set where u achieves its minimum. Indeed, assuming that u is a nonnegative, it follows from Kato’s inequality above (Proposition 6.10) that .u/d > 0 in ¹uO D 0º

(6.5)

in the sense of measures. Such a property was suggested by de la Vallée Poussin [107] in the 1930s and was later extended by Brelot [45]. This is also called Grishin’s property, see [141] and [150]. In the literature, one finds several proofs of variants of Kato’s inequality involving measures, see [16], [97], [141], and [267]. In Chapter 8, we first prove Kato’s inequality when the measure u is diffuse (Proposition 8.12) to focus on how the precise representative enters the proof. We then sketch the proof of Proposition 6.10 in full generality. Compared to the classical version of Kato’s inequality, there is a piece of the inequality missing which concerns the concentrated measure .uC /c (Corollary 6.15), and is related to the inverse maximum principle (Proposition 6.13) that we explain later on.

6.3 Localization properties Given a function u 2 L1loc ./ such that u 2 Mloc ./, we may localize the Sobolev regularity of solutions of the linear Dirichlet problem as follows: Proposition 6.11. If u 2 L1loc ./ is such that u 2 Mloc ./, then (i) for every 1 6 q
0 we may choose K so as to have .A/ 6 .A/ C : Since  is arbitrary, the conclusion follows.



6.4 Inverse maximum principle Maximum principles are based on the idea that if u is nonpositive, then the function u itself should be nonnegative, assuming that some suitable boundary condition is satisfied. The inverse maximum principle we present in this section goes the other way around in the sense that if u is nonnegative, then some part of u must be nonpositive, see Lemma 3.5 in [92] and Theorem 3 in [119]:

6.4. Inverse maximum principle

107

Proposition 6.13. Let u 2 L1 ./ be such that u 2 M./. If u > 0 almost everywhere in , then the concentrated part of u with respect to the W 1;2 capacity satisfies .u/c 6 0: Under the assumptions of the inverse maximum principle, it is not possible to say anything about the diffuse part of u with respect to the W 1;2 capacity. In any case, the boundary values of u are irrelevant. The inverse maximum principle can be better understood by keeping in mind a classical fact in potential theory: measures which are concentrated on sets of zero W 1;2 capacity – polar sets in the language of potential theory – generate unbounded potentials, see Theorem 7.35 in [166]. Whether the potential achieves the value C1 or 1 dictates the sign of the measure. As an example in dimension two, let u 2 L1 ./ be such that u D ˛ıa for some point a 2  and some ˛ 2 R . In a neighborhood of a, the function u behaves like a multiple of the fundamental solution, u.x/ 

˛ log jx 2

aj:

In this case, the measure ˛ıa is concentrated (Proposition 10.3), and if u > 0, then ˛ 6 0. This is a typical behavior one should expect from the inverse maximum principle. We first need the following property, see Proposition 1 in [151]: Lemma 6.14. Let u 2 L1 ./ be such that u 2 M./. If ru 2 L2 ./, then the measure u is diffuse with respect to the W 1;2 capacity. Proof of Lemma 6.14. For every ' 2 Cc1 ./, we have ˆ ˆ ' u D r'
ru: 



Given a compact set K   such that capW 1;2 .K/ D 0, let .'n /n2N be a sequence in Cc1 ./ such that .a/ .'n /n2N converges to 0 in W01;2 ./, .b/ .'n /n2N is bounded in L1 ./, .c/ .'n /n2N converges pointwise to the characteristic function K . By the dominated convergence theorem, we have ˆ ˆ 'n u D lim u D .u/.K/: n!1



K

108

6. Comparison tools

Since ru 2 L2 ./, the convergence of the sequence .'n /n2N in W01;2 ./ implies ˆ r'n
ru D 0: lim n!1



Therefore, .u/.K/ D 0: Since K is an arbitrary compact subset of , by the inner regularity of finite Borel measures (Proposition 2.5), equality holds for every Borel subset of  with zero W 1;2 capacity.  Proof of Proposition 6.13. We first establish the inverse maximum principle when u has compact support in , and in particular is a solution of the Dirichlet problem with density u. Since u is nonnegative, for every  > 0 we have T .u/ D 

.

u/C

almost everywhere in , where T denotes the truncation function at levels ˙. Thus, by Kato’s inequality (Proposition 6.8) applied to the function  u T .u/ 6 .u/C in the sense of distributions in , and so in the sense of measures (Proposition 6.12). This inequality gives a uniform upper bound on the measures T .u/. By the interpolation inequality (Lemma 5.8), we have T .u/ 2 W01;2 ./. Thus, for every Borel set E   such that capW 1;2 .E/ D 0, the measure T .u/ in the left-hand side does not charge E (Lemma 6.14) and, in particular, we have T .u/ 6 0 in the sense of measures on E. We also have T .u/ 6 .u/C in the sense of measures on nE. Using the additivity of measures, we may combine the two inequalities and deduce that T .u/ 6 .u/C bnE in the sense of measures on , and so in the sense of distributions (Proposition 6.12). Thus, for every nonnegative function ' 2 Cc1 ./, we have ˆ ˆ T .u/ ' 6 ' .u/C : 

nE

6.4. Inverse maximum principle

109

As  ! 1 in this inequality, by the dominated convergence theorem we get ˆ ˆ u ' 6 ' .u/C : 

nE

In other words, u 6 .u/C bnE in the sense of distributions in , whence in the sense of measures. In particular, computing both measures on E we find that .u/.E/ 6 .u/C .E n E/ D 0: Since this property holds for every Borel set E   such that capW 1;2 .E/ D 0, we conclude that .u/c 6 0. To consider the case of a function u 2 L1 ./ which need not have compact support, we may proceed as follows. For every nonnegative function ' 2 Cc1 ./, by the localization lemma (Proposition 6.11) the function u' also satisfies the assumptions of the inverse maximum principle. Thus, by the case we have established above, we get ..u'//c 6 0: Given a subdomain ! b , we take ' D 1 in !, whence .u'/ D u in !, and this implies that .u/c 6 0 in !. Since ! is arbitrary, the conclusion follows.  We now return to Kato’s inequality when u is a measure to explain what happens to the concentrated part of the measure uC with respect to the W 1;2 capacity. Corollary 6.15. If u 2 L1 ./ is such that u 2 M./, then uC 2 Mloc ./ and the concentrated part of uC with respect to the W 1;2 capacity satisfies .uC /c D min ¹.u/c ; 0º: Proof. By Kato’s inequality (Proposition 6.8), we have uC > min ¹u; 0º in the sense of distributions in , whence in the sense of measures (Proposition 6.12). Comparing the concentrated parts on both sides, we get .uC /c > .min ¹u; 0º/c D min ¹.u/c ; 0º:

110

6. Comparison tools

Since uC > 0 and uC > u almost everywhere in , by the inverse maximum principle (Proposition 6.13) we have .uC /c 6 0 and .uC /c 6 .u/c : We thus get the reverse inequality, .uC /c 6 min ¹.u/c ; 0º:



Since uC D .uC /d C .uC /c ;

we may assemble the information given by Proposition 6.10 and the corollary above, and write uC > ¹u>0º .u/d C min ¹.u/c ; 0º: Thus, for every nonnegative function ' 2 Cc1 ./, we have ˆ ˆ ˆ uC ' > ' .u/d C ' min ¹.u/c ; 0º: 

¹u>0º



The use of Kato’s inequality in applications relies on the linear structure of the Laplacian. An alternative approach to quasilinear equations – in particular involving the p-Laplacian – is to use a suitable truncation of the solution as test function. This is a delicate issue that has been thoroughly investigated in [23], [97], and [177].

Chapter 7

Balayage

“La physique ne nous donne pas seulement l’occasion de résoudre des problèmes; elle nous aide à en trouver les moyens.”1 Henri Poincaré

We consider the problems of extension and truncation of solutions of the linear Dirichlet problem with measure data. A fundamental tool is the existence of a weak normal derivative on the boundary.

7.1 Weak normal derivative We begin by illustrating how normal derivatives arise in extension and truncation problems: x the extension U W RN ! R defined by Example 7.1. Given a function u 2 C01 ./, ´ u.x/ if x 2 , U.x/ D 0 if x 2 RN n , is continuous in RN . For a smooth bounded open set , we show that U is the finite measure on RN given by U D u HN b

@u N H @n

1

b@

in the sense of distributions in RN , where @u D ru
n @n denotes the outward normal derivative on @. For every function ' 2 Cc1 .RN /, by the divergence theorem in  we have ˆ ˆ ˆ ˆ @u d: U' D u ' D ' u ' N @n R   @ 1 “Physics

not only gives us the opportunity to solve problems; it also helps us to find ways to do it.”

112

7. Balayage

We have the conclusion by identifying the measure arising in the right-hand side as the distribution U over RN . x and a regular value a > 0 of the function u, Example 7.2. Given u 2 C01 ./ C we show that .u a/ is the finite measure on  given by @u N H @n

a/C D u HN b¹u>aº

.u

1

b¹uDaº

in the sense of distributions in , where n is the outward unit normal vector with respect to the set ¹u > aº, namely n D ru=jruj. Firstly, since a is a positive regular value of u, we have that ¹u > aº is a smooth bounded open set whose boundary is ¹u D aº. By the divergence theorem, for every ' 2 Cc1 ./ we then have ˆ ˆ ˆ @u .u a/ ' D ' u ' d: ¹u>aº ¹u>aº ¹uDaº @n Since ˆ

.u ¹u>aº

a/ ' D

ˆ

.u

a/C ';



the conclusion follows. The counterpart of the extension property in Example 7.1 for solutions of the Dirichlet problem with measure data passes through the existence of a weak normal derivative on the boundary, see Theorem 1.2 in [67]: Proposition 7.3. Let  be a smooth bounded open set and let  2 M./. If u is the solution of the linear Dirichlet problem with density , then there exists f 2 L1 .@/ such that kf kL1 .@/ 6 kkM./ and, for every

x 2 C 1 ./, ˆ ru
r 

D

ˆ



d C

ˆ

f d: @

In the statement above, the function f plays the role of the normal derivative of u in a setting where the normal derivative need not exist in the classical sense. For this reason, we call f the weak normal derivative of u, and use the notation f D

@u : @n

7.1. Weak normal derivative

113

This proposition ensures the existence of a force field on the boundary given by the x in the integral identity, we have n. Taking D 1 in  function @u @n ˆ @u d: (7.1) ./ D @ @n One recognizes here Gauss’s law: the net charge inside  equals the flux of the electric field on the boundary. The proposition above carries two distinct properties concerning the normal x becomes derivative. Firstly, an estimate which restricted to functions u 2 C01 ./



@u

6 kukL1 ./ (7.2)

@n 1 L .@/

and, secondly, some regularity information saying that, regardless of the measure data inside the domain, the weak normal derivative is always absolutely continuous with respect to the surface measure on the boundary @. It is worth giving a separate proof of assertion (7.2) for smooth functions: x If u is nonnegative, then u achieves its Proof of inequality (7.2) for u 2 C02 ./. @u minimum on @, whence @n 6 0 on @. This gives ˇ ˇ ˇ @u ˇ ˇ ˇ D @u : ˇ @n ˇ @n

By the divergence theorem, we then get ˆ ˇ ˇ ˆ ˇ @u ˇ @u ˇ ˇ d D d D ˇ @n ˇ @ @ @n

ˆ

u:

(7.3)



Thanks to the pointwise estimate u 6 juj, the inequality follows under the additional assumption that u is nonnegative. Without any hypothesis on the sign of x u, we decompose u as a difference of two nonnegative functions in C02 ./: u D u1

u2 :

Applying the estimate we have established to u1 and to u2 , we then get





@u

@u1

@u2





6 C 6 ku1 kL1 ./ C ku2 kL1 ./ :

@n 1 @n L1 .@/ @n L1 .@/ L .@/

We now take as u1 and u2 the solutions of the linear Dirichlet problems with Lipschitz-continuous densities . u/C and . u/ , respectively. In this case, u1 and u2 are both nonnegative C 2 functions, and we have



@u

6 k. u/C kL1 ./ C k. u/ kL1 ./ D kukL1 ./ :

@n 1 L .@/ x This proves inequality (7.2) for functions in C02 ./.



114

7. Balayage

The previous argument can be adapted to functions u that are C 2 in a neighborhood of @. In this case, identity (7.3) is justified using a partition of the unity x such that ' 2 Cc1 ./ and 'Q 2 C 1 ./ x is supported in a neighbor' C 'Q D 1 in  hood of @ where u is smooth. Indeed, since u satisfies the Poisson equation with density , we have ˆ ˆ 

u ' D

' d;



while by the divergence theorem we have ˆ ˆ ˆ u 'Q D 'Q d C 



@

'Q

@u d: @n

Adding the two identities, since 'Q D 1 on @ we get ˆ ˆ @u d; 0D d C  @ @n whence ˆ

@

@u d D @n

ˆ



d D ./:

The rest of the argument leading to estimate (7.2) remains unchanged. Proof of Proposition 7.3. If u is smooth in a neighborhood of @, then by the observation above we have



@u

6 kkM./ : (7.4)

@n 1 L .@/ To conclude the proof without this additional assumption, let .k /k2N be a sequence of measures converging strongly to  in M./ and such that supp k  . We could take for instance k D b!k , where .!k /k2N is a nondecreasing sequence of open S !k . sets strictly contained in , and such that  D k2N

For each k 2 N, let uk be the solution of the Dirichlet problem with density k . In particular, uk is harmonic in a neighborhood of @ and, using an argument based x we also have on a partition of the unity as above, for every 2 C 1 ./ ˆ ˆ ˆ @uk ruk
r D dk C d: (7.5) @n   @

For every i; j 2 N, estimate (7.4) applied to ui uj yields

@ui @uj

6 ki j kM./ :

@n @n L1 .@/

7.1. Weak normal derivative

115

k / is a Cauchy sequence in L1 .@/, whence converges in L1 .@/ Thus, . @u @n k2N to some function f . By the Sobolev embedding of solutions of the Dirichlet problem (Proposition 5.1), .uk /k2N converges to u in W01;1 ./. Letting k ! 1 in identity (7.5), we deduce that ˆ ˆ ˆ ru
r D d C f d:





@

The proof is complete.



Using Kato’s inequality (6.5), one shows that if u is a nonnegative solution of the linear Dirichlet problem with density  2 M./, then we have @u 6 0 on @ @n

(7.6)

almost everywhere with respect to the surface measure (Lemma 12.15). The proof of Proposition 7.3 yields (7.6) under the stronger assumption that  is nonnegative. Indeed, the approximating sequence .k /k2N is also nonnegative, whence by the weak maximum principle we have uk > 0 in . Since uk is smooth in a neighk borhood of @ and since uk D 0 on @, it follows that @u 6 0 on @. This @n @uk
@u implies (7.6) since the sequence @n k2N converges to @n in L1 .@/.

The existence of the weak normal derivative implies the following property concerning the extension of solutions of the Dirichlet problem with measure data:

Corollary 7.4. Let  be a smooth bounded open set and let  2 M./. If u is the solution of the linear Dirichlet problem with density , and if U W RN ! R denotes the extension of u by zero outside , then U 2 M.RN / and U D b C

@u N H @n

1

b@ :

In particular, we have kU kM.RN / 6 2kkM./ . We now explain the connection between this extension property and the balayage method, which was devised by Poincaré [280] and [281] to prove the existence of solutions of the Dirichlet problem for the Laplace equation with boundary data h 2 C 0 .@/: ´ u D 0 in , uDh

on @.

Poincaré’s strategy is based on the following physical property: given a distribution of charges in space, and some smooth region !, it is possible to move – sweep out – the electric charges from inside ! to the boundary @!, without changing the

116

7. Balayage

electric potential outside !. The new configuration of electric charges on @! is represented by a weak normal derivative (cf. (7.8) below). The balayage method consists in repeating this process infinitely many times over balls inside , so that all electric charges are eventually sent to the boundary @. This construction was extended by de la Vallée Poussin [106] to allow densities of charges described by measures, and not only by smooth functions. He proposed a new approach that emphasizes the role played by the density of charges, instead of focusing on the potentials they generate. Starting with a potential v, the next proposition establishes the existence of the balayage function R! v that describes the new potential after the charges are swept out from !: Proposition 7.5. If v 2 L1loc ./ is such that v 2 Mloc ./, then, for every smooth bounded open subset ! b , there exists a function R! v 2 L1loc ./ such that (i) R! v is harmonic in !,

(ii) R! v D v in  n !,

(iii) .R! v/ 2 Mloc ./. Proof. Denote by  the locally finite measure v in . Then, for every ' 2 Cc1 ./, we have ˆ ˆ v ' D ' d: 



On the other hand, thanks to the property of existence of the weak normal derivative, the solution u of the linear Dirichlet problem in ! with density b! satisfies ˆ ˆ ˆ @u u ' D ' d C ' d: @n ! ! @! Taking the function R! vW  ! R defined for x 2  by ´ v u in !, R! v.x/ D v in  n !,

we thus have

(7.7)

@u d:  n! @! @n We deduce that R! v satisfies the required properties. Indeed, by the integral identity above .R! v/ is the locally finite measure given by ˆ

.R! v/ ' D

ˆ

' d

ˆ

'

@u N 1 H b@! : (7.8) @n In particular, .R! v/ D 0 in the sense of distributions in !, and so by Weyl’s lemma (Proposition 2.14) the function R! v equals almost everywhere a smooth harmonic function in !.  .R! v/ D bn!

7.2. Finite charge up to the boundary

117

Corollary 7.6. If v 2 L1loc ./ is superharmonic, then, for every smooth bounded open subset ! b , the balayage function R! v is also superharmonic and satisfies R! v 6 v almost everywhere in . Proof. By Schwartz’s characterization of nonnegative distributions (see Proposition 2.20),  D v is a nonnegative locally finite measure on . The function u in the proof of Proposition 7.5 thus satisfies a Dirichlet problem with nonnegative density. By the weak maximum principle (Proposition 6.1), u is nonnegative in !, @u satisfies whence by (7.7) we deduce that R! v 6 v in . The normal derivative @n @u Property (7.6), namely @n 6 0 almost everywhere on @!. Since the measure  is nonnegative, we deduce from (7.8) that .R! v/ > 0, and the conclusion follows.  Exercise 7.1. Let  2 Mloc ./, let v 2 L1loc ./ be a nonnegative function such that v D  in the sense of distributions in , and let ! b  be a smooth bounded open set. .a/ Prove that v >  in the sense of .C01 .!// x 0. .b/ Deduce that R! v is also nonnegative in .

7.2 Finite charge up to the boundary Under the assumptions of Kato’s inequality, one might wonder whether the measure uC is finite in , rather than merely locally finite. The answer is negative, even if x see Proposition A.1 in [67]. u is harmonic in  and has a continuous extension to , However, the answer is affirmative if u satisfies the linear Dirichlet problem ´ u D  in , uD0

on @,

in which case we have the explicit estimate kuC kM./ 6 kukM./ : More generally, we have Theorem 1.2 in [67]: Proposition 7.7. Let  be a smooth bounded open set, and let  2 M./. If u is the solution of the linear Dirichlet problem with density , then, for every a 2 R, we have .u a/C 2 M./ and ´ kukM./ if a 6 0, C k.u a/ kM./ 6 2kukM./ if a > 0.

118

7. Balayage

The constant 2 is needed here because if a > 0, then the function .u a/C behaves as if it were compactly supported in . Example 7.8. Let uW . 1; 1/ ! R be the function defined for x 2 . 1; 1/ by u.x/ D 1

jxj:

In the sense of distributions in . 1; 1/, we have u00 D a/C /00 D

..u

2ı0 C ı

a

2ı0 and, for every 0 < a < 1, C ıa :

Therefore, a/C /00 kM..

k..u

1;1//

D 2ku00 kM..

1;1// :

x and a is a Proof of Proposition 7.7. We first prove the estimate when u 2 C01 ./ nonzero regular value of u. By Example 7.2, we have

Since

@u @n

@u N H @n

a/C D u HN b¹u>aº

.u

1

b¹uDaº :

6 0 on ¹u D aº, we compute k.u

C

a/ kM./ D

ˆ

¹u>aº

juj

ˆ

¹uDaº

@u d: @n

For a > 0, the boundary of the set ¹u > aº is ¹u D aº, and by the divergence theorem we have ˆ ˆ @u d D u: ¹u>aº ¹uDaº @n In this case, we get k.u

a/C kM./ D

ˆ

¹u>aº

juj

ˆ

u 6 2 ¹u>aº

ˆ

¹u>aº

juj 6 2

ˆ



juj:

For a < 0, the boundary of the set ¹u > aº is ¹u D aº [ @, so we consider instead ¹u < aº whose boundary is again ¹u D aº. Applying the divergence theorem on ¹u < aº, the orientation on ¹u D aº is reversed, and we get ˆ

¹uDaº

@u d D @n

ˆ

u: ¹uaº

¹u>aº

u ¹u 0 is any constant satisfying the Lipschitz condition of H . Proof. We first consider the case where H is convex, piecewise affine, and such that H 0 D 0 in a neighborhood of 0. We first write the nondecreasing function H 0 almost everywhere in R as H0 D

m X

n X

˛i .ai ;C1/

i D1

ˇj .

j D1

1;bj /

Cc

such that c 2 R, ˛i > 0, ˇj > 0, and am <


< a1 < 0 < b1 <


< bn : The parameters ˛i , ˇj , and c satisfy a linear system of equations and can be explicitly computed in terms of H 0 . We have in particular that m X i D1

˛i C

n X

j D1

ˇj D H 0 .C1/

H 0 . 1/

(7.9)

and c D H 0 .C1/ C H 0 . 1/:

(7.10)

For every t 2 R, we have H.t / D

m X

˛i .t

i D1

ai /C C

n X

ˇj .bj

j D1

t /C C ct C d:

It then follows from Proposition 7.7 that H.u/ 2 M./, and, by the triangle inequality, kH.u/kM./ 6

m X i D1

˛i C

n X

j D1

 ˇj C jcj kukM./ :

7.2. Finite charge up to the boundary

121

Since, for every s; t 2 R, we have max ¹jsj; jt jº D

js

t j C js C t j ; 2

using (7.9) and (7.10) the previous estimate becomes kH.u/kM./ 6 2 max ¹jH 0 .C1/j; jH 0 . 1/jº kukM./ : We thus have the conclusion for convex piecewise affine functions H . The case of a convex Lipschitz-continuous function H follows by approximation of H using a sequence of convex piecewise affine functions .Hn /n2N converging uniformly to H on bounded subsets of R; since 0 is a minimum point of H , we may choose Hn so as to have Hn0 D 0 in a neighborhood of 0.  In contrast with what happens to the composition of Lipschitz-continuous functions with Sobolev functions in W 1;p (see Theorem 2.1 in [12] or Theorem 4.2 [35]) it is not possible to have an explicit formula of H.u/ solely in terms of u and ru. Even when u is a smooth function, H.u/ need not be a function, but a legitimate measure having a singular part (Example 6.7).

Chapter 8

Precise representative

“If the exceptional sets are too large, then it is impossible to discuss derivatives, boundary values, etc., in the normal way.” Nachman Aronszajn and Kennan T. Smith

We identify points that represent well the values of a summable function u in RN using the concept of Lebesgue set. Weak estimates for the maximal function are used to study the size of the exceptional set.

8.1 Lebesgue’s differentiation theorem The Lebesgue differentiation theorem can be stated as follows: Proposition 8.1. If u 2 L1 .RN /, then, for almost every x 2 RN , we have lim

r !0 B.xIr /

ju

u.x/j D 0:

This implies the differentiability almost everywhere of the primitive function associated to a summable function in R, see [192] and [194]: Corollary 8.2. For every u 2 L1 .R/, the function U W R ! R defined by ˆ x U.x/ D u 0

is differentiable for almost every x 2 R, and U 0 D u almost everywhere in R. Proof of Corollary 8.2. Let x 2 R. By the additivity of the integral, for every r > 0, U.x C r/ r

U.x/

u.x/ D

1 r

ˆ

xCr

Œu.y/ x

u.x/ dy:

124

8. Precise representative

Thus, ˇ ˇ U.x C r/ ˇ ˇ r

U.x/

ˇ ˆ ˇ 1 xCr u.x/ˇˇ 6 ju.y/ r x 62

B.xIr /

ju.y/

u.x/j dy u.x/j dy:

We deduce from the Lebesgue differentiation theorem that, for almost every x 2 R, U.x C r/ r r &0 lim

U.x/

D u.x/:

A similar argument gives the limit from below, and the conclusion follows.



In particular, every summable function is the derivative almost everywhere of a continuous function. Lusin’s theorem [215] asserts that such a conclusion holds for every measurable function in R, summable or not; the counterpart in higher dimensions is also true, see [11] and [255]. A useful tool in the proof of the Lebesgue differentiation theorem is the maximal function MuW RN ! Œ0; C1 associated to a function u 2 L1loc .RN /, and defined for every x 2 RN by Mu.x/ D sup

r >0 B.xIr /

juj:

Since the maximal function is the supremum of a family of continuous functions, for every  > 0 the level set ¹Mu > º is open, and so Mu is lower semicontinuous. Using a suitable covering of the open set ¹Mu > º by balls, we prove in the next chapter the following weak L1 estimate for the maximal function (Proposition 9.1): for every  > 0, C j¹Mu > ºj 6 kukL1 .RN / ;  for some constant C > 0 depending on the dimension N . While the Chebyshev inequality 1 j¹juj > ºj 6 kukL1 .RN /  is a trivial consequence of the monotonicity of the Lebesgue measure, the maximal inequality is based on Wiener’s covering lemma (Lemma 9.2). Proof of Proposition 8.1. The conclusion is immediate for continuous functions, in which case the limit holds for every x 2 RN . For an arbitrary function u 2 L1 .RN /, we proceed by approximation with continuous functions with compact support in RN as follows.

8.1. Lebesgue’s differentiation theorem

Given x 2 RN and B.xIr /

ju

125

2 Cc0 .RN /, by the triangle inequality we have

u.x/j 6 B.xIr /

ju

jC

B.xIr /

j

.x/j C j .x/

u.x/j;

for every r > 0. Thus, as r ! 0, we get lim sup r !0

B.xIr /

ju

u.x/j 6 lim sup r !0

B.xIr /

°  M.u

r !0

B.xIr /

/>

j C j .x/

/.x/ C j .x/

6 M.u Hence, for every  > 0, we have ² x 2 RN W lim sup

ju

ju

± ° [ j 2

u.x/j:

u.x/j >  uj >

u.x/j

³

± : 2

By the subadditivity of the Lebesgue measure, we then get ˇ² ³ˇ ˇ ˇ ˇ x 2 RN W lim sup ju u.x/j >  ˇˇ ˇ ˇ° ˇ 6 ˇ M.u

r !0

B.xIr /

/>

 ±ˇˇ ˇˇ° ˇCˇ j 2

uj >

 ±ˇˇ ˇ: 2

By the weak maximal inequality for summable functions (Proposition 9.1), ˇ° ˇ ˇ M.u

/>

and, by the Chebyshev inequality, ˇ° ˇ ˇ j

uj >

 ±ˇˇ 2C ku ˇ6 2   ±ˇˇ 2 ˇ6 k 2 

kL1 .RN / ;

ukL1 .RN / :

Combining these estimates, we deduce that ˇ² ³ˇ ˇ 2.C C 1/ ˇ ˇ6 ˇ x 2 RN W lim sup ju u.x/j >  k ˇ ˇ  r !0 B.xIr /

ukL1 .RN / :

Since Cc0 .RN / is dense L1 .RN /, for every  > 0 we can choose 2 Cc0 .RN / such that the quantity in the right-hand side is less than . Since  > 0 is arbitrary, ˇ² ³ˇ ˇ ˇ ˇ x 2 RN W lim sup ju u.x/j >  ˇˇ D 0: ˇ r !0

B.xIr /

126

8. Precise representative

Applying this conclusion to a non-increasing sequence of positive numbers .n /n2N converging to zero, from the monotone set lemma (Exercise 2.1) we deduce that lim sup r !0

B.xIr /

ju

u.x/j D 0

for almost every x 2 RN .



For every r > 0, we have ˇ ˇ ˇ u ˇ B.xIr /

ˇ ˇ u.x/ˇˇ 6

B.xIr /

ju

u.x/j:

(8.1)

Thus, if x 2 RN satisfies the limit in the conclusion of the Lebesgue differentiation theorem, then lim

r !0 B.xIr /

u D u.x/:

The quantity in the right-hand side of (8.1) gives a stable way of computing limits in terms of other averaging processes, not necessarily over balls (cf. Proposition 8.4). For this purpose, we introduce the notions of Lebesgue point and precise representative motivated by the Lebesgue differentiation theorem: Definition 8.3. Let u 2 L1loc .RN /. A point x 2 RN is a Lebesgue point of u if there exists u.x/ O 2 R such that lim

r !0 B.xIr /

ju

u.x/j O D 0:

The function uW O Lu ! R thus defined on the Lebesgue set Lu of all Lebesgue points is the precise representative of u. Exercise 8.1 (linearity). Let u; v 2 L1loc .RN / and ˛ 2 R. Prove that .a/ Lu \ Lv  LuCv and

1

u C v D uO C vO .b/ Lu  L˛u and

in Lu \ Lv ;

˛c u D ˛ uO in Lu .

Exercise 8.2 (composition with Lipschitz functions). Let u 2 L1loc .RN /. Prove that if H W R ! R is a Lipschitz-continuous function, then Lu  LH.u/ and

1

H .u/ D H.u/ O

in Lu :

8.1. Lebesgue’s differentiation theorem

127

By the Lebesgue differentiation theorem, we know that almost every point x 2 RN is a Lebesgue point of u and u.x/ O D u.x/. In particular, the exceptional set N R n Lu satisfies ˇ ˇ N ˇR n Lu ˇ D 0: Given u 2 L1 .RN / and a sequence of mollifiers .n /n2N , we observe that the sequence .n  u/n2N converges to u in L1 .RN /, whence by the partial converse of the dominated convergence theorem (Proposition 4.9) there exists a subsequence .nk  u/k2N converging to u almost everywhere in RN . This conclusion is unsatisfactory since we are not able to deduce almost everywhere convergence of the entire sequence .n  u/n2N . Assuming that .n /n2N is a sequence of mollifiers of the form n .z/ D r1N . rzn / n for some fixed function  2 Cc1 .B.0I 1// and some sequence of positive numbers .rn /n2N converging to zero, we have that the whole sequence .n  u.x//n2N converges to the precise representative u.x/ O at every Lebesgue point x 2 Lu : 1 N Proposition 8.4. Let u 2 L1 .RN /, and let ´  2 L .R / be a bounded function supported in the ball B.0I 1/ and such that RN  D 1. Then, for every x 2 Lu , we have ˆ x y  1 lim N u.y/ dy D u.x/: O  r !0 r r RN ´ Proof. Let x 2 Lu . Since RN  D 1, for every r > 0 we have ˇ ˇ ˆ ˆ ˇ  ˇ x y  ˇ 1 ˇ 1 ˇ x y ˇ ˇ ˇ u.y/ dy u.x/ O O dy:  6 ˇju.y/ u.x/j ˇ ˇrN ˇ rN r r RN RN

By the assumption on the support of , the integrand vanishes in the complement of the ball B.xI r/. Since  is bounded in RN , we then have ˇ ˇ ˆ x y  ˇ ˇ 1 ˇ ju.y/ u.x/j O dy: u.y/ dy u.x/ O ˇˇ 6 C  ˇrN r B.xIr / RN Letting r ! 0, the conclusion follows.



The concepts of Lebesgue point and precise representative have a natural counterpart for functions u 2 L1loc ./, defined in an open subset   RN , due to the local character of the definition. The identity u.x/ O D lim

r !0 B.xIr /

u

satisfied by every Lebesgue point of  gives a way of recovering – or at least finding the natural candidate for – the precise representative of u. However, the existence of the limit in the right-hand side is not enough to ensure that x is a Lebesgue point:

128

8. Precise representative

Example 8.5. Take uW R ! R defined by u.x/ D 1 if x > 0 and u.x/ D 0 if x < 0. We have 1 lim uD ; r !0 B.0Ir / 2 but 0 is not a Lebesgue point of u since, for every r > 0, we have

B.0Ir /

ˇ ˇ ˇu

1 ˇˇ 1 ˇD : 2 2

8.2 Sobolev functions Compared to exceptional sets of merely summable functions, those associated to Sobolev functions are typically smaller: Proposition 8.6. Let 1 6 q < C1. If u 2 W 1;q .RN /, then the exceptional set RN n Lu satisfies capW 1;q .RN n Lu / D 0: To prove this proposition, we need the following maximal capacitary estimate for Sobolev functions (Proposition 9.3): for every  > 0, capW 1;q .¹Mu > º/ 6

C kukqW 1;q .RN / : q

Another tool is the density of smooth functions in W 1;q .RN /, whose proof is based on a convolution and a cut-off argument, see Theorem 9.2 in [53] or Theorem 6.1.10 in [345]: Proposition 8.7. The set Cc1 .RN / is dense in W 1;q .RN /. For example, if u 2 W 1;q .RN / has compact support, then the approximating sequence can be obtained by convolution with a sequence of mollifiers. Without the additional assumption on the support, we first approximate u by functions of the form u' such that ' 2 Cc1 .RN /. The strategy of the proof of Proposition 8.6 is essentially the same as for the Lebesgue differentiation theorem, although the precise representative need not be u.x/ for quasi-every x 2 RN – i.e., except for a set of zero W 1;q capacity. In this case, it is convenient to characterize Lebesgue points by a Cauchy property, without reference to the value of the precise representative:

8.2. Sobolev functions

129

Lemma 8.8. For every u 2 L1loc .RN /, we have that x 2 RN is a Lebesgue point of u if and only if lim

.r;s/!.0;0/ B.xIr /

B.xIs/

u.z/j dy dz D 0:

ju.y/

O is the precise Proof of Lemma 8.8. If x 2 RN is a Lebesgue point of u and if u.x/ representative of u, then by the triangle inequality B.xIr /

B.xIs/

6 B.xIr /

u.z/j dy dz

ju.y/

ju.y/

u.x/j O dy C

B.xIs/

ju.z/

u.x/j O dz;

for every r; s > 0. Since the right-hand side converges to zero as .r; s/ ! .0; 0/, the direct implication follows. Conversely, given ˛ 2 R, by the triangle inequality ˇ ˇ ˇ ˇ (8.2) u ˛ ˇˇ; ju ˛j 6 ju.y/ u.z/j dy dz C ˇˇ B.xIr /

B.xIr /

B.xIr /

B.xIr /

for every r > 0. By assumption, the first term in the right-hand side converges to zero as r ! 0. We now observe that the limit lim

r !0 B.xIr /

exists. Indeed, for every r; s > 0, we have ˇ ˇ ˇ ˇ ˇ ˇ6 u u ˇ ˇ B.xIr /

B.xIs/

B.xIr /

u

B.xIs/

ju.y/

u.z/j dy dz:

Thus, if the quantity in the right-hand side converges to zero as .r; s/ ! .0; 0/, the average integrals of u satisfy a Cauchy property, so they converge to a limit. To conclude, it now suffices to choose ˛ to be this limit. We deduce from estimate (8.2) that lim

This gives the conclusion.

r !0 B.xIr /

ju

˛j D 0: 

Proof of Proposition 8.6. Given x 2 RN and inequality we have B.xIr /

B.xIs/

6 B.xIr /

j

ju.y/ uj C

2 Cc1 .RN /, by the triangle

u.z/j dy dz

B.xIr /

B.xIs/

j .y/

.z/j dy dz C

B.xIs/

j

uj;

130

8. Precise representative

for every r; s > 0. We upper bound the first and last terms in the right-hand side by M. u/.x/, independently of r and s. Thus, by the continuity of at x, as .r; s/ ! .0; 0/ we get lim sup .r;s/!.0;0/ B.xIr /

B.xIs/

u.z/j dy dz 6 2M.

ju.y/

u/.x/:

Thanks to the monotonicity of the capacity (Proposition A.8), for every  > 0 we have ² ³ capW 1;q x 2 RN W lim sup ju.y/ u.z/j dy dz >  .r;s/!.0;0/ B.xIr /

6 capW 1;q

°

M.

u/ >

 ± 2

B.xIs/

:

Thus, by the maximal capacitary estimate (Proposition 9.3), we get ² ³ N capW 1;q x 2 R W lim sup ju.y/ u.z/j dy dz >  .r;s/!.0;0/ B.xIr /

6

2q C k q

B.xIs/

ukqW 1;q .RN / :

Since Cc1 .RN / is dense in W 1;q .RN / (Proposition 8.7), for every  > 0 we can choose such that the quantity in the right-hand side is less than . We deduce that ² ³ capW 1;q x 2 RN W lim sup ju.y/ u.z/j dy dz >  D 0: .r;s/!.0;0/ B.xIr /

B.xIs/

Applying this conclusion to a non-increasing sequence of positive numbers .n /n2N converging to zero, and using the subadditivity of the capacity (Proposition A.9) it follows that, for quasi-every x 2 RN , we have lim

.r;s/!.0;0/ B.xIr /

B.xIs/

ju.y/

u.z/j dy dz D 0:

We conclude using the Cauchy characterization of Lebesgue points (Lemma 8.8).  The higher order counterpart of Proposition 8.6 for q > 1 can be found in Theorem 3.3.3 in [348]. Precise representatives may be unsuitable in problems involving restrictions over generic lower dimensional sets – e.g., over lines or planes in RN – or inverse images like in co-area formulas. In these cases, other types of representatives should be used, see [154], and this is related to another approach of Sobolev functions via restrictions to lines in connection with the classical notion of absolute continuity, see [73], [109], [256], and [264].

8.3. Potentials

131

8.3 Potentials We investigate the size of the exceptional set of potentials, namely functions u 2 L1loc ./ such that u 2 Mloc ./. The answer involves the W 1;2 capacity, which is related to some capacity cap1 defined in terms of the Laplacian (cf. Propositions 12.1 and 12.2). We begin in the setting of solutions of the Dirichlet problem: Proposition 8.9. Let  be a smooth bounded open set, and let  2 M./. If u is the solution of the linear Dirichlet problem with density , then the exceptional set  n Lu satisfies capW 1;2 . n Lu / D 0: The main ingredient in this case is the following weak maximal inequality (Proposition 9.6): for every  > 0, capW 1;2 .¹Mu > º/ 6

C C kkM./ D kukM./ :  

Since u need not be defined in the whole space RN , the maximal function MuW  ! Œ0; C1 is computed by taking the supremum over balls contained in : Mu.x/ D

sup 0 3, and let  2 M.RN / be a nonnegative measure. .a/ For every x 2 RN , prove that the Newtonian potential N associated to  satisfies lim

r !0 B.xIr /

N D N.x/:

b

.b/ Deduce that LN D ¹N < C1º and N D N in LN . Proof of Proposition 8.9. We first assume that u is superharmonic. By Lemma 8.10, it follows that  n Lu D ¹Mu D C1º: Thus, by the monotonicity of the capacity (Proposition A.8), for every  > 0 we have capW 1;2 . n Lu / 6 capW 1;2 .¹Mu > º/: By the weak maximal estimate for superharmonic functions (Proposition 9.6), we get capW 1;2 . n Lu / 6

C kukM./ : 

Letting  ! 1, we get the conclusion for superharmonic functions. Given u as in the statement, we write u D u1 u2 as a difference of two superharmonic functions in W01;1 ./; for instance, the solutions of the linear Dirichlet problems with densities . u/C and . u/ . Since (Exercise 8.1) Lu1 \ Lu2  Lu ; by the subadditivity of the capacity (Proposition A.9) we deduce that capW 1;2 . n Lu / 6 capW 1;2 . n Lu1 / C capW 1;2 . n Lu2 / D 0:



134

8. Precise representative

We also have the following localized counterpart of Proposition 8.9: Proposition 8.11. If u 2 L1loc ./ is such that u 2 Mloc ./, then the exceptional set  n Lu satisfies capW 1;2 . n Lu / D 0: Proof. Take a sequence of smooth bounded open sets .!n /n2N strictly contained S !n . By the subadditivity of the W 1;2 capacity, in , and such that  D n2N

capW 1;2 . n Lu / 6

1 X

nD0

capW 1;2 .!n n Lu /:

It thus suffices to prove that each of the sets !n n Lu has zero W 1;2 capacity. For this purpose, take 'n 2 Cc1 ./ such that 'n D 1 in !n . By the local character of Lebesgue points, the exceptional points of the functions u and u'n coincide in !n , and thus capW 1;2 .!n n Lu / D capW 1;2 .!n n Lu'n /:

By the localization lemma (Proposition 6.11), we also have u'n 2 W01;1 ./ and .u'n / 2 M./. Hence, u'n satisfies the linear Dirichlet problem with measure datum. It thus follows from Proposition 8.9 that capW 1;2 .!n n Lu / 6 capW 1;2 . n Lu'n / D 0; and this implies the conclusion.



8.4 Kato’s inequality revisited To illustrate how precise representatives arise in applications, we now prove Kato’s inequality when u is a diffuse measure: Proposition 8.12. Let u 2 L1 ./ be such that u 2 M./. If the measure u is diffuse with respect to the W 1;2 capacity, then uC > ¹u>0º u O in the sense of distributions in , where uO is the precise representative of u. The proof follows the same three steps as for the standard Kato’s inequality (Proposition 6.6). The second one is more subtle, and we follow Oloffson’s strategy [267]. An alternative argument (see [65] and [97]) relies on a characterization of diffuse measures as elements of the functional space L1 ./ C .W01;2 .//0 due to Boccardo, Gallouët, and Orsina [32], see Proposition 14.13 below.

8.4. Kato’s inequality revisited

135

Proof of Proposition 8.12. Given a smooth convex function H W R ! R, a nonnegative test function ' 2 Cc1 .RN /, and a sequence of mollifiers .n /n2N such that supp ' supp n b , we have inequality (6.3), which we recall below: ˆ ˆ H.n  u/ ' > H 0 .n  u/.n  u/ ': 



Assuming in addition that n is even, by Fubini’s theorem we can interchange the order of integration in the right-hand side to get ˆ ˆ H.n  u/ ' > n  ŒH 0 .n  u/' u: (8.4) 



We assume for what follows that n .x/ D r1N . rxn / for some fixed function n  2 Cc1 .B.0I 1// and a sequence of positive numbers .rn /n2N converging to zero. Claim. Assume that H 00 is bounded. If x 2  is a Lebesgue point of u, then lim n  ŒH 0 .n  u/'.x/ D H 0 .u.x//'.x/: O

n!1

Proof of the claim. We observe that 0 6 n 6

C1 B.0Irn / ; !N rnN

whence, for every function f 2 L1 .RN / and every ˛ 2 R, jn  f .x/

˛j 6 C1 B.xIrn /

jf

˛j:

In our case, we write this estimate as jn ŒH 0 .n u/'.x/ H 0 .u.x//'.x/j O 6 C1

B.xIrn /

jH 0 .n u/' H 0 .u.x//'.x/j: O

By the triangle inequality, we have jH 0 .n  u/'

H 0 .u.x//'.x/j O

6 k'kL1 .RN / jH 0 .n  u/

H 0 .u.x//j O C jH 0 .u.x//jj' O

'.x/j:

By the mean value theorem, we also have jH 0 .n  u/

H 0 .u.x//j O 6 kH 00 kL1 .RN / jn  u

u.x/j: O

In addition, for every y 2 B.xI rn / we have B.yI rn /  B.xI 2rn /, whence jn  u.y/

u.x/j O 6 C1

B.yIrn /

ju

u.x/j O 6 C1 2N

B.xI2rn /

ju

u.x/j: O

136

8. Precise representative

Therefore, we get

B.xIrn /

jH 0 .n  u/'

6 C2



B.xI2rn /

ju

H 0 .u.x//'.x/j O u.x/j O C

B.xIrn /

j'

 '.x/j :

Since x is a Lebesgue point of u and ' is continuous at x, the right-hand side converges to zero as n ! 1, and this implies lim jn  ŒH 0 .n  u/'.x/

n!1

H 0 .u.x//'.x/j O D 0:

4

We know that the sequence .n  ŒH 0 .n  u/'/n2N converges pointwise to H .u/' O in the Lebesgue set Lu . Since the exceptional set  n Lu satisfies (Proposition 8.11) capW 1;2 . n Lu / D 0; 0

and since the measure u is diffuse with respect to the W 1;2 capacity, we deduce that the sequence .n  ŒH 0 .n  u/'/n2N converges almost everywhere to H 0 .u/' O 0 with respect to u. Assuming that the derivative H is bounded, it follows from inequality (8.4) and the dominated convergence theorem that ˆ ˆ H.u/ ' > H 0 .u/' O u: 



To conclude, it suffices to take a suitable sequence of functions .Hn /n2N converging pointwise to the function R 3 t 7! t C 2 R as in the proof of Kato’s inequality (Proposition 6.6).  We briefly sketch the proof of Kato’s inequality for a general measure u, not necessarily diffuse (Proposition 6.10). In this case, we use the Lebesgue decomposition of the measure u as u D .u/d C .u/c ; in terms of a diffuse part and a concentrated part with respect to the W 1;2 capacity (Proposition 14.12). We then lower bound the integral in the right-hand side of (8.4) by ˆ ˆ ˆ 0 0 n  ŒH .n  u/' u > n  ŒH .n  u/' .u/d C n  ' j.u/c j; 





where C D kH 0 kL1 ./ . Proceeding as in the previous proof, we deduce that H.u/ > H 0 .u/.u/ O d

C j.u/c j;

8.4. Kato’s inequality revisited

137

and then uC > ¹u>0º .u/d O

C j.u/c j;

where both inequalities are considered in the sense of distributions in . The constant C can be chosen to be 1, but this is irrelevant. Since this last inequality is also satisfied in the sense of measures (Proposition 6.12), comparing the diffuse parts from both sides one deduces Proposition 6.10.

Chapter 9

Maximal inequalities

“Almost everywhere convergence is proved as a combination of two parts. The deepest one already contains the essence of the result, and is expressed in terms of a maximal inequality.” Elias M. Stein

We prove weak inequalities satisfied by the maximal operator for three families of functions: summable functions, Sobolev functions, and potentials. Covering lemmata are often important tools in these problems.

9.1 Integral estimate We start with the maximal inequality satisfied by summable functions: Proposition 9.1. If u 2 L1 .RN /, then, for every  > 0, we have j¹Mu > ºj 6 by

3N kukL1 .RN / : 

We recall that the maximal function MuW RN ! Œ0; C1 is defined for x 2 RN Mu.x/ D sup

r >0 B.xIr /

juj:

This maximal inequality was proved by F. Riesz [297] in dimension 1, and then extended by Marcinkiewicz and Zygmund (Lemma 2 in [219]) and by Wiener (Lemma D0 in [344]) to higher dimensions. One cannot expect a strong L1 estimate of the maximal function (cf. Exercise 9.3): Exercise 9.1 (nonsummability in RN ). Let u 2 L1 .RN /. .a/ Prove that ˆ 1 N juj: lim inf jxj Mu.x/ > !N RN jxj!1

.b/ Deduce that if Mu 2 L1 .RN /, then u D 0 almost everywhere in RN .

140

9. Maximal inequalities

The obstruction is also of local nature: Exercise 9.2. Let ˛ > 0, and let uW RN ! R be a measurable function such that, for every x 2 B.0I 21 / n ¹0º, we have u.x/ D

1 jxjN .log 1=jxj/˛

:

.a/ Prove that u 2 L1 .B.0I 12 // if and only if ˛ > 1. .b/ Prove that, for every x 2 B.0I 21 /, we have Mu.x/ >

C jxjN .log 1=jxj/˛ 1

;

for some constant C > 0. Deduce that Mu 62 L1 .B.0I 21 // for ˛ 6 2. We present a proof of Proposition 9.1 based on Wiener’s covering lemma, see Lemma C0 in [344]: Lemma 9.2. Let .A; d / be a metric space. If .B.xi I ri //i 2¹0;:::;nº is a finite covering of A, then there exists a subset of indices J  ¹0; : : : ; nº such that (i) the balls .B.xi I ri //i 2J are disjoint,

(ii) the balls .B.xi I 3ri //i 2J cover A.

The goal here is to extract by induction a maximal family .B.xij I rij //j 2¹0;:::;kº of disjoint balls. We choose in each step a ball whose radius is as large as possible. Proof of Lemma 9.2. We may assume that the balls B.x0 I r0 /;

B.x1 I r1 /;

:::;

B.xn I rn /;

are ordered from the largest to the smallest radius: if i > j , then ri 6 rj . To construct a maximal family having the desired properties, we begin by taking i0 D 0. Suppose that, for some ` 2 ¹0; : : : ; nº, we have already chosen disjoint balls B.xi0 I ri0 /;

:::;

B.xi` I ri` /:

We then take i`C1 to be the first index such that the ball B.xi`C1 I ri`C1 / is disjoint from the balls we have already selected. Since we are given finitely many balls, this process has to stop for some integer ` D k. By construction, the balls .B.xij I rij //j 2¹0;:::;kº are disjoint. To prove the second property, we observe that, for every ball B.xl I rl / from the given covering, there exists B.xij I rij / in the maximal family such that ij 6 l and B.xl I rl / \ B.xij I rij / ¤ ;:

9.1. Integral estimate

141

By the triangle inequality, we then have B.xl I rl /  B.xij I rij C 2rl /: From the relation rij > rl , we deduce that B.xl I rl /  B.xij I 3rij /: Therefore, A

n [

lD0

B.xl I rl / 

k [

j D0

B.xij I 3rij /:

We thus have the conclusion with J D ¹i0 ; : : : ; ik º.



Proof of Proposition 9.1. For every x 2 ¹Mu > º, let r.x/ > 0 be such that B.xIr .x//

juj > 

or, equivalently, !N r.x/

N

1 6 

ˆ

B.xIr .x//

juj:

Given a compact subset K  ¹Mu > º, from the open covering .B.xI r.x///x2K of K we first extract a finite covering .B.xi I r.xi ///i 2¹0;:::;nº . By Wiener’s covering lemma (Lemma 9.2), there exists J  ¹0; : : : ; nº such that the balls .B.xi I r.xi ///i 2J are disjoint and the collection of enlarged balls .B.xi I 3r.xi ///i 2J covers K. In particular, by the monotonicity and subadditivity of the Lebesgue measure we have ˇ X ˇ[ X ˇ ˇ !N r.xi /N : (9.1) !N Œ3r.xi /N D 3N B.xi I 3r.xi //ˇ 6 jKj 6 ˇ i 2J

i 2J

i 2J

Since the balls .B.xi I r.xi ///i 2J are disjoint, by the choice of the radii r.xi / and by the additivity of the integral we have ˆ X1ˆ X 1 !N r.xi /N 6 juj: juj D  B.xIr .xi //  S B.xIr .xi // i 2J

i 2J

i 2J

By the monotonicity of the Lebesgue measure, we then get the estimate ˆ X 1 juj: !N r.xi /N 6  RN i 2J

(9.2)

142

9. Maximal inequalities

Combining inequalities (9.1) and (9.2), we deduce that 3N jKj 6 

ˆ

RN

juj:

We can now apply this estimate to a nondecreasing sequence of compact subsets .Kj /j 2N exhausting the open set ¹Mu > º. Using the monotone set lemma  (Exercise 2.1), we deduce the same upper bound for the measure j¹Mu > ºj. Variants of the maximal inequality can be obtained by applying Proposition 9.1 to some clever choice of function. For instance, we deduce from the maximal inequality applied to v D u¹juj>  º that 2

j¹Mu > ºj 6

2C 

ˆ

¹juj>  2º

juj:

(9.3)

Indeed, from the pointwise comparison juj 6 jvj C =2, we have Mu 6 Mv C =2. Hence, ° ± ¹Mu > º  Mv > : 2 We also have ˆ ˆ RN

jvj D

¹juj>  2º

juj:

Combining the two assertions, we deduce the improved maximal inequality (9.3). In particular, we have lim  j¹Mu > ºj D 0: !1

Exercise 9.3 (Hardy–Littlewood inequality [159]). For every u 2 Lq .RN / such that 1 < q < C1, prove that Mu 2 Lq .RN / and kMukLq .RN / 6 C 0 kukLq .RN / ; for some constant C 0 > 0 depending on N and q. Exercise 9.4 (local summability). Prove that if uW RN ! R is a measurable function in the L log L class, then Mu 2 L1loc .RN /. More precisely, if u log .juj C 1/ 2 L1 .RN /, then, for every R > 0, we have ˆ   juj kMukL1 .B.0IR// 6 2kukL1 .RN / C C juj logC ffl ; RN B.0IR/ juj

for some constant C > 0 depending on N .

9.2. Energy estimate

143

9.2 Energy estimate We now consider the maximal inequality for Sobolev functions of order one: Proposition 9.3. Let 1 6 q < C1. If u 2 W 1;q .RN /, then, for every  > 0, we have C capW 1;q .¹Mu > º/ 6 q kukqW 1;q .RN / ;  for some constant C > 0 depending on N and q. The proof below follows the approach of Federer and Ziemer, see Section 7 in [129] and Section 4.8 in [126]. We apply the following version of the Besicovitch covering lemma for cubic coverings, see Lemmata 1 and 2 in [26]: Lemma 9.4. Let A  RN be a compact set, and let rW A ! R be a positive function such that, for every non-empty compact subset B  A, the supremum supB r is achieved in B. Then, given the covering .Q.xI r.x///x2A of A, there exists a finite subset J  A such that (i) the cubes .Q.xI 21 r.x///x2J are disjoint, (ii) the cubes .Q.xI r.x///x2J cover A, (iii) every point in RN belongs to at most 2N cubes in .Q.xI r.x///x2J . We denote by Q.xI r/ the open cube with respect to the canonical Euclidean basis ¹e1 ; : : : ; eN º centered at x and with side length 2r; we call r the radius of the cube. Denoting by j
j1 the max norm in RN , jzj1 D max ¹jzi jW i 2 ¹1; : : : ; N ºº; we have that y 2 Q.xI r/ if and and only if jx yj1 < r. Our presentation of the Besicovitch covering lemma has been inspired by the book of de Guzmán, see Theorem 1.2 in [104]. The assumption on the radius function rW A ! R is satisfied when r is upper semicontinuous: for every a 2 A, r.a/ > lim sup r.x/; x!a

which is the case in the proof of Proposition 9.3. Another example of interest arises when the range r.A/ is finite, or r.A/ is countably infinite and 0 is the only cluster point of r.A/; for instance, r.A/  ¹2 for some fixed number ˛ > 0.

k

˛W k 2 Nº;

144

9. Maximal inequalities

The use of cubes makes the proof easier than for balls, and also gives a neat estimate on the number of overlaps. The upper bound 2N is the number of quadrants in RN , and arises from the following elementary observation: if Q.xI r.x// and Q.y; r.y// are two cubes containing the origin, and if their centers x and y lie in the same quadrant, then we have jx

yj1 < max ¹r.x/; r.y/º;

(9.4)

whence the largest cube contains the center of the smallest one. Indeed, assuming that both x and y belong to the first quadrant, then, for every i 2 ¹1; : : : ; N º, we have 0 6 xi < r.x/ and 0 6 yi < r.y/. Thus, r.y/ < xi

yi < r.x/;

and this implies estimate (9.4). Proof of Lemma 9.4. We construct a maximal family of cubes Q D .Q.xI r.x///x2J having disjoint centers in the following sense: for every x; y 2 J such that x ¤ y, y 62 Q.xI r.x//: We begin by showing that such a family satisfies Properties .i/ and .iii/. Indeed, for every x; y 2 J such that x ¤ y we have jx yj1 > r.x/. Interchanging the roles of x and y, we deduce that jx

yj1 > max ¹r.x/; r.y/º >

r.y/ r.x/ C ; 2 2

which implies that Q.xI 12 r.x// \ Q.yI 12 r.y// D ;: Thus, any family Q having disjoint centers satisfies the first property. We now show that Q also satisfies the third property. For this purpose, for every z 2 RN we divide the Euclidean space RN into 2N quadrants using the N hyperplanes passing through z, each one orthogonal to one of the elements of the canonical Euclidean basis. It suffices to observe that each quadrant contains the center of at most one cube in Q containing z. Indeed, if x; y 2 J belong to the same quadrant and z 2 Q.xI r.x// \ Q.yI r.y//, then by Property (9.4) we have y 2 Q.xI max ¹r.x/; r.y/º/: Interchanging the roles of x and y if necessary, we may assume that r.x/ > r.y/, and we conclude that y 2 Q.xI r.x//. Since the centers of the covering Q are disjoint, this means that y D x, and so z belongs to at most 2N cubes, one for each quadrant.

9.2. Energy estimate

145

To conclude the proof, it remains to show the existence of a maximal family of cubes with disjoint centers, satisfying the second property. The construction is similar to the proof of Wiener’s covering lemma (Lemma 9.2), and we proceed by induction. We first take x0 2 A such that r.x0 / D sup ¹r.x/W x 2 Aº: Assume now that, for some ` 2 N, we have chosen x0 ; : : : ; x` 2 A such that the cubes .Q.xi I r.xi ///i 2¹0;:::;`º have disjoint centers. If A 6

` [

i D0

Q.xi I r.xi //;

then we take x`C1 in the compact set A n

` S

i D0

Q.xi I r.xi // such that

² ³ ` [ r.x`C1 / D sup r.x/W x 2 A n Q.xi I r.xi // : i D0

Since the point x`C1 does not belong to any of the cubes .Q.xi I r.xi ///i 2¹0;:::;`º , for every i 2 ¹0; : : : ; `º we have jx`C1

xi j1 > r.xi /:

By the choice of the points xi , we also have r.xi / > r.x`C1 /: Therefore, jx`C1

xi j1 > max ¹r.xi /; r.x`C1 /º;

and we deduce that the cubes .Q.xi I r.xi ///i 2¹0;:::;`C1º also have disjoint centers. This construction stops after finitely many steps. Indeed, assume by contradiction that, for every ` 2 N, we had A 6

` [

Q.xi I r.xi //:

(9.5)

1 [

Q.xi I r.xi //:

(9.6)

i D0

We prove that, in this case, A

i D0

146

9. Maximal inequalities

Since A is compact, A would then be covered by finitely many cubes in this union, but this contradicts assumption (9.5) for every ` 2 N. To prove (9.6), observe that since the cubes .Q.xi I 21 r.xi ///i 2N are disjoint and are all contained in some fixed cube containing the bounded set A, by the additivity of the Lebesgue measure the sequence of radii .r.xi //i 2N converges to zero. Thus, given z 2 A, there exists ` 2 N such that r.x`C1 / < r.z/. The point z cannot belong to the class over which r.x`C1 / achieves the maximum of the radius function, and we deduce that z2

` [

i D0

Q.xi I r.xi //:

We have established (9.6), which leads to a contradiction if A were not covered by finitely many cubes. The maximal family Q we obtain after finitely many steps thus satisfies the second property. This concludes the proof of the covering lemma.  The uniform upper bound on the number of overlaps provided by the Besicovitch covering lemma (Assertion .iii/) can be algebraically stated as X Q.xIr .x// 6 2N : x2J

It thus follows that, for every nonnegative function f 2 L1 .RN /, we have the estimate ˆ ˆ X  Xˆ N f: (9.7) Q.xIr .x// f 6 2 f D x2J

Q.xIr .x//

RN

x2J

RN

In the proof of the maximal inequality, we need to replace some smooth function with zero average on a cube by another function with compact support in the whole space: ´ Lemma 9.5. For every v 2 C 1 .QŒaI r/ such that Q.aIr / v D 0 and every ı > 0, there exists vN 2 Cc1 .RN / such that (i) jvN

vj 6 ı in QŒaI r and supp vN  Q.aI 2r/,

(ii) kvk N W 1;q .RN / 6 C kvkW 1;q .Q.aIr // ,

for some constant C > 0 depending on N and q. The proof is based on the Poincaré–Wirtinger inequality on cubes, see Theorem 6.4.9 in [345]: for every u 2 C 1 .QŒaI r/,



u 6 C 0 rkrukLq .Q.aIr // : (9.8) u

q

Q.aIr /

L .Q.aIr //

The explicit dependence on r comes from a scaling and translation argument, starting from the Poincaré–Wirtinger inequality on the unit cube Q.0I 1/.

9.2. Energy estimate

147

Proof of Lemma 9.5. By reflection across the hyperplanes containing the faces of Q.aI r/, one constructs a Lipschitz-continuous extension wW Q.aI 2r/ ! R of v such that kwkLq .Q.aI2r // 6 C1 kvkLq .Q.a;r //

and krwkLq .Q.aI2r // 6 C1 krvkLq .Q.a;r // :

Given ' 2 Cc1 .Q.aI 2r// such that ' D 1 in QŒaI r and given a sequence of mollifiers .n /n2N such that supp ' C supp n b Q.aI 2r/, the function vN n D n  .w'/ belongs to Cc1 .Q.aI 2r//. Moreover, since the sequence .vN n /n2N converges uniformly to w' D v in QŒaI r, there exists n 2 N such that jvN n

vj 6 ı

in QŒaI r.

Assuming that 0 6 ' 6 1 in Q.aI 2r/, we have kvN n kLq .RN / 6 kw'kLq .Q.aI2r // 6 C1 kvkLq .Q.aIr // : Choosing ' such that jr'j 6 C2 =r in Q.aI 2r/, we also have C2 kwkLq .Q.aI2r // r   1 6 C3 krvkLq .Q.aIr // C kvkLq .Q.aIr // : r

kr vN n kLq .RN / 6 krwkLq .Q.aI2r // C

Finally, since

´

Q.aIr /

v D 0, the Poincaré–Wirtinger inequality (9.8) yields

kr vN n kLq .RN / 6 C3 .1 C C 0 /krvkLq .Q.aIr // ; and the proof is complete.



It is possible to construct a smooth – rather than merely Lipschitz-continuous – extension of v to the cube Q.aI 2r/ with control on the W 1;q norm, see [307]. In the proof of Lemma 9.5 in this case there is no need to use a sequence of mollifiers and the resulting function vN coincides with v on Q.aI r/. Proof of Proposition 9.3. We first assume that u is nonnegative and belongs to Cc1 .RN /. For every x 2 ¹Mu > º, let r.x/ > 0 be the largest number such that u > : B.xIr .x//

148

9. Maximal inequalities

Since B.xI r.x//  Q.xI r.x//, we have   uC uC u D Q.xIr .x//

(9.9)

u > ; Q.xIr .x//

where  D !N =2N . One of the two terms in the left-hand side must be greater than or equal to =2. We have in particular the inclusion

where

°  ± [ Ax ; Q.xI r.x//  u > 2 ² Ax D y 2 QŒxI r.x/W u.y/ C

³  ; u> 2 Q.xIr .x// ® ¯ Using u to estimate the capacity of the compact set u >  , we get 2 capW 1;q

® u>

 2

¯


6

2q kukqW 1;q .RN / : ./q

ffl Since the function u C Q.xIr .x// u does not have compact support, we need a separate argument to estimate the capacity of the set Ax . Claim. For every x 2 RN , we have capW 1;q .Ax / 6

C1 kukqW 1;q .Q.xIr .x/// : q

Proof of the claim. By Lemma 9.5 applied to the function u C there exists vx 2 Cc1 .RN / such that    in QŒxI r.x/ vx > uC u 4 Q.xIr .x//

ffl

Q.xIr .x//

u,

and (Exercise 9.5) kvx kW 1;q .RN / 6 C2 kukW 1;q .Q.xIr .x/// : In particular, vx > set Ax , we then get

 4

in Ax . Using the function vx to estimate the capacity of the

capW 1;q .Ax / 6

4q C3 kvx kqW 1;q .RN / 6 q kukqW 1;q .Q.xIr .x/// : q ./ 

4

9.2. Energy estimate

149

Given a compact set K  ¹Mu > º and a finite subset J D ¹x1 ; : : : ; x` º  K such that the cubes .Q.xI r.x///x2J cover K, we have  ± [ [ Axi : K u> 2 °

`

i D1

By the monotonicity and subadditivity of the capacity (Proposition A.3), we get capW 1;q .K/ 6 capW 1;q

` °  ± X u> C capW 1;q .Axi /: 2 i D1

Applying the estimates satisfied by the capacities in the right-hand side, we obtain capW 1;q .K/ 6

` C1 X 2q q kukqW 1;q .Q.x Ir .x /// : C kuk W 1;q .RN / i i ./q q i D1

By the maximality of the choice of the radius function, r is upper semicontinuous in K, and so it attains its supremum on non-empty compact subsets. By the Besicovitch covering lemma (Lemma 9.4), the set J can be chosen such that every point in RN belongs to at most 2N cubes in .Q.xI r.x///x2J . By Property (9.7), we thus have ` X kukqW 1;q .Q.x Ir .x /// 6 2N kukqW 1;q .RN / ; i

i

i D1

and we conclude that

capW 1;q .K/ 6

C4 kukqW 1;q .RN / : q

Taking the supremum over all compact subsets K of the open set ¹Mu > º, we get the estimate for nonnegative functions u 2 Cc1 .RN /. For a nonnegative function u 2 W 1;q .RN /, the conclusion follows by approximation of u by means of nonnegative functions from Cc1 .RN / (cf. Proposition 8.7) and by the lower semicontinuity property of the maximal operator (Lemma 9.8). Finally, for an arbitrary function u 2 W 1;q .RN /, we decompose u as a difference of two nonnegative functions u D uC u , and then derive the estimate applying the subadditivity of the maximal operator (Lemma 9.9): capW 1;q .¹Mu > º/ 6

C5 C5 C q ku kW 1;q .RN / C q ku kqW 1;q .RN / : q  

Since jruC j 6 jruj and jru j 6 jruj almost everywhere in RN (cf. Exercise 5.3), the conclusion follows. 

150

9. Maximal inequalities

Exercise 9.5. Let u 2 Lq .RN /. Frove that, for every a 2 RN and every r > 0,



u 6 2kukLq .Q.aIr // : u

q

Q.aIr /

L .Q.aIr //

Kinnunen [181] has proved that, for every 1 < q < C1 and every u 2 W 1;q .RN /, Mu 2 W 1;q .RN / and jrMuj 6 Mjruj

almost everywhere in RN . The argument relies on the Hardy–Littlewood maximal inequality (Exercise 9.3), which holds for exponents q > 1. In contrast, it is not known whether rMu 2 L1 .RN / for every u 2 W 1;1 .RN /, except in dimension N D 1, see [157], [158], and [188].

9.3 Total charge estimate We now prove a maximal inequality for potentials: Proposition 9.6. Let  be a smooth bounded open set, and let  2 M./. If u is the solution of the linear Dirichlet problem with density , then, for every  > 0, we have C capW 1;2 .¹Mu > º/ 6 kkM./ ;  for some constant C > 0 depending on . We recall that the maximal function MuW  ! Œ0; C1 is computed by taking the supremum over balls contained in : Mu.x/ D

sup 0 º, a regularized version of T .u/= is a natural candidate to estimate the Sobolev capacity. Alternatively, we consider a smooth variant of the interpolation inequality: x and every Lemma 9.7. Let  be a smooth bounded open set. For every v 2 C01 ./ 1 H 2 C .R/ such that H.0/ D 0, we have ˆ H 0 .v/jrvj2 6 kH.v/kL1./ kvkL1 ./ : 

Proof of Lemma 9.7. We have div ŒH.v/rv D H 0 .v/jrvj2 C H.v/v: Since H.v/ D 0 on @, by the divergence theorem we get ˆ ˆ 0 2 H .v/jrvj D H.v/v 6 kH.v/kL1./ kvkL1./ : 





The passage from smooth to arbitrary superharmonic functions is done by approximation. The maximal inequality is preserved due to a lower semicontinuity property of the maximal operator with respect to the L1 convergence: Lemma 9.8. Let .vn /n2N be a sequence in L1loc ./ converging strongly to v in L1loc ./. Then, for every x 2 , we have Mv.x/ 6 lim inf Mvn .x/; n!1

and, for every  > 0, capW 1;q .¹Mv > º/ 6 lim inf capW 1;q .¹Mvn > º/: n!1

Proof of Lemma 9.8. For every x 2  and every 0 < r < d.x; @/, we have B.xIr /

jvj D lim

n!1 B.xIr /

jvn j 6 lim inf Mvn .x/: n!1

Taking the supremum in the left-hand side with respect to r, we get the pointwise inequality for Mv. For every  > 0, such an inequality implies that ¹Mv > º 

1 1 \ [

i D0 nDi

¹Mvn > º:

By the increasing set property (Proposition A.14), capW 1;q

1 1 \ [

i D0 nDi

1 \   ¹Mvn > º D lim capW 1;q ¹Mvn > º : i !1

nDi

The second assertion now follows from the monotonicity of the capacity.



152

9. Maximal inequalities

To deduce the maximal inequality for potentials we decompose any given potential as a difference of superharmonic functions. In this step, we use the subadditivity of the maximal operator: Lemma 9.9. Let v1 ; v2 2 L1loc ./. Then, for every x 2 , we have M.v1 C v2 /.x/ 6 Mv1 .x/ C Mv2 .x/; and, for every  > 0, capW 1;q .¹M.v1 C v2 / > º/

6 capW 1;q .¹Mv1 > =2º/ C capW 1;q .¹Mv2 > =2º/:

Proof of Lemma 9.9. For every x 2  and every 0 < r < d.x; @/, we have B.xIr /

jv1 C v2 j 6

B.xIr /

jv1 j C

B.xIr /

jv2 j 6 Mv1 .x/ C Mv2 .x/:

Taking the supremum in the left-hand side with respect to r, we deduce the subadditivity of the maximal operator. In particular, we have ¹M.v1 C v2 / > º  ¹Mv1 > =2º [ ¹Mv2 > =2º; and the conclusion follows applying the monotonicity and the subadditivity of the W 1;q capacity (Propositions A.8 and A.9).  x and u is superharmonic. Proof of Proposition 9.6. We first assume that u 2 C01 ./ By the weak maximum principle (Corollary 1.10), u is nonnegative. By the mean value property of superharmonic functions (Proposition 1.4), we then have Mu D u: We now take a smooth truncation of the function u. For this purpose, let h 2 C 1 .R/ be such that .a/ for every t 6 21 , h.t / D 0, .b/ for every t > 1, h.t / D 2.

The function h u belongs to Cc1 ./ and satisfies h u > 1 in ¹u > º, whence is admissible to estimate the capacity of the compact set ¹u > º. By the monotonicity of the capacity, we then have

 u  2

: capW 1;2 .¹Mu > º/ 6 capW 1;2 .¹u > º/ 6 h

 W 1;2 ./

9.3. Total charge estimate

153

Thus, by the Poincaré inequality (Proposition 4.5) we have

h  u i 2

capW 1;2 .¹Mu > º/ 6 C1 r h

2 : L ./ 

Applying the interpolation inequality above to the function H 2 C 1 .R/ such that H 0 D .h0 /2 and H.0/ D 0, we then get ˆ h  i2 ˇ ˇ u ˇ ru ˇ2 C2 h0 capW 1;2 .¹Mu > º/ 6 C1 kukL1 ./ : ˇ ˇ 6    

We now prove the maximal inequality for a superharmonic function u satisfying the linear Dirichlet problem for some nonnegative measure  2 M./. For this purx converging to pose, let .n /n2N be a sequence of nonnegative functions in C 1 ./ the nonnegative measure  in the sense of measures on , and such that (Proposition 2.7) lim kn kL1 ./ D kkM./ : n!1

Denoting by un the solution of the linear Dirichlet problem with density n , the estimate we have just proved shows that capW 1;2 .¹Mun > º/ 6

C2 kn kL1 ./ : 

By the linear elliptic L1 estimate (Proposition 3.2), the sequence .un /n2N converges to u in L1 ./. We deduce from the lower semicontinuity property of the maximal operator (Lemma 9.8) that capW 1;2 .¹Mu > º/ 6 lim inf capW 1;2 .¹Mun > º/ 6 n!1

C2 kkM./ : 

For a signed measure  2 M./, we first write u as a difference between the solutions of the linear Dirichlet problems with densities C and  : u D u1 u2 . By the subadditivity of the maximal operator (Lemma 9.9), we have capW 1;2 .¹Mu > º/ 6 capW 1;2 .¹Mu1 > =2º/ C capW 1;2 .¹Mu2 > =2º/: Applying the weak maximal estimate to the superharmonic functions u1 and u2 , we get capW 1;2 .¹Mu > º/ 6

2C2 C 2C2 2C2  ./ C  ./ D kkM./ :   



Chapter 10

Sobolev and Hausdorff capacities

“No complete description of capacity in terms of Hausdorff measure is possible.” Lennart Carleson

Using geometric and potential estimates, we establish some setwise comparison between the Sobolev capacity capW 1;q and the Hausdorff capacity Hıs .

10.1 Comparison properties We summarize the main results that are proved in this chapter. The relation between the Sobolev capacity capW 1;q and the Hausdorff capacity HıN q , defined in Appendices A and B, may be straightforwardly quantified as follows: Proposition 10.1. For every 1 6 q 6 N and every 0 < ı < C1, we have capW 1;q 6 C HıN

q

;

for some constant C > 0 depending on ı, N and q. This kind of inequality is to be understood in the sense of set functions: for every subset A  RN , capW 1;q .A/ 6 C HıN q .A/: In particular, if the Hausdorff measure satisfies HN q .A/ D 0, then capW 1;q .A/ D 0. In view of the extension procedure used to define outer capacities (cf. Sections A.2 and B.3), it suffices to prove the inequality above for compact sets. The dimension N q is naturally associated to the W 1;q capacity since, by a scaling argument, the capacity of a ball decreases at the rate r N q as the radius r ! 0 (Lemma 10.5). For the exponent q D 1, there is a remarkable reverse inequality between the N 1 Sobolev capacity capW 1;1 and the Hausdorff content H1 discovered by Meyers and Ziemer [245]: Proposition 10.2. There exists a constant C 0 > 0, depending on the dimension N , such that N 1 H1 6 C 0 capW 1;1 :

156

10. Sobolev and Hausdorff capacities

We thus recover Fleming’s beautiful conclusion that the Sobolev capacity capW 1;1 and the Hausdorff measure HN 1 vanish on the same sets, see [133]. We complete later on in Chapter 16 the equivalence between capW 1;1 and HıN 1 , for any given gauge 0 < ı < C1, using Maz0 ya’s trace inequality (Proposition 16.12). There is no hope to have a counterpart of Proposition 10.2 for exponents q > 1: Sobolev and Hausdorff capacities are different objects, and the borderline case is detected in dimension N q by sets of finite Hausdorff measure: Proposition 10.3. If 1 < q 6 N , then, for every compact set K  RN such that HN q .K/ < C1, we have capW 1;q .K/ D 0: There is nevertheless a deep counterpart of Proposition 10.2 arising in the context of trace inequalities. The case of interest for us is the following: Proposition 10.4. For every 1 6 q < N and every s > N s H1 6 C 00 .capW 1;q / N

s q

q, we have

;

for some constant C 00 > 0 depending on N , q and s. We deduce from this relation that if capW 1;q .A/ D 0, then Hs .A/ D 0 for every s > N q, whence the Hausdorff dimension of A is at most N q. The proposition above is a corollary of Proposition 10.2 in the case q D 1, and follows from works by Adams [1] and [2] when 1 < q < N . The remaining of the chapter will be devoted to the proofs of Propositions 10.1–10.4.

10.2 Estimating the Sobolev capacity We first prove that the W 1;q capacity of closed balls of radius r decreases at the rate r N q as r ! 0: Lemma 10.5. For every a 2 RN and every 0 < r 6 1, we have capW 1;q .BŒaI r/ 6 capW 1;q .BŒ0I 1/ r N

q

:

Proof. Take a nonnegative function ' 2 Cc1 .RN / such that ' > 1 in BŒ0I 1. The function 'a;r W RN ! R defined for x 2 RN by x a 'a;r .x/ D ' r

10.2. Estimating the Sobolev capacity

157

also has compact support and satisfies 'a;r > 1 in BŒaI r. Thus, capW 1;q .BŒaI r/ 6 k'a;r kqW 1;q .RN / : By a change of variables, we get k'a;r kqW 1;q .RN / D r N

ˆ

RN

j'jq C r N

q

ˆ

RN

jr'jq :

Since r 6 1, we obtain capW 1;q .BŒaI r/ 6 r N

q

k'kqW 1;q .RN / :

Minimizing the right-hand side with respect to ', we have the conclusion.



Proof of Proposition 10.1. We prove the proposition with parameter ı D 1. Given a compact set K  RN , let .B.xi I ri //i 2¹0;:::;`º be a family of balls covering K with radii less than or equal to 1. By the monotonicity and subadditivity of the capacity, we have ` X capW 1;q .K/ 6 capW 1;q .BŒxi I ri /: Thus, by the previous lemma, we get

i D0

capW 1;q .K/ 6 capW 1;q .BŒ0I 1/

` X

riN

q

:

i D0

Minimizing the right-hand side over all coverings of K, we have the conclusion on compact sets with constant C D capW 1;q .BŒ0I 1/=!N q , and this implies the estimate for ı D 1 on arbitrary sets.  The non-equivalence between the Hausdorff measure and the Sobolev capacity was pointed out by Frostman [139] in the case q D 2 and later pursued by Carleson [77] and [79], see also [322]. We first illustrate Proposition 10.3 in the case where K is a single point. In dimension N > 3, this is a matter of scaling: Example 10.6. For every N > 3, capW 1;2 .¹0º/ D 0: To show this, take a nonnegative function ' 2 Cc1 .RN / such that '.0/ > 1, and let .n /n2N be a sequence of positive numbers converging to zero. For each n 2 N, let 'n W RN ! R be the function defined by 'n .x/ D '.x=n /. By a change of variable, we have ˆ ˆ k'n k2W 1;2 .RN / D nN j'j2 C nN 2 jr'j2 : RN

RN

158

10. Sobolev and Hausdorff capacities

Since capW 1;2 .¹0º/ 6 k'n k2W 1;2 .RN / ;

letting n ! 1 the conclusion follows.

The previous argument fails in dimension N D 2 due to the scaling invariance of kr'n kL2 .RN / . The trick to overcome this obstruction consists in taking suitable averages of the functions 'n that converge strongly to 0 in W 1;2 .R2 /. More generally, given a bounded sequence .'n /n2N in W 1;q .RN / converging weakly to 0, let A be the smallest convex subset of W 1;q .RN / containing this sequence: ` ` °X ± X ˛i 'i W AD ˛i D 1 and ˛i > 0 : i D0

i D0

By Mazur’s lemma (see Corollary 3.8 in [53]), the strong closure Ax is weakly closed x Thus, there exists a sequence of finite averages in W 1;q .RN /; in particular, 0 2 A. of .'n /n2N converging strongly to 0 in W 1;q .RN /. To implement this strategy in full generality, we estimate the capacity of a compact set K using functions supported in some small neighborhood of K: Lemma 10.7. Let K  RN be a compact set. For every 0 < ı 6 1 and every  > 0, there exists a nonnegative function 2 Cc1 .RN / such that > 1 in K, supp  K C B.0I 3ı/, and k kqW 1;q .RN / 6 C HıN

q

.K/ C ;

for some constant C > 0 depending on N and q. Proof of Lemma 10.7. Take finitely many balls .B.xi I ri //i 2¹0;:::;`º covering K such that ri 6 ı. For each i 2 ¹0; : : : ; `º, take a nonnegative function 'i 2 Cc1 .RN / such that 'i > 1 in BŒxi I ri . Using a scaling argument as in the proof of Lemma 10.5, we choose 'i such that supp 'i  B.xi I 2ri / and k'i kqW 1;q .RN / 6 C1 riN

q

:

(10.1)

Assuming that the balls B.xi I ri / intersect K, the function max ¹'0 ; : : : ; '` º is supported in the set K C B.0I 3ı/. Next, given  > 0 such that max ¹'0 ; : : : ; '` º > 1 C  in K, take a nonnegative function 2 Cc1 .RN / such that (cf. Lemma A.13) > max ¹'0 ; : : : ; '` º



in RN

and k kqW 1;q .RN / 6

` X i D0

k'i kqW 1;q .RN / :

(10.2)

10.2. Estimating the Sobolev capacity

159

In particular, we have > 1 in K. Since such a function is constructed by convolution, we may assume that is also supported in K C B.0I 3ı/. Combining estimates (10.1) and (10.2), yields k

kqW 1;q .RN /

6 C1

` X

riN

q

:

i D0

To conclude the proof, we take a covering .B.xi I ri //i 2¹0;:::;`º of K such that the sum N q  in the right-side is bounded from above by !N1 q Hı .K/ C =C1 . Proof of Proposition 10.3. Let .ın /n2N be a sequence of positive numbers converging to 0. By Lemma 10.7, for every n 2 N there exists a nonnegative function 1 N n 2 Cc .R / such that .a/

n

> 1 in K,

.b/ supp .c/ k

n  K C B.0I 3ın /, q N q n kW 1;q .RN / 6 C Hın .K/

C 1 6 C HN

q

.K/ C 1.

In particular, . n /n2N is a bounded sequence in W 1;q .RN / that converges pointwise to 0 in RN n K. Since the set K is negligible with respect to the Lebesgue measure and since q > 1, it follows that . n /n2N converges weakly in W 1;q .RN / to 0. Applying Mazur’s lemma, we get a subsequence .n /n2N , obtained by computing finite averages of . n /n2N , which converges strongly to 0 in W 1;q .RN /. Since n is a nonnegative function in Cc1 .RN / such that n > 1 in K, we may apply this  function to estimate capW 1;q .K/. Letting n ! 1, the conclusion follows. Since Mazur’s lemma relies on the Hahn–Banach theorem, we do not know how the averages are being performed. A more explicit construction is based on Cesàro averages, and consists in taking arithmetic means of some suitable subsequence k
P 'ni k2N converges strongly to 0 .'ni /i 2N in such a way that the sequence k1 i D1

in W 1;q .RN /. The possibility of choosing such a subsequence is called the Banach– Saks property. We establish the Banach–Saks property in the setting of Lebesgue spaces, see [19]. The proof below by Okada [266] provides an algorithm to construct the subsequence .'ni /i 2N . Proposition 10.8. Let 1 < q < C1, and let .fn /n2N be a bounded sequence in Lq .RN /. If .fn /n2N converges weakly to f in Lq .RN /, then there exists a subsek
P fni k2N converge quence .fnk /k2N such that the Cesàro averages k1 strongly to f in Lq .RN /.

i D1

160

10. Sobolev and Hausdorff capacities

The explicit choice of the subsequence .fnk /k2N involves the duality map 0

ˆW Lq .RN / ! Lq .RN /; defined by

8 jf jq 2 f ˆ < q 2 ˆ.f / D kf kLq .RN / ˆ : 0

if f ¤ 0, if f D 0.

For every f 2 Lq .RN /, we have ˆ f ˆ.f / D kf kLq .RN / kˆ.f /kLq0 .RN / RN

and kˆ.f /kLq0 .RN / D kf kLq .RN / : The duality map ˆ is uniformly continuous on bounded subsets of Lq .RN /, see Problem 12 (Questions A4 and B2) in [53]. This property is clear for q D 2, since ˆ is the identity map. The general case follows from the uniform convexity of the Lebesgue spaces for any exponent 1 < q < C1; see Theorem 5.4.2 in [345] for an elegant proof of the uniform convexity based on Hanner’s inequalities. We also need below the following inequality satisfied by the duality map that is implicitly stated in [266]: for every g; h 2 Lq .RN /, we have ˆ ˆ 2 2 kgkL khk 6 2 .g h/.ˆ.g/ ˆ.h//C2 .g h/ˆ.h/: (10.3) q .RN / Lq .RN / RN

RN

Proof of inequality (10.3). We first write 2 kgkL q .RN /

2 khkL q .RN /

D .kgkLq .RN /

khkLq .RN / /2 C 2.kgkLq .RN / khkLq .RN /

Since 2 khkL q .RN / D

ˆ

RN

hˆ.h/ D

ˆ

gˆ.h/ RN

ˆ

.g

2 khkL q .RN / /:

h/ˆ.h/;

RN

we get 2 kgkL q .RN /

2 khkL khkLq .RN / /2 q .RN / D .kgkLq .RN /   ˆ C 2 kgkLq .RN / khkLq .RN / gˆ.h/ RN ˆ C2 .g h/ˆ.h/:

RN

10.2. Estimating the Sobolev capacity

161

To conclude the argument, we observe that ˆ .g h/.ˆ.g/ ˆ.h// D .kgkLq .RN / khkLq .RN / /2 RN   ˆ C kgkLq .RN / khkLq .RN / gˆ.h/ RN   ˆ C kgkLq .RN / khkLq .RN / hˆ.g/ : RN

By the Hölder inequality, the last term in parentheses is nonnegative, and we may discard it. Combining the resulting inequality with the previous identity we have the conclusion. 

Proof of Proposition 10.8. Without loss of generality, we may assume that .fn /n2N converges weakly to 0 in Lq .RN /. Given a subsequence .fni /i 2N , to be explicitly chosen below, denote j X Sj D fni : i D1

Applying inequality (10.3) with g D Sj and h D Sj 1 , we get ˆ ˆ 2 2 kSj kLq .RN / 6 kSj 1 kLq .RN / C2 fnj .ˆ.Sj / ˆ.Sj 1 //C2 RN

RN

fnj ˆ.Sj

1 /:

By the boundedness of the sequence .fnk /k2N and by the Hölder inequality, we have ˆ fnj .ˆ.Sj / ˆ.Sj 1 // 6 C kˆ.Sj / ˆ.Sj 1 /kLq0 .RN / : RN

Since the sequence .fn /n2N converges weakly to 0, we extract by induction the subsequence .fnk /k2N using the Banach–Saks criterion (see eq. (6) in [19]): given functions fn1 ; : : : ; fni 1 , we choose fni such that ni > ni 1 and ˆ fni ˆ.Si 1 / 6 1: RN

Combining the above estimates, we get 2 kSj kL q .RN / 6 kSj

2 1 kLq .RN /

C 2C kˆ.Sj /

ˆ.Sj

1 /kLq 0 .RN /

C 2:

We now add these inequalities for j 2 ¹2; : : : ; kº, simplifying common terms in the left and right-hand sides. Dividing both sides by k 2 and using the homogeneity of the duality map ˆ, we get k   1 1 2C X

Sj 2 2 kS k 6 kS k C

ˆ 1 Lq .RN / k Lq .RN / k2 k2 k j D2 k

S  2 j 1 ˆ

q0 N C : L .R / k k

By the uniform continuity of ˆ on bounded subsets of Lq .RN /, the right-hand side converges to zero as k ! 1, and the conclusion follows. 

162

10. Sobolev and Hausdorff capacities

The Banach–Saks property holds for Banach spaces X whose dual X 0 is uniformly convex. Proposition 10.8 is still true for L1 .RN /, whose dual is not uniformly convex, but the proof is based on a different strategy that relies on the equi-integrability of sequences that converge weakly in L1 , see [319].

10.3 Estimating the Hausdorff content N 1 For the proof of the reverse inequality between the Hausdorff content H1 and the Sobolev capacity capW 1;1 (Proposition 10.2), we follow the elegant presentation of Hajłasz and Liu [156]. The essence of the result is contained in the following functional restatement of Gustin’s boxing inequality [152]:

Proposition 10.9. For every ' 2 Cc1 .RN /, we have N 1 H1 .¹' > 1º/ 6 C kr'kL1 .RN / ;

for some constant C > 0 depending on the dimension N . The classical boxing inequality has a more geometric flavor, and says that every compact set K  RN admits a covering of open balls .B.xi I ri //i 2¹0;:::;`º whose total perimeter is controlled by the perimeter of K: ` X

N riN

1

6 C 0 HN

1

.@K/:

i D0

The choice of the radii of the balls covering the set ¹' > 1º is provided by the following lemma: Lemma 10.10. Let ' 2 Cc1 .RN / and s > 0. For every x 2 ¹' > sº, there exist r > 0 and ı > 0 such that, for every s < t 6 s C ı, we have ˆ N 1 00 .t s/r 6C jr'j; ¹s s, consider the truncated function vs;t W RN ! R defined for x 2 RN by

8 ˆ t .

By the Poincaré–Wirtinger inequality (10.5), for every r > 0 we have

B.xIr /

B.xIr /

jvs;t .y/

vs;t .z/j dy dz 6 C 0 r B.xIr /

jrvs;t j:

On the one hand, by the Chebyshev inequality we have

B.xIr /

>

B.xIr /

jvs;t .y/

vs;t .z/j dy dz

j¹' > t º \ B.xI r/j j¹' 6 sº \ B.xI r/j
.t !N r N !N r N

s/:

On the other hand, by composition of Sobolev functions with the truncation operator (cf. Exercise 5.3) we have ˆ 1 jr'j: r jrvs;t j D !N r N 1 ¹s t º \ B.xI r/j j¹' 6 sº \ B.xI r/j
.t !N r N !N r N ˆ C0 6 jr'j: !N r N 1 ¹s sº: We observe that the function .0; C1/ 3 r 7 !

j¹' 6 sº \ B.xI r/j !N r N

is continuous, vanishes in a neighborhood of 0 since '.x/ > s, and tends to 1 as r tends to infinity since s > 0 and ' has compact support in RN . By the intermediate value theorem, there exists r > 0 such that 1 j¹' 6 sº \ B.xI r/j D : !N r N 2 For this choice of radius, the function .s; C1/ 3 t 7 ! tends to have

1 2

j¹' > t º \ B.xI r/j !N r N

as t ! s. Thus, there exists ı > 0 such that, for every s < t 6 s C ı, we

1 j¹' > t º \ B.xI r/j > : !N r N 4 With these choices of r and ı, we thus have, for every s < t 6 s C ı, 1 1 1 j¹' > t º \ B.xI r/j j¹' 6 sº \ B.xI r/j
>
D : N N !N r !N r 4 2 8

Therefore,

C0 s/ 6 !N r N

1 .t 8

1

ˆ

and this proves the lemma.

¹s N 1, it follows from the comparison between Hausdorff contents (Lemma 10.11) that s .H1 .A//

N

1 s

N 1 6 C H1 .A/:

On the other hand, by Proposition 10.2, N 1 H1 .A/ 6 C 0 capW 1;1 .A/:

It now suffices to combine the two inequalities.



The proof of Proposition 10.4 in the case q > 1 is based on an estimate of the integral ˆ ' d; RN

where ' 2 Cc1 .RN / and  is a finite nonnegative Borel measure on RN . For this purpose, we apply the pointwise estimate j'j 6

1 I1  jr'j N

(10.9)

in terms of the Riesz kernel I1 .z/ D 1=jzjN 1 , which follows from the representation formula (Proposition 4.14) ˆ 1 x y dy: '.x/ D r'.y/
N RN jx yjN

10.3. Estimating the Hausdorff content

167

By Fubini’s theorem, we then have ˆ ˆ ˆ 1 1 I1  jr'j d D jr'j.I1  /: ' d 6 N RN N RN RN 0

The next lemma gives an estimate of the Lq norm of the function I1   due to Adams [1] and [2], where q 0 is the conjugate exponent of q: Lemma 10.12. If 1 < q < N and if s > N  2 M.RN / such that

q, then, for every nonnegative measure

s  6 H1 ;

we have kI1  kLq0 .RN / 6 C.RN /1

N q sq

:

s is equivalent to asking that, for every By Proposition B.3, the condition  6 H1 N x 2 R and every r > 0, .B.xI r// 6 !s r s : (10.10)

In terms of Morrey norms (5.1), Adams’s estimate is equivalent to kI1  kLr .RN / 6 C 0 kk1M1.RN / kkMp .RN / ; with r D q 0 , p D NN s and  D Nsq q . Such an inequality thus holds for all exponents p and r in the ranges 1 < p < N and NN 1 < r < C1. The proof of the lemma relies on Cavalieri’s representation of the Riesz potential I1   (Exercise 2.7): ˆ 1 .B.xI r// dr I1  .x/ D .N 1/ : (10.11) rN 1 r 0 Proof of Lemma 10.12. Using the representation formula (10.11) above, by the Minkowski inequality we have ˆ 1 dr 0 k.B.
I r//kLq0 .RN / N : kI1  kLq .RN / 6 .N 1/ r 0 We next observe that, by Fubini’s theorem, ˆ ˆ ˆ .B.xI r// dx D RN

RN

D

ˆ

RN

B.xIr /

ˆ

B.yIr /

D !N r N .RN /:



d.y/ dx 

dx d.y/

168

10. Sobolev and Hausdorff capacities

The quantity we have to integrate is actually .B.xI r// q mate (10.10) satisfied by the measure , it follows that .B.xI r// q

q 1

D .B.xI r// q

1 1

q 1

. From the density esti-

.B.xI r//

6 .min ¹!s r s ; .RN /º/ q

1 1

.B.xI r//:

We apply below the density upper bound !s r s for small values of r, and the total mass .RN / for larger values of r. The previous estimate yields ˆ q q0 k.B.
I r//kL D .B.xI r// q 1 dx q 0 .RN / RN ˆ 1 .B.xI r// dx 6 .min ¹!s r s ; .RN /º/ q 1 RN

N

6 C1 min ¹.R /r q

s

1 CN

It follows that, for every  > 0, we have kI1  kLq0 .RN / 6 .N

1/

C .N 6 C2

ˆ

0

; .RN /q r N º:



dr k.B.
I r//kLq0 .RN / N r 0 ˆ 1 dr 1/ k.B.
I r//kLq0 .RN / N r  ˆ



.RN /

q 1 q

r

dr rN

sCN.q 1/ q

0

C C2

ˆ

1

.RN /r

N.q 1/ q



6 C3 .RN /

q 1 q



s

.N q

q/

dr rN

C C4

.RN / N

 Minimizing the right-hand side with respect to  > 0, we obtain kI1  kLq0 .RN / 6 C5 .RN /1 which is the estimate we wanted to prove.

N q sq

q

:

q

; 

Proof of Proposition 10.4 when q > 1. Given a compact set K  RN , we first show that, for every nonnegative measure  2 M.RN / supported in K and such that s  6 H1 , we have s .K/ 6 C.capW 1;q .K// N q : (10.12) For this purpose, let ' 2 Cc1 .RN / be a nonnegative function such that ' > 1 in K. Then, by the nonnegativity of the measure , the pointwise estimate (10.9), and Fubini’s theorem, we get ˆ ˆ ˆ 1 1 .K/ 6 ' d 6 I1  jr'j d D jr'j.I1  /: N RN N RN RN

10.3. Estimating the Hausdorff content

169

Thus, by the Hölder inequality and the estimate of the Riesz potential of  (see Lemma 10.12), we have .K/ 6

1 1 kr'kLq .RN / kI1  kLq0 .RN / 6 C1 kr'kLq .RN / .RN / N

N q sq

:

Since  is supported in K, .RN / D .K/. For .K/ > 0, we may simplify both sides of the estimate above to get .K/

N

q s

q 6 C1q kr'kL q .RN / :

Minimizing the right-hand side with respect to ', we obtain estimate (10.12). The conclusion is now a consequence of inequality (10.12) and Frostman’s lemma (Proposition B.4). Indeed, for every compact set S  RN there exists a nonnegative s s measure , supported in S , such that  6 H1 and .S / > 3 s H1 .S /. Applying estimate (10.12) to this measure , we get s 3 s H1 .S / 6 .S / 6 C.capW 1;q .S // N

s q

:

The conclusion thus follows for every compact set S , whence for every subset of RN .  In the spirit of Lemma 10.12, one can study the integrability of the Newtonian potential: Exercise 10.2. Let N > 3, and let  2 M.RN / be a nonnegative measure such that N 2

 6 H1

N p

for some p >

N . N 2

.a/ Prove that N 2 Lp .RN /.

.b/ Deduce that the solution of the linear Dirichlet problem with density  in a smooth bounded open set  belongs to Lp ./.

By the Hölder inequality, the density assumption above is satisfied for every pN  2 L N C2p ./. In this case, the Calderón–Zygmund regularity theory (ProposipN tion 5.3) implies that u 2 W 2; N C2p ./, and this Sobolev space injects continuously into Lp ./.

Chapter 11

Removable singularities

“Mon raisonnement était incomplet: il restait à prouver que la fonction limite obtenue, bornée et harmonique, n’avait pas de points singuliers isolés.”1 Henri Lebesgue

We investigate the problem of removable singularities for three classes of functions: bounded (or continuous) functions, Lebesgue Lp functions, and Höldercontinuous functions. The main tools concerning the equivalence of capacities and properties of the solution of the obstacle problem are developed in the next chapter.

11.1 Schwarz’s principle Given a compact set S   and a harmonic function uW  n S ! R, we wish to know whether u is harmonic in . The answer depends on S and on the behavior of u near S . The prototype of removable singularity problems goes back to Schwarz [302]: Proposition 11.1. Let N > 2, a 2 , and uW  n ¹aº ! R be a harmonic function. If u is bounded, then u has a harmonic extension to . Proof. We assume for simplicity that N > 3, a D 0, and B.0I 1/ b . We first prove that if u D 0 on @B.0I 1/, then u D 0 in B.0I 1/ n ¹0º. For this purpose, let M > 0 be such that, for every x 2 B.0I 1/ n ¹0º, we have ju.x/j 6 M: For every  > 0, the function v W B.0I 1/ n ¹0º ! R defined by 1 jxjN v .x/ D M 1 N 1 “My reasoning was incomplete:

2

2

1 1

it still remained to prove that the obtained limit function, which was bounded and harmonic, had no isolated singular points.”

172

11. Removable singularities

is harmonic, vanishes on @B.0I 1/, and equals M on @B.0I /. In particular, we have v 6 u 6 v

in @B.0I 1/ [ @B.0I /.

Applying the weak maximum principle (Corollary 1.10), we deduce that v 6 u 6 v

in BŒ0I 1 n B.0I /.

Given x 2 BŒ0I 1 n ¹0º, for every 0 <  6 jxj we then have ju.x/j 6 v .x/: As we let  ! 0, the right-hand side converges to 0, whence u.x/ D 0. This gives the conclusion when u D 0 on @B.0I 1/. In the case where this condition is not satisfied, we take the harmonic extension hW BŒ0I 1 ! R of the continuous function [email protected]/ . The function u h is bounded and harmonic in B.0I 1/ n ¹0º, and we deduce from the previous argument that u h D 0 in B.0I 1/ n ¹0º.  Schwarz’s removable singularity principle was later pursued by Bouligand [37], who identified the role played by the Newtonian capacity: Proposition 11.2. If S   is a compact set such that capW 1;2 .S / D 0, then every bounded harmonic function uW  n S ! R has a harmonic extension to . The set S in the statement is small: it must have zero Lebesgue measure, and in fact its Hausdorff dimension cannot exceed N 2 (cf. Proposition 10.4). We present two different strategies to prove Bouligand’s result. The first one is based on the following capacitary inequality that will be proved in the next chapter (Proposition 12.1): for every compact set K  RN , 2 1 N inf ¹kr'kL 2 .RN / C k'kL1 .RN / W ' 2 Cc .R /; ' D 1 in a neighborhood of Kº

6 3 capW 1;2 .K/:

First proof of Proposition 11.2. Since u is a bounded harmonic function in  n S , for every ' 2 Cc1 . n S / we have ˆ u ' D 0: 

If capW 1;2 .S / D 0, then by the capacitary inequality above there exists a sequence . n /n2N in Cc1 .RN / such that .a/ for every n 2 N, we have .b/ the sequence .r

n /n2N

.c/ the sequence .

n /n2N

n

D 1 in a neighborhood of S ,

converges to 0 in L2 .RN I RN /,

converges to 0 in L1 .RN /.

In particular, by the Sobolev and Hölder inequalities, the sequence . verges to 0 in L1loc ./.

n /n2N

con-

11.1. Schwarz’s principle

173

For every ' 2 Cc1 ./ and every n 2 N, by Property .a/ we have '.1 n S /. Hence, ˆ

n/

Cc1 .

u Œ'.1



n /

2

D 0:

Since Œ'.1 the sequence .Œ'.1

n /

D .1

n //n2N

n /'

2r'
r

n

'

n;

converges to ' in L1 ./, and we deduce that ˆ u ' D 0: 

Applying Weyl’s lemma (Proposition 2.14), it follows that u has a harmonic extension to .  The second proof strategy is based on an obstacle problem that will be also investigated in the next chapter. To summarize the results that we need, we assume that we are given a bounded open set , a compact subset S   with zero Lebesgue measure, and a bounded harmonic function uW  n S ! R, which plays the role of the obstacle. We consider the class of superharmonic functions above the obstacle u: Ou D ¹v 2 L1 ./W v is superharmonic in  and v > u almost everywhere in º: Since u is bounded, the class Ou is non-empty. In fact, we prove that there exists a smallest element in Ou (Proposition 12.6): Property 11.1. There exists uQ 2 Ou such that, for every v 2 Ou , we have v > uQ almost everywhere in . Taking any constant C 2 R such that u 6 C in  n S , we have uQ 6 C almost everywhere in . Therefore, we may assert that: Property 11.2. Since u is bounded in  n S , the solution uQ of the obstacle problem is also bounded in . We denote by uN the precise representative of u. Q From the characterization of the Lebesgue points of a superharmonic function (Lemma 8.10), we deduce Property 11.3. Since uQ is bounded in , the function uN is defined pointwise in . Using the balayage function (Proposition 7.5), we prove the following fact related to the problem of removable singularities (Proposition 12.8):

174

11. Removable singularities

Property 11.4. Since u is harmonic in nS , the function uN is also harmonic in nS . We conclude that uN is a superharmonic function in  satisfying the assumptions of Proposition 11.2. Using Lebesgue’s barrier argument, we show that uN coincides with the obstacle u on the boundary @ (Proposition 12.9): Property 11.5. If  is a smooth bounded open set and if u has a continuous extension to @, then the function uN also has a continuous extension to @, and both extensions coincide on @. Second proof of Proposition 11.2. We may assume that  is bounded. Let uW N  !R be the precise representative of the smallest superharmonic function which is greater than or equal to u in . We claim that if capW 1;2 .S / D 0, then uN D 0 in the sense of distributions in , and thus by Weyl’s lemma (Proposition 2.14) the function uN is harmonic in . Since uN > 0 in the sense of distributions in , by Schwartz’s characterization of positive distributions (Proposition 2.20), uN is a nonnegative locally finite measure on . Thus, by Property 11.4 above, uN is a finite nonnegative measure supported by the compact set S . For every nonnegative function ' 2 Cc1 ./ such that ' > 1 in S , we then have ˆ 0 6 u.S N /6 ' u: N 

Since uN is bounded (Property 11.2), the localization lemma (Proposition 6.11) 1;2 ./. Therefore, and the interpolation inequality (Lemma 5.8) show that uN 2 Wloc ˆ 0 6 u.S N /6 r uN
r' 6 kr uk N L2 .supp '/ kr'kL2 ./ : 

Let ! b  be an open subset containing the compact set S . Since we have capW 1;2 .S / D 0, there exists a sequence of nonnegative functions .'n /n2N in Cc1 .!/ converging to 0 in W 1;2 .RN / and such that, for every n 2 N, we have 'n > 1 in S . For every n 2 N, we then have 06

u.S N / 6 kr uk N L2 .!/ kr'n kL2 ./ ;

whence as n ! 1 we deduce that u.S N / D 0. Since the measure uN is supported by S , it follows that uN D 0 in the sense of distributions in , and we have the claim.

11.2. Polar sets of second order

175

Similarly, there exists a greatest subharmonic function that is less than or equal to u, and the same argument as above implies that its precise representative u is also harmonic in . To conclude, we may assume that  is also smooth and Nthat u is continuous on @; otherwise it suffices to restrict u to a smooth open subset x of strictly contained in . By Property 11.5 above, the continuous extensions to  the functions uN and u coincide on the boundary @. Thus, by the weak maximum principle (Corollary N1.10), we have uN D u in . Since u 6 u 6 uN in  n S , N . N we deduce that u has a harmonic extension to 

11.2 Polar sets of second order We now consider a problem of removable singularities in which u satisfies a weaker assumption, namely that u 2 Lp ./ for some 1 < p < C1. In this case, the smallness of S is expressed in terms of a Sobolev capacity of order two: Proposition 11.3. If S   is a compact set such that capW 2;p0 .S / D 0, then every harmonic function uW  n S ! R such that u 2 Lp ./ has a harmonic extension to . In this statement, the Hausdorff dimension of S cannot exceed N 2p 0 (cf. Section 10.1). Several authors have contributed to this problem (see [8], [162], [212], and [235]). using equivalent notions of polar sets of order two. We pursue the strategy of the first proof of Proposition 11.2, now based on the following capacitary estimate (Proposition 12.4): for every 1 < q < C1 and every compact set K  RN , inf ¹k'kqW 2;q .RN / W ' 2 Cc1 .RN / and ' D 1 in a neighborhood of Kº 6 C capW 2;q .K/: Proof of Proposition 11.3. Assuming that capW 2;p0 .S / D 0, then by the capacitary estimate above there exists a sequence . n /n2N in Cc1 .RN / converging to 0 0 in W 2;p .RN / and such that, for every n 2 N, n D 1 in a neighborhood of S . 1 For every ' 2 Cc1 ./ and every n 2 N, we have that '.1 n / 2 Cc . n S /. Hence, by the harmonicity of u on  n S we get ˆ u Œ'.1 n / D 0: 

0

p Since the sequence .Œ'.1 n //n2N converges to ' in L ./, we deduce using p the Hölder inequality that if u 2 L ./, then ˆ u ' D 0: 

The conclusion follows from Weyl’s lemma (Proposition 2.14).



176

11. Removable singularities

If q > N2 and capW 2;q .S / D 0, then the Morrey–Sobolev inequality (cf. Corollary 4.16) shows that S is the empty set. The assumption of the previous proposition is thus interesting only when p > NN 2 . This is not surprising since, for every a 2 , the function 1 x7 ! jx ajN 2

is harmonic in  n ¹aº, and belongs to the Lebesgue spaces Lploc ./, for every p < NN 2 . In this example, the set ¹aº is not removable.

11.3 Carleson’s condition The results in the first section also apply to continuous functions, since they are bounded in a neighborhood of the compact set S . We now show that if u satisfies a stronger assumption, namely Hölder continuity, then we are allowed to have bigger removable singular sets S . Our goal is to prove Carleson’s theorem, see [78] and Theorem 7.II in [79]: Proposition 11.4. Let ˛ > 0, and let uW  ! R be a Hölder-continuous function of order ˛. If S   is a compact set such that HN 2C˛ .S / D 0, and if u is harmonic in  n S , then u is harmonic in . The strategy below is reminiscent of the work of Lewy and Stampacchia [205] on obstacle problems, see also [179]. As in the second proof of Proposition 11.2, consider the class Ou D ¹v 2 L1 ./W v is superharmonic in  and v > u almost everywhere in º: Without loss of generality, we may assume that  is a bounded open set, and so u is a bounded function and Ou is non-empty. Following the notation introduced in Section 11.1, we denote by uW N  ! R the precise representative of the smallest element of Ou . By the mean value property for superharmonic functions (Proposition 2.18), uN is the supremum of a family of continuous functions. Hence, uN is lower semicontinuous in , and so is uN u. From the relation ¹uN D uº D ¹uN u 6 0º, we are entitled to assert that: Property 11.6. Since u is continuous, the coincidence set ¹uN D uº is relatively closed in .

11.3. Carleson’s condition

177

By Property 11.4, the measure uN is supported by S . We prove later on some additional information on the support of uN (Lemma 12.10): Property 11.7. The measure uN is supported by the coincidence set ¹uN D uº. The function uN is thus harmonic in the set ¹uN > uº. We also have some control on the modulus of continuity of uN in a neighborhood of the coincidence set (Lemma 12.11): Property 11.8. For every ball B.aI r/ b  whose center satisfies u.a/ N D u.a/, we have sup x2B.aI r2 /

ju.x/ N

u.a/j N 6C

sup x2B.aIr /

ju.x/

u.a/j;

where the constant C > 0 depends on the dimension N . Since u is continuous in , we deduce that the obstacle uN is also continuous in . Another interesting aspect of this inequality is related to the estimate of the density of charges of a superharmonic function in terms of the modulus of continuity: Lemma 11.5. For every superharmonic function v in  and every ball B.aI 2r/ b , we have . v/.B.aI r// 6 C 0 r N

2 B.aI2r /

jv

v.a/j;

where the constant C 0 > 0 depends on the dimension N . Proof of Lemma 11.5. Since v is superharmonic, the distribution v is a locally finite nonnegative measure on  (Proposition 2.20). Given a nonnegative function ' 2 Cc1 ./ such that ' > 1 in B.aI r/, we get ˆ ˆ . v/.B.aI r// 6 ' v D v ': 



Since ' has compact support in , the divergence theorem shows that ˆ ˆ . v/.B.aI r// 6 Œv v.a/ ' 6 k'kL1 ./ jv v.a/j: 

supp '

To conclude we choose ' such that supp '  B.aI 2r/ and k'kL1 ./ 6 C 0 =r 2. 

178

11. Removable singularities

Proof of Proposition 11.4. We may assume that  is a smooth bounded open set, x Let uN be the precise representative of and that u has a continuous extension to . the smallest superharmonic function which is greater than or equal to u. By Properties 11.4 and 11.7, the measure uN is simultaneously supported by the singular set S and by the coincidence set ¹uN D uº in . By the assumption on the size of the compact set S , HN 2C˛ .S \ ¹uN D uº/ D 0: Given a gauge ı > 0, consider a collection of balls .B.xi I ri //i 2¹0;:::;`º covering the set S \ ¹uN D uº such that ri 6 ı. We may assume that each ball B.xi I ri / intersects S \ ¹uN D uº – otherwise we discard the ball – and we take a point ai 2 B.xi I ri / \ S \ ¹uN D uº: In particular, we have B.xi I ri /  B.ai I 2ri /;

and so the balls .B.ai I 2ri //i 2¹0;:::;`º also cover the set S \ ¹uN D uº. By the monotonicity and subadditivity of the measure u, N we have . u/.S N \ ¹uN D uº/ 6

` X i D0

. u/.B.a N i I 2ri //:

Taking 4ı < d.S \ ¹uN D uº; @/, then every ball B.ai I 4ri / is strictly contained in . We may now combine Lemma 11.5 with Property 11.8 above to deduce that N . u/.B.a N i I 2ri // 6 C1 ri

2

sup

ju.x/

u.ai /j:

riN

:

x2B.ai I4ri /

Thus, by the Hölder continuity of u, . u/.S N \ ¹uN D uº/ 6 C2 Taking the infimum in the right-hand side, we get . u/.S N \ ¹uN D uº/ 6 C3 HıN

` X

2C˛

i D0

2C˛

.S \ ¹uN D uº/ D 0:

Since uN is superharmonic and uN is supported by S \ ¹uN D uº, we deduce that uN D 0 in the sense of distributions in , whence by Weyl’s lemma (Proposition 2.14) the function uN is harmonic in . Similarly, if u denotes the precise representative of the greatest subharmonic funcN than or equal to u, then u is also a harmonic function in . Since tion that is smaller N in , x and since uN D u D u on @ uN and u are continuous harmonic functions N N (Property 11.5), we conclude from the weak maximum principle (Corollary 1.10) that u D uN in . Therefore, u coincides with both functions and is harmonic in .  N

11.3. Carleson’s condition

179

We have investigated in this chapter three examples of removable singularity problems for the Laplacian. We explain the optimality of each condition later on in Chapter 13. One may wonder, for example, whether the converse of Proposition 11.4 holds. More precisely, given a compact set S   such that HN 2C˛ .S / is positive – possibly infinite – does there exist a Hölder-continuous function uW  ! R of exponent ˛ which is harmonic in  n S , but is not harmonic in ? The answer is affirmative for every 0 < ˛ < 1 (Proposition 13.5). Several removable singularity results for other operators and different classes of solutions are summarized in Polking’s survey [284].

Chapter 12

Obstacle problems

“Ainsi apparaissent deux méthodes, l’une minimisant une intégrale, l’autre minimisant un potentiel.”1 Marcel Brelot

We present two minimization strategies to achieve the smallest supersolution above some given obstacle. The common solution of the two problems minimizes simultaneously the total charge and the energy. This property is related to an equivalence between first- and second-order capacities.

12.1 Comparison between capacities Here we prove the capacitary estimates that are used to solve the removable singularity problems for bounded and for Lp functions in the previous chapter. The case of bounded functions relies on the following inequality: Proposition 12.1. For every compact set K  RN , we have 2 1 N inf ¹kr'kL 2 .RN / C k'kL1 .RN / W ' 2 Cc .R /; ' D 1 in a neighborhood of Kº

6 3 capW 1;2 .K/: This estimate is reminiscent of the following identity between capacities, see Theorem 4.E.1 in [60]: Proposition 12.2. For every compact set K  RN , we have cap1 .K/ D 2 capNewt .K/; where 2 1 N capNewt .K/ D inf ¹kr'kL 2 .RN / W ' 2 Cc .R / is nonnegative, and ' > 1 on Kº;

cap1 .K/ D inf ¹k'kL1 .RN / W ' 2 Cc1 .RN / is nonnegative, and ' > 1 on Kº: 1 “Thus

two methods appear, one minimizing an integral, the other minimizing a potential.”

182

12. Obstacle problems

The classical Newtonian capacity capNewt was introduced by Wiener [341]. The two proofs can be carried out simultaneously, but we first prove Proposition 12.1, and then provide a separate argument to prove the identity above. An equality between capacities of orders one and two might seem unexpected, but note that by the divergence theorem we have ˆ ˆ ' ' D jr'j2 ; RN

RN

2 for every ' 2 Cc1 .RN /. In this identity, the quantity kr'kL 2 .RN / may be interpreted as an average of the density of charges ' with weight '. The reason for the multiplicative factor 2 in Proposition 12.2 comes from the balance between the negative and positive charges: ˆ ' D 0: (12.1)

RN

A useful tool in the proofs of both propositions is the following case of equality between the energy and the total charge up to the factor 2 (cf. Exercise 12.2): x Lemma 12.3. Given smooth bounded open subsets ! b  in RN , let h 2 C 1 .n!/ be the harmonic function in  n !N such that h D 0 on @ and h D 1 on @!. Then, the function U W RN ! R defined for x 2 RN by 8 ˆ 1 if x 2 !, ˆ < x n !, U.x/ D h.x/ if x 2  ˆ ˆ : x 0 if x 2 RN n , belongs to W 1;2 .RN /, is such that U 2 M.RN /, and the total charge satisfies 2 kU kM.RN / D 2krU kL 2 .RN / :

Proof of Lemma 12.3. We first observe that ´ rh in  n !, x rU D 0 elsewhere, in the sense of distributions in RN . By the harmonicity of h, it also holds that U D

@h N H @n

1

b@!

@h N H @n

1

b@

in the sense of distributions in RN , where n denotes the unit outward normal vector with respect to ! and to , respectively. It thus follows that U 2 W 1;2 .RN / and U 2 M.RN /.

12.1. Comparison between capacities

183

It remains to establish the relation between the total charge and the energy of U . By the divergence theorem and by the harmonicity of h, we have ˆ ˆ ˆ @h d: (12.2) jrU j2 D jrhj2 D N @n R n! x @! In addition, we have kU kM.RN / D jU j.RN / D Since

@h @n

6 0 in @! [ @, we get kU kM.RN / D

ˆ

@

ˆ

ˇ ˇ ˆ ˇ ˇ ˇ @h ˇ ˇ @h ˇ ˇ ˇ d C ˇ ˇ d: ˇ @n ˇ ˇ ˇ @ @! @n

@h d @n

ˆ

@!

@h d: @n

Since h is harmonic, we also have ˆ ˆ ˆ @h @h d d D h D 0; @ @n @! @n n! x whence kU kM.RN / D

2

ˆ

@!

@h d: @n

(12.3)

Combining identities (12.2) and (12.3), the conclusion follows.



Proof of Proposition 12.1. Given smooth bounded open subsets ! b  in RN , the function U W RN ! R defined in the previous lemma belongs to W 1;2 .RN /\Cc0 .RN / and satisfies 2 kU kM.RN / D 2krU kL 2 .RN / : Claim. For every ' 2 Cc1 ./ such that ' > 1 in !, we have krU kL2 .RN / 6 kr'kL2 .RN / : Proof of the claim. We first observe that ˆ 2 krU kL2 .RN / D jrU j2 D RN

ˆ

U U;

RN

which can be justified using an approximation of U by convolution with a sequence of mollifiers. Next, the measure U is supported in @! [ @, is nonnegative on @!, and is nonpositive on @. On the other hand, we have U D 1 6 ' on @! and U D ' D 0 on @. Therefore, U . U / 6 ' . U /

184

12. Obstacle problems

in the sense of measures on RN . We then have ˆ ˆ 2 krU kL2 .RN / 6 ' U D RN

RN

r'
rU:

Thus, by the Hölder inequality we get 2 krU kL 2 .RN / 6 kr'kL2 .RN / krU kL2 .RN / :

Dividing both sides by krU kL2 .RN / , we obtain the estimate.

4

Given a compact set K  RN and a function ' 2 Cc1 .RN / such that ' > 1 on K, take smooth bounded open sets ! and  such that K  ! b ¹' > 1º

and

supp '  :

By the claim, the potential U satisfies 2 kU kM.RN / 6 2kr'kL 2 .RN / :

The function U is not admissible to estimate the capacity. For this reason, we take a convolution kernel  such that K

supp   !:

The function   U thus belongs to Cc1 .RN /, equals 1 in a neighborhood of K, and satisfies the estimates kr.  U /kL2 .RN / 6 krU kL2 .RN / 6 kr'kL2 .RN /

(12.4)

2 k.  U /kL1 .RN / 6 kU kM.RN / 6 2kr'kL 2 .RN / :

(12.5)

and

We deduce that 2 inf¹kr kL 2 .RN / C k kL1 .RN / W

2 Cc1 .RN /;

D 1 in a neighborhood of Kº

2 6 3kr'kL 2 .RN / :

Minimizing the right-hand side with respect to ', the conclusion follows.



12.1. Comparison between capacities

185

Proof of Proposition 12.2. The inequality 6 follows from the previous proof using the test function   U for any constant  > 1. Indeed, this is an admissible test function to estimate cap1 , and by estimate (12.5) we have 2 cap1 .K/ 6 k.  U /kL1 .RN / 6 2kr'kL 2 .RN / :

It thus suffices to let  ! 1, and to minimize the right-hand side with respect to '. To get the reverse inequality >, let T W Œ0; C1/ ! R be the truncation function defined by ´ 1 if s > 1, T .s/ D s if 0 6 s < 1. For every nonnegative function ' 2 Cc1 .RN / such that ' > 1 in K, the function T .'/ is also nonnegative and belongs to W 1;2 .RN / \ Cc0 .RN /. Thus, ˆ ˆ ˆ 2 jrT .'/j D rT .'/
r' D T .'/ ': (12.6) RN

RN

RN

Since, for every s > 0, we have jT .s/ 21 j 6 21 , we deduce from identities (12.1) and (12.6) that ˆ ˆ h 1 1i ' 6 k'kL1 .RN / : T .'/ jrT .'/j2 D N N 2 2 R R

Once again, the function T .'/ is not admissible for estimating the Newtonian capacity, but by using a suitable convolution kernel we get a function   T .'/ which belongs to Cc1 .RN / and satisfies   T .'/ D 1 in a neighborhood of K. Using the admissible function   T .'/ for any constant  > 1, we then get 2 capNewt .K/ 6 kr.  T .'//kL 2 .RN / 6

2 k'kL1 .RN / : 2

Letting  ! 1, and minimizing the right-hand side with respect to ', we obtain the reverse inequality.  We now consider the capacitary estimate that was used to study the removable singularity problem for Lp functions: Proposition 12.4. For every 1 < q < C1 and every compact set K  RN , we have inf ¹k'kqW 2;q .RN / W ' 2 Cc1 .RN / and ' D 1 in a neighborhood of Kº 6 C 0 capW 2;q .K/; for some constant C 0 > 0 depending on N and q.

186

12. Obstacle problems

The main ingredient is the following property of composition with nonnegative test functions due to Maz0 ya, see the proof of Theorem 11 in [231] or Section 3.7 in [234]: Lemma 12.5. Let 1 < q < C1. If H W Œ0; C1/ ! R is a smooth function such that H 00 has compact support in Œ0; C1/, then, for every nonnegative function ' 2 Cc1 .RN /, we have kD 2 H.'/kLq .RN / 6 C kD 2 'kLq .RN / : Proof of Lemma 12.5. We note that D 2 H.'/ D H 0 .'/D 2 ' C H 00 .'/r' ˝ r': If  > 0 is such that supp H 00  Œ0; , then jD 2 H.'/j 6 C1 .jD 2 'j C ¹'6º jr'j2 /: Thus, 2 kD 2 H.'/kLq .RN / 6 C1 .kD 2 'kLq .RN / C kr'kL 2q .¹'6º/ /:

(12.7)

The heart of the matter lies in the following estimate valid for nonnegative functions ': Claim. If ' 2 Cc1 .RN / is nonnegative, then ˆ ˆ jr'j2q 0 6C jD 2 'jq : 'q RN ¹'>0º Proof of the claim. Given  > 0, we prove the estimate ˆ ˆ jr'j2q 0 6C jD 2 'jq ; q RN RN .' C / for some constant C 0 > 0 independent of . For this purpose, we observe that   div .jr'j2q 2 r'/ jr'j2q jr'j2q 2 r' C : D .q 1/ div .' C /q 1 .' C /q .' C /q 1 Since ' has compact support in RN , the divergence theorem then yields ˆ ˆ jr'j2q div .jr'j2q 2 r'/ .q 1/ D : q .' C /q 1 RN .' C / RN

12.1. Comparison between capacities

187

Note that jdiv .jr'j2q

2

r'/j 6 C2 jr'j2q

2

jD 2 'j:

Therefore, ˆ

RN

C2 jr'j2q 6 q .' C / q 1

ˆ

RN

Thus, by the Hölder inequality, we get ˆ

RN

jr'j2q C2 6 .' C /q q 1

ˆ

RN

jr'j2q 2 jD 2 'j : .' C /q 1

jr'j2q .' C /q

 qq 1  ˆ

RN

jD 2 'jq

 q1

:

Simplifying both terms, we get an estimate with a constant independent of . Letting  ! 0, the conclusion follows. 4 It follows from the claim and the Chebyshev inequality that 1 q

ˆ

¹'6º

jr'j2q 6

ˆ

¹'>0º

jr'j2q 6 C0 'q

ˆ

RN

jD 2 'jq :

Thus, 1

2 2 0 q kr'kL 2q .¹'6º/ 6 .C / kD 'kLq .RN / :

Combining with estimate (12.7), the lemma is proved.



The assumption on the sign of ' might seem artificial, but Dahlberg [91] proved that, for any 1 < q < N2 , the estimate kD 2 H.'/kLq .RN / 6 C k'kW 2;q .RN / holds for every ' 2 Cc1 .RN / if and only if H is an affine linear function. As for the interpolation inequality involving the Laplacian (Lemma 5.8), one might expect an interpolation inequality of the type 2 2 kr'kL 2q .¹j'j6º/ 6 C kD 'kLq .RN / ;

but such an inequality cannot hold for every ' 2 Cc1 .RN / in view of Dahlberg’s result. Lemma 12.5 gives a necessary condition for H to operate on nonnegative functions in W 2;q .RN / by left composition. The complete characterization of such functions H has been obtained by Bourdaud, including the borderline cases q D 1 and q D N2 , see Théorèmes 3 and 4 in [39].

188

12. Obstacle problems

Proof of Proposition 12.4. Take a smooth function H W R ! R such that H.0/ D 0 and H.t / D 1, for every t > 1. Given a nonnegative function ' 2 Cc1 .RN /, we have kH.'/kW 1;q .RN / 6 C1 k'kW 1;q .RN / and, by the previous lemma, we also have kD 2 H.'/kLq .RN / 6 C2 kD 2 'kLq .RN / : Assuming in addition that ' > 1 in K, we have H.'/ D 1 in a neighborhood of K. We deduce that inf ¹k kqW 2;q .RN / W

2 Cc1 .RN / and

D 1 in a neighborhood of Kº

6 C3 k'kqW 2;q .RN / : Minimizing the right-hand side with respect to ', the conclusion follows.



We refer the reader to the work of Adams and Polking [8] for a generalization of Proposition 12.4 to higher order capacities, in connection with Bessel capacities.

12.2 Perron–Remak method The pioneer balayage method provides the existence of the smallest supersolution of the linear Dirichlet problem that lies above some given obstacle, see [280] and [281]. Poincaré’s original approach is based on an explicit construction of a non-increasing sequence of potentials generated by electric charges that are gradually swept-out from inside the domain. The Perron–Remak method is an abstract implementation of this idea by directly looking for the smallest function among all supersolutions above the obstacle, see [277] and [292]. More precisely, we consider the class of superharmonic functions above some given obstacle u 2 L1 ./: Ou D ¹v 2 L1 ./W v is superharmonic and v > u almost everywhere in º: The existence of the smallest element in Ou is one of the main ingredients of the Perron–Remak method: Proposition 12.6. If u 2 L1 ./ is such that Ou is non-empty, then there exists uQ 2 Ou such that, for every v 2 Ou , we have v > uQ almost everywhere in .

12.2. Perron–Remak method

189

The proof of this proposition relies on the following property concerning the minimum of superharmonic functions: Lemma 12.7. If v1 and v2 are superharmonic in , then min ¹v1 ; v2 º is also superharmonic in . Proof of Lemma 12.7. For every a; b 2 R, we have min ¹a; bº D a

.a

b/C :

If v1 and v2 are assumed to be smooth in , then by Kato’s inequality (Proposition 6.6) we have  min ¹v1 ; v2 º 6 v1

¹v1 >v2 º .v1

v2 / D ¹v1 6v2 º v1 C ¹v1 >v2 º v2

in the sense of distributions in . Since v1 and v2 are superharmonic functions, we deduce that  min ¹v1 ; v2 º 6 0 in the sense of distributions in . To consider the case where v1 and v2 need not be smooth, take an open subset ! b  and a sequence of mollifiers .n /n2N in Cc1 .RN / such that ! supp n b . Then, n  v1 and n  v2 are smooth superharmonic functions in ! (Lemma 2.22), whence, by the previous computation,  min ¹n  v1 ; n  v2 º 6 0 in the sense of distributions in !. In other words, for every nonnegative function ' 2 Cc1 .!/, we have ˆ min ¹n  v1 ; n  v2 º ' 6 0: 

Letting n ! 1, we deduce that min ¹v1 ; v2 º is superharmonic in !. This gives the conclusion.  Proof of Proposition 12.6. Let I D inf

²ˆ



³ vW v 2 Ou :

Since the infimum is taken over a non-empty set and since u 2 L1 ./, I is finite. We then consider a minimizing sequence: let .vn /n2N be a sequence in Ou such that ˆ vn D I: lim n!1



190

12. Obstacle problems

We construct by induction a non-increasing sequence .un /n2N in Ou having the same property as follows: let u0 D v0 and, for every n 2 N , un D min ¹un

1 ; vn º:

For every n 2 N, we have that un > u and, by the previous lemma, un is superharmonic. Since ˆ ˆ I 6 un 6 vn ; 

we also have lim

n!1

ˆ





un D I:

The pointwise limit uQ of the sequence .un /n2N satisfies the required properties. Firstly, we have uQ > u almost everywhere in . By the monotone convergence theorem, the sequence .un /n2N converges to uQ in L1 ./ and ˆ ˆ I D lim un D u: Q n!1





Since each function un is superharmonic, we deduce that uQ is also superharmonic (Exercise 2.8), and so uQ 2 Ou . It remains to show that uQ is the smallest element of Ou . For this purpose, we observe that, for every v 2 Ou , the sequence .min ¹un ; vº/n2N belongs to Ou , is non-increasing, converges pointwise to min ¹u; Q vº, and for every n 2 N we have ˆ ˆ I 6 min ¹un ; vº 6 un : 



By the monotone convergence theorem, we deduce that ˆ ˆ min ¹u; Q vº D I D u: Q 



Since uQ > min ¹u; Q vº and both integrals coincide, we deduce that uQ D min ¹u; Q vº, whence v > uQ almost everywhere in . This concludes the proof of the proposition.  Throughout this section, we denote by uN the precise representative of the smallest element uQ in the class Ou . We prove that uN satisfies Properties 11.4, 11.5, 11.7, and 11.8, which are used to investigate removable singularity problems for the Laplacian. Property 11.4 follows from the fact that uN is harmonic wherever the obstacle u is harmonic: Proposition 12.8. Under the assumptions of Proposition 12.6, if the obstacle u is harmonic in some open set !  , then uN is also harmonic in !.

12.2. Perron–Remak method

191

Proof. We may assume that ! is a smooth bounded open set strictly contained in , and that the obstacle u is harmonic in some neighborhood V of !. x We denote by R! uQ the balayage of uQ with respect to ! (Proposition 7.5). Since R! uQ is harmonic in !, it suffices to show that R! uQ D uQ almost everywhere in !. By the superharmonicity of u, Q the function R! uQ is also superharmonic in  and satisfies R! uQ 6 uQ almost everywhere in  (Corollary 7.6). To prove that R! uQ > u almost everywhere in , we observe that, by the harmonicity of u in V , we have R! uQ Since uQ

u D R! .uQ

u/

in V .

u is nonnegative in V , we deduce that (Exercise 7.1) R! uQ

u D R! .uQ

u/ > 0

almost everywhere in V , whence R! uQ > u almost everywhere in . Since the function R! uQ is superharmonic, we thus have R! uQ 2 Ou . By the minimality of u, Q we conclude that R! uQ > uQ almost everywhere in . Therefore, equality holds, and the precise representative uN is harmonic in !.  We now consider Property 11.5 and, more generally, the Lewi–Stampacchia continuity principle, see Theorem 2.1 in [204]: the solution uN of the obstacle problem is continuous whenever the obstacle u itself is continuous. x then Proposition 12.9. Let  be a smooth bounded open set. If u 2 C 0 ./, 0 x uN 2 C ./ and uN D u on the boundary @. The reader interested in the proof of Property 11.5 may skip directly to the proof of Proposition 12.9, where we focus on the existence of a continuous extension of uN to the boundary @ using Lebesgue’s barrier argument [195]. The continuity of uN in  is based on a separate analysis for points outside the coincidence set ¹uN D uº (Lemma 12.10) and for points in a neighborhood of the coincidence set (Lemma 12.11). These intermediate steps imply Properties 11.7 and 11.8. By Lemma 8.10, u.x/ N D lim

r !0 B.xIr /

uQ

for every x 2  such that the limit in the right-hand side is finite. It then follows from the mean value property of superharmonic functions (Proposition 2.18) that uN is the supremum of average integrals. Since these average integrals are continuous in the x variable, we deduce that the precise representative uN is lower semicontinuous. By the Lebesgue differentiation theorem (Proposition 8.1), we also have that uN D uQ almost everywhere in , whence u.x/ N D lim

r !0 B.xIr /

u: N

192

12. Obstacle problems

Our first lemma implies Property 11.7: Lemma 12.10. Under the assumptions of Proposition 12.6, if the obstacle u is continuous at a 2  and if u.a/ N > u.a/, then there exists ı > 0 such that uN is harmonic in B.aI ı/. Proof of Lemma 12.10. Given  2 R such that u.a/ <  < u.a/; N by the continuity of u at a and the lower semicontinuity of u, N there exists ı > 0 such that, for every x 2 B.aI 2ı/, u.x/ <  < u.x/: N We denote by RB.aIı/ uN the balayage of uN with respect to the ball B.aI ı/ (Proposition 7.5). It suffices to prove that RB.aIı/ uN D uN almost everywhere in . Since uN is superharmonic, RB.aIı/ uN 6 uN almost everywhere in  (Corollary 7.6). To prove the reverse inequality, we first observe that RB.aIı/ uN D uN > 

in B.aI 2ı/ n B.aI ı/.

Thus, in the ball B.aI 2ı/, the function . RB.aIı/ u/ N C has compact support. Since the balayage RB.aIı/ uN is superharmonic, we have that .

RB.aIı/ u/ N C D max ¹

RB.aIı/ u; N 0º

is a maximum of subharmonic functions, and therefore is also subharmonic in  (cf. Lemma 12.7). Applying the weak maximum principle (Proposition 6.1), we deduce that . RB.aIı/ u/ N C60

almost everywhere in B.aI 2ı/, and this implies that RB.aIı/ uN >  almost everywhere in B.aI 2ı/. Hence, RB.aIı/ uN > u almost everywhere in , and then RB.aIı/ uN 2 Ou . By the minimality of u, N we deduce the reverse inequality RB.aIı/ uN > uN almost everywhere in . Therefore, equality holds and uN is harmonic in B.aI ı/.  The second lemma is a restatement of Property 11.8: Lemma 12.11. Under the assumptions of Proposition 12.6, if the obstacle u is continuous in B.aI r/   and if u.a/ N D u.a/, then sup x2B.aI r2 /

ju.x/ N

u.a/j N 6 2N C1 sup

x2B.aIr /

ju.x/

u.a/j:

12.2. Perron–Remak method

193

N D u.a/, for every x 2 B.aI r/ we Proof of Lemma 12.11. Since uN > u and u.a/ have u.x/ N u.a/ N > u.x/ u.a/;

whence

u.x/ N

u.a/ N >

sup z2B.aIr /

ju.z/

u.a/j:

In order to get the estimate from above, let  <  be such that, for every x 2 B.aI r/,  < u.x/ < : Then, u.x/ N

u.a/ N D u.x/ N

u.a/ 6 u.x/ N

 6 max ¹u.x/; N º

:

(12.8)

Claim. The function max ¹u; N º is subharmonic in B.aI r/, and, for every ball B.xI ı/ b B.aI r/, we have max ¹u; N º.x/ 6

B.xIı/

max ¹u; N º:

Proof of the claim. By the previous lemma, the function uN is harmonic in a neighborhood of the set ¹uN > º \ B.aI r/. In particular, uN is locally bounded in B.aI r/. By the localization lemma (Proposition 6.11) and the interpolation inequality 1;2 (Lemma 5.8), we then have uN 2 Wloc .B.aI r//. Thus, the measure uN is diffuse in B.aI r/ (Lemma 6.14). By Kato’s inequality (Proposition 8.12), we deduce that  max ¹u; N º > ¹u>º uN D 0 N in the sense of distributions in B.aI r/. In other words, the function max ¹u; N º is subharmonic in B.aI r/. By the mean value property of subharmonic functions (cf. Proposition 2.18), for every ball B.xI ı/ b B.aI r/ and every 0 < s 6 ı, we then have

B.xIs/

max ¹u; N º 6

B.xIı/

max ¹u; N º:

On the other hand, by the convexity of the function R 3 t 7! max ¹t; º and by Jensen’s inequality, we may estimate ³ ² u; N  6 max max ¹u; N º: B.xIs/

B.xIs/

Combining the two estimates and letting s ! 0, we then get max ¹u.x/; N º 6

B.xIı/

max ¹u; N º:

4

194

12. Obstacle problems

Applying estimate (12.8) and the claim, for every ball B.xI ı/ b B.aI r/ we have u.x/ N

u.a/ N 6

B.xIı/

max ¹u; N º

D

B.xIı/

max ¹uN

; 

º:

Thus, by the positivity of the function under the integral sign, we get  r N max ¹uN ;  º: u.x/ N u.a/ N 6 ı B.aIr /

Since uN > k and  > k, we have max ¹uN

; 

º 6 uN

C

:

Since the function uN is superharmonic, the mean value property of superharmonic functions yields   uN C  max ¹uN ;  º 6 B.aIr /

B.aIr /

6 u.a/ N

C

:

Since u.a/ N D u.a/ 6 , we deduce that B.aIr /

max ¹uN

; 

º 6 2.

/;

whence

 r N . /: ı Minimizing the right-hand side with respect to  and , we get  r N sup ju.z/ u.a/j: u.x/ N u.a/ N 62 ı z2B.aIr / u.x/ N

u.a/ N 62

Restricting this estimate to x 2 B.aI 2r / and ı D 2r , the conclusion follows.



Proof of Proposition 12.9. The previous lemmata establish the continuity of uN in . We now prove that uN can be continuously extended to @ using Lebesgue’s barrier x ! R be a continuous function argument at a point a 2 @. For this purpose, let hW  such that h.a/ D 0, h.x/ > 0 for x ¤ a, and h is harmonic in . An example of such a function in dimension N > 3 is given by h.x/ D 1

rN 2 jx bjN

2

;

x and r > 0 are taken such that BŒbI r \  x D ¹aº; the existence where b 2 RN n  of b and r is ensured by the smoothness of .

12.3. Total charge and energy minimizations

195

Given  > 0, we may choose M > 0 such that the function v D u.a/ C  C M h belongs to Ou . Indeed, by the continuity of u at a, there exists ı > 0 such that x Since the function Œu u.a/= h is continuous in the u 6 u.a/ C  in B.aI ı/ \ . x compact set  n B.aI ı/, it has a nonnegative upper bound M . The function v thus defined is harmonic in  and satisfies v > u, whence v 2 Ou . By the minimality of u, N we deduce that uN 6 u.a/ C  C M h: Since uN > u, it follows that, for every x 2 , ju.x/ N

u.a/j 6 max ¹ C M h.x/; ju.x/

u.a/jº:

Therefore, uN has a continuous extension at a and lim u.x/ N D u.a/:

x!a



Further results concerning the regularity of obstacle problems were obtained by Brezis and Stampacchia [68] in the framework of variational inequalities and free boundary problems. This setting was also pursued for example by Caffarelli and Kinderlehrer [70] and by Frehse [136]. A connection between potential theory and PDE techniques was made by Lewy and Stampacchia [205], and later by Frehse and Mosco [137] and [138] in the study of irregular obstacles.

12.3 Total charge and energy minimizations We show that the Perron–Remak method can be interpreted as a minimization problem involving the total charge kvkM./ . For this purpose, we consider the following family of potentials above an obstacle u: Pu D ¹v 2 X./W v > u almost everywhere in º; where X./ D ¹v 2 W01;1 ./W v 2 M./º: Proposition 12.12. Let  be a smooth bounded open set. If u 2 L1 ./ is such that Pu is non-empty, then the infimum inf ¹kvkM./ W v 2 Pu º is achieved by the smallest superharmonic function in Pu .

196

12. Obstacle problems

The minimizer need not be unique (Example 12.13), but among all minimizers of the total charge, there exists a smallest one, and this is the smallest superharmonic function in Pu . Example 12.13. Let uW . 2; 2/ ! R be the obstacle 8 ˆ

@n @n

on @;

almost everywhere with respect to the surface measure. Proof of Lemma 12.15. Let V W RN ! R be the function defined by V D

´

v2 0

v1

in , in RN n .

By the extension property (Corollary 7.4), V 2 M.RN / and V D .v2

v1 /b

 @v

2

@n

@v1  N H @n

1

b@ :

Since V is a nonnegative function, it follows from Kato’s inequality (Assertion (6.5)) that the diffuse part of the measure V is nonnegative on the set ¹Vy D 0º, where the precise representative Vy achieves its minimum.

198

12. Obstacle problems

Since the measure HN 1 b@ is diffuse with respect to the W 1;2 capacity (Proposition 10.4), the measure V is diffuse on @ and  @v

.V /d b@ D

2

@n

@v1  N H @n

1

b@ :

Note that the precise representative Vy is defined quasi-everywhere (Proposition 8.9). Since V D 0 in RN n  and  is smooth, we then have that Vy must vanish quasieverywhere on @. In other words, @ is contained in ¹Vy D 0º, except for a set of zero W 1;2 capacity, and we conclude that  @v

2

@n

@v1  N H @n

1

b@ D .V /d b@ > 0:

The lemma follows from this inequality.



Compared to Proposition 10.4, there is a more elementary approach to prove that the measure HN 1 b@ is diffuse. Indeed, it suffices to observe that the classical trace inequality in Sobolev spaces (Proposition 15.1) implies the weaker capacitary estimate (Proposition 16.1) HN

1

b@ 6 C capW 1;2 ;

which is enough for our purposes. Corollary 12.16. Let  be a smooth bounded open set, and let v1 ; v2 2 X./. If v1 6 v2 almost everywhere in , and if v1 is superharmonic, then kv1 kM./ 6 kv2 kM./ : Proof of Corollary 12.16. By the superharmonicity of v1 and by Gauss’s law (7.1), ˆ ˆ @v1 kv1 kM./ D v1 D d:  @ @n A similar conclusion applies to v2 except that, since v2 need not be superharmonic, we get ˆ ˆ @v2 kv2 kM./ > d: v2 D  @ @n Thus, kv1 kM./ 6 kv2 kM./ C

ˆ

@

 @v

2

@n

@v1  d: @n

By the previous lemma, the integrand is nonpositive and the conclusion follows.



12.3. Total charge and energy minimizations

199

Proof of Proposition 12.12. Let .vn/n2N be a minimizing sequence of the total charge in the class Pu . By the Sobolev embedding of solutions of the Dirichlet problem (Proposition 5.1), this sequence is bounded in W01;1 ./, whence by the Rellich– Kondrashov compactness theorem (Proposition 4.8) there exists a subsequence .vnk /k2N converging strongly in L1 ./ to some function w. By the closure lemma in X./ (Lemma 12.14), we have w 2 X./ and kwkM./ 6 lim inf kvnk kM./ : k!1

Since w > u almost everywhere in , we have w 2 Pu . Thus, w is a minimizer of the total charge. We now show that every minimizer of the total charge over the family Pu is superharmonic. Indeed, given a minimizer w, the weak maximum principle (Propositions 6.1 and 6.5) shows that the solution of the Dirichlet problem with density . w/C also belongs to Pu . By the minimality of the total charge that generates w, k. w/C k D kwkM./ : We deduce that . w/ D 0, and so the measure w is nonnegative. Our next goal is to prove that, among all minimizers of the total charge, there exists a smallest element. The argument is similar to the proof of the Perron–Remak method, and relies on the construction of a non-increasing minimizing sequence of the integral ˆ (12.10)

w



in the class of minimizers. We need to be careful not to leave the class Pu . The following claim takes care of this issue: Claim. If w 2 Pu is a minimizer of the total charge, then, for every superharmonic function v 2 Pu , the function min ¹w; vº is superharmonic, belongs to Pu , and also minimizes the total charge. Proof of the claim. For every superharmonic function v 2 Pu , we have that the function min ¹w; vº is also superharmonic (Lemma 12.7). Since min ¹w; vº D w

.w

v/C ;

by the estimate of the total charge up to the boundary (Proposition 7.7), we have that  min ¹w; vº 2 M./. Therefore, min ¹w; vº 2 Pu . Since min ¹w; vº 6 w, it follows from the previous corollary that k min ¹w; vºkM./ 6 kwkM./ : By the minimality property of the total charge of w, equality holds and the conclusion follows. 4

200

12. Obstacle problems

Given a minimizing sequence of the integrals (12.10) in the class of minimizers of the total charge, using the claim we construct by induction a non-increasing sequence .wn /n2N of functions in the same class. By the monotone convergence theorem, .wn /n2N converges in L1 ./ to some function u. Q Since the total charge of the sequence is constant, by the closure property in X./ (Lemma 12.14) we have uQ 2 X./ and kuk Q M./ 6 lim inf kwn kM./ : n!1

We deduce that uQ 2 Pu and that uQ is also a minimizer of the total charge. By construction, uQ is the smallest minimizer. We observe that, by the claim, this function uQ is also the smallest superharmonic function in Pu .  We now consider a variational approach by Lewy and Stampacchia to reach the smallest superharmonic function above an obstacle, see Lemma 1.1 in [204] and Chapter 2, Section 6, in [180], using a strategy by Lions and Stampacchia, see Theorem 6.2 in [211]. For this purpose, consider the family of Sobolev functions that are greater than or equal to u: Su D ¹v 2 W01;2 ./W v > u almost everywhere in º: Proposition 12.17. Let  be a smooth bounded open set. If u 2 L1 ./ is such that Su is non-empty, then the infimum inf ¹krvkL2./ W v 2 Su º is achieved by the smallest superharmonic function in Su . Proof. The existence of a minimizer uQ of the variational problem is obtained using a minimizing sequence .vn /n2N in Su . Indeed, by the Rellich–Kondrashov compactness theorem (Proposition 4.8) there exists a subsequence .vnk /k2N converging strongly to some function uQ in L2 ./, and in particular uQ > u almost everywhere in . Moreover, by the closure lemma (Proposition 4.10), uQ 2 W01;2 ./ and kr uk Q L2 ./ 6 lim inf krvnk kL2 ./ : k!1

In particular, uQ 2 Su , and uQ is a minimizer of the variational problem. To prove that uQ is a superharmonic function, we first establish the Euler–Lagrange equation associated to the variational problem: for every v 2 Su , we have ˆ r uQ
r.v u/ Q > 0: 

12.3. Total charge and energy minimizations

201

The key observation is that Su is a convex set: for every v 2 Su and every 0 6 t 6 1, we have t v C .1 t /uQ 2 Su . Therefore, the function ˆ Œ0; 1 3 t 7 ! jr.t v C .1 t /u/j Q 2 

achieves its minimum at 0, and so it has a nonnegative derivative at this point. Computing the derivative at 0, we find the Euler–Lagrange equation. We now deduce that uQ is superharmonic. Indeed, for every nonnegative function ' 2 Cc1 ./, we have uQ C ' 2 Su . Using v D uQ C ' as a test function in the Euler–Lagrange equation, we find ˆ ˆ 

uQ ' D



r uQ
r' > 0;

whence the conclusion. It remains to show that, for every superharmonic function w 2 Su , we have w > uQ almost everywhere in  – equivalently, .uQ w/C D 0. Indeed, we write (cf. Exercise 5.3) jr.uQ Thus, ˆ

w/C j2 D r.uQ



jr.uQ

C 2

w/
r.uQ

w/ j D

ˆ



w/C D r uQ
r.uQ

r uQ
r.uQ

w/

C

ˆ



w/C

rw
r.uQ

rw
r.uQ

w/C :

w/C :

(12.11)

We show that the first integral in the right-hand side is nonpositive. Indeed, note that .uQ

w/C D min ¹w; uº Q

u; Q

and that min ¹w; uº Q is an admissible test function in the Euler–Lagrange equation since, by assumption, w > u almost everywhere in . We then get ˆ r uQ
r.uQ w/C 6 0: 

We now show that the second integral is nonnegative. Since w is superharmonic, for every nonnegative function ' 2 Cc1 ./ we have ˆ ˆ rw
r' D w ' > 0: 



Since .uQ w/C 2 W01;2 ./ we can approximate this function by a nonnegative .'n /n2N in Cc1 ./ (cf. Proposition 4.4). Applying the previous inequality to ' D 'n and letting n ! 1, we deduce that ˆ rw
r.uQ w/C > 0: 

202

12. Obstacle problems

Combining the two inequalities, it follows from identity (12.11) that ˆ jr.uQ w/C j2 6 0: 

Therefore, r.uQ w/C D 0 almost everywhere in . Since .uQ w/C 2 W01;2 ./, we conclude that .uQ w/C D 0 (Exercise 4.4), and the proof is complete.  The minimizer in the statement of Proposition 12.17 is unique. This is a consequence of the strict convexity of the function RN 3 z 7! jzj2 . Exercise 12.1 (uniqueness). Let u 2 L1 ./. Prove that if uQ 1 and uQ 2 minimize the Dirichlet energy in Su , then uQ 1 D uQ 2 almost everywhere in . We conclude this section by proving that the variational and the potential approaches yield the same solution. This common solution uQ of both obstacle problems reduces simultaneously the Dirichlet energy of the obstacle, kr uk Q L2 ./ 6 krukL2 ./ and its total charge, kuk Q M./ 6 kukM./ :

In addition, the function uQ is superharmonic and inherits the regularity properties from the Perron–Remak method (Section 12.2). Proposition 12.18. Let  be a smooth bounded open set. If u 2 L1 ./ is such that Su \ Pu is non-empty, then the Sobolev function uQ S that minimizes the Dirichlet energy in Su and the smallest minimizer uQ P of the total charge in Pu are equal almost everywhere in . We first consider the case of a bounded obstacle with compact support: Lemma 12.19. The conclusion of Proposition 12.18 holds under the additional assumption that the obstacle u is bounded and has compact support in . Proof of Lemma 12.19. We first prove the inequality uQ S 6 uQ P . For this purpose, we recall that, for any constant  > 0, it follows from the interpolation inequality (Lemma 5.8) that min ¹uQ P ; º 2 W01;2 ./. Since u 2 L1 ./, for any  such that  > u almost everywhere in , we have min ¹uQ P ; º 2 Su . Since the function min ¹uQ P ; º is also superharmonic in  (Proposition 6.9 or Lemma 12.7), by the minimality of uQ S in Su we have uQ S 6 min ¹uQ P ; º

12.3. Total charge and energy minimizations

203

almost everywhere in . Letting  ! 1, we deduce that uQ S 6 uQ P almost everywhere in . To prove the reverse inequality, we observe that, since the distribution uQ S is nonpositive, uQ S is a locally finite measure on  (Proposition 2.19). Using the assumption on the support of the obstacle u, we prove that uQ S is a finite measure on : Claim. The function uQ S is weakly harmonic in  n supp u. Proof of the claim. We may assume that  is connected and that uQ S is nonzero in . Then, uQ S ¤ 0 in . By the quantitative maximum principle (Proposition 2.21), for every open subset ! b  there exists  > 0 such that uQ S >  almost everywhere in !. Taking ! b  n supp u, for every ' 2 Cc1 .!/ and every t 2 R such that jt jk'kL1 .!/ 6  the function uQ S C t ' also belongs to the admissible class Su . Since the function ˆ t7 !



jr.uQ S C t '/j2

achieves its minimum at t D 0, we deduce that ˆ r uQ S
r' D 0; 

and the conclusion follows.

4

We deduce from the claim that the measure uQ S is compactly supported in , whence uQ S 2 M./. Therefore, uQ S 2 Pu , and then, by the minimality of uQ P , uQ P 6 uQ S almost everywhere in . This concludes the proof of the lemma.



Proof of Proposition 12.18. By the weak maximum principle (Exercise 4.8 and Propositions 6.1 and 6.5), the functions uQ S and uQ P are nonnegative. Replacing u by uC if necessary, we may thus assume that the obstacle u is nonnegative. Take a nondecreasing sequence of functions .un /n2N in L1 ./ converging pointwise to u, and such that each un is bounded from above and has compact support in . By the previous lemma, the solutions of the obstacle problems in Sun and in Pun coincide, and we denote them by uQ n .

204

12. Obstacle problems

Given v 2 Su \ Pu , we also have v 2 Sun \ Pun . In particular, by the minimality of uQ n , we get kr uQ n kL2 ./ 6 krvkL2./ and

kuQ n kM./ 6 kvkM./ :

The sequence .uQ n /n2N is also nondecreasing, and bounded from above by v. Thus, .uQ n /n2N converges strongly in L1 ./ to a function w. By the uniform estimates above, we have w 2 W01;2 ./ and w 2 M./ (Proposition 4.10 and Lemma 12.14). Moreover, each function uQ n is superharmonic, whence w is also superharmonic (Exercise 2.8). For every n 2 N, since un 6 u, we also have uQ n 6 uQ S

and uQ n 6 uQ P

almost everywhere in . Thus, letting n ! 1, we deduce that w 6 uQ S

and w 6 uQ P

almost everywhere in . However, since w is superharmonic and belongs to Su \Pu , the reverse inequalities also hold. This concludes the proof.  Exercise 12.2 (capacitary potential). Let ! b  be smooth bounded open subsets of RN , and let U be the function introduced in Lemma 12.3. .a/ Prove that, for every v 2 W01;2 ./ \ L1 ./, we have ˆ ˆ @h vO d; rU
rv D  @! @n where n denotes the exterior normal vector with respect to !. .b/ Prove that if v 2 W01;2 ./ is such that v > 1 almost everywhere in !, then vO > 1 almost everywhere in @! with respect to the surface measure. .c/ Deduce that U is the common solution uQ of the minimization of the total charge and the Dirichlet energy with obstacle u D ! . In particular, we have 2 kuk Q M./ D kr uk Q L 2 ./ :

We have focused on obstacles given in terms of L1 functions, and the obstacle condition is verified in the sense of the Lebesgue measure. A more refined approach is to use the property that the exceptional set of a function in W01;2 ./ or X./ has zero W 1;2 capacity (Propositions 8.6 and 8.9). In this case, one can pursue the minimization problems involving the counterparts of Su and Pu , under the condition that the precise representatives satisfy vO > uO quasi-everywhere, where uW  ! R is an obstacle whose exceptional set also has zero W 1;2 capacity.

Chapter 13

Families of solutions “Le processus que nous avons décrit nous amène donc à définir sur E une véritable répartition de masses potentiantes.”1 Georges Bouligand

We are given a compact set S  RN , and we wish to know whether there is some distribution of charges in S such that the resulting potential is a bounded function, or an Lp function, or a Hölder-continuous function, depending on the size of S .

13.1 Bounded functions The regularity we may expect from a potential u depends on the size of the support supp u of the distribution of charges that generates u. Smaller supports typically give rise to worse behaved solutions. For instance, if supp u D ¹aº is a single point, then u has the same summability as the fundamental solution ® of the ¯Laplacian in a neighborhood of a. In contrast, the function u.x/ D min jxjN1 2 ; 1 is Lipschitzcontinuous and satisfies supp u D @B.0I 1/ in dimension N > 3. We are interested in the intermediate situation between these examples, in connection with the removable singularity problems investigated in Chapter 11. We first prove the converse of the removable singularity property for bounded solutions: Proposition 13.1. Let  be a smooth bounded open set. For every compact set S   such that capW 1;2 .S / > 0, there exists a positive measure  2 M./, supported in S , such that the solution of the linear Dirichlet problem with density  is bounded. The main ingredients are the existence of diffuse measures supported on sets of positive Sobolev capacity (Proposition A.17) and the following boundedness principle by Frostman [139] and Maria [227]: 1 “The process we have described

a potential.”

leads us to prescribe to E a genuine distribution of mass generating

206

13. Families of solutions

Lemma 13.2. Let  be a smooth bounded open set, let  2 M./ be a nonnegative measure, and let uO be the precise representative of the solution of the linear Dirichlet problem with density . If  is diffuse with respect to the W 1;2 capacity and if E   is a Borel set such that uO is bounded on E, then the linear Dirichlet problem with density bE has a bounded solution. Proof of Lemma 13.2. We prove that if v satisfies the linear Dirichlet problem with density bE , then 0 6 v 6 sup uO E

almost everywhere in . Since  is nonnegative, the first inequality follows from the weak maximum principle (Proposition 6.1). The weak maximum principle also gives v 6 u almost everywhere in , which will be useful for proving that v 6 sup u. O

Indeed, by Kato’s inequality (Proposition 8.12), for every s 2 R we have .v

E

s/C > ¹v>sº v O

in the sense of distributions in , where vO denotes the precise representative of v. Thus, .v s/C > ¹v>sº . bE / D b¹v>sº\E : O O Since v 6 u almost everywhere in , vO satisfies the same pointwise inequality vO 6 uO in Lu \ Lv . In particular, for every s > sup uO the set ¹vO > sº \ E is empty, and then E

s/C > 0

.v

in the sense of distributions in . On the other hand, by the Sobolev embedding of solutions of the linear Dirichlet problem (Proposition 5.1) we have v 2 W01;1 ./, and this implies that, for every s > 0, .v s/C 2 W01;1 ./. From the weak maximum principle (Propositions 6.1 and 6.5), we deduce that .v s/C 6 0, whence v 6 s almost everywhere in . Taking s D sup u, O the conclusion follows.  E

The previous lemma raises the question of how large the set E can be. Since the precise representative of a nonnegative superharmonic function coincides pointwise with its maximal function (Exercise 8.3), by the maximal inequality (Proposition 9.6) we have C (13.1) capW 1;2 .¹uO > º/ 6 kkM./ ;  for every  > 0. Thus, taking E D ¹uO 6 º; (13.2) the set  n E D ¹uO > º has W 1;2 capacity as small as we want by choosing  sufficiently large.

13.2. Lebesgue integrable classes

207

Proof of Proposition 13.1. Let S   be a compact set such that capW 1;2 .S / > 0. By the existence of diffuse measures (Proposition A.17), there exists a positive measure  2 M.RN / supported in S and satisfying the setwise inequality 1

 6 .capW 1;2 / 2 : In particular, this measure  is diffuse with respect to the W 1;2 capacity. Given a Borel set E  , by the subadditivity of  and the capacitary upper bound above we have .E/ > ./

. n E/ > ./

1

.capW 1;2 . n E// 2 :

Taking E as the level set (13.2) of the solution of the linear Dirichlet problem with density , it follows from the maximal inequality (13.1) that, for  sufficiently large, we have capW 1;2 . n E/ < ..//2 ;

Thus, the measure bE is nonzero and, by the previous lemma, the solution of the linear Dirichlet problem with density bE is bounded. 

In Chapter 18, we pursue the problem of existence of measures supported in S generating a continuous potential (Corollary 18.11). We need in this case an additional tool: the Evans–Vasilesco continuity principle (Proposition 18.10).

13.2 Lebesgue integrable classes We now show the converse of the problem of removable singularity in the Lp setting: Proposition 13.3. Let 1 < p < C1, and let  be a smooth bounded open set. For every compact set S   such that capW 2;p0 .S / > 0, there exists a positive measure  2 M./, supported in S , such that the solution of the linear Dirichlet problem with density  belongs to Lp ./. The summability of the solution of the Dirichlet problem can be translated in terms of a functional estimate satisfied by the density : Lemma 13.4. Let  be a smooth bounded open set, let  2 M./, and let v be the solution of the linear Dirichlet problem with density . For every 1 < p < C1, we have that v 2 Lp ./ if and only if there exists C > 0 such that, for every x  2 C01 ./, ˇ ˇˆ ˇ ˇ ˇ  dˇˇ 6 C kkW 2;p0 ./ : ˇ 

208

13. Families of solutions

Proof of Lemma 13.4. We begin with the direct implication. If the solution v belongs x by the Hölder inequality we get to Lp ./, then, for every  2 C01 ./, ˇˆ ˇ ˇ ˇ ˇ  dˇˇ 6 kvkLp ./ kkLp0 ./ 6 C1 kvkLp ./ kkW 2;p0 ./ ; ˇ 

and the conclusion follows with C D C1 kvkLp ./ . x the Conversely, if  satisfies the required estimate, then, for every  2 C01 ./, Calderón–Zygmund estimate (Proposition 5.3) yields ˇˆ ˇ ˇ ˇ ˇ v  ˇˇ 6 C kkW 2;p0 ./ 6 C2 kkLp0 ./ : ˇ 

x we apply this estimate to the solution  2 C 1 ./ x of the linear Given f 2 C 1 ./, 0 1 x Dirichlet problem with density f . For every f 2 C ./, we deduce that ˇˆ ˇ ˇ ˇ ˇ vf ˇˇ 6 C2 kf kLp0 ./ : ˇ 

Thus, by the Riesz representation theorem (Proposition 3.3 and Exercise 3.1), we have v 2 Lp ./.  0

x is dense in W 2;p0 ./ \ W 1;p ./ with Since the vector subspace C01 ./ 0 0 respect to the W 2;p norm, the assumption of the lemma amounts to saying that  has 0 0 a unique extension as a continuous linear functional acting on W 2;p ./\W01;p ./. The previous lemma raises the question of how to construct measures  satisfying the functional estimate. This can be achieved using the Hahn–Banach theorem (see Proposition A.17 below). Proof of Proposition 13.3. Let S   be a compact set such that capW 2;p0 .S / > 0. By the existence of diffuse measures (Proposition A.17), there exists a positive measure  2 M.RN / supported in S such that, for every nonnegative function ' 2 Cc1 .RN /, we have ˆ 06 (13.3) ' d 6 k'kW 2;p0 .RN / : RN

x Since S is a compact subset of , we also have, for every  2 C01 ./, ˇˆ ˇ ˇ ˇ ˇ  dˇ 6 C kk 2;p0 : ./ W ˇ ˇ 

13.3. Hölder-continuous solutions

209

x Indeed, given ' 2 Cc1 ./, we have that ' 2 Cc1 ./ for every  2 C01 ./. In general, ' is a signed function, which is not admissible in the functional esti2 Cc1 .RN / such that j'j 6 mate (13.3). To remedy this obstruction, take N in R . Choosing ' D 1 on S , the integrals of  and ' with respect to  are the same. By the nonnegativity of and by Property (13.3), we have ˇ ˆ ˇ ˇˆ ˇˆ ˆ ˇ ˇ ˇ ˇ ˇ ˇ  dˇ D ˇ j'j d 6 d 6 k kW 2;p0 .RN / : ' dˇ 6 ˇ ˇ ˇ 

RN

RN

RN

We choose the nonnegative upper bound

such that (Lemma A.4)

k kW 2;p0 .RN / 6 C1 k'kW 2;p0 .RN / ; and we get

ˇˆ ˇ ˇ ˇ ˇ  dˇ 6 C1 k'k 2;p0 N 6 C kk 2;p0 ; ./ .R / W W ˇ ˇ 

for some constant C > 0 depending on the initial choice of '. In particular,  satisfies the assumptions of the previous lemma, thus the solution of the linear Dirichlet problem with density  belongs to Lp ./. 

13.3 Hölder-continuous solutions In this section, we prove the converse of the removable singularity problem for Hölder-continuous functions due to Carleson [78] and [79]: Proposition 13.5. Let N > 3, let  be a smooth bounded open set, and let 0 < ˛ < 1. For every compact set S   such that HN 2C˛ .S / > 0, there exists a positive measure  2 M./, supported in S , such that the solution of the linear Dirichlet x problem with density  is Hölder-continuous of order ˛ in . We follow Carleson’s argument based on an estimate of the Newtonian potential generated by . For this purpose, we need the Cavalieri representation of the Newtonian potential N in terms of densities on balls (Lemma 2.16): for every x 2 RN , 1 N.x/ D N

ˆ

1 0

.B.xI r// dr : rN 2 r

(13.4)

An alternative proof in the spirit of the Morrey–Sobolev embedding is given in Chapter 16.

210

13. Families of solutions

Proof of Proposition 13.5. Let S   be a compact set such that HN 2C˛ .S / > 0. By Frostman’s lemma (Proposition B.4), there exists a positive measure  2 M.RN /, supported in S , such that N  6 H1

2C˛

:

Thus, for every ball B.aI r/  RN , we have (Proposition B.2) .B.aI r// 6 C r N

2C˛

(13.5)

:

We prove that the Newtonian potential N is Hölder-continuous of order ˛ in R , using its Cavalieri representation (13.4). By the density estimate (13.5), for every z 2 RN we have N.z/ < C1 (Proposition 2.15). By the nonnegativity of the measure , we now estimate the difference N.x/ N.y/ for any ı > 0 by N

1 N.x/ N.y/ 6 N

ˆ

ı 0

Claim. For every ı > 2jx ˆ

1 ı

1 .B.xI r// dr C rN 1 N

ˆ

1 ı

.B.xI r// .B.yI r// dr: rN 1

yj, we have

.B.xI r// .B.yI r// dr 6 C1 jx rN 1

yj

ˆ

1 ı

.B.yI r// dr; rN

for some constant C1 > 0 depending on N . Proof of the claim. Since B.xI r/  B.yI r C jx ˆ

1 ı

.B.xI r// dr 6 rN 1

ˆ

1 ı

Making the change of variable t D r C jx ˆ

1 ı

.B.xI r// dr 6 rN 1

ˆ

yj/, we have

.B.yI r C jx rN 1

yj//

dr:

yj, we then get

1 ıCjx yj

.B.yI t // .t jx yj/N

1

dt:

Switching back to the letter r as the integration variable, we deduce that, for every ı > jx yj, ˆ

1

.B.xI r// .B.yI r// dr rN 1 ı ˆ 1 h 1 6 .B.yI r// .r jx yj/N ı

1

1 i

rN

1

dr:

13.3. Hölder-continuous solutions

By the mean value theorem, for every r > jx 1

1 yj/N 1

jx

.r

whence, for every r > 2jx

1 yj/N

jx

yj there exists 0 <  < 1 such that N 1 jx jx yj/N

.r

yj;

yj,

1 .r

D

rN 1

211

1

rN

1

6

2N .N 1/ jx rN

For every ı > 2jx yj, we then deduce that ˆ 1 .B.xI r// .B.yI r// dr 6 2N .N N 1 r ı

1/jx

yj

yj:

ˆ

1 ı

.B.yI r// dr; rN

which is the estimate we wanted to prove. For every ı > 2jx N.x/

N.y/ 6

4

yj, it now follows from the claim that 1 N

ˆ

ı 0

C1 .B.xI r// dr C jx rN 1 N

yj

ˆ

1 ı

.B.yI r// dr: rN

The density assumption (13.5) on the measure  implies that ˆ

ı 0

.B.xI r// dr 6 C rN 1

ˆ

.B.yI r// dr 6 C rN

ˆ

ı 0

dr r1 ˛

D

C ˛ ı ; ˛

and ˆ

1 ı

1 ı

1 r2 ˛

dr 6

C 1

˛

ı˛

1

:

Therefore, N.x/ Taking ı D 2jx

N.y/ 6 C2 .ı ˛ C jx

yjı ˛

1

/:

yj, we then get N.x/

N.y/ 6 C3 jx

yj˛ :

Switching the roles of x and y, we deduce that N is Hölder-continuous. The Newtonian potential N is smooth in RN n S , and in particular in a neighx ! R of the restriction Nj@ borhood of @. Thus, the harmonic extension hW  1 x belongs to C ./. The function N h satisfies the linear Dirichlet problem with x density  (cf. Example 2.12) and is Hölder-continuous on . 

212

13. Families of solutions

Given a measure  2 M.R2 / supported in a bounded open set   R2 , we have the representation formula (2.4) for the Newtonian potential: N.x/ D

1 2

ˆ

d 0

.B.xI r// dr r

log d .R2 /; 2

x and any fixed d > diam . We can thus proceed as above to establish for every x 2  ˛ x for measures satisfying  6 H1 the Hölder continuity of N in , . We thus have a complete answer to the problem of removable singularities in the class of Hölder-continuous functions of order 0 < ˛ < 1 in any dimension N > 2. An alternative approach, including the Zygmund class investigated by Ullrich [328], can be found in Section 3.8 in [249]. The counterpart for Lipschitz-continuous functions is described in terms of the Lipschitz analytic capacity introduced by Paramonov [271], see also [337]; a difficult issue is to identify sets of zero-capacity in terms of the more familiar HN 1 Hausdorff measure, see [329]. The main result can be summarized as follows: a compact set S  RN such that HN 1 .S / < C1 is removable in the class of Lipschitz functions if and only if S is purely unrectifiable of dimension N 1. By purely unrectifiable we mean that S does not contain any rectifiable subset of dimension N 1 having positive HN 1 measure. When S contains a rectifiable subset of positive HN 1 measure, the construction of a Lipschitz function in RN which is harmonic in RN n S , but not in RN , is explained by Mattila and Paramonov, see Remarks, p. 481, in [229]. The removability of purely unrectifiable sets has been solved in dimension N D 2 by David and Mattila [98], see also [262], and extended to any dimension by Nazarov, Tolsa, and Volberg [261] based on a previous work of Volberg [337] on the Lipschitz analytic capacity ; the easier case HN 1 .S / D 0 is contained in Proposition 11.4. Let us mention another problem where purely unrectifiable sets are removable, that extends Proposition 4.18 above. We rely in this case on the fine properties of functions of bounded variation that were established independently by Federer [128] and Vol0 pert [338]. Proposition 13.6. Let S   be a compact set, and let u 2 C 1 . n S /. If we have that HN 1 .S / < C1 and S is purely unrectifiable of dimension N 1, and if u 2 Lq ./ and ru 2 Lq .I RN /, then u 2 W 1;q ./. Proof. By Lemma 10.7 and the truncation argument in the proof of Lemma 4.19, there exists a uniformly bounded sequence .'n /n2N in Cc1 .RN / such that 'n D 1 in a neighborhood of S , k'n kW 1;q .RN / 6 C HN

1

.S / C 1;

13.3. Hölder-continuous solutions

213

and supp 'n  S C B.0I 1=2n /:

Assuming that u is bounded, the sequence .ur'n /n2N is bounded in L1 .I RN /. Letting n ! 1 in identity (4.8), we deduce from the Riesz representation theorem that ˆ ˆ ˆ ru
ˆ D u div ˆ C ˆ
d; E 





for some finite vector-valued measure  E in . We deduce that the distributional derivative of u is given by ru , E whence u belongs to the class BVloc ./ of functions of (local) bounded variation. By the Federer–Vol0 pert regularity theory of BV functions, the set S cannot carry a jump part of the distributional derivative Du since S is purely unrectifiable, see Theorem 3.78 in [13]; the Cantor part of Du cannot charge S since HN 1 .S / < C1, see Proposition 3.92 in [13]. On the other hand, by the assumption on the support of 'n , the measure  E is singular and supported in S . We conclude that  E D 0, 1;q whence u 2 W ./. If u is not bounded, we may then proceed as in the proof of Proposition 4.18.  Removable singularities for the Laplacian in the class of Lipschitz functions cannot be fully characterized in terms of the HN 1 measure, since there exist compact sets having non  -finite HN 1 measure which are removable, see Corollary 3.2 and Remark in [229]. Since u D div ru, a related question is to describe the removable sets for the divergence operator in the class of bounded vector fields, not necessarily in gradient form. This problem has a complete answer, obtained independently by Moonens [254] and by Phuc and Torres [278]: a compact set S  RN is removable if and only if HN 1 .S / D 0.

Chapter 14

Strong approximation of measures

“Les équations différentielles auxquelles obéissent les phénomènes physiques n’ont été souvent établies que par des raisonnements peu rigoureux ; les résultats expérimentaux, auxquels il s’agit de comparer les conséquences de la théorie, sont eux-mêmes approximatifs.”1 Henri Poincaré

We establish strong approximation properties for measures that are diffuse with respect to the Sobolev capacity capW k;p or to the Hausdorff measure Hs . The main ingredient is an adaptation of the Jordan decomposition theorem. Using Maz0 ya’s trace inequalities, we recover a characterization of diffuse measures in terms of L1 perturbations in dual Sobolev spaces.

14.1 Capacitary and density bounds In problems involving finite Borel measures  in RN , it is straightforward to construct weak approximations .n /n2N of  using smooth functions (cf. Proposition 2.7). On the other hand, the strong convergence of measures, namely lim jn

n!1

j.RN / D 0;

is harder to expect due to analytical obstructions. Indeed, if each n is absolutely continuous with respect to the Lebesgue measure, then  also has such a property, hence  is represented by an L1 function (cf. Exercise 2.5). We now present the main strong approximation results contained in this chapter. We begin with the counterpart for the Sobolev capacity capW k;q for k > 1 and 1 6 q < C1, which is related to the family of trace inequalities introduced by Maz0 ya [230]: 1 “Differential equations obeyed by physical phenomena have often been established by some nonrigorous argument. The experimental results with which one needs to compare the theoretical predictions are themselves approximate.”

216

14. Strong approximation of measures

Proposition 14.1. Let  2 M.RN / be a nonnegative measure. If  is diffuse with respect to the W k;q capacity, then there exists a sequence .n /n2N of nonnegative measures in M.RN / of the form n D bEn such that (i) for every n 2 N, there exists Cn > 0 such that n 6 Cn capW k;q ,

(ii) the sequence .n /n2N is nondecreasing and converges strongly to  in M.RN /. By diffuse we mean that, for every Borel set A  RN such that capW k;q .A/ D 0, we have .A/ D 0. The capacitary upper bound  6 C capW k;q for some given nonnegative measure  2 M.RN / is equivalent to the following trace inequality: for every ' 2 Cc1 .RN /, k'kLq .RN I/ 6 C 0 k'kW k;q .RN / : This estimate will be proved for k D 1 in Chapter 16. The more difficult case k > 2 can be found in the book by Maz0 ya and Shaposhnikova, see Chapter 1 in [239]. The following weaker conclusion for any k > 1 is enough for our purposes: Proposition 14.2. Let  2 M.RN / be a nonnegative measure. If there exists C > 0 such that  6 C capW k;q ; then, for every 1 6 p < q and every ' 2 Cc1 .RN /, we have k'kLp .RN I/ 6 C 00 k'kW k;q .RN / ; for some constant C 00 > 0 depending on C , p, q, and kkM.RN / . Proof. By Cavalieri’s principle (1.5), for every ' 2 Cc1 .RN / we have ˆ

j'jp d D p

RN

ˆ

1 0

tp

1

.¹j'j > t º/ dt:

Given  > 0, we split the integral in the right-hand side in two parts: ˆ

RN

j'jp d D p

ˆ

0



tp

1

.¹j'j > t º/ dt C p

ˆ

1 

tp

1

.¹j'j > t º/ dt:

14.1. Capacitary and density bounds

217

We estimate the first one using the finiteness of the measure  as ˆ  ˆ  p t p 1 .¹j'j > t º/ dt 6 p t p 1 dt kkM.RN / D  p kkM.RN / : 0

0

To estimate the second part, we need the weak capacitary estimate (Lemma A.11): for every t > 0, C1 k'kqW k;q .RN / : tq Thus, by the capacitary upper bound on , we have capW k;q .¹j'j > t º/ 6

.¹j'j > t º/ 6 It then follows that ˆ 1 ˆ p t p 1 .¹j'j > t º/ dt 6 p 

C C1 k'kqW k;q .RN / : tq 1



1 t q pC1

dt C C1 k'kqW k;q .RN /

C2 D q p k'kqW k;q .RN / :  Combining the two estimates, we get ˆ C2 q j'jp d 6  p kkM.RN / C q p k'kW k;q .RN / :  RN Minimizing the right-hand side with respect to  > 0, we obtain the conclusion.



The capacitary estimate from the previous proposition implies in particular that, for every ' 2 Cc1 .RN /, k'kL1 .RN I/ 6 C 00 k'kW k;q .RN / : Since Cc1 .RN / is dense in W k;q .RN /, this amounts to saying that  can be uniquely extended as a continuous linear functional on W k;q .RN /, and we denote this property by  2 .W k;q .RN //0 :

From Propositions 14.1 and 14.2 we deduce that diffuse measures are the strong limits in M.RN / of sequences of measures on the dual space .W k;q .RN //0 , see Théorème 8 in [132], Lemme 4.2 in [20], and Theorem 2.2 in [93]. The original proof of this corollary in the papers by Feyel–de la Pradelle, Baras and Pierre, and Dal Maso is based on the Hahn–Banach theorem. We provide in the next section a different argument due to Mokobodzki [250] relying on tools from measure theory, in the spirit of the Jordan decomposition theorem. This approach also yields the following counterpart of Proposition 14.1 in terms s of the Hausdorff content H1 for 0 6 s < 1:

218

14. Strong approximation of measures

Proposition 14.3. Let  2 M.RN / be a nonnegative measure. If  is diffuse with s respect to the Hausdorff content H1 , then there exists a sequence .n /n2N of nonN negative measures in M.R / of the form n D bEn such that s (i) for every n 2 N, there exists Cn > 0 such that n 6 Cn H1 ,

(ii) .n /n2N is nondecreasing and converges strongly to  in M.RN /. s .A/ D 0, For any set A  RN , we have that Hs .A/ D 0 if and only if H1 therefore the families of diffuse measures with respect to the Hausdorff measure s are the same. Given a nonnegative measure Hs or to the Hausdorff content H1 N  2 M.R /, the equivalence between the condition s  6 C H1

and the density estimate: for every ball B.xI r/  RN , .B.xI r// 6 C 0 r s ; is explained in Section B.1. Under the stronger quantitative property  6 ˛Hs in terms of the Hausdorff measure, we have a more precise control of the density constants: Proposition 14.4. Let  2 M.RN / be a nonnegative measure, and let ˛ > 0. If we have  6 ˛Hs , then there exists a sequence .n /n2N of nonnegative measures in M.RN / such that (i) for every n 2 N, there exist ˛n < ˛ and ın > 0 such that n 6 ˛n Hısn ,

(ii) the sequence .n /n2N is nondecreasing and converges strongly to  in M.RN /. Assuming for example that  is absolutely continuous with respect to the Lebesgue measure on RN , the inequality  6 ˛Hs is satisfied for every s < N and every ˛ > 0. Indeed, we have that either .A/ D 0 or Hs .A/ D C1 for every Borel set A  RN .

For the sake of illustration, we present some regularity properties for the Poisson equation involving the capacitary and density measures above. For this purpose, assume that  2 M.RN / is a nonnegative measure supported in a smooth bounded domain  and that u is the solution of the linear Dirichlet problem in  with density . We then have that

14.2. Radon–Nikodým and Lebesgue decompositions N .a/ u is Hölder-continuous of order 0 < ˛ < 1 if and only if  6 C H1 (Lemma 11.5 and Proposition 13.5);

219 2C˛

0

.b/ u 2 Lp ./ if and only if  2 .W 2;p .RN //0 (Lemma 13.4); .c/ eu 2 L1 ./ provided that  6 ˛HıN u

.d / e 2 W

1;2

2

for some ˛ < 4 (Proposition 17.8);

./ provided that  6 ı capW 1;2 for ı > 0 small (Proposition 17.11).

14.2 Radon–Nikodým and Lebesgue decompositions The proofs of Propositions 14.1 and 14.3 rely on the monotonicity and the semis . A Borel additivity of the Sobolev capacity capW k;p and the Hausdorff content H1 set function having these properties will be called a semimeasure: Definition 14.5. A nonnegative Borel semimeasure T on RN is a Borel set function with values into Œ0; C1 such that (i) T .;/ D 0,

(ii) if A  B are Borel subsets of RN , then T .A/ 6 T .B/,

(iii) there exists M > 0 such that, for every sequence of Borel sets .An /n2N of RN , T

1 [

kD0

1  X Ak 6 M T .Ak /: kD0

A nonnegative Borel measure satisfies conditions .i/–.iii/ with constant M D 1 (cf. Exercise 2.2). We are concerned with measures that do not charge sets which are negligible with respect to T : Definition 14.6. Let T be a nonnegative semimeasure on RN . A measure  2 M.RN / is diffuse with respect to T if, for every Borel set A  RN such that T .A/ D 0, we have jj.A/ D 0: The expression diffuse measure is borrowed from potential theory. This concept is equivalent to the notion of absolute continuity from measure theory: Proposition 14.7. Let T be a nonnegative Borel semimeasure on RN , and let  2 M.RN /. We have that  is a diffuse measure with respect to T if and only if, for every  > 0, there exists ı > 0 such that if A  RN is a Borel set such that T .A/ 6 ı, then jj.A/ 6 :

220

14. Strong approximation of measures

Proof. We begin with the reverse implication. If A  RN is a Borel set such that T .A/ D 0, then, for every  > 0, we have jj.A/ 6 . Thus, jj.A/ D 0 and  is a diffuse measure. For the direct implication, we proceed by contradiction. Given a sequence of positive numbers .˛n /n2N converging to zero, suppose that there exist  > 0 and a sequence .An /n2N of Borel subsets of RN such that, for every n 2 N, we have and jj.An / > :

T .An / 6 ˛n

The sequence .An /n2N need not be non-increasing, but we observe that the sequence 1
S Aj j 2N is non-increasing and satisfies similar estimates. Indeed, for every j 2 nDj

N, by the semi-additivity of the semimeasure T we have

T

1 [

nDj

1 1  X X An 6 M T .An / 6 M ˛n nDj

(14.1)

nDj

and, by the monotonicity of jj, we also have 1 [  jj An > jj.Aj / > : nDj

Let A be the limit of the sequence

1 S

nDj

AD

Aj




j 2N

1 1 [ \

, namely

An :

j D0 nDj

In particular, by the monotone set lemma (Exercise 2.1), we have jj.A/ > . Choosing from the beginning a summable sequence .˛n /n2N and letting j ! 1, we deduce from (14.1) and the monotonicity of T that T .A/ D 0. This contradicts the assumption that  is diffuse with respect to T .  We prove the following characterization of diffuse measures due to Mokobodzki [250] that implies Propositions 14.1 and 14.3: Proposition 14.8. Let T be a nonnegative Borel semimeasure on RN , and let  2 M.RN / be a nonnegative measure. Then,  is diffuse with respect to T if and only if, for every  > 0, there exist C > 0 and a Borel set E  RN such that bE 6 C T

and .RN n E/ 6 :

14.2. Radon–Nikodým and Lebesgue decompositions

221

When T is the Lebesgue measure on RN this proposition implies that every measure which is absolutely continuous with respect to the Lebesgue measure can be strongly approximated by bounded functions. In this case, this is a consequence of the classical Radon–Nikodým Theorem, see Theorem 3.8 in [134]. The main tool is the following extraction lemma reminiscent of the Jordan decomposition theorem (Proposition 2.3): Lemma 14.9. Let T be a nonnegative Borel semimeasure on RN . Then, for every nonnegative measure  2 M.RN /, there exists a Borel set E  RN such that bE 6 T

and T .RN n E/ 6 M.RN n E/;

where M is the constant satisfying the semi-additivity property of T . When T is a finite measure, the extraction lemma follows from the Jordan decomposition theorem applied to the measure T , in which case the set E may be chosen so that bE 6 T and T bRN nE 6 : The main idea of the proof of the extraction lemma lies in the following observation: if the inequality  6 T does not hold on every Borel set, then there exists some Borel set F  RN such that T .F / < .F /, and we try to choose F so that .F / is as large as possible. Since  is a finite measure, we eventually exhaust the part of  that prevents the inequality  6 T from holding. Proof of Lemma 14.9. Let F0 D ; and 0 <  < 1. We construct by induction a sequence .Fn /n2N of disjoint Borel subsets of RN such that, for every n 2 N , we have .a/ Fn  RN n

nS1

Fk and T .Fn / 6 .Fn /,

kD0

.b/ .Fn / > n , where n ° ± [1 N n D sup .F /W F  R n Fk and T .F / 6 .F / : kD0

By the semi-additivity of T and the additivity of , we have T

1 [

kD0

1 1 1   [ X X Fk 6 M T .Fk / 6 M Fk : .Fk / D M kD0

kD0

kD0

(14.2)

222

14. Strong approximation of measures

We now claim that b

RN n

1 S

Fk

6 T:

(14.3)

kD0

Assume by contradiction that this inequality is not true. Then, there exists a Borel set G  RN such that 1   [ T .G/ <  G n Fk : kD0

Let

DDGn

1 [

Fk :

kD0

By the monotonicity of T and the assumption on the set G, we have T .D/ 6 T .G/ < .D/: In particular, .D/ > 0. Since D is an admissible set in the definition of the numbers n , for every n 2 N, we have .D/ 6 n : This is not possible, since by the additivity of  we have 

1 X

kD1

k 6

1 X

kD1

.Fk / D 

1 [

kD1

 Fk 6 .RN / < C1:

Thus, the sequence .n /n2N converges to 0, but this contradicts the fact that .n /n2N is bounded from below by the positive number .D/. In view of properties (14.2) and (14.3), we thus have the conclusion of the lemma by choosing N

EDR n

1 [

Fk :



kD0

The extraction lemma above was inspired by Lemma 2 in [22], which was used to obtain some strong approximation of measures such that  6 4HN 2 . Proof of Proposition 14.8. Given  > 0, let E  RN be the Borel set given by the extraction lemma above with measure  and semimeasure T . More precisely, bE 6 T and T .RN n E / 6 M.RN n E / 6 M.RN /:

In particular, since  is a finite measure,

lim T .RN n E / D 0:

!C1

14.2. Radon–Nikodým and Lebesgue decompositions

223

By the absolute continuity of  with respect to T (Proposition 14.7), the conclusion of the direct implication follows by taking C D  sufficiently large. For the reverse implication, given  > 0, let E  RN be the Borel set and C > 0 be the constant satisfying conditions (i) and (ii). Then, for every Borel set A  RN , we have .A/ D .A \ E/ C .A n E/ 6 C T .A/ C : In particular, if T .A/ D 0, then we deduce that .A/ 6 . Since  is nonnegative and  > 0 is arbitrary, it follows that .A/ D 0, and so the measure  is diffuse.  The following corollary contains Proposition 14.1 and Proposition 14.3 by taking s T D capW k;p and T D H1 , respectively: Corollary 14.10. Let T be a nonnegative Borel semimeasure on RN , and let  2 M.RN / be a nonnegative measure. If  is diffuse with respect to T , then there exists a sequence .n /n2N of nonnegative measures in M.RN / of the form n D bEn such that (i) for every n 2 N, there exists Cn > 0 such that n 6 Cn T ,

(ii) the sequence .n /n2N is nondecreasing and converges strongly to  in M.RN /. Proof. Let .n /n2N be a sequence of positive numbers converging to zero. By Proposition 14.8, for every n 2 N there exist a Borel set En  RN and a constant Cn0 > 0 such that bEn 6 Cn0 T and .RN n En / 6 n : Since the sequence .En /n2N need not be nondecreasing, we consider the sequence n
S Ek n2N . By the subadditivity of , of Borel sets kD0

b

n S

6 Ek

kD0

n X

bEk 6

kD0

n X

kD0

 Ck0 T;

and, by the monotonicity of , k

b

n S

kD0

Ek

n   [ kM.RN / D  RN n Ek 6 .RN n En / 6 n : kD0

The sequence .n /n2N defined by n D b

n S

Ek

thus has the required properties.

kD0



224

14. Strong approximation of measures

We conclude this section by proving the counterpart of the Lebesgue decomposition of a finite Borel measure with respect to semimeasures. The meaning of concentrated measure transfers immediately to the setting of semimeasures: Definition 14.11. Let T be a nonnegative Borel semimeasure on RN . A measure  2 M.RN / is concentrated with respect to T if there exists a Borel set F  RN such that T .F / D 0 and jj.RN n F / D 0: The strategy of the proof of the Lebesgue decomposition theorem is standard, see Lemma 4.A.1 in [60]: Proposition 14.12. Let T be a nonnegative Borel semimeasure on RN . For every  2 M.RN /, there exist unique measures d ; c 2 M.RN / such that  D d C c ; where d is diffuse with respect to T and c is concentrated with respect to T . Proof. We first prove the existence of the decomposition when the measure  is nonnegative. For this purpose, let ˛ D sup ¹.A/W A 2 B.RN / and T .A/ D 0º: Since the measure  is finite, ˛ is finite. We claim that the supremum is achieved. Indeed, let .An /n2N be a sequence in B.RN / such that, for every n 2 N, we have T .An / D 0 and lim .An / D ˛: n!1

By the semi-additivity of T , we have T

1 S

kD0

for every n 2 N we then get .An / 6 

1 [

kD0


Ak D 0. By the monotonicity of ,  Ak 6 ˛:

Letting n ! 1, we deduce that .F / D ˛, where F D

1 [

Ak :

kD0

Thus, F achieves the supremum above and, in particular, the measure bF is concentrated with respect to T .

14.3. Perturbation of diffuse measures

225

Since, by the additivity of , we have  D bRN nF C bF ; it now suffices to show that the measure bRN nF is diffuse. For this purpose, take a Borel set A  RN such that T .A/ D 0. We also have T .A [ F / D 0, whence by the additivity of the measure  and the maximality of the set F , .A n F / D .A [ F /

.F / 6 ˛

˛ D 0:

Since  is nonnegative, we deduce that .AnF / D 0, and the decomposition follows for nonnegative measures. The case of a signed measure  is obtained using the Jordan decomposition theorem (Proposition 2.3), since we can apply the previous argument to the measures C and  . To prove the uniqueness of such a decomposition, we observe that if  D d C c

and  D Q d C Q c

are two decompositions of  in terms of diffuse and concentrated parts, then the measure d Q d D Q c c is simultaneously diffuse and concentrated, and so must be identically zero. 

14.3 Perturbation of diffuse measures We present here a characterization of diffuse measures due to Boccardo, Gallouët, and Orsina as an L1 perturbation of finite measures in .W k;q .RN //0 , see Theorem 2.1 in [32] and Remarque, p. 200 in [20]: Proposition 14.13. Let 1 < q < C1, and let  2 M.RN /. If  is diffuse with respect to the W k;q capacity, then there exist a signed measure  2 .W k;q .RN //0 and f 2 L1 .RN / such that DCf in the sense of measures on RN .

The argument is based on a trick by Gallouët and Morel [144]. We begin with the strong convergence of the convolution product in dual Sobolev spaces: Lemma 14.14. Let 1 < q < C1. If S 2 .W k;q .RN //0 , then, for every sequence of mollifiers .n /n2N , we have lim kn  S

n!1

S k.W k;q .RN //0 D 0:

226

14. Strong approximation of measures

Proof of Lemma 14.14. By the Hahn–Banach theorem (Proposition 3.4), the linear functional defined for u 2 W k;q .RN / by .u; Du; : : : ; D k u/ 7 ! S Œu can be extended as a continuous linear functional in a Cartesian product of Lebesgue Lq spaces. Thus, by the Riesz representation theorem, for every i 2 ¹0; : : : ; kº there 0 i exists Gi 2 Lq .RN I RN / such that, for every u 2 W k;q .RN /, we have S Œu D

k ˆ X i D0

RN

D i uŒGi ;

where we use the convention D 0 uŒG0  D uG0 . Assume for simplicity that each n is an even function. By the definition of the convolution with the functional S and by Fubini’s theorem, we have .n  S /Œu D S Œn  u D

k ˆ X i D0

RN

D i uŒn  Gi ;

and we deduce that kn  S

S k.W k;q .RN //0 6 C

k X i D0

kn  Gi

Gi kLq0 .RN / :

As n ! 1, the right-hand side converges to zero and the conclusion follows.



Proof of Proposition 14.13. We may assume that  is nonnegative; in this case, the function f is chosen to be nonnegative. We consider any decomposition of  as a strongly convergent series of nonnegative measures, D

1 X

k ;

kD0

such that, for every n 2 N, we have n 2 .W k;q .RN //0 . We may take for instance 0 D 0 and, for n > 1, n D n n 1 , where .n /n2N is a nondecreasing sequence satisfying the conclusion of Proposition 14.1; see also Proposition 14.2. Given a sequence .n /n2N of positive numbers, Gallouët and Morel’s trick consists in writing each measure n in the form n D .n

n  n / C n  n :

14.4. Precise density bound

227

Let .˛n /n2N be a summable sequence of positive numbers. For each n 2 N, by the previous lemma we may choose n > 0 so that kn and therefore the series

1 P

.k

kD0

n  n k.W k;q .RN //0 6 ˛n ; k  k / converges in .W k;q .RN //0 . On the other

hand, by Fubini’s theorem we have

kn  n kL1 .RN / 6 kn kM.RN / D n .RN /; and so

1 P

kD0

k  k converges in L1 .RN /, and

M.RN /. We have the conclusion with D

1 X

.k

kD0

k  k /

1 P

.k

kD0

and f D

1 X

kD0

k  k / converges in

k  k :



The Boccardo–Gallouët–Orsina decomposition behaves linearly, in the sense that if 1 and 2 are diffuse measures written as  1 D  1 C f1

and 2 D 2 C f2 ;

then, for every ˛1 ; ˛2 2 R, ˛1 1 C ˛2 2 D .˛1 1 C ˛2 2 / C .˛1 f1 C ˛2 f2 / gives a decomposition of the same nature for the measure ˛1 1 C ˛2 2 . In contrast, the construction of the decomposition provided by the proof of Proposition 14.13 is nonlinear with respect to . One might ask whether there is a linear construction underneath or, more precisely, if there exists a continuous linear functional from the Banach space of diffuse measures with respect to the W 1;2 capacity into the space .W 1;2 .RN //0  L1 .RN / which gives such a decomposition. However, Ancona [15] gave a negative answer.

14.4 Precise density bound We prove in this section the more precise version of the strong approximation property for measures satisfying the estimate  6 ˛Hs (Proposition 14.4). For this purpose, we need the following counterpart of Proposition 14.8:

228

14. Strong approximation of measures

Proposition 14.15. Let  2 M.RN / be a nonnegative measure, and let ˛ > 0. If we have  6 ˛Hs , then, for every  > 0, there exists a Borel set E  RN such that (i) for every ˇ > ˛, there exists ı > 0 such that bE 6 ˇHıs , (ii) .RN n E/ 6 . This proposition is reminiscent of the uniform convergence of the Hausdorff capacities Hıs to the Hausdorff measure Hs on sets of finite Hausdorff measure (Proposition B.14). The argument – taken mostly from [287] and [288] – relies on the extraction lemma (Lemma 14.9), but is more involved compared to the proof of Proposition 14.8. We bypass the lack of additivity of the Hausdorff capacity Hıs using the following result: Lemma 14.16. Let  2 M.RN / be a nonnegative measure, let  > 0, and let F1 ; : : : ; Fn be disjoint Borel subsets of RN . If, for every k 2 ¹1; : : : ; nº, we have bFk 6 ˛Hs ; then, for every  > 0, there exist 0 < ı 6  and a Borel set F  bF 6

˛Hıs

and 

n [

kD1

 Fk n F 6 :

n S

Fk such that

kD1

Proof of Lemma 14.16. It suffices to establish the statement with ˛ D 1. For each i 2 ¹1; : : : ; nº, let Ki  Fi be a compact subset. By the subadditivity of , for every Borel set A  RN we have b S n

Ki

.A/ 6

i D1

n X i D1

.A \ Ki / 6

n X i D1

Hs .A \ Ki /:

Let 0 < ı 6  be such that, for every i; j 2 ¹1; : : : ; nº and i ¤ j , d.Ki ; Kj / > 2ı. In particular, we have d.A \ Ki ; A \ Kj / > 2ı: By the metric additivity of the Hausdorff capacity (Lemma B.12), we get n X i D1

Hıs .A \ Ki / D Hıs

n [

 .A \ Ki / :

i D1

14.4. Precise density bound

229

Since Hs 6 Hıs , we deduce that, for every Borel set A  RN , we have b S n

i D1

Ki

.A/ 6

n X i D1

Hs .A \ Ki / 6 n S

Thus, the compact set F D

n X i D1

Hıs .A \ Ki / D Hıs

n [

 .A \ Ki / 6 Hıs .A/:

i D1

Ki satisfies the first property of the statement.

i D1

We now explain how to choose the compact sets Ki to have the second property. Since n n n  [  [ [ .Fi n Ki /; Ki D Fi n i D1

i D1

the subadditivity of the measure  shows that 

n  [

i D1

i D1

n n  X  [ .Fi n Ki /: Ki 6 Fi n i D1

i D1

Given  > 0, by the inner regularity of the measure  (Proposition 2.5) we can choose Ki  Fi such that  .Fi n Ki / 6 : n Thus, n n  [  [   Fi n  Ki 6 n D : n This proves the lemma.

i D1

i D1



Proof of Proposition 14.15. We begin by proving the existence of a Borel set E  RN depending on ˇ > ˛: Claim. For every  > 0 and every ˇ > ˛, there exist a Borel set E  RN and ı > 0 such that bE 6 ˇHıs and .RN n E/ 6 : Proof of the claim. Let .n /n2N be a sequence of positive numbers converging to zero. Given ˇ > ˛, we construct below a sequence of Borel sets .Fn /n2N in RN such that .a/ bF0 6 ˇHs 0 ; .b/ for every n 2 N , we have b

Fn n

nS1

Fk

6 ˇHs n ;

kD0

.c/ for every n 2 N, we have ˇHs n .RN n Fn / 6  RN n

n S

kD0


Fk :

230

14. Strong approximation of measures

We proceed by induction on n 2 N. Let F0  RN be a Borel set satisfying the conclusion of the extraction lemma (Lemma 14.9) with semimeasure T D ˇHs 0 . Given n 2 N and Borel sets F0 ; : : : ; Fn 1 , we apply the extraction lemma with measure b and semimeasure T D ˇHs n . We then obtain a Borel set nS1 RN n

Fk

kD0

Fn  RN such that

b Fn n

nS1

Fk

6 ˇHs n

kD0

and

n   [ ˇHs n .RN n Fn / 6  RN n Fk : kD0

This sequence satisfies Properties .a/, .b/ and .c/. We now observe that lim  RN n

n!1

where G D RN n

1 S

n [

kD0


Fk D .G/;

Fk . For every n 2 N, by monotonicity of the Hausdorff

kD0

capacity ˇHs n and by Property .c/ above, we also have ˇHs n .G/

6

ˇHs n .RN

n   [ N n Fn / 6  R n Fk : kD0

Letting n ! 1 , we get

ˇHs .G/ 6 .G/:

By the upper bound of  in terms of the Hausdorff measure, we deduce that .ˇ

˛/.G/ 6 0:

Thus, .G/ D 0, whence 

N

lim  R n

n!1

For every Borel set E 

n S

n [

kD0

 Fk D .G/ D 0:

(14.4)

Fk , by the subadditivity of the measure  we have

kD0

n n  [   [ .RN n E/ 6  RN n Fk C  Fk n E : kD0

kD0

14.4. Precise density bound

231

Given  > 0, the limit (14.4) above shows that there exists n 2 N such that 

N

 R n

n [

kD0

  Fk 6 : 2

By Properties .a/ and .b/, and the weak additivity of the Hausdorff capacity (see nS1 Lemma 14.16) applied to the disjoint sets F0 ; F1 n F0 ; : : : ; Fn n Fk , there exist kD0

0 < ı 6 min ¹0 ; : : : ; n º and E  such that bE 6 ˇHıs

and 

n [

kD0

In particular, we have .RN n E/ 6

n [

Fk

kD0

  Fk n E 6 : 2

  C D ; 2 2

and this concludes the proof of the claim.

4

Let .n /n2N and .ˇn /n2N be two sequences of positive numbers to be chosen below. For every n 2 N, by the previous claim there exist a Borel set En  RN and a gauge ın > 0 such that bEn 6 ˇn Hısn Let E D

1 T

kD0

and

.RN n En / 6 n :

Ek . For every n 2 N, the monotonicity of  implies that bE 6 bEn 6 ˇn Hısn :

Choose a sequence .ˇn /n2N converging to ˛ from above. Then, for every ˇ > ˛, there exists n 2 N such that ˇn 6 ˇ and we have that bE 6 ˇn Hısn 6 ˇHısn : Therefore, Property .i/ holds with ı D ın . By the subadditivity of the measure , we also have 1 1 X X N N .R n E/ 6 .R n Ek / 6 k : kD0

kD0

Choosing from the beginning a summable sequence .n /n2N such that we deduce Property (ii). The proof of the proposition is complete.

1 P

k 6 ,

kD0



232

14. Strong approximation of measures

Proof of Proposition 14.4. Let .n /n2N be a sequence of positive numbers converging to zero, and let En  RN be a Borel set satisfying the conclusion of Proposition 14.15 with parameter n . Thus, .a/ for every n 2 N and every ˇ > ˛, there exists ın > 0 such that bEn 6 ˇHısn , .b/ the sequence .bEn /n2N converges strongly to  in M.RN /. Claim. The sequence .En /n2N of Borel sets may be chosen to be nondecreasing. Proof of the claim. For every n 2 N, we have k

b

1 T

Ek

kM.RN /

kDn

1   \ N D R n Ek kDn

and RN n

1 \

kDn

Ek D

1 [

.RN n Ek /:

kDn

Thus, by the subadditivity of the measure , k

b

1 T

kM.RN / 6

Ek

kDn

1 X

kDn

.RN n Ek / 6

1 X

k :

kDn

Choosing a summable sequence .n /n2N , it follows that lim k

n!1

b

1 T

Ek

kM.RN / D 0:

kDn

Thus, the sequence

1 T

kDn

Ek




n2N

of Borel sets also satisfies Properties (a) and (b),

and is nondecreasing. If necessary, we may thus replace En by

1 T

Ek .

kDn

4

By the claim, we can suppose that .En /n2N is a nondecreasing sequence of Borel sets. Let .n /n2N be an increasing sequence of positive numbers converging to 1. We show that the sequence .n bEn /n2N satisfies the conclusion of the proposition. Indeed, this sequence is nondecreasing and, for every n 2 N, we have k

n bEn kM.RN / D .1

n /.En / C .RN n En /:

14.4. Precise density bound

233

Thus, .n bEn /n2N converges strongly to  in M.RN /. It remains to show that Property (i) is satisfied. For this purpose, let n < n < 1. By Property (a), there exists ın > 0 such that ˛ s H ; bEn 6 n ın whence Property (i) holds with ˛n D

n ˛. n

The proof of the proposition is complete. 

The next corollary to Proposition 14.15 gives the analogue of the absolute continuity for measures which do not charge sets of finite Hausdorff measure Hs (cf. Proposition 14.7): Corollary 14.17. For every  2 M.RN /, the following assertions are equivalent: (i) for every Borel set A  RN such that Hs .A/ < C1, we have jj.A/ D 0I (ii) for every  > 0 and every  > 0, there exists ı > 0 such that if A  RN is a Borel set such that Hıs .A/ 6 , then jj.A/ 6 : To illustrate this corollary, assume that  is a nonnegative measure such that .A/ D 0, for every Borel set A satisfying HN

1

.A/ < C1:

We conclude in this case that  has uniformly small mass on flat cylinders having a fixed .N 1/-dimensional base volume. Indeed, denote by C.r; h/ any cylinder in RN whose base is an .N 1/-dimensional ball Br0 of radius r and whose height is h. Since HN 1 .Br0 / D !N 1 r N 1 , it follows that, for every 0 < h 6 h.ı/ sufficiently small, we have HıN 1 .C.r; h// 6 2!N 1 r N 1 : Given  > 0 and  D 2!N We thus have that

1r

N 1

, choose a gauge ı > 0 as in Assertion .ii/.

.C.r; h// 6 ; for any cylinder C.r; h/ such that 0 < h 6 h.ı/. Unlike in the case of Proposition 14.7, we have no direct proof of Corollary 14.17. If that were the case, we would then have a simpler proof of Proposition 14.15 along the lines of the proof of Proposition 14.8.

234

14. Strong approximation of measures

Proof of Corollary 14.17. It suffices to prove the statement for a nonnegative measure . We first consider the direct implication. Since  does not charge sets of finite Hausdorff measure Hs , for every ˛ > 0 we have  6 ˛Hs . Given a Borel set E  RN , for every Borel set A  RN we have .A/ D .A \ E/ C .A n E/ 6 bE .A/ C .RN n E/:

(14.5)

Given  > 0, let E  RN be a Borel set satisfying the conclusion of Proposition 14.15 with parameter 2 . It thus follows from estimate (14.5) that, for every ˇ > 0, there exists ı > 0 such that  .A/ 6 ˇHıs .A/ C : 2 For  > 0, choose ˇ > 0 such that ˇ 6 2 . If Hıs .A/ 6 , we then have .A/ 6 ˇ C

 6 : 2

To prove the reverse implication, take a Borel set A  RN such that Hs .A/ < C1. Given  > 0 and  D Hs .A/, take ı > 0 satisfying the assumption. Then, Hıs .A/ 6 Hs .A/ D ; whence .A/ 6 . Since  > 0 is arbitrary, we deduce that .A/ D 0.



For the sake of completeness, let us also mention the following companion to Corollary 14.17 for measures that are diffuse with respect to the Hausdorff measure Hs (cf. Proposition 14.7): Exercise 14.1 (absolute continuity with respect to Hıs ). For every  2 M.RN / and 0 < ı 6 C1, prove that the following assertions are equivalent: (i) for every Borel set A  RN such that Hs .A/ D 0, we have jj.A/ D 0;

(ii) for every  > 0, there exists  > 0 such that if A  RN is a Borel set such that Hıs .A/ 6 , then jj.A/ 6 . Other consequences of Propositions 14.8 and 14.15, and connections to the problem of removable singularities for the divergence operator have been investigated in [287].

Chapter 15

Traces of Sobolev functions “Often the genuine difficulty is shifted to the final task of ascertaining in what sense a result obtained in terms of ideal functions is indeed expressible by ordinary functions.” Richard Courant

The concept of trace inequality arises in the problem of giving a meaning to boundary values of Sobolev functions. Fractional Sobolev spaces on the boundary are related to the range of the trace operator.

15.1 Existence of the trace We begin with the classical trace inequality, whose short proof below relies on the divergence theorem, see [252] or Exercise 4.2 in [348]. Proposition 15.1. Let 1 6 q < C1, and let  be a smooth bounded open set. Then, x we have for every u 2 C 1 ./, kukLq .@/ 6 C kukW 1;q ./ ; for some constant C > 0 depending on q and . x Given F 2 C 1 .RN I RN /, we apply Proof. We first assume that u is positive in . the divergence theorem to the vector field uq F to get ˆ ˆ q u F
n d D div .uq F /: @



Note that div .uq F / D uq div F C quq

1

ru
F 6 C1 .uq C uq

1

jruj/

in :

236

15. Traces of Sobolev functions

When q > 1, we apply the Young inequality to estimate uq 1 jruj. For every q > 1, we then get div .uq F / 6 C2 .uq C jrujq / in : Choosing the vector field F such that F
n > 1 on @, we deduce that ˆ ˆ uq d 6 C2 .uq C jrujq /: @



x not necessarily positive, we apply this For an arbitrary function u 2 C 1 ./, 1 estimate to the function un D .u2 C n / 2 , where .n /n2N is a sequence of positive numbers converging to zero. As n ! 1, the conclusion follows.  x is dense in the For smooth bounded open sets , the vector subspace C 1 ./ 1;q Sobolev space W ./ equipped with its intrinsic norm (4.1), see Section 6.3 in [345] or Section 9.2 in [53]. From the trace inequality above, we deduce that the linear operator x 3 u 7 ! uj@ 2 Lq .@/ C 1 ./ has a unique continuous linear extension to W 1;q ./, which we denote by TrW W 1;q ./ 7 ! Lq .@/: This gives a meaning to the notion of boundary value or, to be more precise, of trace on @ for every Sobolev function from the point of view of functional analysis. The trace of an arbitrary function u 2 W 1;q ./ is thus computed by taking any x converging strongly to u in W 1;q ./. By the trace sequence .ui /i 2N in C 1 ./ inequality above, the restriction to the boundary .ui j@ /i 2N is a Cauchy sequence in Lq .@/. The limit of this sequence in Lq .@/ is the trace of u on @, and the trace does not depend on the initial choice of the approximating sequence .ui /i 2N . Exercise 15.1. Let  be a smooth bounded open set. x RN /, the following .a/ Prove that, for every u 2 W 1;q ./ and every ˆ 2 C 1 .I integration by parts formula holds: ˆ ˆ ˆ u div ˆ D ru
ˆ C .Tr u/ ˆ
n d: 



@

.b/ Deduce that the kernel of the trace operator Tr in W 1;q ./ is the vector subspace W01;q ./. From the point of view of measure theory, it is possible to identify the trace in terms of precise representatives:

15.1. Existence of the trace

237

Exercise 15.2. Let u 2 W 1;q .RN / and let  be a smooth bounded open set. .a/ Prove that the trace of the function uj on @ is given by the restriction of the precise representative uO to the boundary: Tr .uj / D uj O @ :

.b/ Deduce that lim

r !0 B.xIr /\

u D Tr .uj /.x/

for almost every x 2 @ with respect to the surface measure. The Sobolev inequality (Corollary 4.12) has a counterpart on smooth bounded domains: for every exponent 1 6 q < N and every u 2 W 1;q ./, we have

(15.1)

kukLq ./ 6 C kukW 1;q ./ ;



where q is the Sobolev exponent. This inequality follows from its counterpart in RN or from the existence of a continuous linear operator that extends any function in W 1;q ./ as a function with compact support in W 1;q .RN /, see Theorem 9.7 in [53] or Theorem 6.3.1 in [345]. As a consequence of (15.1), the trace operator TrW W 1;q ./ ! Lq .@/ is never surjective for exponents q > 1, since the function Tr u satisfies a better summability property: Proposition 15.2. Let 1 < q < N , and let  be a smooth bounded open set. Then, for every u 2 W 1;q ./, we have Tr u 2 L kTr uk

L

.N 1/q N q

.@/

.N 1/q N q

.@/ and

6 C kukW 1;q ./ ;

for some constant C > 0 depending on N , q, and . x and F 2 C 1 .RN I RN /, we estimate Proof. Given a positive function u 2 C 1 ./ the divergence of the vector field up F for some exponent p > 1 to be chosen later on. By the Hölder inequality, we have ˆ p 1 krukLq ./ : up 1 jruj 6 kukL .p 1/q 0 ./ 

Thus, ˆ



p p 1 div .up F / 6 C1 .kukL p ./ C kuk .p L

1/q 0 ./

krukLq ./ /:

We now choose p such that .p 1/q 0 D q  . In other words, p D .NN 1/q and, q in particular, p < q  . It thus follows from the Hölder inequality and the Sobolev inequality (15.1) that ˆ div .up F / 6 C2 kukpW 1;q ./ : 

238

15. Traces of Sobolev functions

Choosing the vector field F so that F
n > 1 on @, we deduce from the divergence theorem that ˆ ˆ up d 6

@



div .up F / 6 C2 kukpW 1;q ./

for every positive smooth function u. The argument for signed smooth functions and then for arbitrary Sobolev functions is the same as in the proof of Proposition 15.1 and the definition of the trace.  From the trace inequality, we deduce that the surface measure  D HN satisfies, for every ' 2 Cc1 .RN /, the estimate k'k

L

.N 1/q N q

.RN I/

1

b@

6 C k'kW 1;q .RN / :

More generally, a nonnegative Borel measure  in RN has a trace inequality if there exist exponents p and q such that, for every ' 2 Cc1 .RN /, we have k'kLp .RN I/ 6 C k'kW 1;q .RN / : In Chapter 16, we relate this estimate to capacitary and density estimates satisfied by . We now prove the following property of the trace operator for exponents q > 1 in connection with fractional Sobolev spaces (Section 15.2): Proposition 15.3. Let 1 < q < C1, and let  be a smooth bounded open set. Then, for every u 2 W 1;q ./, the trace of u on @ satisfies ˆ ˆ jTr u.x/ Tr u.y/jq d .x/ d .y/ < C1 jx yjqCN 2 @ @ and, more precisely, we have kTr ukLq .@/ C

ˆ

@

ˆ

@

jTr u.x/ Tr u.y/jq d .x/ d .y/ jx yjqCN 2

 q1

6 C kukW 1;q ./ ;

for some constant C > 0 depending on N , q and . To simplify the exposition, we prove the counterpart of this statement for the upper half-space N 1 RN  RW t > 0º; C D ¹.x; t / 2 R N 1 and we identify the boundary @RN . The proof relies on the 1-dimensional C with R Hardy inequality, which is a consequence of the fundamental theorem of calculus:

15.1. Existence of the trace

239

Lemma 15.4. If q > 1, then, for every smooth bounded function f W Œ0; C1/ ! R such that f .0/ D 0, we have ˆ 1 ˆ 1 jf .r/jq dr 6 C jf 0 jq ; rq 0 0 for some constant C > 0 depending on q. Proof of Lemma 15.4. For every r > 0, we have d  jf .r/jq  jf .r/jq 2 f .r/f 0 .r/ D q dr r q 1 rq 1

.q

1/

jf .r/jq : rq

Since f .0/ D 0 and f is smooth and bounded, it follows from the fundamental theorem of calculus that ˆ 1 ˆ 1 jf .r/jq jf .r/jq 2 f .r/f 0 .r/ .q 1/ dr D q dr: rq rq 1 0 0 By the Hölder inequality, ˆ

1 0

jf .r/jq

2

f .r/f 0 .r/

rq

1

dr 6

ˆ

1 0

jf .r/jq dr rq

 qq 1  ˆ

1 0

0 q

jf j

 q1

:

From the assumptions on f , the first integral in the right-hand side is finite. Combining this estimate with the previous identity, we have the conclusion.  Proof of Proposition 15.3 using  D RN C . We estimate the double integral for funcN tions u 2 Cc1 .RN C /. For every x; y 2 R xCy jx yj and .y; 0/ with the value at . 2 ; 2 /:

u.y; 0/jq ˇ ˇ 6 2q 1 ˇu.x; 0/

ju.x; 0/

1

, we compare the values of u at .x; 0/

 x C y jx yj ˇq ˇ  x C y jx yj  ˇ ˇ u ; ; ˇ C ˇu 2 2 2 2

ˇq  ˇ u.y; 0/ˇ :

(15.2)

We focus on the integral estimate of the second term in the right-hand side; the first one satisfies a similar estimate interchanging the roles of x and y. For this purpose, write xCy D y C x 2 y . Making an affine change of variable z D x y with respect 2 to the variable x, we have ˇq ˇ  x C y jx yj  ˇ ˇ ˆ u.y; 0/ˇ ; ˇu 2 2 dx jx yjqCN 2 RN 1 ˇq ˇ  z jzj  ˇ ˇ ˆ u.y; 0/ˇ ˇu y C ; 2 2 dz: D jzjqCN 2 RN 1

240

15. Traces of Sobolev functions

Thus, by the integration formula in polar coordinates, ˇq ˇ  x C y jx yj  ˇ ˇ ˆ ; u.y; 0/ˇ ˇu 2 2 dx jx yjqCN 2 RN 1 ˇq ˇ  r r ˇ ˇ  ˆ 1ˆ u.y; 0/ˇ ˇu y C ; 2 2 D d ./ dr; rq 0 SN 2

where SN 2 denotes the .N 2/-dimensional unit sphere in RN 1 . Since q > 1, we can estimate the right-hand side using Hardy’s inequality (Lemma 15.4). Indeed, the function f W Œ0; C1/ ! R defined by  r r f .r/ D u y C ; u.y; 0/ 2 2 is smooth, bounded, and satisfies f .0/ D 0. Since ˇ  r r ˇˇ ˇ jf 0 .r/j 6 ˇru y C ; ˇ; 2 2

for every  2 SN

2

, Fubini’s theorem and Hardy’s inequality yield ˇ  x C y jx yj  ˇq ˇ ˇ ˆ ; u.y; 0/ˇ ˇu 2 2 dx jx yjqCN 2 RN 1  ˆ 1ˆ ˇ  r r ˇˇq ˇ 6 C1 ˇru y C ; ˇ d ./ dr: 2 2 0 SN 2

We now integrate both sides with respect to y. Applying Fubini’s theorem and using the affine change of variables z D y C 2r  with respect to y, we obtain ˇq ˇ  x C y jx yj  ˇ ˇ  ˆ ˆ ˆ ; u.y; 0/ˇ ˇu 2 2 dx dy 6 C jrujq : 2 qCN 2 N N 1 N 1 jx yj RC R R

Estimating in a similar way the first term in the right-hand side of (15.2), we deduce Gagliardo’s inequality for smooth functions: ˆ  q1 ˆ ju.x; 0/ u.y; 0/jq dx dy 6 C3 krukLq .RN / : C jx yjqCN 2 RN 1 RN 1

The case of an arbitrary function u 2 W 1;q .RN C / follows from Fatou’s lemma and 1;q the fact that Cc1 .RN .R N C / is dense in W C /.



The previous result gives a stronger estimate for the trace compared to Proposition 15.2 (cf. Proposition 15.5), and also fully characterizes all possible traces arising from W 1;q functions for q > 1 (see Proposition 15.12).

15.2. Fractional Sobolev embedding

241

15.2 Fractional Sobolev embedding The fractional Sobolev space W s;q .RN / can be defined for every order 0 < s < 1 as the set of functions u 2 Lq .RN / having finite Gagliardo seminorm ŒuW s;q .RN / given by  ˆ ˆ  ju.x/ u.y/j q dx dy q ŒuW s;q .RN / D : (15.3) jx yjs jx yjN RN RN

We establish the counterpart of the Sobolev embedding for W s;q .RN / into the Lebesgue space Lp .RN / for the critical exponent p given by the identity 1 1 D p q

s : N

(15.4)

We learned the following beautiful elementary proof in a personal communication with H. Brezis in 2002: N s

Proposition 15.5. If 1 6 q < u 2 W s;q .RN /, we have kuk

Nq sq

LN

Nq sq

, then W s;q .RN /  L N

.RN /

.RN / and, for every

6 C ŒuW s;q .RN / ;

for some constant C > 0 depending on N and q. Proof. Let u 2 W s;q .RN /. For every x; y 2 RN , by the triangle inequality we have ju.x/j 6 ju.x/

u.y/j C ju.y/j:

Integrating with respect to y over a ball B.xI r/, we get ju.x/j 6

B.xIr /

ju.x/

u.y/j dy C

B.xIr /

ju.y/j dy:

Using the Hölder inequality, we then estimate the first integral in the right-hand side as follows:   q1 ju.x/ u.y/j dy 6 ju.x/ u.y/jq dy B.xIr /

B.xIr /

 6 r sqCN

B.xIr /

ju.x/ u.y/jq dy jx yjsqCN

Again by the Hölder inequality, for every 1 6 p < C1 we have B.xIr /

ju.y/j dy 6



p

B.xIr /

ju.y/j dy

 p1

:

 q1

:

242

15. Traces of Sobolev functions

We deduce that  q1  ˆ ju.x/ u.y/jq 1 s dy Cr ju.x/j 6 r !N RN jx yjsqCN



N p

1 !N

ˆ

p

RN

ju.y/j dy

 p1

:

Assume that u is bounded, hence u 2 Lp .RN / for every q < p < C1. Since u 2 W s;q .RN /, the right-hand side is then finite for almost every x 2 RN . Minimizing the expression above with respect to r > 0, we get ju.x/j 6 C1

ˆ

q

RN

ju.x/ u.y/j dy jx yjsqCN



1 pq s C1 p N

ˆ

RN

ju.y/jp dy



s pN s C1 p N

:

Choose p given by identity (15.4); with this choice, the power of the first integral becomes p1 . Raising both sides to the exponent p, and integrating with respect to x, we get ˆ ˆ sq  ˆ ˆ N ju.x/ u.y/jq p p p ju.x/j dx 6 C1 ju.y/j dy dy dx : sqCN yj RN RN RN jx RN

This gives the estimate we sought for bounded functions. The case of an arbitrary function u 2 W s;q .RN / can be obtained using the truncated functions T .u/, and then letting the parameter  ! 1.  Other proofs of the fractional Sobolev inequality can be found in Theorem 6.5 in [115] and Theorem 14.29 in [202]. Given a smooth bounded open set , we define the fractional Sobolev space W s;q .@/ in analogy with the case RN , where N is to be replaced by the dimension N 1 of the boundary @. The Gagliardo seminorm becomes  ˆ ˆ  ju.x/ u.y/j q d .x/ d .y/ ŒuqW s;q .@/ D : jx yjs jx yjN 1 @ @

Using this notation, one identifies the quantity appearing in Proposition 15.3 as the 1 Gagliardo seminorm of W 1 q ;q .@/. Thus, for q > 1, the trace operator is contin1 uous from the Sobolev space W 1;q ./ into the Gagliardo trace space W 1 q ;q .@/, and for every u 2 W 1;q ./ we have kTr ukLq .@/ C ŒTr u

W

1

1 q ;q .@/

6 C kukW 1;q ./ :

1

On the other hand, since Tr u 2 W 1 q ;q .@/, by the counterpart of the fractional 1 Sobolev inequality in W 1 q ;q .@/ for exponents q < N , we have kTr uk

L

.N 1/q N q

.@/

6 C 0 .kTr ukLq .@/ C ŒTr u

W

1

1 q ;q .@/

/:

Combining the two inequalities, we recover the estimate from Proposition 15.2.

15.2. Fractional Sobolev embedding

243

We conclude this section with a discussion of independent interest intended to motivate the fractional Sobolev spaces in connection with W 1;q .RN /. The Gagliardo seminorm gives an intermediate object between the Lq and W 1;q norms: Proposition 15.6. For every 0 < s < 1 and every u 2 W 1;q .RN /, we have ŒuW s;q .RN / 6

C Œs.1

s/

1 q

1 s s kukL q .RN / krukLq .RN / ;

for some constant C > 0 depending on N and q. Proof. By approximation, it suffices to establish the inequality for u 2 Cc1 .RN / (Proposition 8.7). For every r > 0, we write ˆ ˆ ˆ ˆ ˆ ˆ ju.x/ u.y/jq q dx dy D ŒuW s;q .RN / D C : yjsqCN RN RN RN RN jx RN RN jx yj>r

jx yj6r

From the estimate ju.x/

u.y/jq 6 2q

1

.ju.x/jq C ju.y/jq /

and Fubini’s theorem, we get ˆ ˆ C1 ju.x/ u.y/jq dx dy 6 r sqCN yj s RN RN jx

sq

ˆ

RN

jujq :

jx yj>r

Using the fundamental theorem of calculus and the Hölder inequality, we now estimate ˆ 1 q ju.x/ u.y/j 6 jru.y C t .x y//jq dt jx yjq ; 0

which implies that ˆ ˆ

ju.x/ u.y/jq C2 .1 r dx dy 6 sqCN yj 1 s RN jx

s/q

RN jx yj6r

ˆ

RN

jrujq :

Combining the two inequalities, we get ŒuqW s;q .RN / 6

C1 r s

sq

q kukL q .RN / C

C2 .1 r 1 s

s/q

q krukL q .RN / :

Minimizing the right-hand side with respect to r, we deduce that ŒuqW s;q .RN /

6 C3

 kukq q

 L .RN / 1 s s

 krukq q

L .RN /

1

s

s

:

Since t t D et log t 6 C4 for every 0 < t < 1, the interpolation estimate follows.



244

15. Traces of Sobolev functions

One is tempted to use the Gagliardo seminorm with s D 1 to retrieve the classical Sobolev space W 1;q .RN / without relying on weak gradients. Such an approach is hopeless, since if u 2 Lq .RN / satisfies ˆ ˆ  ju.x/ u.y/j q dx dy < C1; (15.5) jx yj jx yjN RN RN

then u D 0 almost everywhere in RN (Exercise 15.3 below). This apparent defect of 1 the Gagliardo seminorm can be fixed by introducing the correction factor .1 s/ q as s ! 1. Exploiting this idea, Bourgain, Brezis, and Mironescu [42] (see also [52] and [285]) obtained the following elegant characterization of Sobolev functions of order one: Proposition 15.7. Let q > 1, and let u 2 Lq .RN /. Then, u 2 W 1;q .RN / if and only if 1 lim inf .1 s/ q ŒuW s;q .RN / < C1: s%1

In this case, we have lim .1

s%1

1

s/ q ŒuW s;q .RN / D KkrukLq .RN / ;

for some constant K > 0 depending on N and q. Proof. The direct implication follows from the interpolation inequality for the Gagliardo seminorm. To prove the reverse implication, note that every smooth function vW RN ! R satisfies ˆ  jv.x/ v.y/j q dy lim .1 s/ D K q jrv.x/jq (15.6) s s%1 jx yj jx yjN B.xIr / for every x 2 RN and r > 0, for some constant K > 0 depending on N and q. Thus, by Fatou’s lemma and the monotonicity of the integral, we have KkrvkLq .B.xIr // 6 lim inf .1 s%1

1

s/ q ŒvW s;q .B.xIr // 6 lim inf .1 s%1

1

s/ q ŒvW s;q .RN / :

Since this holds for every r > 0, by the monotone convergence theorem we get KkrvkLq .RN / 6 lim inf .1 s%1

1

s/ q ŒvW s;q .RN / :

(15.7)

To apply this information, we now use the fact that the convolution decreases the Gagliardo seminorm. More precisely, given a sequence of mollifiers .n /n2N , by Jensen’s inequality we have Œn  uW s;q .RN / 6 ŒuW s;q .RN / :

(15.8)

15.2. Fractional Sobolev embedding

245

We deduce from (15.7) and (15.8) that 1

Kkr.n  u/kLq .RN / 6 lim inf .1

s/ q ŒuW s;q .RN / :

s%1

Since q > 1, from the closure lemma (cf. Proposition 4.10) we have u 2 W 1;q .RN / and 1 KkrukLq .RN / 6 lim inf .1 s/ q ŒuW s;q .RN / : s%1

To prove the reverse inequality with the limsup, we use the density of Cc1 .RN / in W 1;q .RN / (Proposition 8.7). Since the limit in (15.6) is uniform on compact subsets of RN , for every ' 2 Cc1 .RN / we have lim .1

s%1

1

s/ q Œ'W s;q .RN / D Kkr'kLq .RN / :

Moreover, for every 12 6 s < 1, by the triangle inequality and the interpolation inequality satisfied by the Gagliardo seminorm we have ŒuW s;q .RN / 6 Œ'W s;q .RN / C Œu 6 Œ'W s;q .RN / C Multiplying both sides by .1 lim sup .1 s%1

'W s;q .RN / C1

.1

1

s/ q

ku

'kW 1;q .RN / :

1

s/ q and letting s ! 1, we get

1

s/ q ŒuW s;q .RN / 6 Kkr'kLq .RN / C C1 ku

'kW 1;q .RN / :

Applying this estimate to a sequence .'n /n2N converging strongly to u in W 1;q .RN /, and letting n ! 1, we obtain lim sup .1

1

s/ q ŒuW s;q .RN / 6 KkrukLq .RN / :

s%1

The conclusion follows.



Exercise 15.3 (characterization of constant functions [52]). Prove that if uW RN ! R is a locally summable function satisfying (15.5) for some exponent 1 6 q < C1, then u is almost everywhere constant in RN . The use of Jensen’s inequality in the previous proof was suggested by E. Stein; another proof of Proposition 15.7, based on rearrangements, can be found in [331]. The case of exponent q D 1 involves functions of bounded variation, and has been independently solved by Ambrosio and Dávila [99]. The limit of the Gagliardo

246

15. Traces of Sobolev functions

seminorm as s ! 0 has been investigated by Maz0 ya and Shaposhnikova, see Theorem 3 in [238]. These results have been explained by Milman [246] in the context of real interpolation of Banach spaces. In view of Proposition 15.7, the elementary proof of the fractional Sobolev embedding above is not satisfactory, since the constant obtained does not depend on s, and so does not allow to recover the classical Sobolev embedding (Corollary 4.12) as s ! 1. This is indeed possible, but requires a more difficult argument which has been implemented by Maz0 ya and Shaposhnikova [238]. One might wonder whether there is a fractional differential object of order 0 < s < 1 that would lead to the classical gradient as s ! 1. For this purpose, we restrict our attention to functions ' 2 Cc1 .RN /. We thus consider the fractional gradient of order s as the vector field r s 'W RN ! RN defined for x 2 RN by ˆ '.x/ '.y/ x y dy s r '.x/ D ; s jx yj jx yj jx yjN RN see [243] and [27]. We then have that lim .1

s%1

s/r s '.x/ D

1 r'.x/ N

and kr s 'kL1 .RN / 6 Œ'W s;1 .RN / :

Although r s ' need not have compact support, it satisfies the pointwise estimate (cf. Exercise 15.5 .a/) C jr s '.x/j 6 ; (15.9) .1 C jxj/N Cs

from which we deduce that r s ' 2 L1 .RN / \ L1 .RN /. We may recover the function ' from its fractional gradient r s ' using an integral representation formula, see Theorem 1.12 in [309] (cf. Proposition 4.14 below):

Proposition 15.8. Let N > 2 and 0 < s < 1. For every ' 2 Cc1 .RN / and every x 2 RN , we have ˆ x y dy; '.x/ D A r s '.y/
N jx yjN sC1 R for some constant A > 0 depending on N and s. We first establish the connection between the fractional gradient and the classical gradient through M. Riesz’s kernels I˛ .z/ D 1=jzjN ˛ . This is reminiscent of an approach investigated by Horváth [169] based on the classical representation formula involving the gradient.

15.2. Fractional Sobolev embedding

247

Lemma 15.9. Let 0 < s < 1. For every ' 2 Cc1 .RN / and every x 2 RN , we have ˆ r'.y/ 1 dy: r s '.x/ D N 1 C s RN jx yjN Cs 1 Proof of Lemma 15.9. Given e 2 RN , consider the vector field ˆW RN n ¹xº ! RN defined by ˆ.y/ D

'.x/ '.y/ e: jx yjN Cs 1

We apply the divergence theorem to ˆ in the annulus B.xI R/ n B.xI r/ for any radii 0 < r < R. Since the function div ˆ.y/ D .N C s

1/

'.x/ '.y/ x jx yjN Cs jx

y
e yj

jx

r'.y/ yjN Cs

1


e

is integrable in RN , and since lim r N

r !0

1

lim RN

kˆkL1 [email protected] // D 0 and

1

R!C1

kˆkL1 [email protected]// D 0;

we get ˆ

RN

div ˆ D 0:

This gives the identity we want, in the direction e.



We also need the following identity related to the semigroup property of the Riesz kernel, see p. 118 in [317]: Lemma 15.10. For every ˛; ˇ > 0 such that ˛ Cˇ < N , and for every y 2 RN n¹0º, we have ˆ y z 1 0 dz D A ; N ˛C1 jy zjN ˇ jyjN .˛Cˇ /C1 RN jzj for some constant A0 > 0 depending on ˛, ˇ, and N .

Proof of Lemma 15.10. For every y ¤ 0, denote the integral in the left-hand side by G.y/. We deduce the formula for G.y/ using suitable changes of variable in the integral. For example, making the change of variable z D jyj and extracting the term jyj from the integrand, we get that G.y/ D

1 jyjN

.˛Cˇ /

G.y=jyj/:

248

15. Traces of Sobolev functions

We may now assume that jyj D 1; we show in this case that G.y/ D A0 y for some constant A0 > 0. For this purpose, let T W RN ! RN be a linear orthogonal transformation such that T .eN / D y, where eN denotes the last element of the canonical Euclidean basis. Making now the change of variable z D T ./ in the integral, we get G.y/ D T .G.eN //; where G.eN / D For every j 2 ¹1; : : : ; N

ˆ

 RN

jjN ˛C1

1º, the function

R 3 j 7 !

jeN

j jjN ˛C1 jeN

1 jN

1 jN

ˇ

d:

ˇ

is odd, whence by Fubini’s theorem the j th component of G.eN / vanishes. We then deduce that ˆ  N 1 G.eN / D d eN : N ˛C1 je jN ˇ N RN jj Using the notation  D .0 ; N / 2 RN the variable N as ˆ N 1 dN N ˛C1 jeN jN ˇ R jj D

ˆ

1 0

N jjN

˛C1



j.0 ; 1

1

 R, we write the integral with respect to

1 N /jN

ˇ

1 j.0 ; 1 C N /jN

ˇ



dN :

Since the integrand is positive, we deduce that G.eN / D A0 eN with A0 > 0, whence G.y/ D T .G.eN // D T .A0 eN / D A0 y: This gives the conclusion.



Proof of Proposition 15.8. Denote the integral in the right-hand side by F .x/. By the relation between r s ' and the Riesz kernel I1 s (Lemma 15.9), we have  ˆ ˆ 1 x y 1 F .x/ D r'.z/
dz dy: N C s 1 RN jx yjN sC1 jy zjN 1Cs RN

15.2. Fractional Sobolev embedding

249

Since .N s/ C .N 1 C s/ > N , the integrand belongs to L1 .RN  RN /. Making the change of variables  D x y and  D x z, and interchanging the order of integration, we get ˆ  ˆ 1  1 r'.x /
d d: F .x/ D N sC1 j N C s 1 RN jN 1Cs RN jj By the semigroup property of the Riesz kernel (Lemma 15.10), we deduce that ˆ A0  F .x/ D r'.x /
N d: N C s 1 RN jj According to the representation formula involving the gradient (Proposition 4.14), the integral in the right-hand side equals N '.x/.  Using the representation formula given by Proposition 15.8, we explain how to establish the Sobolev inequality for the fractional gradient (cf. Proposition 15.5): k'k

Nq sq

LN

.RN /

6 C kr s 'kLq .RN /

for every 1 6 q < Ns and every ' 2 Cc1 .RN / in dimension N > 2. The case q > 1 is covered by the classical estimate involving the Riesz kernel I˛ , which Nq acts by convolution from Lq .RN / into L N ˛q .RN / for every 0 < ˛ < N , see Theorem V.1 in [317]: kI˛  f k

Nq ˛q

LN

.RN /

6 C 0 kf kLq .RN / :

(15.10)

Inequality (15.10) fails for q D 1, but has a true counterpart for curl-free vector fields fE 2 L1 .RN I RN / in dimension N > 2 that can be applied in particular to fE D r s '. Such an estimate involves the vector-valued Riesz kernel IE˛ .x/ D x=jxjN ˛C1 . More precisely, we have kIE˛  fEk

N

LN

˛

.RN /

6 C 00 kfEkL1 .RN / ;

(15.11)

provided that @fEi @fEj D @xj @xi in the sense of distributions in RN , for every i; j 2 ¹1; : : : ; N º. This critical estimate has been established by Schikorra, Spector, and Van Schaftingen [306], and is closely related to the study of div-curl systems in the seminal work of Bourgain and Brezis [40] and [41]. The proof of (15.11) relies on a duality argument and on a strategy introduced by Van Schaftingen [330] based on Fubini’s theorem.

250

15. Traces of Sobolev functions

Estimate (15.11) is false in dimension N D 1, since every vector field is curl-free and its scalar counterpart (15.10) cannot hold, see p. 119 in [317]. The argument that leads to the integral representation formula ˆ x y dy (15.12) '.x/ D A r s '.y/
jx yj2 s R requires some small modifications:

Exercise 15.4 (representation formula in dimension 1). Let 0 < s < 1 and ' 2 Cc1 .R/. .a/ Write ˆ ˆ 1 s y r '.y/ r s '. y/ s r '.y/ 2 s dy D dy: jyj y1 s R 0 .b/ Prove that, for every z ¤ 0, we have ˆ 1  1  1 1 dy D A0 sgn z; y 1 s jy zjs jy C zjs 0 for some constant A0 > 0 depending on s. .c/ Deduce that (15.12) holds for x D 0, and then for every x 2 R.

We have adopted a self-contained proof of the integral representation formula based on elementary properties of the Lebesgue integral and the divergence theorem; Proposition 15.8 can alternatively be proved using the Riesz and the Fourier transforms, see [309]. In this respect, one should observe that, in terms of the Fourier transform, we have that (cf. Lemma 15.9)

b

r s './ D c i

 jjs './; O jj

for some constant c 2 R. Further properties of fractional Sobolev spaces and fractional gradients, and their connection with the interpolation theory between the spaces Lq and W 1;q can be found in Chapter 7 in [9], [115], [246], [309], and [321]. Our presentation of the fractional gradient benefited from discussions with D. Spector, and has been inspired from his works with Mengesha [243] and Shieh [309]. A companion notion is that of fractional divergence of order s, see [117]: Exercise 15.5. Given a vector field ˆ 2 Cc1 .RN I RN /, define divs ˆW RN ! R for every x 2 RN by ˆ dy ˆ.x/ ˆ.y/ x y
: divs ˆ.x/ D s jx yj jx yj jx yjN RN .a/ Prove that, for every x 2 RN , we have jdivs ˆ.x/j 6

C : .1 C jxj/N Cs

15.3. Range of the trace operator

251

.b/ Prove the following analogue of the divergence theorem in RN : ˆ divs ˆ D 0: RN

.c/ Prove that, for every ' 2 Cc1 .RN /, we have ˆ ˆ s r '
ˆD ' divs ˆ: RN

RN

15.3 Range of the trace operator We now identify the elements w in Lq .@/ which can be extended as functions in W 1;q ./ in the sense of traces: Tr u D w for some u 2 W 1;q ./. The full characterization of the vector subspace Tr .W 1;q .// has been obtained by Gagliardo [142]. We begin with the surjectivity of the trace operator from W 1;1 ./ into L1 .@/, following an elegant argument by Mironescu [248]: Proposition 15.11. Let  be a smooth bounded open set. For every w 2 L1 .@/, there exists u 2 W 1;1 ./ such that Tr u D w on @ and kukW 1;1 ./ 6 C kwkL1 .@/; for some constant C > 0. 1 N Proof using  D RN C . We first consider the case of boundary datum ' 2 Cc .R

We take a smooth extension

uW RN C

1

/.

! R of the form

u.x; t / D '.x/.t /; where W Œ0; C1/ ! R is a nonnegative smooth function with compact support such that .0/ D 1. Claim. For each ' 2 Cc1 .RN

1

/, the function  can be chosen such that

kukW 1;1 .RN / 6 2k'kL1 .RN C

1/

:

252

15. Traces of Sobolev functions

Proof of the claim. By Fubini’s theorem, we have ˆ 1   k'kL1 .RN kukL1 .RN / D C

0

1/

:

To estimate the quantity krukL1 .RN / , we assume in addition that  is non-increasing. C We then have jru.x; t /j 6  0 .t /j'.x/j C .t /jr'.x/j:

By Fubini’s theorem and the fundamental theorem of calculus, we get ˆ 1  krukL1 .RN / 6 k'kL1 .RN 1 / C  kr'kL1 .RN 1 / : C

0

We obtain the estimate we want by choosing  such that ˆ 1   kr'kL1 .RN 1 / 6 k'kL1 .RN 0

1/

4

:

We now prove the proposition for an arbitrary function w 2 L1 .RN 1 / using the claim. Since Cc1 .RN 1 / is dense in L1 .RN 1 /, we can write w as a strongly convergent series in L1 .RN 1 / in terms of functions 'k 2 Cc1 .RN 1 / (Exercise 15.6 below): wD

1 X

'k

and

kD0

1 X

kD0

k'k kL1 .RN

1/

6 C1 kwkL1.RN

1/

:

(15.13)

By the claim, every function 'k has a smooth extension uk W RN C ! R with compact support such that kuk kW 1;1 .RN / 6 2k'k kL1 .RN 1 / : C

The function u D

1 P

kD0

uk belongs to W 1;1 .RN C / and satisfies the required properties. 

Exercise 15.6. Prove that if X is a normed vector space and if .xn /n2N is a sequence in X converging to x ¤ 0, then, for every  > 0, there exists a subsequence .xnk /k2N such that 1 X x D xn0 C .xnk xnk 1 / kD1

and

kxn0 k C

1 X

kD1

kxnk

xnk

1

k 6 .1 C /kxk:

15.3. Range of the trace operator

253

The previous proposition holds for any fixed constant C > 1, independently of , which can be seen by a straightforward adaptation of the proof of the claim, see Remark 2.17 in [147]. Thus, for every w 2 L1 .@/, there exists u 2 W 1;1 ./, depending on C , such that Tr u D w and krukL1 ./ 6 C kwkL1.@/ : One might wonder whether such a conclusion is valid with C D 1. This is indeed the case if  is a ball and w is continuous, see [318], but fails if  is the 2-dimensional disk and w is the characteristic function of some Cantor set on the circle, see [314]. We now consider the case q > 1: Proposition 15.12. Let 1 < q < C1, and let  be a smooth bounded open set. Then, for every w 2 W s;q .@/, there exists u 2 W 1;q ./ such that Tr u D w on @ and kukW 1;q ./ 6 C.kwkLq .@/ C ŒwW s;q .@/ /; for some constant C > 0 depending on N , q and .

1 N Proof using  D RN C . We fix a mollifier  2 Cc .R N 1 q N 1 B.0I 1/ in R . Given w 2 L .R /, define

1

/ supported in the unit ball

vW RN C !R by v.x; t / D  t  w.x/ D where  t .z/ D

1 tN

1

ˆ

RN

1

 t .x

y/w.y/ dy;

. zt /. By Jensen’s inequality, for every t > 0 we have kv.
; t /kLq .RN

and v.
; t / converges to w in Lq .RN Claim. If w 2 W s;q .RN

1

1/

1

6 kwkLq .RN

1/

(15.14)

;

/ as t ! 0.

/, then rv 2 Lq .RN C / and

krvkLq .RN / 6 C 0 ŒwW s;q .RN C

1/

:

Proof of the claim. Since rz  t has compact support in RN ˆ rz  t D 0; RN

1

, we have

1

and since the integral of  t over RN 1 is constant with respect to t , we also have ˆ ˆ d t d  t D 0: D dt RN 1 RN 1 dt

254

15. Traces of Sobolev functions

Denoting by D t the full derivative of the function .z; t / 7!  t .z/, we then have ˆ rv.x; t / D D t .x y/w.y/ dy RN

ˆ

D

1

RN

1

D t .x

y/Œw.x/

w.y/ dy:

Since  is supported in the unit ball, the integrand above is supported in B.xI t /. Thus, by the Hölder inequality we have jrv.x; t /j 6 kD t kL

q0

.RN

1/

ˆ

B.xIt/

jw.x/

q

w.y/j dy

 q1

;

where q 0 is the conjugate exponent of q. Raising to the power q and integrating with respect to x, we get  ˆ ˆ ˆ q q q jw.x/ w.y/j dy dx: jrv.x; t /j dx 6 kD t kLq0 .RN 1 / RN

RN

1

1

B.xIt/

Since  t is supported in the ball B.0I t /, we have  z i C C1 h  z  2 jD t j 6 N  C jrj 6 N B.0It/ ; t t t t whence

q kD t kL q 0 .RN

We deduce that ˆ jrv.x; t /jq dx 6 RN

1

C3 t qCN

1

ˆ

1/

RN

6

1

C3 : qCN 1 t

ˆ

B.xIt/

jw.x/

q



w.y/j dy dx:

Integrating with respect to t and applying Fubini’s theorem, we then get ˆ ˆ 1   ˆ dt q q jw.x/ w.y/j krvkLq .RN / 6 C3 dy dx qCN 1 C RN 1 RN 1 jx yj t ˆ ˆ jw.x/ w.y/jq dx dy: 4 D C4 yjqCN 2 RN 1 RN 1 jx Since the function v need not belong to Lq .RN C /, we take a fixed smooth function W Œ0; C1/ ! R with compact support such that .0/ D 1, and define uW RN C !R by u.x; t / D v.x; t /.t /:

15.3. Range of the trace operator

255

By estimate (15.14) and the claim, kukLq .RN / C krukLq .RN / 6 C5 .kwkLq .RN C

C

1/

C ŒwW s;q .RN

1/

/:

1;q Since u is smooth in RN .R N C , we deduce that u 2 W C /. To conclude, it remains to prove that Tr u D w. For this purpose, we take a sequence of positive numbers .i /i 2N converging to zero. The sequence of smooth functions ui W RN C ! R

defined by

ui .x; t / D u.x; t C i /

converges to u in W 1;q .RN C / as i ! 1. On the other hand, since each function ui is a convolution of u, the sequence .ui jRN 1 /i 2N converges to w in Lq .RN 1 /. By the continuity of the trace operator, we deduce that Tr u D w.  The proof of Proposition 15.12 is based on a linear construction with respect to the boundary datum w. In contrast, there exists no continuous linear operator from L1 .@/ into W 1;1 ./ that sends a given boundary value w 2 L1 .@/ to a function in W 1;1 ./ whose trace is w, see [275]. In other words, the trace operator TrW W 1;1 ./ ! L1 .@/ has no continuous right inverse. In the proof of Proposition 15.11, the decomposition of w 2 L1 .RN 1 / in terms of a series of functions in Cc1 .RN 1 / satisfying (15.13) depends nonlinearly on w. We have restricted our presentation to smooth bounded open sets. We refer the reader to the book of Maz0 ya and Poborchi [236] for counterexamples to the trace problem in Sobolev spaces on nonsmooth domains.

Chapter 16

Trace inequality

“Il metodo di dimostrazione non fa uso della teoria del potenziale e della nozione di ‘soluzione fondamentale’. Quindi, anche quando si ritrovano risultati già noti, la dimostrazione mi sembra di tipo nuovo.”1 Sergio Campanato

We present the formalism of trace inequalities and their connection to capacitary estimates. As an application, we establish Hölder estimates of solutions of the Poisson equation, and the equivalence between the Sobolev capacity capW 1;1 and the Hausdorff capacity HıN 1 .

16.1 Capacitary, geometric and pointwise interpretations A trace inequality implies a strong relation with the Sobolev capacity: Proposition 16.1. Let 1 6 p; q < C1. If  2 Mloc .RN / is a nonnegative measure such that, for every ' 2 Cc1 .RN /, we have k'kLp .RN I/ 6 C k'kW 1;q .RN / ; then  6 C p .capW 1;q /p=q : Proof. Assuming that the trace inequality holds, then, for every compact set K  RN and every nonnegative function ' 2 Cc1 .RN / such that ' > 1 in K, we have ˆ .K/ 6 ' p d 6 C p k'kpW 1;q .RN / : RN

Minimizing the right-hand side with respect to ', we obtain the capacitary inequality  6 C p .capW 1;q /p=q for K. By the inner regularity of  (Proposition 2.5), this inequality then holds for open subsets of RN , and, by outer regularity of the capacity (Definition A.7), we deduce the same inequality for every Borel subset of RN .  1 “The proof does not require any potential theory nor the notion of fundamental solution.

if the results obtained are already known, the proof seems to be new.”

Thus, even

258

16. Trace inequality

The capacitary upper bound in the previous proposition is equivalent to the trace inequality for p > q. This is the celebrated formulation of the trace inequality due to Maz0 ya, see Theorem 1 in [230], Corollary 3 in [231], and [233]: Proposition 16.2. Let 1 6 q 6 p < C1. If  2 Mloc .RN / is a nonnegative measure such that p  6 C.capW 1;q / q ; then, for every ' 2 Cc1 .RN /, we have k'kLp .RN I/ 6 C 0 k'kW 1;q .RN / ; for some constant C 0 > 0 depending on C , p, and q. We postpone for a moment the proof of this proposition, which is based on Maz0 ya’s strong capacitary estimate (Lemma 16.4). When the strict inequality p > q holds, the capacitary assumption above can be restated in a more geometric flavor, in terms of Hausdorff contents. This is a beautiful contribution by D. Adams [1] and [2] based on the following setwise estimate (Proposition 10.4): s H1 6 C.capW 1;q / N

for every s > N

s q

(16.1)

;

q, see also Section 7.2 in [7].

Proposition 16.3. Let 1 6 q < N , and let  2 M.RN / be a nonnegative measure. Then, for every p > q, the following assertions are equivalent: p

(i) there exists C > 0 such that  6 C.capW 1;q / q ; .N q/ p q

(ii) there exists C 0 > 0 such that  6 C 0 H1

:

Proof of Proposition 16.3. The reverse implication follows from estimate (16.1) above. Conversely, for every ball B.xI r/  RN such that r 6 1, by a scaling and translation argument we have (Lemma 10.5) p

p

.B.xI r// 6 C.capW 1;q .BŒxI r// q 6 .capW 1;q .BŒ0I 1// q r .N

q/ p q

:

On the other hand, if r > 1, then .B.xI r// 6 .RN / 6 .RN /r .N

q/ p q

:

Therefore, the measure  satisfies a density estimate of order .N q/ pq for every ball B.xI r/, and this is equivalent to the upper bound in terms of the Hausdorff content (Proposition B.3). 

16.1. Capacitary, geometric and pointwise interpretations

259

The equivalence between assertions .i/ and .ii/ is not true in the critical case p D q, for every q > 1. Indeed, taking a compact set S  RN such that 0 < HN

q

.S / < C1;

we have (Proposition 10.3) capW 1;q .S / D 0:

Hence, .S / D 0 for every nonnegative measure satisfying  6 C capW 1;q . However, by Frostman’s lemma (Proposition B.4), there exists a positive measure  supported in S such that N q  6 H1 :

In Chapter 17, we investigate this lack of equivalence in the case p D q D 2 from the point of view of the Poisson equation. Using the Maz0 ya–Adams trace inequalities, we can revisit the trace theory of Sobolev functions (Chapter 15). Indeed, a smooth bounded open set  satisfies HN

1

.B.xI r/ \ @/ 6 C !N

1r

N 1

;

for every ball B.xI r/  RN and some constant C > 1, whence (Proposition B.3) HN

1

N 1 b@ 6 C H1 :

Combining Propositions 16.2 and 16.3, we deduce that, for every 1 6 q < N and every ' 2 Cc1 .RN /, we have k'k

L

.N 1/q N q

.@/

6 C 0 k'kW 1;q .RN / :

This inequality is obtained in Section 15.1 using the divergence theorem and the Sobolev inequality (Proposition 15.2). We now prove Maz0 ya’s strong capacitary estimate, see Theorem 3 in [231]: Lemma 16.4. Let 1 6 q < C1. For every ' 2 Cc1 .RN /, we have ˆ 1 t q 1 capW 1;q .¹j'j > t º/ dt 6 C k'kqW 1;q .RN / ; 0

where the constant C > 0 depends on q. Proof of Lemma 16.4. We begin with the following variant of the weak capacitary estimate (Lemma A.11): Claim. For every ' 2 Cc1 .RN / and every t > 0, we have   ˆ 1 q 0 t jr'j : capW 1;q .¹j'j > t º/ 6 C j¹j'j > 2 ºj C q t t ¹ 6j'j6tº 2

260

16. Trace inequality

Proof of the claim. Let H W R ! R be a nonnegative smooth function such that .a/ H.s/ D 0 for every s 2 R such that jsj 6 12 , .b/ H.s/ D 2 for every s 2 R such that jsj > 1.

' t

Thus, for every ' 2 Cc1 .RN / and every t > 0, we have H ' t

H




> 1 in ¹j'j > t º:




2 Cc1 .RN / and

By the definition of capacity of the compact set ¹j'j > t º, we get capW 1;q .¹j'j > t º/ 6 kH We have ˆ

RN

and ˆ

RN

' t

jH ' t

jrH

' t




kqW 1;q .RN / :

ˇ ˇ
q j 6 C1 ˇ¹j'j > 2t ºˇ


q C2 j 6 q t

ˆ

t ¹ 6j'j6tº 2

jr'jq :

Combining these estimates, the claim follows.

4

By the previous claim, we have ˆ 1 t q 1 capW 1;q .¹j'j > t º/ dt 0

6C

0

ˆ

1 0

t

q 1

ˇ ˇ ˇ¹j'j > t ºˇ dt C C 0 2

ˆ

1 0

ˆ

q

t ¹ 6j'j6tº 2

jr'j



dt : t

By Cavalieri’s principle (1.5), we get ˆ 1 ˆ ˇ ˇ 2q t q 1 ˇ¹j'j > 2t ºˇ dt D j'jq : N q 0 R

By Fubini’s theorem, we also have ˆ

1 0

ˆ

q

t ¹ 6j'j6tº 2

jr'j



dt D t

ˆ

q

RN

jr'.x/j

D .log 2/

ˆ

RN

ˆ

2j'.x/j j'.x/j

dt t



dx

jr'jq :

Inserting these identities in the estimate above, the conclusion follows.



16.1. Capacitary, geometric and pointwise interpretations

261

The strong capacitary estimate bears a striking resemblance with Cavalieri’s principle (1.5) in the Lebesgue Lq setting. The integral in the left-hand side is, by definition, the Choquet integral of j'jq with respect to the Sobolev capacity capW 1;q , up to the multiplicative factor q, see [17], Section 48.1 in [84], and [101]. In this formalism, we may rewrite the estimate as ˆ j'jq d capW 1;q 6 C 0 k'kqW 1;q .RN / : RN

Proof of Proposition 16.2. Let ' 2 Cc1 .RN /. By Cavalieri’s principle (1.5), ˆ ˆ 1 j'jp d D p t p 1 .¹j'j > t º/ dt: RN

0

By the assumption on the measure , for every t > 0 we have p

.¹j'j > t º/ 6 C.capW 1;q .¹j'j > t º// q and, by the weak capacitary estimate (Lemma A.11), we have capW 1;q .¹j'j > t º/ 6

1 k'kqW 1;q .RN / : tq

Combining the two inequalities, we get p

.¹j'j > t º/ 6 C.capW 1;q .¹j'j > t º// q 6

C tp q

1

capW 1;q .¹j'j > t º/

p q k'kW 1;q .RN / capW 1;q .¹j'j > t º/:

Thus, ˆ

RN

p q j'jp d 6 Cpk'kW 1;q .RN /

ˆ

1 0

tq

1

capW 1;q .¹j'j > t º/ dt:

By the strong capacitary estimate (Lemma 16.4), the conclusion follows.



Maz0 ya’s truncation argument that is used in the proof of the strong capacitary estimate is a powerful tool to prove the equivalence between strong and weak trace inequalities, see [155]: Exercise 16.1. Let 1 6 q 6 p < C1, and let  2 Mloc .RN / be a nonnegative measure. Prove that the following inequalities are equivalent: (i) for every ' 2 Cc1 .RN /, k'kLp .RN I/ 6 C kr'kLq .RN / ;

262

16. Trace inequality

(ii) for every ' 2 Cc1 .RN / and every t > 0, 1

t.¹j'j > t º/ p 6 C 0 kr'kLq .RN / : We are now interested in the meaning of the integral ˆ u d RN

when  satisfies the trace inequality and u 2 W 1;q .RN / is not necessarily continuous. The reason is the following. By the trace inequality and by the denseness of Cc1 .RN / in W 1;q .RN /, the linear functional LW Cc1 .RN / ! R defined for ' 2 Cc1 .RN / by ˆ L.'/ D

' d

RN

x W 1;q .RN / ! R. As in the case of traces has a unique continuous linear extension LW of Sobolev functions (Exercise 15.2), measure theory provides a tool to explicitly x evaluate L.u/ as an integration with respect to : the notion of precise representative u. O This is the content of the proposition below: Proposition 16.5. Let 1 6 p; q < C1. If  2 M.RN / is a nonnegative measure such that, for every ' 2 Cc1 .RN /, we have k'kLp .RN I/ 6 C k'kW 1;q .RN / ; then (i) for every u 2 W 1;q .RN /, the precise representative uO belongs to Lp .RN I / and satisfies kuk O Lp .RN I/ 6 C kukW 1;q .RN / I (ii) the linear functional W

1;q

N

.R / 3 u 7 !

ˆ

RN

uO d 2 R

is well defined and continuous. Proof. Let .'n /n2N be a sequence in Cc1 .RN / converging strongly to u in W 1;q .RN / and converging quasi-everywhere to uO with respect to the W 1;q capacity (cf. Propositions 8.4, 8.6 and 8.7). By the trace inequality, the measure  is diffuse (Proposition 16.1), whence .'n /n2N also converges almost everywhere to uO with respect to . Since, for every n 2 N, we have k'n kLp .RN I/ 6 C k'n kW 1;q .RN / ;

16.2. Hölder continuity revisited

263

it follows from Fatou’s lemma that uO 2 Lp .RN I / and kuk O Lp .RN I/ 6 lim inf k'n kLp .RN I/ 6 C lim k'n kW 1;q .RN / D C kukW 1;q .RN / : n!1

n!1

By the Hölder inequality, we then have 1 0

p kuk O Lp .RN I/ 6 C 0 kukW 1;q .RN / ; kuk O L1 .RN I/ 6 kkM. RN /

whence the linear functional above is well defined and continuous.



The interested reader is invited to check the literature for personal accounts on the trace inequality by Maz0 ya [233] and Adams [6]. We have focused on first-order trace inequalities. The higher order counterpart in terms of the Sobolev spaces W k;q .RN / involves nontrivial technical issues, see [4], [7], and [239]. Related questions concerning the evaluation of distributions satisfying some trace inequality have been investigated by Brezis and Browder [55] and [56].

16.2 Hölder continuity revisited As an example of how trace inequalities can be used to deduce estimates in the setting of elliptic PDEs, we revisit the proof of the Hölder continuity of solutions of the Poisson equation u D : In Section 13.3, we follow Carleson’s strategy based on the Newtonian potential. We now consider a different approach based on the next estimate: Proposition 16.6. Let N > 2, and let  2 M.RN / be a nonnegative measure such that, for every ' 2 Cc1 .RN /, we have k'kLp .RN I/ 6 C k'kW 1;2 .RN / ; for some exponent 1 < p < C1. If u 2 W 1;2 ./ is a solution of the Poisson equation with density  in , then, for every ball B.xI r/  , we have 8 N ˆ 0 N 2 .p 1/ ˆ , if p < 1 C 1 C . N 2

264

16. Trace inequality

By the Poincaré–Wirtinger inequality (10.5), we have

B.xIr /

B.xIr /

ju.y/

We then deduce that, for p < 1 C B.xIr /

B.xIr /

C 00

u.z/j dy dz 6 r

N

2 2

krukL2 .B.xIr // :

N , N 2

ju.y/

u.z/j dy dz 6 C 000 r

N

2 2

.p 2/

:

Assuming in addition that p > 2, the double average integral thus converges locally uniformly to zero with order N 2 2 .p 2/. We then get the local Hölder continuity of the precise representative u, O based on the Campanato–Meyers property (cf. Proposition 16.9). The trace inequality k'kLp .RN I/ 6 C k'kW 1;2 .RN / implies that  2 .W 1;2 .RN //0 . By Propositions 16.1, 16.2, and 16.3, the trace in.N 2/ p 2

equality above for p > 2 is equivalent to the density condition  6 C 0 H1 and the latter can be restated as (Proposition B.3) .B.xI r// 6 C 00 r .N

2/ p 2

:

;

(16.2)

We then recover the assumption that is used in the proof of Proposition 13.5 to establish the Hölder continuity of solutions of the Poisson equation. In the proof of Proposition 16.6, we follow the strategy implemented by Rakotoson and Ziemer (Theorem 3.7 in [291]) and by Lieberman (Lemma 5.1 in [209]); see also [179]. We first establish the main estimate when u satisfies the Dirichlet problem: Lemma 16.7. Under the assumptions of Proposition 16.6, if  is a smooth bounded open set and if u 2 W01;2 ./, then we have 1

krukL2 ./ 6 C 0 ..// p0 : Proof of Lemma 16.7. For every ' 2 Cc1 ./, we have ˆ ˆ ru
r' D ' d: 



Thus, by the Hölder inequality and the trace inequality satisfied by , we have ˇ ˇˆ ˇ ˇ 1 1 ˇ ru
r' ˇ 6 ..// p0 k'kLp .I/ 6 C..// p0 k'kW 1;2 ./ : ˇ ˇ 

16.2. Hölder continuity revisited

265

Since  is bounded, the Poincaré inequality (Proposition 4.5) yields ˇˆ ˇ ˇ ˇ 1 ˇ ru
r' ˇ 6 C1 ..// p0 kr'kL2 ./ : ˇ ˇ 

We now apply this estimate to a sequence .'n /n2N in Cc1 ./ converging to u in W01;2 ./. As n ! 1, we deduce that 1

2 p 0 kruk 2 krukL 2 ./ 6 C1 ..// L ./ ;

and this gives the conclusion.



The connection between Proposition 16.6 and Lemma 16.7 will be made using the harmonic extension of u on balls. We also need Campanato’s fundamental lemma, see Lemma 6.II in [75]: Lemma 16.8. Let ˛; ˇ > 0 with ˛ ¤ ˇ, and d > 0. If f W .0; d  ! R is a function such that, for every 0 < s < t 6 d , f .s/ 6 C t ˛ C then, for every 0 < r 6 d ,

 s ˇ t

f .r/ 6 C 0 .d ˛ C sup f / .d 2 ;d 

f .t /;

 r min ¹˛;ˇ º d

:

Proof of Lemma 16.8. We first rewrite the estimate satisfied by f as t˛ f .t / f .s/ 6 C C ˇ ; ˇ ˇ s s t and denote, for 0 < z 6 d , g.z/ D

f .z/ : zˇ

Given 0 < r 6 d2 , we apply this estimate with s D 2i r and t D 2i C1 r. More precisely, let k 2 N be such that d2 < 2k r 6 d . For i 2 ¹0; : : : ; k 1º, we have g.2i r/ 6 C 2˛ .2i r/˛ ˇ C g.2i C1 r/: Adding these inequalities with respect to i 2 ¹0; : : : ; k 1º, and simplifying common terms in the left- and right-hand sides, we get g.r/ 6 C 2˛ r ˛

ˇ

k X1 i D0

2i.˛

ˇ/

C g.2k r/:

266

16. Trace inequality

Multiplying this inequality by r ˇ , we obtain f .r/ 6 C 2˛ r ˛

k 1 X

2i.˛

C

d 2

< 2k r 6 d , we have

i D0

This gives the estimate of f .r/. Indeed, since

f .2k r/ ˇ r : .2k r/ˇ

ˇ/

f .2k r/  2 ˇ 6 sup f: d .2k r/ˇ . d ;d  2

Concerning the first term, for ˛ < ˇ we have, for every k 2 N , k 1 X

2i.˛

ˇ/

6

i D0

1 2˛ ˇ

1

;

while, for ˛ > ˇ, we have k 1 X

2i.˛

ˇ/

6

i D0

1 2˛ ˇ

1

2k.˛

ˇ/

6

1 2˛ ˇ

 d ˛

ˇ

1 r

:

In both cases, we have the conclusion.



Proof of Proposition 16.6. Given a ball B.xI d /   and 0 < t 6 d , let v be the harmonic function in B.xI t / such that v u 2 W01;2 .B.xI t //. This function v is obtained by minimizing the Dirichlet energy ˆ jrzj2 B.xIt/

in the class u C W01;2 .B.xI t //. By Lemma 16.7 applied to the ball B.xI t / and the function u 0 < s 6 t we have kr.u

v/kL2 .B.xIs// 6 kr.u

v, for every 1

v/kL2.B.xIt// 6 C1 ..B.xI t /// p0 :

By the triangle inequality, it follows that 1

krukL2 .B.xIs// 6 C1 ..B.xI t /// p0 C krvkL2 .B.xIs// :

(16.3)

Since v is harmonic in B.xI t /, the function jrvj2 is subharmonic. Hence, by the mean value property of subharmonic functions (cf. Proposition 1.4) and by the minimality of the Dirichlet energy of v over B.xI t /, we have krvkL2.B.xIs// 6

 s  N2 t

krvkL2.B.xIt// 6

 s  N2 t

krukL2 .B.xIt// :

16.2. Hölder continuity revisited

267

Inserting the density estimate (16.2) and the estimate above into (16.3), we deduce that  s  N2 N 2 krukL2 .B.xIt// : (16.4) krukL2 .B.xIs// 6 C2 t 2 .p 1/ C t For p < 1 C NN 2 , the first exponent is smaller than the second one. Thus, by Campanato’s fundamental lemma (Lemma 16.8) we deduce that, for every 0 < r 6 d, krukL2 .B.xIr // 6 C3 .d 6 C4 r

N

2 2

N

2 2

.p 1/

.p 1/

C krukL2 .B.xId // /

:

 r  N2 2 .p

1/

d

For p > 1 C NN 2 , the second exponent in the estimate (16.4) is smaller than the first one, and we conclude as above. In both cases, the constant C4 depends on d , and hence on the distance d.x; @/.  We now investigate the zeroth-order version of Morrey’s estimate due to Campanato [74] and Meyers [244] that is needed to obtain the Hölder regularity of solutions. For convenience, we focus on the case of functions defined in RN . Proposition 16.9. Let ˛ > 0, and let u 2 L1loc .RN /. If B.xIr /

B.xIr /

ju.y/

u.z/j dy dz 6 C r ˛ ;

for every open ball B.xI r/  RN , then the precise representative uO is defined everywhere in RN and is Hölder-continuous of order ˛: for every y; z 2 RN , u.z/j O 6 C 0 jy

ju.y/ O

zj˛ :

for some constant C 0 > 0 depending on C , ˛ and N . The proof of Proposition 16.9 relies on the following lemma (cf. Lemma 8.8): Lemma 16.10. Let ˛ > 0, let u 2 L1loc .RN / and x 2 RN . If, for every r > 0, we have ju.y/ u.z/j dy dz 6 C r ˛ ; B.xIr /

B.xIr /

then, for every 0 < s 6 r, we have

B.xIr /

B.xIs/

ju.y/

u.z/j dy dz 6 C 00 r ˛ ;

for some constant C 00 > 0 depending on C , ˛ and N .

268

16. Trace inequality

Proof of Lemma 16.10. For every 0 <  < , we have ˇ ˇ ˇ ˇ ˇ ju.y/ u.z/j dy dz u uˇˇ 6 ˇ B.xI/

B.xI/

B.xI/

6

B.xI/

  N 

B.xI/

B.xI/

u.z/j dy dz:

ju.y/

It is thus convenient to estimate differences of averages when the radii are comparable. Now, given 0 < s 6 r, take k 2 N such that r

r : 2k

0 we get

B.xIr /

B.xIs/

ju.y/

u.z/j dy dz 6 C r ˛ C 2N 00 ˛

k X

C

i D0

 r ˛ 2i

C 2N C s ˛

6C r : This proves the lemma.



Proof of Proposition 16.9. By the previous lemma, for every x 2 RN we have lim

.r;s/!.0;0/ B.xIr /

B.xIs/

ju.y/

u.z/j dy dz D 0:

Thus, by the Cauchy property of Lebesgue points (Lemma 8.8), x is a Lebesgue point of u and, in particular, the precise representative can be explicitly computed by using averages on balls: u.x/ O D lim

s!0 B.xIs/

u:

By the previous lemma (cf. Exercise 10.1), for every 0 < s 6 r we also have

B.xIr /

Thus, as s ! 0, we get

ˇ ˇ ˇu.y/ ˇ

B.xIr /

B.xIs/

ˇ ˇ uˇˇ dy 6 C 00 r ˛ :

u.x/j O dy 6 C 00 r ˛ :

ju.y/

(16.5)

We now show that this estimate implies the Hölder continuity of u. O For this purpose, let x; z 2 RN . By the triangle inequality, for every y 2 RN we have ju.x/ O

u.z/j O 6 ju.y/

u.x/j O C ju.y/

u.z/j: O

Taking the average over y 2 B. xCz I r/ for some r > 0, we get 2 ju.x/ O

u.z/j O 6

B. xCz 2 Ir /

ju.y/

u.x/j O dy C

B. xCz 2 Ir /

ju.y/

u.z/j O dy:

270

16. Trace inequality

To estimate the first integral in the right-hand side using (16.5), we need an average integral over a ball centered at x. For this purpose, it suffices to note that x C z

  jx I r  B xI r C

ju.y/

u.x/j O dy

B

2

2

zj 

:

By the monotonicity of the integral and by Property (16.5), we then have

B. xCz 2 Ir /

6

rC

jx

2

r

6 C 00

rC

zj !N

jx

2

r

B.xIr C jx 2 zj /

zj !N



rC

jx

2

ju.y/ zj ˛

u.x/j O dy

:

A similar estimate holds for the second integral. Therefore,

ju.x/ O

u.z/j O 6 2C 00

Taking for instance r D

jx zj , 2

rC

jx r

2

zj !N



rC

jx

2

zj ˛

:

the conclusion follows.



Combining the Poincaré–Wirtinger inequality (10.5) and the Campanato–Meyers property (Proposition 16.9), we obtain Morrey’s first-order embedding: 1;1 Corollary 16.11. Let ˛ > 0, and let u 2 Wloc .RN /. If, for every open ball B.xI r/  RN , we have

r B.xIr /

jruj 6 C r ˛ ;

then the precise representative uO is defined everywhere in RN and is Hölder-continuous of order ˛. Exercise 16.2 (Morrey–Sobolev embedding). Let q > N and u 2 W 1;q .RN /.

.a/ Prove that the precise representative uO is defined everywhere in RN and, for every y; z 2 RN , we have ju.y/ O

u.z/j O 6 C krukLq .RN / jy

zj1

N q

:

.b/ Deduce that uO is uniformly continuous in RN and lim u.x/ O D 0. jxj!1

16.3. Delocalization of capacitary measures

271

16.3 Delocalization of capacitary measures Using the trace inequality, we prove the equivalence between the Sobolev capacity capW 1;1 and the Hausdorff capacity HıN 1 , see Lemma 3 in [80]: Proposition 16.12. For every 0 < ı < C1, there exist constants C > 0 and C 0 > 0 such that C HıN 1 6 capW 1;1 6 C 0 HıN 1 : The trace inequality arises in the following delocalization property, where a local capacitary inequality becomes a global one: Lemma 16.13. Let 1 6 q < C1, and let  2 Mloc .RN / be a nonnegative measure. If there exist ı > 0 and C > 0 such that, for every x 2 RN , we have bB.xIı/ 6 C capW 1;q ; then there exists C 0 > 0, depending on C , N , q, and ı, such that  6 C 0 capW 1;q : p Denoting by .xn /n2N the vertices of a regular grid in RN of width ı= N , we have that [ p RN  Q.xn I ı= N /; n2N

p and every point x 2 RN belongs to at most 2N cubes Q.xn I ı= N /. In the proof of the lemma, we use a smooth partition of unity . n /n2N in RN with the following properties: p .a/ supp n  Q.xn I ı= N / for every n 2 N; .b/ jr

nj

6 M=ı for every n 2 N;

.c/ for every x 2 R, there exists n 2 N such that

n .x/

The last assertion is a consequence of the fact that

> 1=2N .

1 P

nD0

are at most 2N terms which do not vanish in the series.

n .x/

D 1 and that there

Proof of Lemma 16.13. Applying Maz0 ya’s trace inequality with measure bB.xIı/ (Proposition 16.2) we deduce that, for every x 2 RN and every ' 2 Cc1 .B.xI ı//, k'kLq .RN I/ D k'kLq .RN IbB.xIı/ / 6 C1 k'kW 1;q .RN / :

(16.6)

272

16. Trace inequality

Take a sequence of points .xn /n2N as above and a smooth partition of unit . satisfying Properties .a/–.c/. Given ' 2 Cc1 .RN /, by Property .c/ we have j'jq 6 2N q

1 X

nD0

j'

n /n2N

q nj :

By Property .a/, each function ' n is supported in a ball of radius ı. Thus, by the trace inequality (16.6) for the measure bB.xn Iı/ we have ˆ

q

RN

Nq

j'j d 6 2

1 ˆ X

nD0

RN

j'

q

nj

d 6 C2

1 ˆ X

nD0

.j' RN

q nj

C jr.'

q n /j /:

We then deduce from Property p .b/ and from the uniform control on the number of overlaps of the cubes .Q.xn I ı= N //n2N that (cf. (9.7)) ˆ ˆ 1 ˆ X q q N .j'j C jr'j / 6 C 2 .j'jq C jr'jq /: j'jq d 6 C3 3 p RN

nD0

Q.xn Iı= N /

RN

The trace inequality thus holds for every ' 2 Cc1 .RN /, whence, by Proposition 16.1, the measure  satisfies the global capacitary inequality.  We also have the following localization property of density measures: Lemma 16.14. Let s > 0, and let  2 Mloc .RN / be a nonnegative measure. If there exist 0 < ı < C1 and C > 0 such that  6 C Hıs ; then, for every x 2 RN , we have s bB.xIı/ 6 C H1 :

Proof of Lemma 16.14. For every compact set A  RN , we claim that bB.xIı/ .A/ D .A \ B.xI ı// 6 C Hıs .A \ BŒxI ı/: Since the compact set A \ BŒxI ı is contained in a closed ball of radius ı, we claim that s Hıs .A \ BŒxI ı/ D H1 .A \ BŒxI ı/: (16.7)

Assuming this identity, it follows from the monotonicity of the Hausdorff content that s s bB.xIı/ .A/ 6 C H1 .A \ BŒxI ı/ 6 C H1 .A/:

This gives the conclusion on compact sets. We observe that (16.7) also holds for an arbitrary Borel set A, and the same argument leads to the conclusion.

16.3. Delocalization of capacitary measures

273

We now establish identity (16.7) for a compact set A. For this purpose, it suffices to prove the inequality 6. Take a sequence of balls .B.xn I rn //n2N covering A \ BŒxI ı. We have that either (1) rn 6 ı for every n 2 N, and then Hıs .A \ BŒxI ı/ 6

1 X

!s rns ;

nD0

or (2) rn > ı for some n 2 N, and then, by the monotonicity of Hıs and Proposition B.2, we have Hıs .A

\ BŒxI ı/ 6

Hıs .BŒxI ı/

s

6 !s ı 6

1 X

!s rns :

nD0

In both cases, we get the same upper bound. Minimizing the common right-hand side with respect to all coverings .B.xn I rn //n2N , we obtain (16.7). This concludes the proof of the lemma.  The last ingredient is a localized version of Frostman’s lemma (Proposition B.4): Lemma 16.15. Let ı > 0. Then, for every compact set S  RN , there exists a nonnegative measure  2 M.RN /, supported in S , such that s .S / > 3 s H3ı .S / and  6 Hıs :

The proof of this variant of Frostman’s lemma can be carried out as in Section B.2. The only difference is to apply the Hahn–Banach theorem to the modified sublinear function Pı defined for  2 C 0 .S / by Pı ./ D inf

²X ` i D0

˛i !s ris W  6

` X i D0

³ ˛i B.xi Iri / ; ˛i > 0 and ri 6 ı :

By restricting the sizes of the radii ri , the conclusion of Lemma B.5 becomes s H3ı .S /

63

s

` X

˛i !s ris ;

i D0

where the gauge 3ı comes from Wiener’s covering lemma (Lemma 9.2). Therefore, s H3ı .S / 6 3s Pı .S /;

and one deduces Lemma 16.15 along the lines of the proof of Proposition B.4.

274

16. Trace inequality

Proof of Proposition 16.12. The second inequality follows from Proposition 10.1. To establish the first inequality, we begin by proving the following claim: Claim. For every nonnegative measure  2 M.RN / such that  6 HıN

1

, we have

 6 C 00 capW 1;1 : Proof of the claim. By the localization of densities measures (Lemma 16.14), for every x 2 RN we have N 1 bB.xIı/ 6 H1 :

Thus, by Proposition 10.2, bB.xIı/ 6 C1 capW 1;1 : We then apply the delocalization property of capacitary measures (Lemma 16.13) to conclude. 4 Take a compact set S  RN . By the local version of Frostman’s lemma above, there exists a nonnegative measure  2 M.RN /, supported in S , such that .S / > 3

.N 1/

N 1 H3ı .S / and

 6 HıN

1

:

Evaluating on S the capacitary estimate given by the claim, we then get 3

.N 1/

N 1 H3ı .S / 6 .S / 6 C 00 capW 1;1 .S /:

Since this inequality holds with an arbitrary compact set S  RN , we deduce that 3

.N 1/

N H3ı

1

6 C 00 capW 1;1 :

We may now replace ı by ı=3 to get the inequality we sought to prove.



Chapter 17

Critical embedding

“There are moments in a match when the ball hits the top of the net, and for a split second it can either go forward or fall back. With a little luck, it goes forward and you win. Or maybe it doesn’t, and you lose.” Woody Allen N 2 The non-equivalence between the Hausdorff content H1 and the Sobolev capacity capW 1;2 yields different embeddings for solutions of the linear Dirichlet problem. This is related to the failure of the Calderón–Zygmund estimates in the L1 setting.

17.1 Bounded mean oscillation N 2C˛ In contrast with the condition  6 C H1 , which gives Hölder continuity of solutions of the Poisson equation with nonnegative density  for 0 < ˛ < 1, the assumption N 2  6 C H1 (17.1)

is not strong enough to imply continuity, nor even local boundedness.

Example 17.1. The function uW R3 ! R defined for .x1 ; x2 / ¤ .0; 0/ by

satisfies

1 u.x1 ; x2 ; x3 / D log q x12 C x22 u D 2H1 b¹x1 Dx2 D0º

1 in the sense of distributions in R3 . Note that H1 b¹x1 Dx2 D0º 6 H1 , since, for every 3 ball B.aI r/  R , we have

H1 b¹x1 Dx2 D0º .B.aI r// 6 2r: We next show that condition (17.1) is related to the concept of bounded mean oscillation introduced by John and Nirenberg [173]:

276

17. Critical embedding

Definition 17.2. A function u 2 L1loc .RN / has bounded mean oscillation if there exists C > 0 such that, for every ball B.xI r/  RN , we have B.xIr /

B.xIr /

u.z/j dy dz 6 C:

ju.y/

We adopt here the equivalent formulation of functions of bounded mean oscillation from the work of Brezis and Nirenberg [62] on degree theory and BMO spaces (cf. Exercise 10.1). We denote in this case ŒuBMO D

sup B.xIr /RN

B.xIr /

B.xIr /

u.z/j dy dz:

ju.y/

A first connection with condition (17.1) arises in the study of the Newtonian potential, see Proposition 3.3 in [3] and proof of Theorem 2 in [171]: Proposition 17.3. Let N > 3. If  2 Mloc .RN / is a nonnegative measure such that N 2  6 C H1 ;

then the Newtonian potential N has bounded mean oscillation. From the proof of Proposition 13.5, for every y; z 2 RN and every ı > 2jy zj we have that ˆ ı ˆ 1 .B.yI t // C1 .B.zI t // 1 dt C jy zj dt: N.y/ N.z/ 6 N 0 tN 1 N tN ı We deduce Proposition 17.3 from this estimate: Proof of Proposition 17.3. Given a ball B.xI r/  RN , for every y; z 2 B.xI r/ we have jy zj 6 2r. We may thus apply the previous estimate with ı D 4r: ˆ 4r ˆ 1 C1 .B.yI t // .B.zI t // 1 C dt C 2r dt: ŒN.y/ N.z/ 6 N 1 N 0 t N tN 4r By the density assumption on , we have (Proposition B.3) .B.zI t // 6 C !N Thus, ˆ

1 4r

.B.zI t // dt 6 C !N tN

2

2t

N 2

ˆ

1

4r

:

1 C2 : dt D 2 t r

We deduce that ŒN.y/

N.z/C 6

1 N

ˆ

4r 0

.B.yI t // dt C C3 : tN 1

(17.2)

17.1. Bounded mean oscillation

277

By Fubini’s theorem, for every 0 < t 6 4r we have  ˆ ˆ ˆ .B.yI t // dy D d./ dy B.xIr /

B.xIr /

D

ˆ

B.xIr Ct/

B.yIt/

ˆ

B.It/\B.xIr /



dy d./

6 !N t N .B.xI 5r//; and then ˆ

B.xIr /

ˆ

4r 0

 ˆ 4r .B.yI t // dt dy 6 !N .B.xI 5r// t dt 6 C4 r N : tN 1 0

Integrating both sides of (17.2) with respect to y, we get ˆ ŒN.y/ N.z/C dy 6 C5 r N ; B.xIr /

for every z 2 B.xI r/. It now suffices to integrate this estimate with respect to z over B.xI r/. By symmetry with respect to the variables y and z, the conclusion follows.  An alternative argument in the spirit of the proof of Proposition 16.6, based on Caccioppoli’s estimates, can be found in Theorem 1.8 in [247]. The counterpart of Proposition 17.3 in dimension N D 2 holds for finite measures  having compact support (cf. remark following the proof of Proposition 13.5). Conversely, we prove that superharmonic functions having bounded mean oscillation are generated by measures satisfying condition (17.1), see Proposition 3.4 in [3]: Proposition 17.4. Let  2 Mloc .RN / be a nonnegative measure, and let u be a solution of the Poisson equation with density  in RN . If u has bounded mean oscillation, then N 2  6 C 0 ŒuBMO H1 : Proof. Since u is superharmonic, by Lemma 11.5 we have that, for every z 2 RN and every r > 0, ˆ C1 .B.zI 2r// 6 2 ju.y/ u.z/j dy: r B.zI4r / Given x 2 RN , for every z 2 B.xI r/ we have B.xI r/  B.zI 2r/ and B.zI 4r/  B.xI 5r/:

278

17. Critical embedding

Hence, by the monotonicity of  and of the Lebesgue measure, we get ˆ C1 ju.y/ u.z/j dy: .B.xI r// 6 2 r B.xI5r / Integrating both members with respect to z over B.xI r/, we deduce that ˆ ˆ C1 !N r N .B.xI r// 6 2 ju.y/ u.z/j dy dz: r B.xI5r / B.xI5r / Since u has bounded mean oscillation, this implies that .B.xI r// 6 C2 ŒuBMO r N

2

:

The conclusion follows from the characterization of density measures on balls (see  Proposition B.3). We can combine the conclusions of Propositions 17.3 and 17.4 to deduce that, for N every nonnegative measure  in the Morrey space M 2 .RN /, we have A ŒNBMO 6 kk

M

N 2

.RN /

6 A0 ŒNBMO ;

for some constants A; A0 > 0 and dimension N > 3. Exercise 17.1 (Liouville–Riesz’s theorem). .a/ Prove that if w is a harmonic function in RN and has bounded mean oscillation, then w is constant. .b/ Let N > 3, and let  2 M.RN /. Under the assumptions of Proposition 17.4, deduce that there exists ˛ 2 R such that u D N C ˛ almost everywhere in RN .

17.2 Brezis–Merle inequality and beyond Condition (17.1) yields the exponential integrability of solutions of the Dirichlet problem with nonnegative density  2 M./: ´ u D  in , uD0

on @.

Indeed, the Newtonian potential N is a supersolution of this problem (Lemma 17.6) and has bounded mean oscillation (Proposition 17.3). By the weak maximum principle (Proposition 6.1), we thus have that 0 6 u 6 N:

17.2. Brezis–Merle inequality and beyond

279

Applying the John–Nirenberg inequality (see [173] or Section IX.13 in [113]), we deduce that eˇ u 2 L1 ./ (17.3) for some constant ˇ > 0 depending on ŒNBMO , which can be estimated in terms of the constant C > 0 verifying (17.1). In the study of absorption problems involving exponential nonlinearities (Section 21.4), we need to take ˇ as large as possible. For this purpose, we present an alternative proof of the exponential integrability, which gives an explicit upper bound of ˇ in terms of C . More precisely, we prove that (17.3) holds for every ˇ < 4=C ; the exponential integrability need not hold for ˇ D 4=C (cf. Example 17.1). We begin with the estimate in dimension two. The following inequality is established by Brezis and Merle, see Theorem 1 in [61], and is based on Jensen’s inequality: Proposition 17.5. Let N D 2, and let  2 M.R2 / be a nonnegative measure with compact support. If .R2 / < 4; then eN 2 L1loc .R2 / and, for every ball B.aI R/  R2 , we have keN kL1 .B.aIR// 6 C; for some constant C > 0 depending on .R2 / and R. Proof. For every x 2 R2 , we write N.x/ D

ˆ

R2

 1  d.y/ .R2 / log D 2 jx yj .R2 /

ˆ

log R2



1 jx

R2 /  .2 d.y/ : yj .R2/

Since =.R2 / is a probability measure and log is a concave function, by Jensen’s inequality for concave functions we have N.x/ 6 log

ˆ



R2

Thus, N.x/

e

6

ˆ

R2

By Fubini’s theorem, we get ˆ

B.aIR/

N.x/

e

dx 6

ˆ

R2



1 jx 1

jx

ˆ

 R2 /  .2 d.y/ : yj .R2 /

R2 /  .2 d.y/ : yj .R2 /

B.aIR/



1 jx

yj

R2 /  .2

dx



d.y/ : .R2/

280

Since

17. Critical embedding .R2 / 2

ˆ

< 2, for every y 2 R2 we have

B.aIR/

Thus,



1 jx

yj

R2 /  .2

ˆ

R2 /  1  .2 dx 6 dz D C < C1: B.0IR/ jzj

ˆ

eN.x/ dx 6 C

B.aIR/

ˆ

R2

d.y/ D C: .R2/



The previous estimate is inherited by solutions of the linear Dirichlet problem with density , based on the following property (cf. Example 6.2 and Proposition 20.2): Lemma 17.6. Let  be a bounded open set, and let  2 M./. If u 2 L1 ./ satisfies u >  in the sense of distributions in , and if u is nonnegative in , then u > 

x 0. in the sense of .C01 .//

x and a nonnegative convex smooth function H W R ! R Proof. Given  2 C01 ./ such that H D 0 in a neighborhood of 0, it follows that ' D H./ is a nonnegative function in Cc1 ./; by the convexity of H , we have ' D H 0 ./  C H 00 ./jrj2 > H 0 ./ : Using ' as a test function and the nonnegativity of u, we deduce that ˆ ˆ 0 uH ./  > H./ d: 



We now apply an approximation procedure, replacing the test function  by =n for some sequence .n /n2N of positive numbers converging to zero. Multiplying both sides of the inequality by n , we get ˆ ˆ uH 0 .=n /  > n H.=n / d: 

Taking H such that .a/ H.t / D 0 for every t 6 1,

.b/ H 0 .t / D 1 for every t > 2,



17.2. Brezis–Merle inequality and beyond

and letting n ! 1, we deduce that ˆ

u  >

¹>0º

ˆ

281

 d: ¹>0º

This gives the conclusion if  > 0 in . x we may conclude taking some For an arbitrary nonnegative function  2 C01 ./, 1 x fixed function w 2 C0 ./ such that w > 0 in . For every ı > 0, we have  C ıw > 0 in , whence ˆ ˆ u . C ıw/ > . C ıw/ d: 



Letting ı ! 0, we have the conclusion with .



Corollary 17.7. Let N D 2, and let  be a smooth bounded open set. If  2 M./ is such that, for every x 2 , .¹xº/ < 4, then the solution u of the linear Dirichlet problem with density  satisfies eu 2 L1 ./. In particular, if  is an L1 function, then we always have that eu 2 L1 ./. This is however a qualitative information, since keu kL1 ./ cannot be estimated solely in terms of kkL1 ./ when kkL1 ./ > 4. The proof below follows the strategy of Corollary 1 in [61]. Proof of Corollary 17.7. By the outer regularity of C (Proposition 2.5), there exists  > 0 such that, for every a 2 , C .B.aI / \ / < 4: Given a 2 , let u1 be the Newtonian potential generated by C bB.aI/ , and let u2 be the Newtonian potential generated by C bnB.aI/ . Hence, NC D u1 C u2 : Claim. The solution u satisfies u 6 NC

C ./ 1 log 2 d

almost everywhere in , where d D diam . Let us temporarily assume the claim and conclude the proof. Since u2 is a harmonic function in B.aI /, it is bounded in B.aI / for 0 <  < 1. We thus have C

eu 6 C1 e N 6 C2 eu1

in B.aI / \ .

282

17. Critical embedding

By the Brezis–Merle inequality (Proposition 17.5), we have that eu1 2 L1 ./, whence eu 2 L1 .B.aI / \ /:

To conclude the argument we cover  with finitely many balls of radius  and we deduce that eu 2 L1 ./. It remains to prove the claim: Proof of the claim. For every x; y 2 , we have jx yj 6 d , and then ˆ ˆ 1 1 1 1 1 C ./ dC .y/ > log log : NC .x/ D log dC D 2  jx yj 2 d  2 d

By Lemma 17.6, we thus have   NC

C ./ 1 > C log 2 d

x 0 . Since u satisfies the Dirichlet problem with density  and in the sense of .C01 .// C since  > , we conclude that   C ./ 1  NC log u >0 2 d 1 x 0 in the sense of .C0 .// . By the weak maximum principle (Proposition 6.1), the claim follows. 4 The proof of the corollary is complete.



A precise version of the Brezis–Merle inequality has been investigated by Cassani, Ruf and Tarsi using radial rearrangements, see Corollary 2 in [83]. The counterpart of the Brezis–Merle inequality in higher dimensions has been established by Bartolucci, Leoni, Orsina, and Ponce, see Theorem 2 in [22]: Proposition 17.8. Let N > 3, and let  2 M.RN / be a nonnegative measure. If  6 ˛HıN

2

for some coefficient ˛ < 4 and some gauge 0 < ı 6 C1, then eN 2 L1loc .RN / and, for every ball B.aI R/  RN , we have keN kL1 .B.aIR// 6 C; for some constant C > 0 depending on N , ˛, ı, .RN /, and R. The proof of the Proposition 17.8 is based on Cavalieri’s representation of the Newtonian potential in terms of densities on balls (Lemma 2.16): ˆ 1 .B.xI r// dr 1 : N.x/ D N 0 rN 2 r

17.2. Brezis–Merle inequality and beyond

283

This formula reduces the problem to the following 1-dimensional estimate: Lemma 17.9. Let > 0, s > 0, and let f W Œ0; 1 ! R be a nondecreasing function. If, for every 0 6 r 6 1, we have 0 6 f .r/ 6 r s ; then ˆ

1 0

  ˆ 1 f .r/ dr f .r/ dr 0 6 log 1 C C ;

Cs r rs r 0 r

for some constant C 0 > 0 depending on s and f .1/.

We temporarily assume the lemma and proceed with the proof of Proposition 17.8: Proof of Proposition 17.8. We establish the estimate for the gauge ı D 1; by Proposition B.3, the assumption  6 ˛H1N 2 is equivalent to the density condition on balls of radii at most 1: for every x 2 RN and every 0 < r 6 1, we have .B.xI r// 6 ˛!N

2r

N 2

(17.4)

:

We begin by writing the Newtonian potential for every x 2 RN as ˆ 1 1 .B.xI r// N.x/ D dr D v1 .x/ C v2 .x/; N 0 rN 1 where 1 v1 .x/ D N

ˆ

1 0

1 and v2 .x/ D N

.B.xI r// dr rN 1

ˆ

1 1

Thanks to the finiteness of the measure , we have ˆ 1 ˆ 1 1 1 .B.xI r// 1 N v2 .x/ D dr 6 . R / N N 1 rN 1 N r 1

.B.xI r// dr: rN 1

1

dr 6 C1 :

Thus, eN.x/ 6 eC1 ev1 .x/ : To estimate ev1 .x/ , we apply the previous lemma with exponent s D N function f W Œ0; 1 ! R defined for r 2 Œ0; 1 by f .r/ D

1 .B.xI r//: N

By the density estimate (17.4) and the identity N D N !N D 2!N f .r/ 6

2 and

1 ˛!N N

2r

N 2

D

˛ N r 2

2

:

2,

we have

284

17. Critical embedding

Applying Lemma 17.9, we obtain ev1 .x/ 6 1 C C 0

ˆ

1

.B.xI r// ˛

0

r 2 CN

1

dr:

Integrating both sides with respect to x over a ball B.aI R/, by Fubini’s theorem we get  ˆ ˆ 1ˆ 1 v1 .x/ N 0 dr: e dx 6 !N R C C .B.xI r// dx ˛ CN 1 2 r 0 B.aIR/ B.aIR/ By the nonnegativity of the measure  and by Fubini’s theorem, we have ˆ ˆ .B.xI r// dx 6 .B.xI r// dx RN

B.aIR/

Since

˛ 2

D

ˆ

D

ˆ

RN

RN

ˆ

B.xIr /

ˆ

B.zIr /

 d.z/ dx 

dx d.z/ D !N r N .RN /:

< 2, it then follows that

ˆ

B.aIR/

ev1 .x/ dx 6 !N RN C C 0 !N .RN /

ˆ

0

1

1 r

˛ 2

1

dr < C1;

and this gives the conclusion.



We first prove Lemma 17.9 in the case s D 0, which contains the main idea of the proof. We then explain how the general case reduces to this one. Proof of Lemma 17.9 for s D 0. Assume that f vanishes in a neighborhood of 0 and is smooth in Œ0; 1. Further, by integration by parts, we have ˆ 1 ˆ 1 dr 1 f .r/ log f 0 .r/ dr: D r r 0 0 Since f is nondecreasing and f .0/ D 0, the function f 0 =f .1/ is a probability density in Œ0; 1. We rewrite the previous identity as ˆ 1 ˆ 1 ˆ 1 dr 1  f 0 .r/ 1  f 0 .r/ f .r/ D dr D dr: f .1/ log log f .1/ r r f .1/ f .1/ r 0 0 0

Hence, by Jensen’s inequality for concave functions, we get  ˆ 1   ˆ 1 0 ˆ 1 1 1 f 0 .r/ f .r/ dr dr : 6 log dr D log f .r/ f .1/ f .1/ r f .1/ 0 r f .1/ 0 r 0

(17.5)

17.2. Brezis–Merle inequality and beyond

285

By integration by parts, since f vanishes in a neighborhood of 0 we have ˆ 1 ˆ 1 0 f .r/ f .r/ dr D f .1/ C f .1/ dr: f .1/ f .1/C1 0 r 0 r Since s D 0, we have that f .1/ 6 . We then obtain   ˆ 1 ˆ 1 dr f .r/ f .r/ 6 log 1 C dr :

C1 r 0 0 r The conclusion follows when f vanishes in a neighborhood of 0 and is smooth in Œ0; 1. These assumptions can be removed using an approximation argument on f .  We would like to deduce the case s > 0 directly from s D 0, but the function r7 !

f .r/ rs

need not be nondecreasing. For this reason, we apply estimate (17.5) to the nondecreasing function f .t / : r 7 ! sup s 06t6r t We need in this case an additional tool: Lemma 17.10. Given a continuous function F W Œ0; 1 ! R, let GW Œ0; 1 ! R be the function defined for r 2 Œ0; 1 by G.r/ D sup F .t /: 06t6r

If F 2 W

1;1

..0; 1//, then G 2 W

1;1

..0; 1//, and we have

0 6 G 0 6 max ¹F 0 ; 0º almost everywhere in .0; 1/. Proof of Lemma 17.10. Let .xn /n2N be a dense sequence in Œ0; 1. Define by induction a sequence of continuous functions .Fn /n2N in Œ0; 1 by F0 D F and, for n 2 N , ´ Fn 1 in Œ0; xn 1 /, Fn D max ¹Fn 1 ; Fn 1 .xn 1 /º in Œxn 1 ; 1.

Since F is continuous and .xn /n2N is dense in Œ0; 1, the sequence .Fn /n2N converges uniformly to G in Œ0; 1. In addition, for every n 2 N we have that Fn 2 W 1;1 ..0; 1// and min ¹F 0 ; 0º 6 Fn0 6 max ¹F 0 ; 0º:

286

17. Critical embedding

In particular, the sequence .Fn0 /n2N is bounded in L1 ..0; 1// and is equi-integrable. Letting n ! 1, we deduce that G 2 W 1;1 ..0; 1// and min ¹F 0 ; 0º 6 G 0 6 max ¹F 0 ; 0º:

Since G is nondecreasing, we have that G 0 > 0, and the conclusion follows.



Proof of Lemma 17.9. We first assume that f vanishes in a neighborhood of 0 and is smooth in Œ0; 1. Let gW Œ0; 1 ! R be the function defined for r 2 Œ0; 1 by g.r/ D sup

06t6r

f .t / : ts

Note that g is bounded and nondecreasing, and ˆ 1 ˆ 1 f .r/ dr dr 6 g.r/ : s r r r 0 0 By Lemma 17.10, we have g 2 W 1;1 ..0; 1//. Observe that estimate (17.5) is satisfied by g, whence   ˆ 1 0 ˆ 1 1 g .r/ dr 6 log g.r/ dr : r g.1/ 0 r g.1/ 0 For almost every r 2 .0; 1/, we also have ° d f .r/ ± 0 6 g 0 .r/ 6 max ;0 : dr r s Since f is nonnegative and f 0 > 0, we get 0 6 g 0 .r/ 6 We then have

f 0 .r/ : rs

  ˆ 1 0 1 f .r/ dr f .r/ 6 log dr : rs r g.1/ 0 r g.1/Cs 0 We now proceed as in the case s D 0. By integration by parts, we obtain   ˆ ˆ 1 f .1/  s  1 f .r/ f .r/ dr 6 log C 1 C dr : rs r g.1/ g.1/ 0 r g.1/CsC1 0 ˆ

1

Since f .1/ 6 g.1/ and g.1/ 6 , we get   ˆ ˆ 1  s  1 f .r/ f .r/ dr 6 log 1 C 1 C dr : rs r f .1/ 0 r CsC1 0

This establishes the inequality when f vanishes in a neighborhood of 0 and is smooth in Œ0; 1. The general case follows by approximation of f by a sequence of functions .fn /n2N of this type. This approximation can be done so that .fn /n2N converges almost everywhere to f in Œ0; 1 and, for every n 2 N and every 0 6 r 6 1, we have that 0 6 fn .r/ 6 r s . The proof of the estimate is complete. 

17.3. Exponential Sobolev embedding

287

The function f W Œ0; 1 ! R defined for r 2 Œ0; 1 by f .r/ D r s is a natural candidate to apply Lemma 17.9, but both sides of the inequality are infinite. A second attempt is to consider, for every parameter 0 <  < 1, the function f W Œ0; 1 ! R defined for r 2 Œ0; 1 by ´ 0 if 0 6 r < , f .r/ D s

r if  6 r 6 1. We now have ˆ

1

f .r/ dr 1 D log : rs r 

ˆ

1

1 f .r/ dr D

Cs r r 

0

On the other hand,

and thus

0

 ˆ log 1 C C

0

1

f .r/ dr r Cs r



1;

D log

1 C O.1/; 

where O.1/ denotes a quantity that remains bounded as  ! 0. Thus, both terms in the inequality tend to infinity at the same rate.

17.3 Exponential Sobolev embedding Under the stronger capacitary assumption  6 C capNewt ;

(17.6)

the solution of the linear Dirichlet problem with density  satisfies an exponential Sobolev estimate, see [131] and Theorem 3.14 in [172]: Proposition 17.11. Let  be a bounded open set. Then, there exists ı > 0 such that if  2 M./ is a nonnegative measure such that  6 ı capNewt ; and if u 2 W01;2 ./ satisfies the linear Dirichlet problem with density , then we have that eu 2 W 1;2 ./ and kr.eu /kL2 ./ 6 C krukL2 ./ :

288

17. Critical embedding

The capacitary assumption  6 ı capNewt is equivalent to the following trace inequality (cf. Propositions 16.1 and 16.2): for every ' 2 Cc1 .RN /, k'kL2 .RN I/ 6 C 0 ıkr'kL2 .RN / :

(17.7)

We have in particular that  2 .W01;2 .//0 (cf. Proposition 16.5). The proof of Proposition 17.11 relies on this trace inequality and the following estimate based on Moser’s iteration technique: Lemma 17.12. Let  be a bounded open set. If u 2 W01;2 ./ is a nonnegative function such that, for every v 2 W01;2 ./ \ L1 ./, ˆ ˆ 2 ru
r.v / 6 C jrvj2 



for some constant 0 < C < 2, then eu 2 W 1;2 ./ and kr.eu /kL2 ./ 6

1 1

C 2

krukL2 ./ :

Proof of Lemma 17.12. We begin with the following: Claim. For every k 2 N , we have ˆ  C k ˆ k 2 jruj2 : u jruj 6 .k C 1/Š 4  

Proof of the claim. We first establish the estimate assuming that u is bounded. Note that 1 ru
r.ukC1 / uk jruj2 D kC1 and  k C 1 2 kC1 uk 1 jruj2 : jr.u 2 /j2 D 2 kC1

Applying the assumption with the test function v D u 2 , we then deduce that ˆ ˆ C k 2 u jruj 6 .k C 1/ uk 1 jruj2 : 4  

The claim follows by induction on k when u is bounded. Assuming that u is not bounded, we repeat the same argument using the truncated kC1 test functions v D ŒTs .u/ 2 for s > 0 (Exercise 5.3). In this case, we get ˆ  C k ˆ jrTs .u/j2 : ŒTs .u/k jrTs .u/j2 6 .k C 1/Š 4  

As s ! 1, the claim follows.

4

17.3. Exponential Sobolev embedding

289

By the claim, for every k 2 N we have ˆ



 C k .2u/k jruj2 6 .k C 1/ kŠ 2

ˆ



jruj2 :

Thus, for every n 2 N we get

ˆ hX n n hX  C k i ˆ .2u/k i 2 jruj 6 .k C 1/ jruj2 : kŠ 2   kD0

kD0

Since 0
0 depending on ı. For every v 2 W01;2 ./ \ L1 ./, we have v 2 2 W01;2 ./. Using v 2 as a test function in the formulation of the Dirichlet problem in W01;2 ./ (Proposition 4.6), we get ˆ 

ru
r.v 2 / D Œv 2 :

(17.8)

By the trace inequality (17.7) and by the approximation property (Proposition 4.3) applied to the function v 2 , we then have the functional estimate ˆ ˆ 2 2 2 ru
r.v / D Œv  6 C1 ı jrvj2 : 



It now suffices to choose ı > 0 so that C1 ı 2 < 2, and the conclusion follows from Lemma 17.12.  The exponential Sobolev integrability need not be true under the weaker assumption N 2  6 C H1 :

1;2 Indeed, we have that e˛u 62 Wloc .R3 /, for every ˛ > 0, where uW R3 ! R is the function given by Example 17.1.

290

17. Critical embedding

Examples of measures satisfying the capacitary condition (17.6) can be obtained from bounded superharmonic functions: Proposition 17.13. Let  2 Mloc .RN / be a nonnegative measure, and let u be a solution of the Poisson equation with density  in RN . If u is bounded, then  6 2kukL1 .RN / capNewt : Proof. Given a compact set K  RN , let ' 2 Cc1 .RN / be a nonnegative function such that ' > 1 in K. Since the measure  is nonnegative, we have ˆ ˆ .K/ 6 ' d D u ' 6 kukL1 .RN / k'kL1 .RN / : RN

RN

Minimizing the right-hand side with respect to ', we get .K/ 6 kukL1 .RN / cap1 .K/: By the equivalence between the capacities capNewt and cap1 (Proposition 12.2), the conclusion follows for compact sets.  The Newtonian potential generated by a measure satisfying the capacitary condition (17.6) need not be bounded nor even locally bounded. We provide such an example taken from Section B.5.3 in [118] and Example 7.3 in [172], which relies on Hardy’s inequality (cf. Lemma 15.4): for every N > 3 and ' 2 Cc1 .RN /, we have ˆ ˆ j'.x/j2 dx 6 C jr'j2 : jxj2 RN RN Example 17.14. By Hardy’s inequality in dimension N > 3, any nonnegative measure  2 Mloc .RN / such that  6 jxj1 2 HN satisfies the capacitary condition (17.6) (cf. Proposition 16.1). We prove that the Newtonian potential generated by the finite measure 1 D B.0I1/ HN jxj2

is unbounded in a neighborhood of 0. For this purpose, we apply Cavalieri’s representation of the Newtonian potential N (Lemma 2.16). We first observe that, for every 0 < t 6 1, .B.0I t // D

ˆ

B.0It/

dx D jxj2

ˆ

t 0

N r N r2

1

dr D N

ˆ

0

t

rN

3

dr D

N N t N 2

2

:

17.4. W 1;2 and W 2;1 capacities

Next, for x 2 RN and r > 2jxj, we have B.xI r/  B.0I r For every x 2 B.0I 12 /, it thus follows that N.x/ >

1 N

ˆ

2

2jxj

291

 r jxj/  B 0I : 2

1 .B.xI r// dr > rN 1 N

ˆ

2 2jxj

.B.0I 2r // 1 : dr D C log rN 1 jxj

Maz0 ya and Verbitsky [240], [241], and [242] have made some important progress concerning elliptic estimates for densities  satisfying the trace inequality (17.7), and more generally a stronger estimate of the form k'kL2 .RN I/ 6 ıkr'kL2 .RN / C Cı k'kL2 .RN / ; for every ı > 0. We refer the reader to the book of Maz0 ya and Shaposhnikova [239] on the theory of Sobolev multipliers. The connection of this type of condition with the existence of bounded solutions of elliptic PDEs has been initiated by Trudinger [326]. There is nevertheless a functional characterization of measures generating bounded solutions based on the Riesz representation theorem (cf. Proposition 18.1): Exercise 17.2. Let  be a bounded open set, and let  2 M./. Prove that the linear Dirichlet problem with density  has a bounded solution if and only if there exists x we have C > 0 such that, for every  2 C01 ./, ˇ ˇˆ ˇ ˇ ˇ  dˇ 6 C kkL1 ./ : ˇ ˇ 

17.4 W 1;2 and W 2;1 capacities In the previous sections, we investigate properties of solutions of the Poisson equation and its Dirichlet problem with nonnegative densities  satisfying N  6 C1 H1

2

or  6 C2 capNewt :

In view of the setwise inequality (cf. Proposition 10.1) N 2 capNewt 6 C H1 ;

it is natural to have some stronger embedding under the latter assumption.

(17.9)

292

17. Critical embedding

We recall that the Newtonian capacity capNewt coincides with the capacity cap1 associated to the quantity k'kL1 .RN / , up to a multiplicative factor (Proposition 12.2): cap1 D 2 capNewt : We now introduce the following capacity related to the full second-order derivative D 2 ', defined for every compact set K  RN by capD 2;1 .K/ D inf ¹kD 2 'kL1 .RN / W ' 2 Cc1 .RN / is nonnegative, and ' > 1 on Kº: By the triangle inequality, we have that cap1 6 C 0 capD 2;1 :

(17.10)

One might wonder whether the capacities cap1 and capD 2;1 are equivalent, despite the fact that the norms kD 2 'kL1 ./ and k'kL1 ./ are not equivalent in x for bounded open sets  (see remark after Proposition 5.3). The answer is C01 ./ N 2 negative, and the reason lies in the equivalence between the Hausdorff content H1 and the capacity capD 2;1 , see p. 122 in [5]: Proposition 17.15. For every N > 3, there exist constants C; C 0 > 0 such that N C H1

2

N 6 capD 2;1 6 C 0 H1

2

on every compact subset of RN . Inequalities (17.9) and (17.10) are then equivalent, and this establishes an unexpected connection with the failure of the Calderón–Zygmund estimates in the L1 setting. We also deduce that the capacities capW 1;2 and capW 2;1 are not equivalent: Corollary 17.16. Let N > 3. Then, for every compact set K  RN such that 0 < HN we have (i) capW 1;2 .K/ D cap1 .K/ D 0; (ii) capW 2;1 .K/ > capD 2;1 .K/ > 0.

2

.K/ < 1;

17.4. W 1;2 and W 2;1 capacities

293

Proof of Corollary 17.16. Since HN 2 .K/ < 1, it follows from the non-equivalence between the W 1;2 capacity and the HN 2 Hausdorff measure that capW 1;2 .K/ D 0; see Proposition 10.3. Thus, cap1 .K/ D 2 capNewt .K/ 6 2 capW 1;2 .K/ D 0: N 2 On the one hand, since HN 2 .K/ > 0, the Hausdorff content H1 satisfies the N 2 same property. By the equivalence above between H1 and capD 2;1 , we then have N 2 .K/ > 0: capW 2;1 .K/ > capD 2;1 .K/ > C H1



Proposition 17.15 relies on a second-order trace inequality by Maz0 ya, see [232] and Section 1.4.3 in [234]: given a nonnegative measure  2 Mloc .RN / such that N 2  6 H1 , we have k'kL1 .RN I/ 6 C kD 2 'kL1 .RN / ;

(17.11)

for every ' 2 Cc1 .RN /. This inequality has a counterpart for higher orders: Proposition 17.17. Let k 2 ¹1; : : : ; N 1º. If  2 Mloc .RN / is a nonnegative measure such that N k  6 H1 ; then, for every ' 2 Cc1 .RN /, we have

k'kL1 .RN I/ 6 C kD k 'kL1 .RN / ; for some constant C > 0 depending on N and k. We have excluded from this statement the case k D N , which relies on the following elementary argument. For every ' 2 Cc1 .RN / and every x 2 RN , we have that ˆ x1 ˆ xN @N ' '.x/ D


; 1 1 @x1 : : : @xN

whence

k'kL1 .RN /



@N '

6 6 C kD N 'kL1 .RN / : @x1 : : : @xN L1 .RN /

Thus, for every nonnegative measure  2 M.RN / such that .RN / 6 1, we deduce that k'kL1 .RN I/ 6 C kD N 'kL1 .RN / :

294

17. Critical embedding

In a similar way, one deduces Proposition 17.17 for product measures of the form  D 0  H N

k

;

where 0 2 M.Rk / and k 2 ¹1; : : : ; N 1º. To prove Proposition 17.17, we use a property of the Riesz kernel I1 due to Adams, see Theorem 6 in [1]: Lemma 17.18. If 0 < s < N such that

1, then, for every nonnegative measure  2 Mloc .RN / s  6 H1 ;

we have

sC1 I1   6 C 0 H1 ;

for some constant C 0 > 0 depending on N and s.

The conclusion of the lemma is meant in the sense of Borel set functions, where I1   is understood to be the Lebesgue measure with density I1  . By Proposition B.3, this is equivalent to proving that, for every ball B.xI r/  RN , we have ˆ sC1 I1   6 C 0 H1 .B.xI r// D C 0 !sC1 r sC1 : B.xIr /

Denoting p D NN s , Adams’s result yields a Sobolev inequality in Morrey spaces, which can be stated as kI1  k

Np p

MN

.RN /

6 C 00 kkMp .RN / ;

for every 1 < p < N and every  2 Mp .RN /. Proof of Lemma 17.18. Given r > 0, we split Cavalieri’s representation of the Riesz potential (Exercise 2.7) into two parts: ˆ r ˆ 1 .B.yI t // .B.yI t // I1  .y/ D .N 1/ dt C .N 1/ dt: N t tN 0 r Since s < N ˆ 1 r

1, by the density assumption on  (Proposition B.3) we have ˆ 1 ˆ 1 .B.yI t // dt !s t s dt 6 dt D !s D C1 r sC1 N : N N N t t t s r r

We deduce that I1  .y/ 6 .N

1/

ˆ

r 0

.B.yI t // dt C C2 r sC1 tN

N

:

(17.12)

17.4. W 1;2 and W 2;1 capacities

By Fubini’s theorem, for every 0 < t 6 r we have ˆ ˆ ˆ .B.yI t // dy D B.xIr /

B.xIr /

D

ˆ

B.xIr Ct/

295



d.z/ dy

B.yIt/

ˆ



dy d.z/

B.zIt/\B.xIr /

6 !N t N .B.xI 2r//; and then, by the density assumption on ,   ˆ r ˆ r ˆ ˆ dt .B.yI t // dt dy D .B.yI t // dy N 6 C3 r sC1 : N t t 0 B.xIr / B.xIr / 0 Integrating both sides of (17.12) over B.xI r/, we get ˆ I1   6 C4 r sC1 ; B.xIr /

and the conclusion follows from Proposition B.3.



Proof of Proposition 17.17. We proceed by induction on k; we begin with the case k D 1. Given ' 2 Cc1 .RN /, by Cavalieri’s principle (Proposition 1.7) and by the assumption on , we have ˆ ˆ 1 ˆ 1 N 1 j'j d D .¹j'j > t º/ dt 6 H1 .¹j'j > t º/ dt: RN

0

0

The trace inequality of order k D 1 follows from the next claim, which gives an improvement of the boxing inequality (Proposition 10.9) based on Maz0 ya’s truncation argument (cf. proof of Lemma 16.4): Claim. For every ' 2 Cc1 .RN /, we have ˆ 1 N 1 H1 .¹j'j > t º/ dt 6 C kr'kL1 .RN / : 0

Proof of the claim. Fix a smooth function H W R ! R such that .a/ H.s/ D 0 for every s 2 R such that jsj 6 21 , .b/ H.s/ D 1 for every s 2 R such that jsj > 1.

In particular,

¹j'j > t º  ¹H. 't / > 1º:

By the boxing inequality applied to the function H. 't /, we then have ˆ ˆ C2 ' N 1 jrH. t /j 6 H1 .¹j'j > t º/ 6 C1 jr'j: t ¹ 2t 6j'j6tº RN

296

17. Critical embedding

Thus, ˆ

1 0

N 1 H1 .¹j'j > t º/ dt 6 C2

ˆ

1 0

The claim now follows from Fubini’s theorem.

 ˆ

 dt jr'j : t t ¹ 2 6j'j6tº 4

Assume that N > 3, and we have established the trace inequality of order k for some k 6 N 2. Let  2 M.RN / be a nonnegative measure such that N  6 H1

.kC1/

:

For every ' 2 Cc1 .RN /, by the pointwise estimate j'j 6 .1=N /I1  jr'j satisfied by the Riesz potential (cf. Proposition 4.14) and by Fubini’s theorem, we have ˆ ˆ 1 I1  jr'j d k'kL1 .RN I/ D j'j d 6 N RN RN ˆ 1 1 kr'kL1 .RN II1 / : jr'j.I1  / D D N RN N Since 0 < N

.k C 1/ < N

1, the previous lemma yields N k I1   6 C3 H1 :

By the induction hypothesis, we can apply the trace inequality of order k with the measure I1  . We thus have kr'kL1 .RN II1 / 6 C4 kD k .r'/kL1 .RN / D C4 kD kC1 'kL1 .RN / : Combining the two estimates, we deduce the trace inequality of order k C 1, and the conclusion follows.  Proof of Proposition 17.15. The second inequality N capD 2;1 6 C 0 H1

2

is a consequence of scaling on balls. Indeed, for every closed ball BŒxI r  RN we have capD 2;1 .BŒxI r/ D capD 2;1 .BŒ0I 1/r N 2 : (17.13)

Given a compact set K  RN , let .B.xi I ri //i 2¹0;:::;`º be a family of balls covering K. By the monotonicity and subadditivity of the capacity (cf. Assertion .a/ after Proposition A.3), we have capD 2;1 .K/ 6

` X i D0

capD 2;1 .BŒxi I ri / 6 capD 2;1 .BŒ0I 1/

` X i D0

We then minimize the right-hand side with respect to the covering.

riN

2

:

17.4. W 1;2 and W 2;1 capacities

297

The first inequality relies on the second-order trace inequality (17.11) and on Frostman’s lemma (Proposition B.4). We begin by proving that, for every nonnegaN 2 tive measure  2 M.RN / such that  6 H1 , we have  6 C 00 capD 2;1 :

(17.14)

Indeed, for every compact set K  RN and every nonnegative function ' 2 Cc1 .RN / such that ' > 1 in K, by the second-order trace inequality we have ˆ .K/ 6 ' d 6 C 00 kD 2 'kL1 .RN / : RN

Minimizing the right-hand side with respect to ', we obtain estimate (17.14) on K. Now, for every compact set S  RN , Frostman’s lemma yields a nonnegative N 2 N 2 measure , supported in S , such that  6 H1 and .S / > 3 .N 2/ H1 .S /. Applying the capacitary estimate (17.14) on the set S , we get 3

.N 2/

N 2 H1 .S / 6 .S / 6 C 00 capD 2;1 .S /;

and the conclusion follows.



Exercise 17.3 (counterpart of Proposition 17.15 in R2 ). .a/ For every x 2 R2 and every r > 0, prove that capD 2;1 .BŒxI r/ D capD 2;1 .¹xº/: .b/ For every ' 2 Cc1 .R2 /, prove that k'kL1 .R2 / 6 kD 2 'kL1 .R2 / : .c/ Deduce that where ˛ D capD 2;1 .¹0º/ > 0.

0 ; capD 2;1 D ˛H1

We refer the reader to Maz0 ya’s book (Section 1.4.3 in [234]) for additional results on higher order trace inequalities. Proposition 17.15 has a counterpart for the equivalence between the W k;1 and HıN k capacities, with k 2 N and k < N , in the spirit of Proposition 16.12 above; see Lemma 3 in [80].

Chapter 18

Quasicontinuity

“The Dirichlet problem fails to be soluble, not by the non-existence of a harmonic function corresponding to the boundary conditions, but by the fact that this harmonic function fails to assume continuously the desired value.” Norbert Wiener

We characterize measures generating continuous potentials. We then apply the Evans–Vasilesco principle to investigate properties of diffuse measures on connection with continuous potentials.

18.1 Continuous potentials We begin with the following characterization of measures generating continuous potentials due to Aizenman and Simon, see Theorem 4.14 in [10], based on a functional condition introduced by Kovalenko and Semenov [184] (cf. Exercise 17.2): Proposition 18.1. Let  be a smooth bounded open set, and let  2 M./. x if and The linear Dirichlet problem with density  has a continuous solution in  1 x only if, for every  > 0, there exists C > 0 such that, for every  2 C0 ./, we have ˇ ˇˆ ˇ ˇ ˇ  dˇ 6 kkL1 ./ C C kkL1 ./ : ˇ ˇ 

x Proof of Proposition 18.1: “ H) ”. To prove the direct implication, let u 2 C00 ./ 1 x be the solution of the Dirichlet problem with density , and take 2 C0 ./ to be x we have chosen below. By the divergence theorem, for every  2 C01 ./ ˆ ˆ ˆ ˆ  d D u  D . u/   : 







Thus, ˇ ˇˆ ˇ ˇ ˇ  dˇ 6 k ˇ ˇ 

ukL1 ./ kkL1 ./ C k kL1 ./ kkL1 ./ :

300

18. Quasicontinuity

x such that k Given  > 0, we choose 2 C01 ./ the conclusion with C D k kL1 ./ .

ukL1 ./ 6 . We then have 

The argument of Aizenman and Simon to prove the sufficiency part relies on the Dirichlet problem ´ u C au D  in , (18.1) u D 0 on @, where a > 0. The existence of a solution for every  2 M./ can be established as in the case a D 0 (cf. proofs of Propositions 3.2 and 5.1). Lemma 18.2. Let  be a smooth bounded open set, and let  2 M./. Assume that, for every  > 0, the measure  satisfies the functional estimate of Proposition 18.1. If .an /n2N is a sequence of positive numbers diverging to infinity, and if un satisfies the Dirichlet problem (18.1) with coefficient an , then un 2 L1 ./ and lim kun kL1 ./ D 0:

n!1

x let n 2 C 1 ./ x be the solution of the Proof of Lemma 18.2. Given h 2 C 1 ./, 0 Dirichlet problem (18.1) with density h. We have ˆ ˆ ˆ un h D un . n C an n / D n d: 





Given  > 0, by the assumption on  we then get the estimate ˇˆ ˇ ˇ ˇ ˇ un hˇ 6 kn kL1 ./ C C kn kL1 ./ : ˇ ˇ 

By the contraction estimate (Lemma 3.8), we have

an kn kL1 ./ 6 khkL1 ./ ; and then kn kL1 ./ 6 2khkL1 ./ : Inserting these two inequalities into (18.2), we deduce that ˇˆ ˇ   ˇ ˇ ˇ un hˇ 6 2 C C khkL1 ./ : ˇ ˇ an 

(18.2)

18.1. Continuous potentials

301

x and since C 1 ./ x is dense in L1 ./, Since this estimate holds for every h 2 C 1 ./, 1 we conclude that un 2 L ./ and kun kL1 ./ 6 2 C Letting n ! 1, we get

C : an

lim sup kun kL1 ./ 6 2: n!1

Since this inequality holds for every  > 0, the lemma follows.



Proof of Proposition 18.1: “ (H ”. Take a sequence of positive numbers .an /n2N diverging to infinity, and, for each n 2 N, let un be the solution of the Dirichlet problem (18.1) with coefficient an . By Lemma 18.2, we have an un 2 L1 ./. Thus, the solution wn of the Dirichlet problem ´ wn D an un in , wn D 0

on @,

x Note that the function wn C un satisfies the linear Dirichlet probbelongs to C00 ./. lem with density : ´ .wn C un / D  in , wn C un D 0

on @,

By the uniqueness of the solution of the Dirichlet problem (Proposition 3.5), for every m; n 2 N we thus have wm wn D un um : Since .un /n2N converges to 0 in L1 ./, we deduce that .wn /n2N is a Cauchy x which converges uniformly to the solution of the linear Dirichlet sequence in C00 ./ problem with density . The proof of the proposition is complete.  A first-order counterpart of Proposition 18.1 concerning distributions that can be written as the divergence of a continuous vector field has been investigated by De Pauw and Pfeffer [110] (see also [111]). x in In the proof of the direct implication, we relied on the denseness of C01 ./ 0 x the space of continuous functions C0 ./. A stronger property holds in the setting of potentials: x is such that Proposition 18.3. Let  be a smooth bounded open set. If u 2 C00 ./ 1 x u 2 M./, then, for every  > 0, there exists 2 C0 ./ such that k

ukL1 ./ 6 

and k kL1 ./ 6 2kukM./ :

302

18. Quasicontinuity

Proof. Given a sequence of mollifiers .n /n2N , the sequence .n  u/n2N converges x Since the function n  u need not vanish on @, we take a uniformly to u in . x The function harmonic extension hn of .n  u/j@ in . n

D n  u

hn ;

x By the triangle inequality and the classical weak maximum prinbelongs to C01 ./. ciple (Corollary 1.10), k

n

ukL1 ./ 6 kn  u 6 kn  u

ukL1 ./ C khn kL1 ./ ukL1 ./ C kn  ukL1 .@/ :

As n ! 1, the right-hand side tends to zero. It thus suffices to take n for n sufficiently large to obtain the first estimate. The second estimate is satisfied by all functions n . Denote by U W RN ! R x By the extension property the extension of u by zero in the complement of . N (Corollary 7.4), it follows that U 2 M.R / and kU kM.RN / 6 2kukM./ : By Fubini’s theorem, we have that (Lemma 2.22) .n  u/ D n  U

in RN ,

and then k.n  u/kL1 .RN / 6 kU kM.RN / 6 2kukM./ : Since 

n

D .n  u/ in , the conclusion follows.



The following classical condition involving the fundamental solution characterizes the continuity of the Newtonian potential generated by a nonnegative measure. The proof relies on Dini’s theorem about the uniform convergence of monotone sequences of continuous functions on compact sets. Proposition 18.4. Let N > 3, and let  2 M.RN / be a nonnegative measure. The Newtonian potential N is continuous in RN if and only if the limit ˆ d.y/ lim D0 r !0 B.xIr / jx yjN 2 is uniform on every compact subset of RN .

18.2. Lusin property

303

Proof. Given a non-increasing sequence of positive numbers .rn /n2N converging to zero, let .Hn /n2N be a nondecreasing sequence of continuous functions Hn W Œ0; C1/ ! Œ0; C1/ such that .a/ Hn .t / D t if t 6 .b/ Hn .t / D 0 if t >

1 .2rn /N 1 .rn /N

2

2

,

.

For every n 2 N, the function un W RN ! R defined by ˆ  1 1 un .x/ D Hn .N 2/N RN jx yjN

2



d.y/

is continuous, and by properties .a/ and .b/ we have ˆ ˆ d.y/ d.y/ 06 6 .N 2/N ŒNu.x/ un .x/ 6 N 2 yj yjN B.xIrn / jx B.xI2rn / jx

2

:

If the quantity in the right-hand side converges uniformly to zero on a compact set K as n ! 1, then the sequence .un /n2N converges uniformly to Nu on K, whence Nu is continuous on K. Conversely, if Nu is continuous on a compact set K, then by Dini’s theorem the sequence of continuous functions .un /n2N converges uniformly to Nu on K, and so the integral in the left-hand side converges uniformly to zero on K.  Measures satisfying the assumption of the previous proposition are sometimes called Kato measures [10].

18.2 Lusin property To motivate the notion of quasicontinuity, let us begin with the following property discovered by Lusin [214], see also Theorem 7.10 in [134]: Proposition 18.5. Let vW A ! R be a Borel measurable function defined on a Borel set A  RN . Then, for every ı > 0, there exists a Borel subset E  A such that the restriction vjE is continuous in E and jA n Ej 6 ı. Lusin’s proof uses a combination of two tools: Egorov’s theorem concerning the uniform convergence of Cauchy sequences in L1 except for a set of small measure, and the density of continuous functions in L1 . A simple argument based solely on the outer regularity of the Lebesgue measure (Proposition 2.5) can be found in [130].

304

18. Quasicontinuity

Definition 18.6. Let A  RN be a Borel set. A function vW A ! R is quasicontinuous with respect to the W 1;q capacity if, for every ı > 0, there exists a Borel subset E  A such that the restriction vjE is continuous in E and capW 1;q .A n E/ 6 ı. We prove the following counterpart of Proposition 18.5 adapted to Sobolev functions: Proposition 18.7. For every u 2 W 1;q .RN /, the precise representative uO is quasicontinuous with respect to the W 1;q capacity. Proof. Let .un /n2N be a sequence in Cc1 .RN / converging strongly to u in W 1;q .RN / and converging pointwisely to uO in the Lebesgue set Lu (cf. Propositions 8.4, 8.6, and 8.7). Given a summable sequence of positive numbers .i /i 2N , and a subsequence .uni /i 2N to be chosen below, consider the set E D Lu \

1 \

¹juni C1

i D0

uni j < i º:

Since .i /i 2N is summable, the subsequence .uni /i 2N converges uniformly to uO in E, and so the function uj O E is continuous in E. To estimate the capacity of the set Lu n E, we observe that Lu n E 

1 [

i D0

¹juni C1

uni j > i º:

By the subadditivity of the capacity (Proposition A.3) and by the weak capacitary estimate (Lemma A.11), we then have capW 1;q .Lu n E/ 6 6

1 X

i D0 1 X i D0

capW 1;q .¹juni C1 1 kuni C1 iq

uni j > i º/

uni kqW 1;q .RN / :

Given ı > 0, we have the conclusion by choosing the subsequence .uni /i 2N such that the series in the right-hand side is smaller than ı.  Lusin’s theorem has also a counterpart for potentials: Proposition 18.8. Let  be a smooth bounded open set, and let  2 M./. If u satisfies the linear Dirichlet problem with density , then the precise representative uO is quasicontinuous with respect to the W 1;2 capacity.

18.2. Lusin property

305

The proof is based on the decomposition of the function u as u D T .u/ C S .u/ for some  > 0, where T W R ! R is the truncation function at levels ˙, and S W R ! R is defined for t 2 R by S .t / D t T .t /. Both functions T and S are Lipschitz-continuous, hence every Lebesgue point of u is also a Lebesgue point of T .u/ and S .u/ (Exercise 8.2). Moreover, in the Lebesgue set Lu , the precise representatives of T .u/ and S .u/ coincide with T .u/ O and S .u/, O respectively. Therefore, we have uO D T .u/ O C S .u/ O in Lu . (18.3) Proof of Proposition 18.8. By the interpolation inequality (Lemma 5.8), the function T .u/ belongs to W01;2 ./, whence by the previous proposition we have that T .u/ O is quasicontinuous in Lu with respect to the W 1;2 capacity. On the other hand, ¹S .u/ O ¤ 0º D ¹juj O > º  ¹Mu > º; where Mu denotes the maximal function associated to u. By the maximal inequality for potentials (Proposition 9.6), we then have C kkM./ :  Given a Borel set E  Lu such that the restriction T .u/j O E is continuous, it follows from decomposition (18.3) that the function uj O E \¹S .u/D0º is continuous. O Since Lu n .E \ ¹S .u/ O D 0º/  .Lu n E/ [ ¹S .u/ O ¤ 0º; capW 1;2 .¹S .u/ ¤ 0º/ 6 capW 1;2 .¹Mu > º/ 6

the subadditivity of the capacity (Proposition A.3) implies that

O ¤ 0º/ O D 0º// 6 capW 1;2 .Lu n E/ C capW 1;2 .¹S .u/ capW 1;2 .Lu n .E \ ¹S .u/ C 6 capW 1;2 .Lu n E/ C kkM./ :  Given ı > 0, take  > 0 such that C kkM./ 6 2ı , and then, using the quasicontinuity of T .u/, O choose E such that capW 1;2 .Lu n E/ 6 2ı . Therefore, we have O D 0º// 6 capW 1;2 .Lu n .E \ ¹S .u/ and this gives the conclusion.

ı ı C D ı; 2 2 

Corollary 18.9. Let  be a smooth bounded open set, let  2 M./, and let uO be the precise representative of the solution of the linear Dirichlet problem with density . If  is diffuse with respect to the W 1;2 capacity, then, for every  > 0, there exists a compact set K   such that the restriction uj O K is continuous in K and jj. n K/ 6 .

306

18. Quasicontinuity

Proof. By the quasicontinuity of the precise representative, given ı > 0 there exists a Borel set E  Lu such that uj O E is continuous in E and capW 1;2 .Lu n E/ 6 ı. Since the exceptional set  n Lu has zero W 1;2 capacity (Proposition 8.9), and since  is a diffuse measure, by the additivity of  we have jj. n E/ D jj.Lu n E/: Thus, jj. n K/ D jj.Lu n E/ C jj.E n K/:

for every compact set K  E. By the absolute continuity of  with respect to the W 1;2 capacity (Proposition 14.7), we can take ı > 0 such that capW 1;2 .Lu n E/ 6 ı implies  jj.Lu n E/ 6 : 2 Then, using the inner regularity of finite Borel measures (Proposition 2.5), we take a compact set K  E such that jj.E n K/ 6

 : 2

Therefore, jj. n K/ 6 , and we have the conclusion.



18.3 Continuity principle We consider a remarkable continuity principle between a potential and the measure generating this potential (cf. Corollary 18.9): Proposition 18.10. Let  be a smooth bounded open set, let  2 M./ be a nonnegative measure, and let uO be the precise representative of the solution of the linear Dirichlet problem with density . If  is diffuse with respect to the W 1;2 capacity and if K   is a compact set such that the restriction uj O K is continuous in K, then x the linear Dirichlet problem with density bK has a continuous solution in . This principle was proved by G. Evans (see Lemma II in [123], and Section II.5 in [124]) and by Vasilesco (see [332]). We first give three applications of Proposition 18.10. We begin with the converse of the removable singularity property for continuous solutions (cf. Proposition 13.1): Corollary 18.11. Let  be a smooth bounded open set. For every compact set S   such that capW 1;2 .S / > 0, there exists a positive measure  2 M./, supported in S , such that the solution of the linear Dirichlet problem with density  is continuous x in .

18.3. Continuity principle

307

Proof of Corollary 18.11. Since capW 1;2 .S / > 0, there exists a positive diffuse measure  2 M.RN / supported in S (Proposition A.17). Let uO be the precise representative of the solution of the linear Dirichlet problem with density . By Corollary 18.9, given  > 0 there exists a compact set K   such that the restriction uj O K is continuous in K and . n K/ 6 : By the continuity principle above, the Dirichlet problem with density bK has a continuous solution. For any  < ./, the measure bK is positive and the conclusion follows.  We also have the following functional characterization of diffuse measures (cf. Proposition 18.1): Corollary 18.12. Let  be a smooth bounded open set, and let  2 M./. If  is diffuse with respect to the W 1;2 capacity, then, for every  > 0, there exists C > 0 x we have such that, for every  2 C01 ./, ˇˆ ˇ ˇ ˇ ˇ  dˇ 6 .kkL1 ./ C kkL1 ./ / C C kkL1 ./ : ˇ ˇ 

Proof of Corollary 18.12. Let uO be the precise representative of the solution of the linear Dirichlet problem with density , and let K   be a compact set such that x we have the restriction uj O K is continuous in K. For every  2 C01 ./, ˇˆ ˇ ˇˆ ˇ ˇ ˇ ˇ ˇ ˇ  dˇ 6 ˇ  dbK ˇ C . n K/kkL1./ : ˇ ˇ ˇ ˇ 



By the quasicontinuity of potentials (Corollary 18.9), the compact set K can be chosen so that . n K/ 6 . By the continuity principle, the Dirichlet problem with density bK has a continuous solution. We may thus apply the direct implication of Proposition 18.1 to estimate the integral in the right-hand side.  We now prove a decomposition of diffuse measures as L1 perturbations of measures generated by continuous potentials, see Proposition 4.D.1 in [60] (cf. Proposition 14.13): Corollary 18.13. Let  be a smooth bounded open set, and let  2 M./. If  is x and f 2 L1 ./ diffuse with respect to the W 1;2 capacity, then there exist v 2 C00 ./ such that  D v C f in the sense of distributions in .

308

18. Quasicontinuity

Proof of Corollary 18.13. We may assume that  is nonnegative. Denote by uO the precise representative of the solution of the linear Dirichlet problem with density . Applying Corollary 18.9 inductively, we may decompose  as a strongly convergent series in M./ of the form (see Lemme 7 in [82] and [94]) D

1 X

bKj ;

j D0

such that the compact sets Kj are disjoint and uj O Kj is continuous in Kj . For every j 2 N, by the continuity principle the linear Dirichlet problem with density bKj x Take j 2 C 1 ./, x and write has a solution uj 2 C00 ./. 0 n X

j D0

bKj D

n X

j D0

. uj / D

Choose the functions

j



n hX

n i X .  / C j

.uj

1 P

j D0

vD

1 X

.uj

(18.4)

j D0

j D0

such that the series

In particular, the function

j /:

kuj

j kL1 ./

converges.

j/

j D0

x By Proposition 18.3, we may assume in addition that belongs to C00 ./. k

j kL1 ./ 1 P

6 2kbKj kM./ D 2.Kj /: j/

belongs to L1 ./ and, letting n ! 1

in (18.4), we get the decomposition of .



Hence, the function f D

. 

j D0

Measures that do not charge Borel sets of finite HN 1 Hausdorff measure have a characterization in the spirit of Corollaries 18.12 and 18.13 involving the divergence operator, see Theorem 1.4 and Corollary 1.5 in [287]. The argument in this case relies on the strong approximation of measures that are diffuse with respect to HN 1 (Proposition 14.3), and also on works by De Pauw and Pfeffer [110] and Phuc and Torres [278] on the characterization of the divergence of continuous vector fields. Proof of Proposition 18.10. Denote by uK the solution of the linear Dirichlet problem with density bK ; to simplify the notation, we systematically use precise representatives in the proof. By the boundedness principle (Lemma 13.2), uK is bounded, and so is defined at every point of  (Lemma 8.10 and Exercise 8.3). Since uK is harmonic in  n K, we only need to prove the continuity of uK at every point a 2 K. For this purpose, given r > 0 we write i e uK D uK C uK ;

18.3. Continuity principle

309

as a sum of the solutions of the linear Dirichlet problems with densities bK\B.aIr / and bKnB.aIr / . By the triangle inequality, for every x 2  we have juK .x/

i i e uK .a/j 6 uK .x/ C uK .a/ C juK .x/

e uK .a/j:

e is harmonic in B.aI r/, the last term in the right-hand side Since the function uK converges to zero as x ! a. Thus,

lim sup juK .x/ x!a

i i uK .a/j 6 lim sup uK .x/ C uK .a/:

(18.5)

x!a

To estimate the above lim sup, as for the solution uK we write u D ui C ue ; as a sum of the solutions of the linear Dirichlet problems with densities bB.aIr / and bnB.aIr / . By the weak maximum principle (Proposition 6.1), i uK 6 ui

almost everywhere in . Since the precise representative ui is defined at every point of K, this inequality holds everywhere in K. Since the restriction ujK is continuous in K, and ue is harmonic in B.aI r/, we estimate ui .z/, for every z 2 K, as ui .z/ D u.z/

ue .z/ 6 ju.z/

u.a/j C ui .a/ C jue .z/

ue .a/j:

Thus, lim sup ui .z/ 6 ui .a/:

(18.6)

z!a z2K

The connection between estimates (18.5) and (18.6) is established using a Harnack inequality: Claim. For every 0 < ı 6 14 d.a; @/, we have i sup uK 6 2N

B.aIı/

sup

ui :

B.aI2ı/\K

i Proof of the claim. It suffices to estimate uK at a point x 2 B.aI ı/nK. Since a 2 K and B.aI 4ı/  , we have that

d.x; K/ < ı

and d.x; @/ > 3ı:

i Thus, for every 0 < t < d.x; K/, we have B.xI t / b  n K. Since uK is harmonic in  n K, by the mean value formula for harmonic functions (Proposition 1.4) we have i i uK .x/ D uK : (18.7) B.xIt/

310

18. Quasicontinuity

Let r D d.x; K/, and take z 2 K such that jx

zj D d.x; K/ D r:

i In particular, we have B.zI 2r/ b . Since uK is nonnegative and superharmonic in , the monotonicity of the integral and the mean value property for superharmonic functions (Proposition 2.18 and Exercise 8.3) yield ˆ ˆ i i i uK 6 uK 6 !N .t C r/N uK .z/: (18.8) B.xIt/

B.zItCr /

Combining (18.7) and (18.8), we get i uK .x/ 6

Letting t ! r, we deduce that

 t C r N i uK .z/: t

i i uK .x/ 6 2N uK .z/ 6 2N ui .z/:

Since z 2 B.aI 2ı/, the claim follows.

4

We deduce from the claim that i lim sup uK .x/ 6 2N lim sup ui .z/

(18.9)

z!a z2K

x!a

We now combine estimates (18.5), (18.6) and (18.9). Making explicit the dependence of the function ui on the parameter r > 0, we deduce that lim sup juK .x/ x!a

uK .a/j 6 .2N C 1/uir .a/:

(18.10)

Assuming that N > 3, the Newtonian potential NbB.aIr / is nonnegative in , whence NbB.aIr / > bB.aIr / 1 x 0 in the sense of .C0 .// (Lemma 17.6). By the weak maximum principle (Proposition 6.1), we then have uir 6 NbB.aIr / almost everywhere in . This property holds quasi-everywhere (Exercise 8.4), and in particular at the Lebesgue point a. Since N.a/ < C1, it follows from the dominated convergence theorem that lim sup uir .a/ 6 lim NbB.aIr / .a/ D 0: r !0

r !0

Letting r ! 1 in (18.10), we get

lim sup juK .x/ x!a

and this concludes the proof.

uK .a/j D 0; 

Chapter 19

Nonlinear problems with diffuse measures

“En prenant des fonctions de moins en moins rapidement croissantes, telles que f 1C , f log f , f log log f , etc. on peut former une échelle de criteriums de plus en plus précis de continuité absolue de l’intégrale.”1 Charles de la Vallée Poussin

We characterize measures that are diffuse with respect to the W 1;2 capacity using the nonlinear equation with absorption u C g.u/ D : We do not assume any growth assumption on the nonlinearity g at infinity.

19.1 Unconditional existence We show in this section that the nonlinear Dirichlet problem with absorption ´ u C g.u/ D  in ; uD0

on @;

(19.1)

always has a solution if the measure  is diffuse, see Theorem 1.2 in [269]: Proposition 19.1. Let  be a smooth bounded open set, and let  2 M./. If the measure  is diffuse with respect to the W 1;2 capacity, then, for every continuous function gW R ! R satisfying the sign condition (Definition 4.20), the nonlinear Dirichlet problem with density  has a solution. Diffuse measures are thus suitable for existence purposes, and this is related to the trace inequality: for every ' 2 Cc1 ./, k'kL1 .Ijj/ 6 C k'kW 1;2 ./ : 1 “By taking functions growing less and less rapidly, like f 1C , f

(19.2)

log f , f log log f , etc., we can form a scale of increasingly precise critera for the absolute continuity of the integral.”

312

19. Nonlinear problems with diffuse measures

Indeed, .a/ measures  satisfying the trace inequality above belong to the dual space .W01;2 .//0 , and are dense in the class of diffuse measures on M./ (Propositions 14.1 and 14.2); .b/ for every  2 .W01;2 .//0 , the nonlinear Dirichlet problem with absorption has a variational solution u 2 W01;2 ./ satisfying the Euler–Lagrange equation (Propositions 4.21 and 4.23): for every v 2 W01;2 ./ \ L1 ./, ˆ



ru
rv C

ˆ



g.u/v D Œv:

(19.3)

Summarizing, the nonlinear Dirichlet problem with absorption has a solution for data  in a dense subset of the class of diffuse measures. The proof of Proposition 19.1 is based on a trick by Gallouët and Morel for L1 data, see Theorem 1 in [144], which we adapt here to diffuse measures using Lemma 19.2 below. Assume that  satisfies the trace inequality (19.2), whence  2 .W01;2 .//0 . Denoting the evaluation of  on a Sobolev function v 2 W01;2 ./ by Œv, then by the pointwise interpretation of the trace inequality (Proposition 16.5) we have that the precise representative vO belongs to L1 .I jj/, and Œv can be computed as an integral with respect to : ˆ Œv D vO d: 

This useful identity allows us to pass from the formalism of functional analysis to that of measure theory. Lemma 19.2. Let  be a bounded open set, let gW R ! R be a continuous function satisfying the sign condition, and let  2 M./ be a measure satisfying the trace inequality (19.2). If u 2 W01;2 ./ solves the Euler–Lagrange equation (19.3), then, for every s > 0, we have ˆ jg.u/j 6 jj.¹juj O > sº/: ¹juj>sº

This is the counterpart of estimate (4.12) for measure data. Taking in particular s D 0 in the estimate above, we deduce the absorption estimate kg.u/kL1 ./ 6 jj./:

(19.4)

19.1. Unconditional existence

313

Proof of Lemma 19.2. As in the proof of estimate (4.12), we apply the Euler– Lagrange equation with a test function of the form v D H.u/. Given s > 0 take a smooth nondecreasing function H W R ! R such that H.0/ D 0 and H.t / D sgn t for jt j > s. By composition with Sobolev functions (Exercise 4.6), we know that H.u/ 2 W01;2 ./. Since H is nondecreasing, we have ru
rH.u/ D H 0 .u/jruj2 > 0: By the sign condition on the nonlinearity g, we also have g.u/H.u/ D jg.u/jjH.u/j > jg.u/j¹juj>sº :

Applying the Euler–Lagrange equation with test function H.u/ 2 W01;2 ./, we then get ˆ ¹juj>sº

jg.u/j 6 ŒH.u/:

Since the function H is Lipschitz-continuous, the precise representative of H.u/ equals H.u/ O on the Lebesgue set Lu (Exercise 8.2). By the interpretation of ŒH.u/ as an integration with respect to , we then have ˆ ˆ jg.u/j 6 H.u/ O d: ¹juj>sº



We now apply this estimate to a sequence of smooth functions .Hn /n2N as above converging pointwise to zero in . s; s/. By the dominated convergence theorem, as n ! 1 we get ˆ ˆ jg.u/j 6 sgn uO d 6 jj.¹juj O > sº/: ¹juj>sº

¹juj>sº O

This gives the conclusion for s > 0, and the case s D 0 follows from the previous estimate as s ! 0.  Proof of Proposition 19.1. Applying the strong approximation property of diffuse measures to the positive and negative parts of  (Proposition 14.1), we find a sequence of measures .n /n2N such that .a/ for every n 2 N, jn j 6 Cn capW 1;2 , .b/ .n /n2N converges strongly to  in M./. Each measure n is obtained by contraction of  to some Borel subsets of  and, in particular, we have jn j 6 jj:

Since n 2 .W01;2 .//0 (Proposition 14.2), we have that the Dirichlet problem (19.1) with density n has a variational solution un 2 W01;2 ./ satisfying the Euler– Lagrange equation (19.3).

314

19. Nonlinear problems with diffuse measures

Claim. For every s > 0, there exists C1 > 0 such that, for every Borel set E   and every n 2 N, we have ˆ jg.un /j 6 C1 jEj C jj.¹juO n j > sº/: E

We postpone the proof of the claim, and conclude the proof of the proposition by proving that the sequence .g.un //n2N is equi-integrable. By the weak capacitary estimate (Proposition 9.6), for every s > 0 we have capW 1;2 .¹juO n j > sº/ 6

C2 kn s

g.un /kM./ :

Thus, by the triangle inequality and the absorption estimate (19.4), we get capW 1;2 .¹juO n j > sº/ 6

2C2 jj./: s

This estimate combined with the claim imply that the sequence .g.un //n2N is equiintegrable. Indeed, for every  > 0, by the absolute continuity of diffuse measures (Proposition 14.7) and by the capacitary estimate above, there exists s > 0 such that, for every n 2 N,  jj.¹juO n j > sº/ 6 : 2 For this given s > 0, it follows that, for every Borel set E   such that jEj 6 =2C1 , we have ˆ   jg.un /j 6 C1 C D : 2C 2 1 E Hence, the sequence .g.un //n2N is equi-integrable. By the triangle inequality and the absorption estimate (19.4), for every n 2 N we have kun kM./ 6 jn j./ C jn j./ 6 2jj./: It then follows from the compactness of solutions of the linear Dirichlet problem (Proposition 5.9) that there exists a subsequence .unk /k2N which converges in L1 ./ and almost everywhere in  to some function u. Since the sequence .g.unk //k2N is equi-integrable and converges almost everywhere to g.u/, we conclude by Vitali’s theorem (Exercise 19.1) that .g.unk //k2N converges to g.u/ in L1 ./. Therefore, u is a solution of the nonlinear Dirichlet problem with density . It remains to establish the claim: Proof of the claim. For every Borel set E   and every s > 0, we have ˆ ˆ ˆ jg.un /j 6 jg.un /j C jg.un /j: E

E \¹jun j6sº

¹jun j>sº

19.2. Measures must be diffuse

315

Since g is bounded on the compact set Œ s; s, the first integral in the right-hand side can be estimated as follows: ˆ jg.un /j 6 C1 jE \ ¹jun j 6 sºj 6 C1 jEj; E \¹jun j6sº

where C1 > 0 is any upper bound of jgj over Œ s; s. To estimate the second integral, we apply the a priori estimate (Lemma 19.2): ˆ jg.un /j 6 jn j.¹juO n j > sº/: ¹jun j>sº

Since jn j 6 jj, the claim follows. The proof of the proposition is complete.

4 

Exercise 19.1 (Vitali’s theorem). Let  be a nonnegative finite measure on X. Prove that if .fn /n2N is an equi-integrable sequence of functions in L1 .XI / converging almost everywhere to f , then .fn /n2N converges to f in L1 .XI /. Proposition 19.1 stems from the pioneering works by Brezis and Strauss [69] for L1 data and by Brezis and Browder [54] for .W01;2 /0 data, see Propositions 3.7 and 4.23 above. The two results can be combined to give the conclusion for an arbitrary diffuse measure using its decomposition in .W01;2 /0 CL1 by Boccardo, Gallouët, and Orsina, see Theorem 4.B.4 in [60].

19.2 Measures must be diffuse Proposition 19.1 gives a large class of measures for which the nonlinear Dirichlet problem always has a solution, regardless of the growth rate of the nonlinearity. One might wonder whether there are other measures which have such a property. Brezis, Marcus, and Ponce gave a negative answer to this question, see Theorem 4.14 in [60]: Proposition 19.3. Let  2 M./. If, for every continuous nondecreasing function gW R ! R such that g.0/ D 0, the nonlinear Poisson equation u C g.u/ D  has a solution in the sense of distributions in , then the measure  is diffuse with respect to the W 1;2 capacity.

316

19. Nonlinear problems with diffuse measures

The proof of this result relies on the following observation by de la Vallée Poussin, see Remarque 23 in [105]: Lemma 19.4. For every nonnegative function f 2 L1 ./, there exists a nonnegative nondecreasing continuous function hW Œ0; C1/ ! R such that h.0/ D 0, h.t / D C1; t!C1 t lim

and h.f / 2 L1 ./. Proof of Lemma 19.4. We take h of the form h.t / D .t /t; where W Œ0; C1/ ! R is a nondecreasing continuous function diverging to infinity. Given an increasing sequence .n /n2N of positive numbers to be chosen later on, by additivity of the integral to write ˆ ˆ 1 ˆ X h.f / D h.f / C h.f /: ¹f 0

is a nonnegative nondecreasing continuous function such that h .0/ D 0. The Legendre transform is defined to be the smallest function h W Œ0; C1/ ! R satisfying Young’s inequality: for every s > 0 and t > 0, st 6 h.s/ C h .t /: 0

For example, if h.s/ D s p =p for some p > 1, then h .t / D t p =p 0 , where p 0 is the conjugate exponent of p. Proof of Lemma 19.5. We only prove that the Legendre transform h is continuous; the other properties follow directly from the definition. We actually prove that h is Lipschitz-continuous in every interval of the form Œ0; . We first shrink the set where the supremum is computed. Indeed, by the superlinearity of h, for every  > 0 there exists M > 0 such that if s > M , then h.s/ > s: Thus, for every t 2 Œ0; , we have h .t / D sup ¹st

(19.7)

h.s/º:

06s6M

Indeed, for every s > M , st

h.s/ 6 st

s D .t

/s 6 0:

Since h .t / is nonnegative, we may thus discard the parameters s > M , without changing the value of the supremum. We now prove the Lipschitz continuity of the Legendre transform h on Œ0;  with Lipschitz constant M . For this purpose, let t;  2 Œ0;  be such that t 6  . For every 0 6 s 6 M , we have h .t / > st

h.s/ D s

h.s/

s.

t / > s

h.s/

M.

t /:

318

19. Nonlinear problems with diffuse measures

Maximizing the right-hand side with respect to s, we get h .t / > h . /

M.

t /:

Since h is nondecreasing, we deduce that 0 6 h . /

h .t / 6 M.

t /:

Therefore, h is Lipschitz-continuous in Œ0; . Since  > 0 is arbitrary, the conclusion follows.  Proof of Proposition 19.3. On the half line Œ0; C1/, take g to be the Legendre transform of some nonnegative continuous function h such that h.0/ D 0 to be chosen below; we extend g to R as an odd function. Denoting by u a solution of the nonlinear Poisson equation with nonlinearity g, then, for every ' 2 Cc1 ./, we have ˆ ˆ ˆ ' d D u ' C g.u/': 





We can estimate the first integrand in the right-hand side using Young’s inequality: ju 'j 6 jg.u/j C h.j'j/: We now explain how the function h is chosen. Given a compact set K   such that capW 1;2 .K/ D 0, the equivalence of capacities (cf. Proposition 12.2) shows that there exists a sequence .'n /n2N of nonnegative functions in Cc1 ./ such that .a/ .'n /n2N converges pointwise to the characteristic function K , .b/ .'n /n2N is bounded in L1 ./, .c/ .'n /n2N converges to 0 in L1 ./. On the one hand, by the dominated convergence theorem, we have ˆ ˆ jg.u/j'n D 0: 'n d D .K/ and lim lim n!1



n!1

(19.8)



On the other hand, since the sequence .'n /n2N converges in L1 ./, by the partial converse of the dominated convergence theorem (Proposition 4.9), there exist a subsequence .'nk /k2N and f 2 L1 ./ such that, for every k 2 N, j'nk j 6 f and .'nk /k2N converges almost everywhere to 0. By Lemma 19.4, there exists a nonnegative continuous superlinear function hW Œ0; C1/ ! R such that h.0/ D 0 and h.f / 2 L1 ./. Then, for every k 2 N, we have ju 'nk j 6 juf j 6 jg.u/j C h.f /:

19.2. Measures must be diffuse

319

Since the sequence .'nk /k2N converges almost everywhere to 0, by the dominated convergence theorem we have ˆ lim u 'nk D 0: (19.9) k!1



Combining limits (19.8) and (19.9), we deduce that .K/ D 0 for every compact set K   such that capW 1;2 .K/ D 0. By the inner regularity of Borel measures (Proposition 2.5), this implies that  is a diffuse measure.  A further question would be whether there exists some very large nonlinearity g such that the nonlinear Dirichlet problem (19.1) has a solution if and only if  is diffuse with respect to the W 1;2 capacity. The answer is negative, see Theorem 1 in [286]: for every continuous nondecreasing function gW R ! R, there exists a positive measure  concentrated in a compact set of zero W 1;2 capacity such that the nonlinear Dirichlet problem has a solution for this nonlinearity g. An example of nonlinear Dirichlet problem for which diffuse measures are the only good measures, but with an explicit dependence on the gradient, is ´ u C ujruj2 D  in ; uD0

on @:

This problem has been studied by Boccardo, Gallouët, and Orsina [33]; see also [258] and [63] for further extensions and related results.

Chapter 20

Extremal solutions “All animals are equal, but some animals are more equal than others.” George Orwell

We investigate the existence of the smallest and largest solutions of the nonlinear Dirichlet problem using the method of sub- and supersolutions. We also prove that the vanishing average condition lim

!0

¹d@ 0, we have ˆ juj 6 C  2 kkM./ ; ¹d@ 0 depending on . We denote by d@ W  ! R the distance to the boundary, defined for x 2  by d@ .x/ D d.x; @/: Since the open set  is assumed to be smooth and bounded, there exist constants C 0 ; C 00 > 0 such that, for every 0 <  6 1, C 0  6 j¹d@ < ºj 6 C 00 :

322

20. Extremal solutions

According to the proposition above, the solution u thus satisfies the vanishing average condition: lim

!0

¹d@ 0, the function  W  x ! R defined by  2 C01 ./.    D H  x whence is an admissible test function. In the proof, we require belongs to C01 ./, the following further properties on H and : .a/ H is bounded, nondecreasing, concave, and such that H 00 .0/ < 0, .b/  is positive and superharmonic in . Using  as a test function in the Dirichlet problem, by the boundedness of H we get ˆ ˆ u  D  d 6 C1 kkM./ : 



Claim. The above properties on H and  ensure that  is superharmonic in , and there exists N > 0 such that, for every 0 <  6 N , we have  >

c 

in ¹d@ < º,

for some constant c > 0. Proof of the claim. We have  D H 0

 

  1  C H 00 jrj2 :   

Since H is nondecreasing and  is superharmonic in , we then get 1 00    jrj2 : H  >  

(20.1)

Due to the concavity of H , the right-hand side is nonnegative, whence  is superharmonic in . Since H 00 .0/ < 0 and H 00 is continuous, there exist c1 > 0 and ı > 0 such that H 00 > c1 in Œ0; ı.

20.1. Boundary data revisited

323

Since  is positive and superharmonic in , by the Hopf lemma (see Section 6.4.2 in [125]) we deduce that jrj > 0 on @. Thus, there exist N > 0 and c2 > 0 such that jrj > c2 in ¹d@ 6 N º. We deduce from (20.1) that

 >

c 

in ¹= 6 ıº \ ¹d@ 6 N º.

Since  vanishes on @, the mean value inequality yields  6 krkL1 ./ d@ : Choosing  such that krkL1 ./ 6 ı;

it follows that, for every 0 <  6 N and every x 2  such that d@ .x/ < , we have .x/= 6 ı

and d@ .x/ 6 N ;

whence and the claim follows.

¹d@ < º  ¹= 6 ıº \ ¹d@ 6 N º; 4

By the claim, since u is nonnegative, we have ˆ cu 6 C1  2 kkM./ ; ¹d@ N , the conclusion follows from the linear elliptic L1 estimate (Proposition 3.2) kukL1 ./ 6 C kkM./ : If  is a signed measure, then we may proceed as follows. By the weak maximum principle (Proposition 6.1), the solution v of the linear Dirichlet problem with density jj is nonnegative and satisfies v 6 u 6 v: Thus, for every  > 0, we get ˆ ˆ juj 6 ¹d@ 0. Therefore, uC D max ¹u; 0º 6 v: Thus, for every  > 0, we get 06

ˆ

C

u 6 ¹d@ t º is smooth and strictly contained in . Applying Green’s identity on ¹ > t º with test function D  t , we get ˆ ˆ ˆ @ u  6 . t / d u d: @n ¹>tº ¹>tº @¹>tº For every x 2 @¹ > t º, we have @ .x/ D @n

jr.x/j:

326

20. Extremal solutions

Since t is a regular value of , we also have @¹ > t º D ¹ D t º. Thus, ˆ ˆ ˆ @ ujrj d 6 uC jrj d; u d D @n ¹Dtº ¹Dtº @¹>tº

which gives

ˆ

u  6 ¹>tº

ˆ

. ¹>tº

t / d C

ˆ

¹Dtº

uC jrj d:

(20.3)

x is such that  > 0 in  and @ < 0 on @, then Claim. If  2 C01 ./ @n ˆ uC jrj d D 0: lim inf t!0

¹Dtº

Proof of the claim. By the co-area formula (see Theorem 1.23 in [147]), for every ˛ > 0 and every  > 0 we have  ˆ ˛  ˆ ˆ C u jrj d dt D uC jrj2 : ¹Dtº

0

¹0 0 such that  > ˛d@ . Thus, for every  > 0 we get ¹0 <  < ˛º  ¹d@ < º:

We then have the estimate ˆ ˛  ˆ 0

C

¹Dtº



u jrj d dt 6 C

ˆ

uC : ¹d@ 0 in , x we proceed as For an arbitrary test function  2 C01 ./ 1 x N follows. Let  2 C0 ./ be any function satisfying the assumptions of the claim. For every ı > 0, the function  C ı N also satisfies these assumptions, and we get ˆ ˆ N 6 . C ı / N d: u . C ı / 

Letting ı ! 0, the conclusion follows.





20.2. Method of sub- and supersolutions

327

There is an alternative proof of the implication (ii) H) (i) based on the linear Dirichlet problem ´ u D  in , (20.4) u D  on @,

where  is a finite measure on @. The measure  is called the boundary trace of u, see Theorem 2.13 in [335] or Lemma 1.4 in [226]. Problem (20.4) was first studied by Brezis in 1972 in an unpublished work when  2 L1 .@/, see Lemma 4.1 in [148] or Lemma 2.5 in [335]. The existence of solutions with boundary measure data follows from an estimate of the solutions in Lp ./, for 1 6 p < NN 1 , using the Poisson kernel. In this case, the gradient ru need not belong to L1 ./, but some estimates of ru in L1 spaces with weights are available [112]. Concerning the proof of Proposition 20.2, one first shows that there exist nonpositive measures  2 M.@/ and  2 Mloc ./ such that d@  2 M./ and, for every x we have nonnegative function  2 C01 ./, ˆ ˆ ˆ ˆ @ d: u  D  d C  d @n    @ The existence of the measure  is a consequence of Schwartz’s characterization of positive distributions (Proposition 2.20), while the existence of  is a consequence of Herglotz’s theorem concerning the boundary trace of positive harmonic functions, see Theorem 2.13 in [335]. One may then conclude using the strategy of the proof of Lemma 1.5 in [222].

20.2 Method of sub- and supersolutions The method of sub- and supersolutions relies on the idea that there should be a solution between a subsolution and a supersolution of the nonlinear Dirichlet problem ´ u C g.u/ D  in ; uD0

on @:

The notion of subsolution is inspired by Littman–Stampacchia–Weinberger’s concept of weak solution: Definition 20.4. Let gW R ! R be a continuous function, let  be a smooth bounded open set, and let  2 M./. A function v 2 L1 ./ is a subsolution of the nonlinear x Dirichlet problem if g.v/ 2 L1 ./ and, for every nonnegative function  2 C01 ./, we have ˆ ˆ ˆ v  C g.v/ 6  d: 





328

20. Extremal solutions

Thus, x 0. v C g.v/ 6  in the sense of .C01 .//

The boundary condition v 6 0 on @ is implicitly encoded in this weak formulation. Indeed, it follows from Proposition 20.2 that a function v is a subsolution of the nonlinear Dirichlet problem if and only if in the sense of distributions in 

v C g.v/ 6 

and the positive part v C satisfies the vanishing average condition lim

!0

¹d@ f x 0 , then in the sense of .C01 .// uC > ¹u>0º f x 0. in the sense of .C01 .// Proof of Lemma 20.8. By Kato’s inequality (Proposition 6.9), uC > ¹u>0º f in the sense of distributions in . Applying the direct implication of Proposition 20.2, we also have lim

!0

¹d@ g.v/

g.v/ N

x 0 . We now apply Kato’s inequality up to the boundary in the sense of .C01 .// (Lemma 20.8) to the function u v, N .u

v/ N C > ¹u>vº N Œg.v/

g.v/ N

x 0 . In the set ¹u > vº, in the sense of .C01 .// N we have v D F .v/ D v, N and it follows that .u v/ N C>0 x 0 . By the weak maximum principle (Proposition 6.1), in the sense of .C01 .// we deduce that .u v/ N C D 0 almost everywhere in , and the claim follows. 4 We claim that F satisfies the assumptions of Schauder’s fixed point theorem: .a/ the map F is continuous with respect to the L1 norm, .b/ the set F .K/ is relatively compact in L1 ./.

20.2. Method of sub- and supersolutions

331

To show this, take a sequence .vn /n2N in K converging to v in L1 ./, and denote by un the solution of the Dirichlet problem (20.5) associated to the function vn . By the definition of F as a pointwise truncation, for every n 2 N we have jF .vn /

F .v/j 6 jun

uj

in ,

whence kF .vn /

F .v/kL1 ./ 6 kun

ukL1 ./ :

Since un u is the solution of the linear Dirichlet problem with density g.v/ g.vn/, by the elliptic L1 estimate (Proposition 3.2) we get kF .vn /

F .v/kL1 ./ 6 C1 kg.vn /

g.v/kL1./ :

By the integrability condition (Lemma 20.7), there exists h 2 L1 ./ such that, for every n 2 N, jg.vn /j 6 h

almost everywhere in . Since .vn /n2N converges to v in L1 ./, it follows from the dominated convergence theorem and its partial converse (Proposition 4.9) that the sequence .g.vn //n2N converges to g.v/ in L1 ./ (cf. Exercise 4.9). The sequence .F .vn //n2N thus converges to F .v/ in L1 ./, whence F is continuous. To prove that F .K/ is contained in a compact subset of L1 ./, take a sequence .vn /n2N in K. By the Sobolev embedding of solutions of the linear Dirichlet problem (Proposition 5.1), for every n 2 N we have kun kW 1;1 ./ 6 C2 k

g.vn /kM./ :

Using the triangle inequality and the integrability condition, we deduce that kun kW 1;1 ./ 6 C2 .kkM./ C khkL1 ./ /: The sequence .un /n2N is thus bounded in W01;1 ./. By the Rellich–Kondrashov compactness theorem (Proposition 4.8), there exists a subsequence .uni /i 2N converging strongly in L1 ./. Since kF .vni /

F .vnj /kL1 ./ 6 kuni

unj kL1 ./ ;

the sequence .F .vni //i 2N is Cauchy in L1 ./, and so it converges in L1 ./. It thus follows from Schauder’s fixed point theorem that F has a fixed point v 2 K. By the claim, v satisfies the nonlinear Dirichlet problem with density . 

332

20. Extremal solutions

The method of sub- and supersolutions is reminiscent of works by Picard [279], using a monotone iteration scheme, and by Peano [272] and [273]. Perron [276] and Scorza Dragoni [305] used sub- and supersolutions to prove the existence of solutions of first- and second-order ordinary differential equations (ODEs). Different aspects of this method in the setting of ODEs are illustrated in the book [100]. Concerning elliptic partial differential equations, Nagumo [259] proved the method of sub- and supersolutions using Schauder’s fixed point theorem, and this is the strategy that we have adopted, see [86] and Theorem 1.1 in [253]. Such an approach separates the issue of existence of solutions from the question of regularity, which can be established using the elliptic regularity theory. It also applies to nonlinearities gW   R ! R satisfying Carathéodory’s condition following the same argument; the counterpart of Lemma 20.7 in this case can be found in [185] and Lemma 17.2 in [186].

20.3 Nonlinear Perron–Remak method We establish the existence of a largest solution of the nonlinear Dirichlet problem using the following counterpart of the Perron–Remak method (cf. Proposition 12.6): Proposition 20.9. Let  be a smooth bounded open set, let gW R ! R be a continuous function, and let  2 M./. If the nonlinear Dirichlet problem has a subsolution v and a supersolution vN such that v 6 vN almost everywhere in , and if the nonlinN N earity g satisfies the integrability condition with respect to v and v, N then N (i) among all subsolutions v such that v 6 vN almost everywhere in , there exists a largest subsolution, which we denote by u, N

(ii) the function uN is a solution of the nonlinear Dirichlet problem with density . The proof is based on a property about the maximum of subsolutions of the nonlinear Dirichlet problem (cf. Lemma 12.7): Lemma 20.10. Let  be a smooth bounded open set, let gW R ! R be a continuous function, and let  2 M./. If v1 and v2 are subsolutions of the nonlinear Dirichlet problem with density , then max ¹v1 ; v2 º is also a subsolution. Granting this statement, we now establish the existence of a largest subsolution in the lines of the linear Perron–Remak method:

20.3. Nonlinear Perron–Remak method

333

Proof of Proposition 20.9. Assertion (i). Let 8 9 ˇ ˇ v 6 vN almost everywhere in , and f1

and u2 > f2

in the sense of distributions in , then  max ¹u1 ; u2 º > ¹u1 >u2 º f1 C ¹u2 >u1 º f2 C ¹u1 Du2 º

f1 C f2 2

in the sense of distributions in . We give a direct proof based on a smooth approximation of the function max. This statement can also be deduced from Kato’s inequality for measures (Proposition 6.10 and Corollary 6.15) by noticing that, for every a1 ; a2 2 R, we have max ¹a1 ; a2 º D

a1 C a2 C .a1

a2 /C C .a2 2

a1 /C

:

20.3. Nonlinear Perron–Remak method

335

Proof of Proposition 20.11. Given a smooth function H W R ! R, let wW  ! R be the function defined by wD

u1 C u2 C H.u1 2

u2 /

:

We first assume that u1 and u2 are smooth. If H is convex, then w >

1 C H 0 .u1 2

u2 /

u1 C

1

H 0 .u1 2

u2 /

u2 :

Suppose that 1 6 H 0 .t / 6 1 for every t 2 R. In this case, the coefficients of u1 and u2 in the inequality above are nonnegative. By the assumption on u1 and u2 , we deduce that w >

1 C H 0 .u1 2

u2 /

f1 C

1

H 0 .u1 2

u2 /

f2 :

(20.6)

We now prove the same inequality, assuming that u1 and u2 are merely L1 functions. For this purpose, let ! b  be an open set, and let .n /n2N be a sequence of mollifiers such that ! supp n b . We have (Lemma 2.22) .n  u1 / > n  f1

and .n  u2 / > n  f2

pointwise in !, whence, by estimate (20.6), the function wn W ! ! R defined by wn D

n  u1 C n  u2 C H.n  u1 2

n  u2 /

satisfies wn >

1 C H 0 .n  u1 2

n  u2 /

 n  f1 C

1

H 0 .n  u1 2

n  u2 /

 n  f2

pointwise in !. Letting n ! 1, we deduce that (20.6) is satisfied in the sense of distributions in !. Since ! b  is arbitrary, this inequality holds in the sense of distributions in . To get the conclusion, we consider an approximation of the absolute value by a sequence of smooth convex functions .Hn /n2N such that .a/

1 6 Hn0 .t / 6 1 for every t 2 R,

.b/ lim Hn0 .t / D n!1

1 for every t < 0,

.c/ lim Hn0 .t / D 1 for every t > 0, n!1

.d / Hn0 .0/ D 0 for every n 2 N. Applying estimate (20.6) with Hn , and letting n ! 1, we conclude using the dominated convergence theorem. 

336

20. Extremal solutions

Proof of Lemma 20.10. Let w be the solution of the linear Dirichlet problem ´ w D  in , wD0

on @.

We have .v1

w/ > g.v1 / and .v2

w/ > g.v2 /

x 0 , hence in the sense of distributions in . Moreover, in the sense of .C01 .// max ¹v1 ; v2 º D max ¹v1

w; v2

wº C w:

Applying Kato’s inequality (Proposition 20.11) to the functions v1 we get  max ¹v1 w; v2 wº > g.max ¹v1 ; v2 º/;

w and v2

w,

and then

 max ¹v1 ; v2 º C g.max ¹v1 ; v2 º/ 6 

in the sense of distributions in . To show that max ¹v1 ; v2 º is a subsolution of the nonlinear Dirichlet problem, it remains to study the behavior of this function near the boundary of . Note that 0 6 .max ¹v1 ; v2 º/C 6 .v1 /C C .v2 /C : Thus, for every  > 0, we have 06 ¹d@ 3, and let gW R ! R be a convex function. If g satisfies the integral assumption of Proposition 21.2, then, for every nonnegative measure  2 M.RN /, the Newtonian potential N satisfies g.N/ 2 L1loc .RN /. Proof of Lemma 21.3. We may assume that .RN / > 0, and we write ˆ d.y/  N.x/ D ; N 2 .RN / N jx yj R

21.1. Subcritical case

339

where  D .RN /=.N 2/N . Since =.RN / is a probability measure and g is a convex function, by Jensen’s inequality we have ˆ  d.y/   : (21.3) g.N.x// 6 g jx yjN 2 .RN / RN By the monotonicity of g and the integration formula in polar coordinates, for every R > 0 we have ˆ ˆ       g dx 6 dz g jx yjN 2 jzjN 2 B.0IR/ B.0IR/ ˆ R    g N 2 r N 1 dr: D N r 0

Making the change of variables t D =r N 2 , we then get ˆ 1 ˆ    g.t / dt dx 6 C1 : g N N 2  jx yj B.0IR/ tN 2 t RN 2 Since the right-hand side is bounded from above independently of y, it follows from estimate (21.3) and Tonelli’s theorem that g.N/ 2 L1 .B.0I R//.  Proof of Proposition 21.2: “ (H ”. Let 1 D max ¹; 0º and 2 D min ¹; 0º. The function N1 is nonnegative. Thus, by the sign condition, g.N1 / is also nonnegative. By the previous lemma, we have g.N1 / 2 L1 ./. Since N1 is a nonnegative supersolution of the nonlinear Poisson equation with density , we deduce that N1 is a supersolution of the nonlinear Dirichlet problem (Lemma 17.6). By Property (21.2), we also have that g.N2 / 2 L1 ./. Arguing as above, it follows that N2 is a nonpositive subsolution of the nonlinear Dirichlet problem with density . Since g is nondecreasing, we may apply the method of sub- and supersolutions (Proposition 20.5) to conclude that the nonlinear Dirichlet problem with density  has a solution u such that N2 6 u 6 N1 :



To prove the direct implication of Proposition 21.2, we first study the behavior of spherical averages of potentials around a point. Lemma 21.4. Let N > 3. If u 2 L1loc ./ is such that u 2 Mloc ./, then, for every a 2 , 1 u.¹aº/: lim r N 2 u d D r !0 .N 2/N @B.aIr /

340

21. Absorption problems

Proof of Lemma 21.4. We temporarily assume that u is smooth. By the divergence theorem, we have (cf. proof of Lemma 1.5) ˆ d 1 @u d D u d D u: ds @B.aIs/ N s N 1 B.aIs/ @B.aIs/ @n By the fundamental theorem of calculus, for every 0 < r <  < d.a; @/ we then get  ˆ  ˆ ds 1 (21.4) u N 1 : u d D u d  s N r B.aIs/ @B.aIr / @B.aI/ When u is merely L1loc ./ and such that u 2 Mloc ./, this identity holds for almost every r; this can be verified using an approximation argument (cf. Lemma 2.22). Since ˆ lim u D u.¹aº/; s!0 B.aIs/

we have lim .N

r !0

2/r

N 2

Multiplying identity (21.4) by r

ˆ

 r

N 2

 ds u N 1 D u.¹aº/: s B.aIs/

ˆ

, and letting r ! 0, the conclusion follows.



Proof of Proposition 21.2: “ H) ”. Let u be the solution of the nonlinear Dirichlet problem with density ıa , for some a 2 . By the integration formula in polar coordinates, for every 0 <  < d.a; @/ we have    ˆ ˆ  ˆ ˆ  N 1 g.u/ D r g.u/ d dr D N g.u/ d dr: B.aI/

0

0

@B.aIr /

@B.aIr /

Since g is convex, by Jensen’s inequality we have  g.u/ d > g

@B.aIr /

@B.aIr /

 u d :

Given 0 <  < 1=.N 2/N , the previous lemma shows that there exists  > 0 such that, for almost every 0 < r 6 , u d > @B.aIr /

Since g is nondecreasing, we deduce that ˆ ˆ g.u/ > N B.aI/

 rN 2



rN 0

Making the change of variables t D =r N

2

1

g

:

  rN

2



dr:

, we have the conclusion.



21.2. Contraction and stability

341

Exercise 21.1 (Section 5 in [333]). Let   R2 . Prove that if u 2 L1loc ./ is such that u 2 Mloc ./, then, for every a 2 , we have 1 r !0 log 1=r lim

@B.aIr /

u d D

1 u.¹aº/: 2

Exercise 21.2 (Corollary 1 in [333]). Let   R2 be a smooth bounded open set, and let gW R ! R be a convex function satisfying the sign condition. Prove that the nonlinear Dirichlet problem has a solution for every  2 M./ if and only if, for every ˛ > 0, we have ˆ 1

g.t / e

˛t

0

dt < 1:

21.2 Contraction and stability It is convenient to have a contraction estimate available for arbitrary solutions of the nonlinear Dirichlet problem with absorption, not necessarily in the variational setting (cf. Lemmata 3.8 and 19.2): Proposition 21.5. Let  be a smooth bounded open set, and let gW R ! R be a continuous function satisfying the sign condition. If u is a solution of the nonlinear Dirichlet problem with density  2 M./, then we have kg.u/kL1 ./ 6 kkM./ : The estimate is formally obtained using sgn u as a test function. Since this is not a legitimate test function, we rely on a Lipschitz approximation of the sign function. Proof of Proposition 21.5. Let .fn /n2N be a sequence in L2 ./ that converges to g.u/ in L1 ./, and let .n /n2N be a sequence in L2 ./ that converges weakly to  in the sense of measures on  and such that (Proposition 2.7) lim kn kL1 ./ D kkM./ :

n!1

Given n 2 N, let un 2 W01;2 ./ be the solution of the linear Dirichlet problem ´ un D n fn in ; un D 0

on @:

Take  > 0, and let S W R ! R be the function defined for t 2 R by 8 ˆ < 1 if t < ; S .t / D t = if  6 t 6 ; ˆ : 1 if t > :

342

21. Absorption problems

Using S .un / 2 W01;2 ./ as a test function in the linear Dirichlet problem above, we deduce that ˆ ˆ ˆ jrun j2 S0 .un / C fn S .un / D S .un /n : 





Since the first integrand is nonnegative, we get ˆ ˆ fn S .un / 6 jn j D kn kL1 ./ : 

(21.5)



By the compactness of solutions of the linear Dirichlet problem (Proposition 5.9), the sequence .un /n2N has a subsequence .unk /k2N that converges in L1 ./ to some function v. Since u and v satisfy the same linear Dirichlet problem with density  g.u/, by the uniqueness of solution of the linear problem (Proposition 3.5) we have u D v almost everywhere in . This implies that the entire sequence .un /n2N converges to u in L1 ./. By the dominated convergence theorem and its partial converse (Proposition 4.9), the sequence .fn S .un //n2N converges to g.u/S .u/ in L1 ./. Letting n ! 1 in estimate (21.5), we obtain ˆ g.u/S .u/ 6 lim kn kL1 ./ D kkM./ : n!1



Thus, letting  ! 0 we get ˆ



g.u/ sgn u 6 kkM./ :

The estimate follows using the sign condition.



A second tool in the study of nonlinear Dirichlet problems with absorption is the stability property under strong convergence of measures: Proposition 21.6. Let  be a smooth bounded open set, and let gW R ! R be a continuous function satisfying the sign condition and the integrability condition below. If .n /n2N is a nondecreasing sequence of measures in M./ converging strongly to , and if, for every n 2 N, the nonlinear Dirichlet problem with density n has a nonnegative supersolution, then the nonlinear Dirichlet problem with density  has a nonnegative solution. We say that a continuous function gW R ! R satisfies the integrability condition if, for any functions v; v; vN 2 L1 ./ such that N v 6 v 6 vN almost everywhere in  N and g.v/; g.v/ N 2 L1 ./; we have N g.v/ 2 L1 ./:

21.2. Contraction and stability

343

This property is automatically satisfied when g is nondecreasing, or satisfies the assumptions of Propositions 21.7 and 21.8 below. Proof of Proposition 21.6. For each n 2 N, denote by vN n a nonnegative supersolution of the nonlinear Dirichlet problem with density n . We begin by constructing a nondecreasing sequence of supersolutions: Claim 1. For every n 2 N, the function inf ¹vN k W k > nº is a supersolution of the nonlinear Dirichlet problem with density n . Proof of Claim 1. Since the sequence .n /n2N is nondecreasing, for each m > n the function vN m is a supersolution of the nonlinear Dirichlet problem with density n , hence so is the function min ¹vN n ; : : : ; vN m º (cf. Lemma 20.10). By the integrability condition with respect to 0 and vN n (Lemma 20.7), there exists hn 2 L1 ./ such that, for every m > n, jg.min ¹vN n ; : : : ; vN mº/j 6 hn almost everywhere in . As m ! 1, by the dominated convergence theorem we deduce that the sequence .g.min ¹vN n ; : : : ; vN mº//m>n converges to g.inf ¹vN k W k > nº/ in L1 ./. Therefore, the function inf ¹vN k W k > nº is a supersolution of the nonlinear Dirichlet problem with density n . 4 We thus have that inf ¹vN k W k > nº is a nonnegative supersolution and 0 is a subsolution of the nonlinear Dirichlet problem with density n . By the nonlinear Perron– Remak method (Proposition 20.9), the largest subsolution un between 0 and inf ¹vN k W k > nº solves the nonlinear Dirichlet problem with density n . Claim 2. The sequence .un /n2N is nondecreasing and bounded in L1 ./. Proof of Claim 2. For every m > n, we have that un is a subsolution of the nonlinear Dirichlet problem with density m and 0 6 un 6 inf ¹vN k W k > mº in . By the maximality of the subsolution um , we then have um > un in . By the contraction estimate (Proposition 21.5), for every n 2 N we have kg.un /kL1 ./ 6 kn kM./ ;

(21.6)

whence kun kM./ D kg.un /

n kM./ 6 2kn kM./ :

Thus, the sequence .un /n2N is bounded in M./, and the claim follows from the 4 linear elliptic L1 estimate (Proposition 3.2).

344

21. Absorption problems

Denoting by u the pointwise limit of the sequence .un /n2N , it follows from the contraction estimate (21.6) and Fatou’s lemma that g.u/ 2 L1 ./. By the integrability condition with respect to 0 and u (Lemma 20.7), there exists h 2 L1 ./ such that, for every n 2 N, jg.un /j 6 h

almost everywhere in . It thus follows from the dominated convergence theorem that the sequence .g.un //n2N converges to g.u/ in L1 ./, and so u satisfies the nonlinear Dirichlet problem with density . 

21.3 Polynomial growth By the counterexample of Bénilan and Brezis (Proposition 3.9), the nonlinear Dirichlet problem does not always have a solution if the nonlinearity has power growth with some exponent p > NN 2 (cf. Proposition 21.2). The main result in this supercritical case is due to Baras and Pierre, see Theorem 4.1 in [20]: Proposition 21.7. Let  be a smooth bounded open set, and let gW R ! R be a continuous function satisfying the sign condition and such that, for every jt j > 1, we have C jt jp 6 jg.t /j 6 C 0 jt jp ; for some exponent p > 1 and for some constants 0 < C 6 C 0 . Then, the nonlinear Dirichlet problem with density  2 M./ has a solution if and only if  is diffuse 0 with respect to the W 2;p capacity.

This statement is consistent with Bénilan and Brezis’s existence result for 0 p < NN 2 , since in this case every measure is diffuse with respect to the W 2;p capacity. Indeed, we have that p 0 > N2 , and the empty set is the only set with zero 0 W 2;p capacity; this is a consequence of the Morrey–Sobolev embedding theorem (Corollary 4.16). On the other hand, if p > NN 2 , then, for every a 2 RN , we have capW 2;p0 .¹aº/ D 0. Hence, a diffuse measure cannot charge points, and in particular Dirac masses are not allowed (cf. Proposition 3.9). Proof of Proposition 21.7: “ H) ”. Assume that the nonlinear Dirichlet problem with density  has a solution u. In particular, we have D

u C g.u/

in the sense of distributions in . By the growth assumption on g at infinity, we have u 2 Lp ./ and then, for every ' 2 Cc1 ./, ˇ ˇˆ ˇ ˇˆ ˇ ˇ ˇ ˇ ˇ g.u/' ˇ: ˇ ' dˇ 6 kukLp ./ k'k p0 (21.7) L ./ C ˇ ˇ ˇ ˇ 



21.3. Polynomial growth

345

Given a compact set K   such that capW 2;p0 .K/ D 0, let .'n /n2N be a sequence of nonnegative functions in Cc1 ./ such that (cf. Proposition 12.4) .a/ .'n /n2N converges pointwise to the characteristic function K , .b/ .'n /n2N is bounded in L1 ./, 0

.c/ .'n /n2N converges to 0 in W 2;p ./. Applying estimate (21.7) to this sequence, and letting n ! 1, it follows from the dominated convergence theorem that .K/ D 0. The inner regularity of  (Proposition 2.5) yields the conclusion.  The proof of the reverse implication of Proposition 21.7 is based on the stability property (Proposition 21.6). Starting with a nonnegative measure , we look for supersolutions given by the linear Dirichlet problem ´ v D  in ; vD0

on @:

x By Lemma 13.4, we have that v 2 Lp ./ if and only if, for every  2 C01 ./, we have ˇ ˇˆ ˇ ˇ ˇ 6 C 00 kk 2;p0 : ˇ  d ./ W ˇ ˇ 

Another ingredient is the strong approximation of diffuse measures, and the connection of capacitary measures with trace inequalities (Propositions 14.1 and 14.2). Proof of Proposition 21.7: “ (H ”. We first assume that  is nonnegative. Extending  by zero in RN n , we may suppose that  2 M.RN /. By Proposition 14.1, there exists a nondecreasing sequence .n /n2N of nonnegative measures on M.RN / that converges strongly to  in M.RN / and such that, for every n 2 N, n 6 Cn capW 2;p0 : By Proposition 14.2, we have that each measure n verifies the trace inequality: for every ' 2 Cc1 .RN /, ˇˆ ˇ ˇ ˇ 0 ˇ ˇ ' d (21.8) n ˇ 6 Cn k'kW 2;p0 .RN / : ˇ RN

We may further assume that each measure n is compactly supported in . x We prove in this case that there exists Cn00 > 0 such that, for every  2 C01 ./, ˇ ˇˆ ˇ ˇ ˇ  dn ˇ 6 C 00 kk 2;p0 : n ./ W ˇ ˇ 

346

21. Absorption problems

Indeed, take n 2 Cc1 .RN / such that supp x we have Then, for every  2 C01 ./, ˆ ˆ  dn D  

On the other hand, the function  k

n

n

  and

n

D 1 in supp n .

dn :



n

is supported in  and

n kW 2;p0 .RN /

6 Cn000 kkW 2;p0 ./ :

Thus, by the trace inequality (21.8) in RN , we get ˇ ˇ ˇˆ ˇˆ ˇ ˇ ˇ ˇ ˇ  dn ˇ D ˇ  n dn ˇ 6 C 0 k n k 2;p0 N 6 C 0 C 000 kk 2;p0 : n n n ./ .R / W W ˇ ˇ ˇ ˇ 



We deduce that the solution of the linear Dirichlet problem with density n belongs to Lp ./ (Lemma 13.4), and so is a nonnegative supersolution of the nonlinear Dirichlet problem with density n . By the stability property (Proposition 21.6), we deduce that the nonlinear Dirichlet problem with nonnegative density  has a solution. 0 Given a signed measure  which is diffuse with respect to the W 2;p capacity, the previous argument implies that the nonlinear Dirichlet problem with nonnegative density max ¹; 0º has a nonnegative solution u: N this is a supersolution of the nonlinear Dirichlet problem with density . Similarly, the nonlinear Dirichlet problem with density min ¹; 0º has a nonpositive solution u: this is a subsolution of the nonlinear N Dirichlet problem with density . The conclusion now follows from the method of  sub- and supersolutions (Proposition 20.5). We can also consider the nonlinear Dirichlet problem with boundary measure data: ´ u C g.u/ D 0 in ; uD

on @:

The study of this problem when g is a power nonlinearity with subcritical growth was initiated by Gmira and Véron [148] and has vastly expanded in recent years; see the papers of Marcus and Véron [221], [222], [223], [224], and [225], and their book [226]. Motivations coming from the theory of probability – and the use of probabilistic methods – have reinvigorated the whole subject; see the pioneering papers of Le Gall [198] and [199], the books of Dynkin [120] and [121], and the numerous references therein. One of the reasons for studying separately the nonlinear Dirichlet problem with interior measure data and boundary measure data is that these conditions uncouple, see Theorem 6 in [66], which means that we may investigate each type of datum separately.

21.4. Exponential growth

347

21.4 Exponential growth We now consider absorption problems with exponential behavior at infinity: Proposition 21.8. Let  be a smooth bounded open set, and let gW R ! R be a continuous function satisfying the sign condition and such that, for every jt j > 1, we have C et 6 jg.t /j 6 C 0 et ; for some constants 0 < C 6 C 0 . Then, the nonlinear Dirichlet problem has a solution for every density  2 M./ such that  6 4HN 2 . This result in dimension N D 2 is due to Vázquez (Theorem 2 in [333]) and, in higher dimensions, to Bartolucci, Leoni, Orsina, and Ponce (Theorem 1 in [22]). The inequality  6 4HN 2 is meant in the sense of measures: for every Borel set A  , .A/ 6 4HN 2 .A/: (21.9) Since HN 2 is not a finite measure – not even a  -finite measure – the right-hand side of (21.9) may be infinite, and this is always the case for a non-empty open set A  RN . Every L1 function trivially satisfies inequality (21.9), since if HN 2 .A/ is finite, then A is negligible with respect to the Lebesgue measure, and so the lefthand side of (21.9) vanishes. More generally, every measure which is diffuse with respect to the W 1;2 capacity also satisfies this inequality since if HN 2 .A/ is finite, then capW 1;2 .A/ D 0 (Proposition 10.3). We thus recover two cases for which the Dirichlet problem has a solution regardless of the nonlinearity g (Propositions 3.7 and 19.1). In dimension N D 2, by the subadditivity of , assumption (21.9) amounts to saying that, for every x 2 , .¹xº/ 6 4H0 .¹xº/ D 4; and we recover Vázquez’s condition. We may decompose the measure  in terms of .a/ a non-atomic part  such that, for every x 2 , .¹xº/ D 0, P .b/ an atomic part ˛j ıaj , formed by countably many disjoint Dirac masses, j 2J

DC

X

j 2J

˛j ıaj :

348

21. Absorption problems

Using this notation, the Dirichlet problems with exponential nonlinearities g.t / D et

1 or

g.t / D et=2 .et=2

1/

(21.10)

have a solution provided that ˛j 6 4, for every j 2 J . The later nonlinearity arises in the Chern–Simons scalar equation, see [346]. The proof of Proposition 21.8 is based on the stability property (Proposition 21.6) and the exponential estimate satisfied by the solution of the linear Dirichlet problem ´ v D  in , vD0

on @.

In dimension N D 2, we have ev 2 L1 ./ provided that .¹xº/ < 4, for every x 2  (Corollary 17.7). The same conclusion holds in dimension N > 3, under the assumption that  6 ˛HıN 2 , for some ˛ < 4 and ı > 0 (Proposition 17.8). Proof of Proposition 21.8. We first assume that  is nonnegative. In dimension N D 2, let .n /n2N be a nondecreasing sequence of measures converging strongly to  such that, for every n 2 N and every x 2 , we have n .¹xº/ < 4. For instance, we may take n D ˛n ; where .˛n /n2N is an increasing sequence of numbers converging to 1. In dimension N > 3, by the property of strong approximation of Hausdorff measures (Proposition 14.4), there exists a nondecreasing sequence of measures .n /n2N converging strongly to  such that, for every n 2 N, we have n 6 ˛n HıNn

2

where ˛n < 4 and ın > 0. By the exponential estimates (Corollary 17.7 in dimension two or Proposition 17.8 in higher dimensions), the solution of the linear Dirichlet problem with density n is a nonnegative supersolution of the nonlinear Dirichlet problem. By the stability property (Proposition 21.6), we deduce that the nonlinear Dirichlet problem with density  has a solution. When  is a signed measure, we apply the previous argument to the positive part max ¹; 0º to obtain a nonnegative supersolution of the nonlinear Dirichlet problem with density . Since g is bounded from below, the solution of the linear Dirichlet problem with negative density min ¹; 0º is a subsolution of the nonlinear Dirichlet problem with density . By the method of sub- and supersolutions (Proposition 20.5), the conclusion follows. 

21.4. Exponential growth

349

To understand the role of the quantity 4, we recall that the fundamental solution of the Laplacian in R2 is given by v.x/ N D

1 1 log : 2 jxj

For every ˛ > 0, we would expect the function ˛ vN to be a supersolution of the nonlinear Dirichlet problem with density ˛ı0 : ´ u C .eu 1/ D ˛ı0 in B.0I 1/, on @B.0I 1/.

uD0

This is indeed the case when ˛ < 4, since the function N e˛v.x/ D

1 ˛

jxj 2

;

belongs to L1 .B.0I 1// if and only if ˛ < 4. By the method of sub- and supersolutions (Proposition 20.5), we deduce the existence of a solution of the nonlinear Dirichlet problem with density ˛ı0 for any ˛ < 4. The function 4 vN is not a legitimate supersolution with density 4ı0 , but, by the stability property (Proposition 21.6), we still have the existence of a solution in this case. The situation is nevertheless hopeless for ˛ > 4 due to an analytical obstruction: Proposition 21.9. Let N D 2, and let u 2 L1loc ./ be such that u > ˛ıa C f in the sense of distributions in , for some function f 2 L1 ./ and some a 2 . If eu 2 L1loc ./, then ˛ 6 4. Proof. Assume that eu 2 L1loc ./. For every 0 <  < d.a; @/, by the integration formula in polar coordinates and by Jensen’s inequality we have  ˆ  ffl ˆ ˆ   eu d dr > 2 r e @B.aIr/ u d dr: eu D 2 r B.aI/

@B.aIr /

0

0

For every  < ˛, there exists 0 <  < 1 such that, for almost every 0 < r 6 , we have (Exercise 21.1) u d > @B.aIr /

We deduce that

ˆ

u

ˆ

 1 log : 2 r 

1

dr: 1 r In particular, the integral in the right-hand side is finite, whence estimate holds for every  < ˛, it follows that ˛ 6 4. B.aI/

e > 2

0

 2

 2

< 2. Since this 

350

21. Absorption problems

In dimension N > 3, the condition given by Proposition 21.8 does not characterize all measures for which the nonlinear Dirichlet problem has a solution, even for explicit nonlinearities like in (21.10). Indeed, there exists a positive measure , supported in a compact set of zero Hausdorff measure HN 2 , such that the solution v of the linear Dirichlet problem with density  satisfies ev 2 L1 ./, see Theorem 3 in [286]. In this case, the inequality  6 4HN 2 fails on the support of , but, by the method of sub- and supersolutions (Proposition 20.5), the nonlinear Dirichlet problem with density  has a solution. Véron [336] has implemented a different strategy, by investigating the connection of the Dirichlet problem with an Orlicz capacity of order L log L.

Chapter 22

The Schrödinger operator

“The customary method does not seem to work owing to the high singularity of the potential.” Tosio Kato

The Schrödinger operator CV is associated to a force field of the form rV . We establish a strong maximum principle for nonnegative smooth functions satisfying u C V u > 0 in ,

where the potential V merely belongs to the Lebesgue space Lp ./ for some 1 6 p 6 C1. The proof relies on the existence of solutions of the Dirichlet problem for the Schrödinger operator involving measures.

22.1 Strong maximum principle The classical strong maximum principle for the Laplacian (Lemma 1.11) implies that if  is a connected open set and if uW  ! R is a nonnegative smooth function such that u > 0 in ; then either u D 0 in  or u > 0 in . Such a property is also satisfied by Schrödinger operators with bounded potentials: Proposition 22.1. Let  be a connected open set, and let V 2 L1 ./. If uW  ! R is a nonnegative smooth function such that u C V u > 0 in , and if there exists a 2  such that u.a/ D 0, then u D 0 in . Proof. Given  > 0, consider the nonnegative function    UW  !R ; 2 2 defined by U.x; s/ D u.x/ cos .s/:

352

22. The Schrödinger operator

Computing the Laplacian of U with respect to the variable .x; s/ we get U.x; s/ D Œu.x/

2 u.x/ cos .s/:

Since cos s > 0, by the assumption on u we get U.x; s/ 6 ŒV .x/

2 U.x; s/:

1=2 Choosing  > kV kL 1 ./ , the function U is superharmonic. Now, if u.a/ D 0 for some a 2 , then U.a; 0/ D 0, and it follows from the strong maximum principle for superharmonic functions (Lemma 1.11) that U is identically zero. This implies the conclusion for the function u. 

By the Harnack inequality (see Theorem 5 in [308], Corollaire 8.1 in [316], or Theorem 5.2 in [326]) based on Moser’s iteration technique [257], the same conclusion remains true for potentials V 2 Lp ./ for some exponent p > N2 . Below this threshold, nonnegative supersolutions of the Schrödinger operator may vanish without being identically zero: the function uW B.0I 1/ ! R defined by u.x/ D jxj2 satisfies the equation u C V u D 0 in B.0I 1/

with V .x/ D 2N =jxj2 . In this case, we have V 2 Lp .B.0I 1// for every 1 6 p < N2 , but V 62 LN=2 .B.0I 1//. When the supersolution u vanishes on a sufficiently large set, one would still hope to conclude that u D 0 in  for badly behaved potentials like V 2 L1 ./, see Theorem C.1 in [24], Theorem 5.2 in [87], or Theorem in [327]. Ancona beautifully identified the role played by the W 1;2 capacity to detect the size of the set ¹u D 0º, see Theorem 9 in [14]: Proposition 22.2. Let  be a connected open set, and let V 2 L1 ./. If uW  ! R is a nonnegative smooth function such that u C V u > 0 in , and if u vanishes in a compact subset of positive W 1;2 capacity, then u D 0 in . Ancona’s proof relies on tools from potential theory. We present an alternative strategy based on the following estimate, see [64] and [327]: Lemma 22.3. Let V 2 L1 ./, and let uW  ! R be a nonnegative smooth function. If u C V u > 0 in , then, for every ' 2 Cc1 ./, we have ˆ ˆ jr log.1 C u/j2 ' 2 6 C .V C ' 2 C jr'j2 /: 



22.1. Strong maximum principle

353

Proof of Lemma 22.3. Given ' 2 Cc1 ./, we multiply the differential inequality by ' 2 =.1 C u/ to get '2 Vu 2 u 6 ' 6 V C'2: 1Cu 1Cu Note that  '2 2' jruj2 2 '2  D u C ru
r' ' : div ru 1Cu 1Cu 1 C u .1 C u/2 Thus,

jruj2 2 2' ' 6 V C ' 2 C ru
r' .1 C u/2 1Cu

 '2  : div ru 1Cu

(22.1)

On the other hand, for every  > 0 we have (Exercise 4.14) ru
r'

2' jruj2 2 1 6 ' C jr'j2 : 1Cu .1 C u/2 

Inserting this inequality in (22.1), we get .1

/

jruj2 2 1 ' 6 V C ' 2 C jr'j2 2 .1 C u/ 

 '2  div ru : 1Cu

Integrating both sides over , it follows from the divergence theorem that ˆ ˆ ˆ 1 jruj2 2 C 2 ' 6 V ' C .1 / jr'j2 : 2 .1 C u/     We deduce the estimate by taking any fixed 0 <  < 1.



We need the following variant of the Poincaré inequality for functions vanishing on a set of positive W 1;2 capacity: Lemma 22.4. Let  be a connected smooth bounded open set, and let K   be a compact set. If capW 1;2 .K/ > 0, then there exists C > 0 such that, for every function x vanishing in K, we have 2 C 1 ./ k kL2 ./ 6 C kr kL2 ./ : Proof of Lemma 22.4. Arguing by contradiction, if the inequality is not true, then x such that, for every n 2 N , there exists a sequence . n /n2N in C 1 ./ .a/ .b/ k

n

.c/ kr

D 0 in K,

n kL2 ./

D 1,

n kL2 ./

6 n1 .

354

22. The Schrödinger operator

Thanks to the smoothness of , we may extend n as a function supported in a ball B.0I R/ c , with uniform control of the W 1;2 norm. Thus, by the Rellich– Kondrashov compactness theorem (Proposition 4.8), there exists a subsequence . nk /k2N converging strongly in L2 ./ to some function u such that kukL2 ./ D 1: Since .r nk /k2N converges to 0 in L2 .I RN /, we deduce that u 2 W 1;2 ./ and krukL2 ./ D 0. By the connectedness of , we then have u D ˛ almost everywhere 1

in  for some ˛ 2 R. Since j˛j jj 2 D 1, we have that ˛ ¤ 0. Given a nonnegative function ' 2 Cc1 ./ such that ' > 1 in K, for each k 2 N 1;2 the function .1 capacity of K, nk =˛/ ' is admissible in the definition of the W whence 2 capW 1;2 .K/ 6 k.1 nk =˛/ 'kW 1;2 ./ : As k ! 1, the quantity in the right-hand side converges to zero, and we deduce that  capW 1;2 .K/ D 0. This is a contradiction. Proof of Proposition 22.2. For every ı > 0, the function u=ı satisfies the assumptions of Lemma 22.3. Thus, for every smooth bounded open subset ! b , there exists a constant C1 > 0, independent of ı > 0, such that ˆ ˇ  u ˇˇ2 ˇ ˇr log 1 C ˇ 6 C1 : ı ! Since the function log.1 C u=ı/ vanishes in a compact subset of positive W 1;2 capacity in , we may choose a connected smooth bounded open set ! b  having the same property. Thus, by the Poincaré inequality above, we have ˆ ˇ ˆ ˇ   u ˇˇ2 u ˇˇ2 ˇ ˇ ˇr log 1 C ˇlog 1 C ˇ 6 C2 ˇ 6 C2 : ı ı ! !

By the Chebyshev inequality and the monotonicity of the logarithm, for every t > 0, ˇ  t ˇˇ2 ˇ j! \ ¹u > t ºj ˇlog 1 C ˇ 6 C2 : ı

Letting ı ! 0, we deduce that j! \ ¹u > t ºj D 0. Thus, 0 6 u 6 t in !, for every t > 0, whence u D 0 in !.  It follows from Proposition 22.2 that either u D 0 in , or the set ¹u D 0º has Hausdorff dimension at most N 2 (cf. Proposition 10.4). The W 1;2 capacity gives a more precise information in the sense that the following converse of the strong maximum principle holds, see Proposition 6.3 in [270]: for any compact set K  RN

22.1. Strong maximum principle

355

such that capW 1;2 .K/ D 0, there exist a potential V 2 L1 .RN / and a nonnegative smooth function uW RN ! R such that K D ¹u D 0º and u C V u D 0 in RN :

(22.2)

For potentials V having some better integrability property like V 2 Lp ./, for some p > 1, one would expect the level set ¹u D 0º to be typically smaller. This is indeed the case, and the bridge between the strong maximum principles for potentials in L1 ./ and L1 ./ is provided by the W 2;p capacity, see Theorem 1 in [270]: Proposition 22.5. Let  be a connected open set, and let V 2 Lp ./ for some 1 < p < C1. If uW  ! R is a nonnegative smooth function such that u C V u > 0 in , and if u vanishes in a compact subset of positive W 2;p capacity, then u D 0 in . This statement gives an affirmative answer to a question of Bénilan and Brezis, see Open Problem 4 in [24]. The case N2 < p < C1 is fully covered, since by the Morrey–Sobolev inequality every non-empty set has positive capacity in this range. In particular, if u.a/ D 0 for some a 2 , we then have capW 2;p .¹u D 0º/ > capW 2;p .¹aº/ > 0; and by the proposition above we deduce that u D 0 in . For any 1 < p < C1, the condition capW 2;p .¹u D 0º/ > 0 ensures that there exists a positive measure , supported in ¹u D 0º, such that  is diffuse with respect to the W 2;p capacity (Proposition A.17). Since V 2 Lp ./, we also have that the Dirichlet problem associated to the Schrödinger operator  C V C has a solution for this measure  (Proposition 22.8). These facts will be used in Section 22.3 to prove Proposition 22.5. We systematically assume in this chapter that the supersolution u of the Schrödinger operator is smooth. The conclusions of Propositions 22.2 and 22.5 remain unchanged if u 2 L1 ./ is such that V u 2 L1 ./, and the differential inequality is satisfied in the sense of distributions in , see [270]. The argument in this case requires an additional regularization step based on Lemma 2.22. The set ¹u D 0º is understood in the sense of the precise representative u; O since u is nonnegative, we have that u.x/ O D 0 if and only if lim

r !0 B.xIr /

u D 0:

356

22. The Schrödinger operator

22.2 Existence of solutions with measure data We investigate the existence of solutions of the Dirichlet problem ´ z C V z D  in , zD0

on @,

(22.3)

associated to the Schrödinger operator  C V . The meaning of the solution is the following: Definition 22.6. Let  be a bounded open set, let V W  ! R be a Borel measurable function, and let  2 M./. A function z 2 L1 ./ is a solution of the Dirichlet problem for the Schrödinger operator with density  if V z 2 L1 ./ and, for every x we have  2 C01 ./, ˆ ˆ z .  C V / D  d: 



In the same spirit, one introduces the notions of subsolution and supersolution of the Dirichlet problem for the Schrödinger operator. The following counterpart of the method of sub- and supersolutions provides a convenient tool to establish the existence of a solution: Proposition 22.7. Let  be a smooth bounded open set, let V W  ! R be a Borel measurable function, and let  2 M./. If the Dirichlet problem for the Schrödinger operator has a subsolution v and a supersolution vN such that v 6 vN almost everyN . where in , then there existsN a solution z such that v 6 z 6 vN in N The proof is based on Schauder’s fixed point theorem, along the lines of the proof of Proposition 20.5. One considers as before the closed convex set K D ¹v 2 L1 ./W v 6 v 6 vN almost everywhere in º: N

For every v 2 K, we have

jV vj 6 max ¹jV vj; jV vjº N N

and, in particular, V v 2 L1 ./. Hence, there exists a unique solution u 2 L1 ./ of the linear Dirichlet problem for the Laplacian ´ u D  V v in ; uD0

on @:

22.2. Existence of solutions with measure data

357

One then defines the map F W K ! K at v as follows: 8 ˆ v.x/ if u.x/ < v.x/; ˆ v.x/: N

As in the proof of Proposition 20.5, one then shows that F is continuous with respect to the L1 norm, and the set F .K/ is relatively compact in L1 ./. By Schauder’s fixed point theorem, the map F thus has a fixed point v 2 K. Using Kato’s inequality up to the boundary (Lemma 20.8), it follows that v D u almost everywhere in , whence v satisfies the Dirichlet problem for the Schrödinger operator. We leave the details to the reader. There is some analogy between the Dirichlet problem for the Schrödinger operator  C V with a nonnegative potential V and the semilinear problem  C g for nonlinearities g satisfying the sign condition. We have for instance the following companion of Proposition 21.7 that involves power nonlinearities: Proposition 22.8. Let  be a smooth bounded open set. If V is a nonnegative function in Lp ./ for some 1 < p < C1, then the Dirichlet problem for the Schrödinger operator has a solution for every measure  2 M./ that is diffuse with respect to the W 2;p capacity. By the linearity of the equation, it suffices to prove the existence of solutions for nonnegative measures . In this case, the zero function provides a subsolution of the Dirichlet problem, and we can focus on the construction of a nonnegative supersolution. The following lemma is a basic tool in the proofs of the strong maximum principle (Proposition 22.5) and the existence of solutions of the Dirichlet problem for potentials V 2 Lp ./ (cf. Lemma 13.4): Lemma 22.9. Let  be a smooth bounded open set, and let V be a nonnegative function in Lp ./, for some 1 < p < C1. If  2 M./ is a nonnegative measure with compact support such that, for every nonnegative function ' 2 Cc1 ./, ˆ 06 ' d 6 C k'kW 2;p ./ ; 

then the Dirichlet problem for the Schrödinger operator has a nonnegative solution 0 in Lp ./. Proof of Lemma 22.9. Proceeding as in the proof of Proposition 13.3, for every x we have  2 C01 ./ ˇ ˇˆ ˇ ˇ ˇ  dˇ 6 C 0 kkW 2;p ./ : (22.4) ˇ ˇ 

358

22. The Schrödinger operator

Then, by Lemma 13.4, it follows that the solution of the linear Dirichlet problem ´ u D  in , uD0

on @,

0

belongs to Lp ./. By the weak maximum principle (Proposition 6.1), we also have u > 0 almost everywhere in . Since V u > 0 in , we deduce that u is a supersolution of the Dirichlet problem for the Schrödinger operator. By the method of sub and supersolutions (Proposition 22.7), we have the conclusion. Two additional ingredients in the proof of Proposition 22.8 are .a/ the strong approximation of diffuse measures on connection with the trace inequality (Propositions 14.1 and 14.2); .b/ the absorption estimate kV zkL1 ./ 6 kkM./ ;

(22.5)

satisfied by the solution of the Dirichlet problem for nonnegative potentials V , which can be established as in the proof of Proposition 21.5 by using an approximation of sgn z. Proof of Proposition 22.8. We may assume that  is a nonnegative measure. Take a sequence of nonnegative measures .n /n2N converging strongly to  in M./ and such that (22.6) n 6 Cn capW 2;p : Since  2 M./, we may assume in addition that each measure is compactly supported in . By the capacitary estimate (22.6), the trace inequality holds: for every ' 2 Cc1 .RN /, k'kL1 .RN In / 6 Cn0 k'kW 2;p .RN / : By Lemma 22.9, the Dirichlet problem for the Schrödinger operator with density n has a solution zn . By the absorption estimate (22.5) and the linearity of the equation, we then have kV zm V zn kL1 ./ 6 km n kM./ : It follows from the strong convergence of the sequence .n /n2N that .V zn /n2N is a Cauchy sequence in L1 ./. Thus, the sequence of measures .zn /n2N converges strongly in M./. By the linear elliptic L1 estimate (Proposition 3.2), the sequence .zn /n2N is Cauchy in L1 ./ and converges strongly to a function z. In particular, the sequence .V zn /n2N converges in L1 ./ to the function V z. Therefore, z satisfies the Dirichlet problem for the Schrödinger operator with density . 

22.2. Existence of solutions with measure data

359

Exercise 22.1 (Dirichlet problem for L log L potentials). Let N > 3. Prove that if  is a smooth bounded open set, and if V W  ! R is a nonnegative Borel measurable function such that V log .V C 1/ 2 L1 ./, then the Dirichlet problem for the Schrödinger operator has a solution for every measure  2 M./ which is diffuse with respect to the HN 2 Hausdorff measure. Applying the Fréchet–Riesz representation theorem, one obtains variational solutions of the Dirichlet problem for the Schrödinger operator, see Theorem 2.4 in [95] (cf. Proposition 4.6): Proposition 22.10. Let  be a bounded set, let V W  ! R be a nonnegative Borel measurable function, and let  2 .W01;2 .//0 . Then, there exists a unique function z 2 W01;2 ./ \ L2 .I V / such that, for every v 2 W01;2 ./ \ L2 .I V /, we have ˆ .rz
rv C V zv/ D Œv: 

Exercise 22.2 (weak maximum principle). Under the assumptions above, prove that if  > 0, then z > 0 almost everywhere in . When V 2 L1 ./, we are allowed to use test functions in Cc1 ./ and, more genx in the variational formulation, see Section 3 in [96]. This functional erally, in C01 ./ setting thus provides a solution in the sense of Definition 22.6 for finite measures  such that  2 .W01;2 .//0 . Exercise 22.3 (Dirichlet problem for L1 potentials, Theorem 1.2 in [269]). Prove that if  is a smooth bounded open set, and if V 2 L1 ./ is a nonnegative function, then the Dirichlet problem for the Schrödinger operator has a solution for every measure  2 M./ that is diffuse with respect to the W 1;2 capacity. In the distributional setting, one should be careful when dealing with potentials V that are not summable, see Theorem 1.4 in [269]: Example 22.11. The Dirichlet problem 8 z < z C D 1 in B.0I 1/, jx1 j : z D 0 on @B.0I 1/,

associated to the potential V .x/ D 1=jx1 j, has a variational solution in the sense of Proposition 22.10. Note that every test function  2 C01 .BŒ0I 1/ such that V  2 2 L1 .B.0I 1// vanishes on the set ¹x1 D 0º. The variational solution thus satisfies two independent Dirichlet problems on the half-balls contained in ¹x1 > 0º and ¹x1 < 0º. Using a Hopf Lemma (see Proposition A.1 in [269]), one shows that there exists no solution in the sense of Definition 22.6 in the whole domain B.0I 1/.

360

22. The Schrödinger operator

22.3 The bridge The proof of Proposition 22.5 relies on the existence of nonnegative solutions of the Dirichlet problem for the Schrödinger operator, for some suitable choice of signed datum: Lemma 22.12. Let  be a smooth bounded open set, and let V 2 L1 ./ be a nonnegative function. If  2 M./ is a positive measure such that the Dirichlet problem for the Schrödinger operator with density  has a solution z, then there exists f 2 L1 ./ such that (i) f > 0 almost everywhere in ; (ii) the solution of the Dirichlet problem for the Schrödinger operator with density  f is nonnegative. For every f 2 L1 ./, there exists a solution v of the Dirichlet problem for the Schrödinger operator with density f (Proposition 22.10). If, in addition, f is nonnegative, then v is also nonnegative (Exercise 22.2), and so v is a subsolution of the Dirichlet problem ´  D kf kL1 ./ in , on @.

D0

By the weak maximum principle (Proposition 6.1), we deduce that 0 6 v 6 kkL1 ./ 6 C kf kL1 ./ almost everywhere in . In particular, v 2 L1 ./. Proof of Lemma 22.12. Given  > 0, let v be the solution of the Dirichlet problem for the Schrödinger operator with bounded density ¹z>º , and let C > 0 be such that 0 6 v 6 C

in .

In particular, this gives 0 6 v 6 C z

in ¹z > º.

Claim. We have 0 6 v 6 C z

in .

(22.7)

22.3. The bridge

361

Proof of the claim. Since the measure  is nonnegative, we have .v

C z/ > V .v

C z/

¹z>º

x 0 . It thus follows from Kato’s inequality up to the boundary in the sense of .C01 .// (Lemma 20.8) that .v

C z/C > ¹v >C zº V .v

C z/

¹v >C zº ¹z>º

x 0 . The first term in the right-hand side is nonnegative, while in the sense of .C01 .// the second one vanishes by estimate (22.7). Hence, .v

C z/C > 0

x 0 . By the weak maximum principle (Proposition 6.1), we in the sense of .C01 .// conclude that .v C z/C 6 0 almost everywhere in , and the claim follows.

4

By the claim, given a sequence of positive numbers .n /n2N converging to zero we have 1 1 X 1 X n vn z > DW 2w in . 2z D 2n C nD0 2n nD0

Defining

f D

1 1 X n ¹z>n º ; 2C nD0 2n

we have that f 2 L1 ./, and the solution z w of the Dirichlet problem for the Schrödinger operator with density  f is nonnegative. By the strong maximum principle for the Schrödinger operator with potentials in L1 ./ (cf. Proposition 22.2), z > 0 almost everywhere in . Thus, f > 0 almost everywhere in , and the conclusion follows.  In the previous proof, it is also possible to choose f D .T1 .z//2 =2C , where T1 denotes the truncation at level 1. Indeed, taking n D 1=2n we get ˆ 1 1 X X n ¹z>n º > 2 2n nD0 nD0

1 2n 1 2nC1

t¹z>tº dt D 2

ˆ

1 0

t¹z>tº dt D .T1 .z//2 :

Thus, the solution of the Dirichlet problem for the Schrödinger operator with density  .T1 .z//2 =2C is also nonnegative.

362

22. The Schrödinger operator

Proof of Proposition 22.5. Let K   be a compact set such that K  ¹u D 0º and capW 2;p .K/ > 0. Replacing  by any connected smooth bounded open subset x By the existence ! b  containing K if necessary, we may assume that u 2 C 1 ./. of diffuse measures (Proposition A.17), there exists a positive measure  supported by K such that, for every nonnegative function ' 2 Cc1 ./, ˆ 06 ' d 6 k'kW 2;p ./ : 

C

Replacing V by V , we may assume that V is nonnegative. For this measure , the Dirichlet problem for the Schrödinger operator thus has a solution (Lemma 22.9). We deduce from Lemma 22.12 that there exists a function f 2 L1 ./ such that f > 0 almost everywhere in , and the solution of the Dirichlet problem ´ w C V w D  f in , on @,

wD0

is nonnegative. Since  is supported in ¹u D 0º, and the equation above is satisfied in the sense of measures in  (Proposition 6.12), we have ˆ ˆ ˆ ˆ uf D u .f / D u w uV w: (22.8) 







Since w has a weak normal derivative @w in L1 .@/ (Proposition 7.3), we get @n ˆ ˆ ˆ @w d: u w D w u C u   @ @n The nonnegativity of w implies that nonnegative, we get ˆ

@w @n

6 0 on @ (Lemma 12.15). Since u is also

u w 6



ˆ

w u: 

Inserting this inequality into identity (22.8), we deduce that ˆ ˆ uf 6 w .u V u/ 6 0: 



By nonnegativity of the integrand in the left-hand side, it follows that uf D 0 almost everywhere in . Since f > 0 almost everywhere in , we conclude that u D 0 in .  For every exponent 1 < p < C1, Proposition 22.5 also has a converse: given a compact set K  RN such that capW 2;p .K/ D 0, there exist V 2 Lp .RN / and a nonnegative smooth function uW RN ! R such that K D ¹u D 0º and equation (22.2) holds, see Proposition 6.1 in [270]. Since the W 1;2 and W 2;1 capacities

22.3. The bridge

363

are not equivalent (Corollary 17.16), we have an unexpected jump between the assumptions of the strong maximum principle in the cases p D 1 and p > 1. Using a capacity capp explicitly defined in terms of the Laplacian, one can merge Propositions 22.2 and 22.5 in a single statement. More precisely, given a smooth bounded open set  and a compact subset K  , for every p > 1 let p 1 capp .KI / D inf ¹k'kL p ./ W ' 2 Cc ./ nonnegative and ' > 1 in Kº:

Such a capacity vanishes at the same sets as capW 2;p by the Calderón–Zygmund estimates for 1 < p < C1, and vanishes at the same sets as capW 1;2 for p D 1 (cf. Proposition 12.2). This is not the end of the story for the W 2;1 capacity, which is related to potentials V in the Zygmund class L log L: Exercise 22.4 (strong maximum principle for L log L potentials). Let  be a connected open set, let V W  ! R be a nonnegative measurable function such that V log .V C 1/ 2 L1 ./, and let uW  ! R be a nonnegative smooth function such that u C V u > 0 in . Prove that if u vanishes in a compact subset of positive W 2;1 capacity (or, equivalently, of positive HN 2 Hausdorff measure), then u D 0 in .

Appendices

Appendix A

Sobolev capacity

“La capacité est une mesure dont la détermination effective est très difficile. En général il faut se contenter de décider si elle est nulle ou non.”1 Otto Frostman

We present the fundamental notion of capacity with respect to Sobolev norms, and we prove the existence of positive diffuse measures supported on sets of positive capacity.

A.1 Finite semi-additivity We begin by defining the capacity on compact sets with respect to a W k;q Sobolev norm such that k 2 N and 1 6 q < C1: Definition A.1. Given a compact set K  RN , the W k;q capacity of K is defined as capW k;q .K/ D inf ¹k'kqW k;q .RN / W ' 2 Cc1 .RN / is nonnegative, and ' > 1 in Kº; where q k'kqW k;q .RN / D k'kL q .RN / C

k X

q kD j 'kL q .RN / :

j D1

By the Chebyshev inequality, we have jKj 6 capW k;q .K/; but it may happen that capW k;q .K/ > 0, even if K is negligible for the Lebesgue measure. 1 “Capacity

is a measure whose actual value is very difficult to compute. In general we must accept to determine only whether it is null or not.”

368

A. Sobolev capacity

Example A.2. For every smooth bounded open set   RN , capW k;q .@/ > 0: Indeed, by the classical trace inequality (Proposition 15.1), for every ' 2 Cc1 .RN / we have k'kLq .@/ 6 C k'kW 1;q ./ : Hence, if ' > 1 on @, we then get  .@/ 6 C q k'kqW k;q ./ ; where  denotes the surface measure. Minimizing the right-hand side with respect to ', we deduce that  .@/ 6 C q capW k;q .@/: Monotonicity of the capacity readily follows from the definition: for every compact sets K  L, we have capW k;q .K/ 6 capW k;q .L/: Another fundamental property is the finite semi-additivity: Proposition A.3. For every finite family of compact subsets .Kj /j 2¹1;:::;nº of RN , we have n n [  X capW k;q capW k;q .Kj /; Kj 6 C j D1

j D1

for some constant C > 1 depending on N , k and q, but not on n. This estimate holds with constant C D 1 in some cases, and we then talk about finite subadditivity: .a/ exponent q D 1: whose proof below relies on the triangle inequality; .b/ order k D 1: as a consequence of the strong subadditivity of the W 1;q capacity (Proposition A.12). Proof of Proposition A.3 when q D 1. For each j 2 ¹1; : : : ; nº, take a nonnegative n P 'j is admissible in function 'j 2 Cc1 .RN / such that 'j > 1 in Kj . The function j D1

A.1. Finite semi-additivity

the definition of the capacity of the compact set

n S

369

Kj , and by the triangle inequality,

j D1

capW k;1

n [

j D1

Kj



n

X

'j 6 j D1

W k;1 .RN /

6

n X

k'j kW k;1 .RN / :

j D1

Minimizing the right-hand side with respect to 'j , we have the conclusion.



The proof of the semi-additivity property when 1 < q < C1 is based on the following lemma: Lemma A.4. If 1 < q < C1, then, for any '1 ; : : : ; 'n 2 Cc1 .RN /, there exists 2 Cc1 .RN / such that max ¹'1 ; : : : ; 'n º 6 and k

kqW k;q .RN /

6C

n X

j D1

k'j kqW k;q .RN / ;

for some constant C > 0 depending on N , k, and q. The proof of Lemma A.4 for every order k 2 N is based on the isomorphism between the Sobolev space W k;q .RN / and the Lebesgue space Lq .RN / in terms of Bessel potentials. For a background on Bessel potentials we refer the reader to Chapter 7 in [249] and Chapter V in [317]. An alternative strategy in the case k D 1, including the exponent q D 1, is studied in Section A.3 below. We recall that the Bessel kernel Gk W RN ! R is a positive summable function yk is given by whose Fourier transform G yk ./ D .1 C 4 2 kk2 / G where y k ./ D G

ˆ

e

i 2x


RN

k 2

;

Gk .x/ dx:

The Bessel kernel Gk can be recovered by computing the inverse Fourier transform yk : of G ˆ y k ./ d: Gk .x/ D ei 2x
 G RN

By Calderón’s theorem [71] (see also Theorem 7.2.2 in [249] or Theorem V.3 in [317]), for every exponent 1 < q < C1 and every ' 2 W k;q .RN / there exists a unique function f 2 Lq .RN / such that ' D Gk  f;

370

A. Sobolev capacity

and we have C 0 k'kW k;q .RN / 6 kf kLq .RN / 6 C 00 k'kW k;q .RN / :

(A.1)

In our case, we are dealing with a function ' 2 Cc1 .RN /, which belongs to the Schwartz class S.RN / of functions with fast decay at infinity. It thus follows that the potential f also belongs to S.RN /. Proof of Lemma A.4. For each j 2 ¹1; : : : ; nº, let fj be the potential in the Schwartz class S.RN / such that 'j D Gk  fj . Taking h D max ¹f1 ; : : : ; fn º; then, for every j 2 ¹1; : : : ; nº, we have 'j 6 Gk  h, whence max ¹'1 ; : : : ; 'n º 6 Gk  h

in RN :

By the subadditivity of the Lebesgue measure, we also have q khkL q .RN / 6

n X

q kfj kL q .RN / :

j D1

Thus, by the isomorphism (A.1) of the Bessel potential, we get 1 q khkL q .RN / C0 n n 1 X C 00 X q 6 0 kfj kL 6 k'j kqW k;q .RN / : q .RN / C j D1 C 0 j D1

kGk  hkqW k;q .RN / 6

Since Gk is summable and h is bounded and uniformly continuous, the function Gk  h is also bounded and uniformly continuous. To get a function in Cc1 .RN / having the desired properties, we first take a nonn S supp 'j , for some  > 0. negative function  2 Cc1 .RN / such that  >  in We next take a mollifier  2 Cc1 .RN / such that

j D1

max ¹'1 ; : : : ; 'n º 6   .Gk  h/ C  We then take  2 Cc1 .RN / such that  D 1 in

n S

supp 'j . The function

j D1

D Œ  .Gk  h/ C  belongs to Cc1 .RN / and satisfies max ¹'1 ; : : : ; 'n º 6

in RN :

in RN :

A.2. Outer capacity and pointwise convergence

371

Since Gk  h 2 W k;q .RN /, the conclusion follows by choosing ,  and  as above, and such that k kW k;q .RN / 6 2kGk  hkW k;q .RN / :  Proof of Proposition A.3 when 1 < q < C1. Let 'j 2 Cc1 .RN / be a nonnegative function such that 'j > 1 in Kj , for every j 2 ¹1; : : : ; nº. We then have max ¹'1 ; : : : ; 'n º > 1 in

n [

Kj :

j D1

2 Cc1 .RN / be a function satisfying the conclusion of the previous lemma. n S Kj , we deduce that Since is nonnegative and > 1 in Let

j D1

capW k;q

n [

j D1



Kj 6 k

kqW k;q .RN /

6C

n X

k'j kqW k;q .RN / :

j D1

Taking the infimum of the right-hand side over the functions 'j , the conclusion follows. 

A.2 Outer capacity and pointwise convergence The notion of capacity was introduced by Wiener [341] for compact sets and later extended by de la Vallée Poussin [106] for closed sets by inner regularity – i.e., as a supremum on compact subsets. Brelot followed a different strategy by first extending the capacity to open sets, see p. 321 in [44], and §6 in [81]: Definition A.5. For every open set U  RN , the W k;q capacity of U is defined as capW k;q .U / D sup ¹capW k;q .K/W K  RN is compact and K  U º: The capacity on compact sets is outer regular in the following sense: Proposition A.6. For every compact set K  RN and every  > 0, there exists an open set U  RN such that K  U and capW k;q .U / 6 capW k;q .K/ C :

372

A. Sobolev capacity

Proof. Given  > 0, let ' 2 Cc1 .RN / be a nonnegative function such that ' > 1 on K and k'kqW k;q .RN / 6 capW k;q .K/ C : The conclusion of the proposition is satisfied by the open set U D ¹' > 1º. Indeed, for every compact set L  ¹' > 1º, the function ' is also admissible in the definition of the W k;q capacity of L, whence capW k;q .L/ 6 k'kqW k;q .RN / 6 capW k;q .K/ C : We then take the supremum of the left-hand side with respect to L.



It follows from the previous proposition that, for every compact set K  RN , we have capW k;q .K/ D inf ¹capW k;q .U /W U  RN is open and K  U º: Therefore, there is no ambiguity in defining the capacity of an arbitrary subset of RN by outer regularity: Definition A.7. For every A  RN , the W k;q capacity of A is defined as capW k;q .A/ D inf ¹capW k;q .U /W U  RN is open and A  U º: The capacity is monotone: Proposition A.8. For every A; B  RN such that A  B, we have capW k;q .A/ 6 capW k;q .B/: An advantage of such a definition in two steps is that the capacity inherits the semi-additivity property (Proposition A.3): Proposition A.9. For every sequence .An /n2N of subsets of RN , we have capW k;q

1 [

nD0

1  X An 6 C capW k;q .An /; nD0

for the same constant C > 1 satisfying the finite semi-additivity property on compact sets. Proof. We first prove the proposition for a sequence .An /n2N of open subsets of RN . 1 S An , there exist finitely many open sets An1 ; : : : ; An` Given a compact set K  such that

nD0

K  An1 [


[ An` :

A.2. Outer capacity and pointwise convergence

373

For each i 2 ¹1; : : : ; `º, take a compact set Ki  Ani such that (Exercise A.1) K  K1 [


[ K` :

By the monotonicity and finite semi-additivity on compact sets (Proposition A.3), we have ` X capW k;q .Ki /: capW k;q .K/ 6 C i D1

Thus, by the definition of capacity on open sets, we have capW k;q .K/ 6 C

` X

1 X

capW k;q .Ani / 6 C

capW k;q .An /:

nD0

i D1

Since this estimate holds for every compact subset of

1 S

An , the conclusion holds

nD0

when all sets An are open. Assume now that .An /n2N is an arbitrary sequence of sets in RN . Given a sequence of positive numbers .n /n2N , for every n 2 N let Un  RN be an open set containing An , and such that capW k;q .Un / 6 capW k;q .An / C n : By the definition of capacity of the set

1 S

An , we have

nD0

capW k;q

1 [

nD0

1  [  An 6 capW k;q Un : nD0

By the semi-additivity property of the capacity for open sets that we have proved above, and by the choice of .Un /n2N , we get 1 1 1 1 [  X X X capW k;q capW k;q .Un / 6 C An 6 C capW k;q .An / C C n : nD0

nD0

nD0

nD0

Given  > 0, it suffices to choose a summable sequence .n /n2N such that C

1 X

n 6 

nD0

and this gives the conclusion.



Exercise A.1. Given a compact set K and open sets .Ai /i 2¹1;:::;`º of RN such that ` S Ai , prove that, for every i 2 ¹1; : : : ; `º, there exists a compact set Ki  Ai K i D1

such that K 

` S

i D1

Ki .

374

A. Sobolev capacity

The abstract formalism of capacities introduced by Choquet [84] requires a constant C D 1 in the semi-additivity formula – which is valid for exponent q D 1 or for order k D 1 – and in this case we speak of subadditivity of the capacity. We now prove a counterpart of the partial converse of the dominated convergence theorem (Proposition 4.9) in the setting of Sobolev spaces: Proposition A.10. Let 1 < q < C1. If .'n /n2N is a sequence in Cc1 .RN / converging strongly in W k;q .RN /, then there exist a subsequence .'ni /i 2N and h 2 W k;q .RN / such that (i) .'ni /i 2N converges pointwise in RN n F for some Borel set F  RN such that capW k;q .F / D 0; (ii) for every i 2 N, we have j'ni j 6 h in RN . We need a weak capacitary estimate in the spirit of the Chebyshev inequality for Lebesgue spaces: 1 q j¹j'j > ºj 6 q k'kLq .RN / :  Lemma A.11. If 1 < q < C1, then, for every ' 2 Cc1 .RN / and every  > 0, we have C0 capW k;q .¹j'j > º/ 6 q k'kqW k;q .RN / ;  0 for some constant C > 0 depending on N , k and q. Proof of Lemma A.11. By Lemma A.4, there exists

2 Cc1 .RN / such that

j'j D max ¹'; 'º 6 and k kqW k;q .RN / 6 2C k'kqW k;q .RN / : For any  > 1, we have  = > 1 in the set ¹j'j > º. By the definition of capacity on compact sets, we then get capW k;q .¹j'j > º/ 6





k kW k;q .RN /

q

6

C0 k'kqW k;q .RN / : q



The weak capacitary estimate would be a trivial consequence of the definition if we could take j'j= as test function. This is almost true when k D 1 since j'j 2 W 1;q .RN /; we may then proceed as in the second part of the proof of Lemma A.4 to construct a function 2 Cc1 .RN / such that j'j 6 and k kW 1;q .RN / 6 C k'kW 1;q .RN / :

A.2. Outer capacity and pointwise convergence

375

This argument does not rely on the Bessel potential and includes the exponent q D 1 (cf. Proposition A.12). However, for orders k > 2 such an approach seems hopeless, since we typically have j'j 62 W k;q .RN / for signed functions '. Proof of Proposition A.10. Given a summable sequence of positive numbers .i /i 2N and a subsequence .'ni /i 2N to be chosen later on depending on .i /i 2N , let Fj D

1 [

i Dj

¹j'ni C1

'ni j > i º

and F D

1 \

Fj :

j D0

For every x 2 RN n Fj and every ˛; ˇ 2 N satisfying j 6 ˛ < ˇ, we have ˇ 1

j'nˇ .x/

'n˛ .x/j 6

X i D˛

ˇ 1

j'ni C1 .x/

'ni .x/j 6

X

i :

i D˛

By the summability of .i /i 2N , we then have that .'ni .x//i 2N is a Cauchy sequence in R. Hence, .'ni /i 2N converges pointwise in RN n F . We now estimate the W k;q capacity of the set F . By the monotonicity and semiadditivity of the capacity, for every j 2 N we have capW k;q .F / 6 capW k;q .Fj / 6 C

1 X

capW k;q .¹j'ni C1

'ni j > i º/:

i Dj

By the weak capacitary estimate of the sets ¹j'ni C1 capW k;q .F / 6 C1

1 X 1 q k'ni C1  i Dj i

'ni j > i º, we then have 'ni kqW k;q .RN / :

(A.2)

To get a pointwise upper bound of the subsequence .'ni /i 2N , we write j'nj j 6 j'n0 j C

j 1 X i D0

j'ni C1

'ni j 6 j'n0 j C

1 X i D0

j'ni C1

Applying Lemma A.4 to the pairs of functions ˙'0 and ˙.'ni C1 we obtain a sequence . i /i 2N in Cc1 .RN / such that j'nj j 6 j'n0 j C

1 X i D0

j'ni C1

'ni j 6

1 X

'ni j: 'ni / for i 2 N,

i

i D0

and 1 X i D0

k

1  X k 6 C k C k' k'ni C1 k;q N k;q N i W 2 n0 W .R / .R / i D0

 'ni kW k;q .RN / : (A.3)

376

A. Sobolev capacity

In view of estimates (A.2) and (A.3), we now choose a subsequence .'ni /i 2N such that both series 1 1 X X 1 q k'ni C1 'ni kW k;q .RN / and k'ni C1 'ni kW k;q .RN / q i D0 i D0 i

converge. Letting j ! 1 in (A.2), we deduce that capW k;q .F / D 0. By the completeness of the Sobolev spaces W k;q .RN / and by estimate (A.3), the function 1 P k;q .RN /.  hD j belongs to W j D0

For every j 2 N, the subsequence .'ni /i 2N above converges uniformly in the set RN n Fj . The pointwise limit u.x/ D lim 'ni .x/ i !1

defined for every x 2 R n F is thus quasicontinuous with respect to the W k;q capacity: for every  > 0, there exists a Borel set E  RN n F such that the restriction ujE is continuous in E and capW k;p .RN n E/ 6 . N

A.3 Strong additivity The W 1;q capacity is strongly additive: Proposition A.12. For every A1 ; A2  RN , we have capW 1;q .A1 [ A2 / C capW 1;q .A1 \ A2 / 6 capW 1;q .A1 / C capW 1;q .A1 /: By induction, the W 1;q capacity satisfies the finite subadditivity property, and so it is subadditive with constant C D 1. If 'i 2 Cc1 .RN / is a nonnegative function such that 'i > 1 in Ai , we then have max ¹'1 ; '2 º > 1 in A1 [ A2

and

min ¹'1 ; '2 º > 1 in A1 \ A2 :

The proof of Proposition A.12 for compact sets relies on a smooth perturbation of the functions max ¹'1 ; '2 º and min ¹'1 ; '2 º. We first prove the following analogue of Lemma A.4: Lemma A.13. Let '1 and '2 be nonnegative functions in Cc1 .RN /. Then, for every  > 0, there exist nonnegative functions and  in Cc1 .RN / such that j

max ¹'1 ; '2 ºj C j

min ¹'1 ; '2 ºj 6 

in RN ;

and k kqW 1;q .RN / C kkqW 1;q .RN / 6 k'1 kqW 1;q .RN / C k'2 kqW 1;q .RN / :

A.3. Strong additivity

377

For any regular value ı of '2 '1 , the set ¹'2 '1 D ıº is a compact smooth manifold of dimension N 1, and, in particular, is negligible for the Lebesgue measure. Since, for every ˛; ˇ 2 R, we have max ¹˛; ˇº D

˛ C ˇ C j˛ 2

ˇj

and

min ¹˛; ˇº D

˛Cˇ

2

j˛

ˇj

;

it follows that, for every ı > 0, the functions max ¹'1 ; '2 ıº and min ¹'1 C ı; '2 º are Lipschitz-continuous, they have compact support in RN , and in the open set ¹'2 '1 ¤ ıº they are differentiable and their gradients are given by ´ r'1 if '2 '1 < ı, r max ¹'1 ; '2 ıº D r'2 if '2 '1 > ı, and r min ¹'1 C ı; '2 º D

´

r'2 r'1

if '2

'1 < ı,

if '2

'1 > ı.

Proof of Lemma A.13. For every ı > 0, we have ıºjq C jmin ¹'1 C ı; '2 ºjq D j'1 jq C j'2 jq ;

jmax ¹'1 ; '2

and, assuming that ı is a regular value of '2 jr max ¹'1 ; '2

'1 , we also have

ıºjq C jr min ¹'1 C ı; '2 ºjq D jr'1 jq C jr'2 jq

almost everywhere in RN . Therefore, kmax ¹'1 ; '2

ıºkqW 1;q .RN / C kmin ¹'1 C ı; '2 ºkqW 1;q .RN /

D k'1 kqW 1;q .RN / C k'2 kqW 1;q .RN / : The conclusion follows using a regularization argument by convolution. On the one hand, we have 0 6 max ¹'1 ; '2 º

max ¹'1 ; '2

ıº 6 ı

in RN .

On the other hand, given a sequence of mollifiers .n /n2N in Cc1 .RN /, by Jensen’s inequality we have kn  max ¹'1 ; '2

ıºkW 1;q .RN / 6 kmax ¹'1 ; '2

ıºkW 1;q .RN / :

Since a similar property holds for the minimum min ¹'1 C ı; '2 º, we have the conclusion with D n  max ¹'1 ; '2

ıº and  D n  min ¹'1 C ı; '2 º;

for 2ı <  and n 2 N sufficiently large.



378

A. Sobolev capacity

Proof of Proposition A.12. We first prove the strong additivity for compact sets A1 ; A2  RN . For this purpose, let 'i 2 Cc1 .RN / be a nonnegative function such that 'i > 1 in Ai , and let  > 0 be such that 'i > 1 C  in Ai . Taking and  as in the previous lemma, we have > 1 in A1 [ A2 and  > 1 in A1 \ A2 . Thus, capW 1;q .A1 [ A2 / C capW 1;q .A1 \ A2 / 6 k'1 kqW 1;q .RN / C k'2 kqW 1;q .RN / : Minimizing the right-hand side with respect to '1 and '2 , we deduce the strong subadditivity for compact sets. We now assume that A1 and A2 are open subsets of RN . Given compact sets L1  A1 [ A2 and L2  A1 \ A2 , we first take compact sets Ki  Ai such that (Exercise A.1) L1  K 1 [ K 2 :

Replacing Ki by the larger compact set Ki [ L2 which is also contained in Ai , we may also assume that L2  K 1 \ K 2 : By the monotonicity and strong additivity of the capacity on compact sets, we get capW 1;q .L1 / C capW 1;q .L2 / 6 capW 1;q .K1 [ K2 / C capW 1;q .K1 \ K2 / 6 capW 1;q .K1 / C capW 1;q .K2 /:

It follows from the definition of capacity on open sets that capW 1;q .L1 / C capW 1;q .L2 / 6 capW 1;q .A1 / C capW 1;q .A2 /: Maximizing the left-hand side with respect to L1 and L2 , we deduce the strong additivity on open sets. Given arbitrary sets A1 ; A2  RN , let Ui be an open set containing Ai . Thus, by the definition of capacity and by the strong subadditivity on open sets, we obtain capW 1;q .A1 [ A2 / C capW 1;q .A1 \ A2 /

6 capW 1;q .U1 [ U2 / C capW 1;q .U1 \ U2 / 6 capW 1;q .U1 / C capW 1;q .U2 /:

Minimizing the right-hand side with respect to U1 and U2 , the conclusion follows.  The strong additivity is an important tool in the proof of the increasing set lemma: Proposition A.14. For every nondecreasing sequence .An /n2N of subsets of RN , we have 1 [  capW 1;q Ai D lim capW 1;q .An /: i D0

n!1

A.3. Strong additivity

379

Proof. By the monotonicity of the capacity, it suffices to prove the inequality 6. We first prove the property for a nondecreasing sequence .An /n2N of open subsets 1 S Ai . By the monotonicity of of RN . For this purpose, take a compact set K  i D0

.An /n2N , there exists ` 2 N such that K  A` . We then have

capW 1;q .K/ 6 capW 1;q .A` / 6 lim capW 1;q .An /: n!1

Maximizing the left-hand side with respect to K, we have the conclusion for a sequence of open sets. Given an arbitrary nondecreasing sequence of sets .An /n2N , for each n 2 N we take an open set Un  An . The sequence of open sets .Un /n2N need not be nondecreasing. For this reason, we estimate the capacity of the elements of the nonn
S Ui n2N . To achieve this goal, we use the strong additivity decreasing sequence i D0

of the capacity to prove Choquet’s fundamental estimate: Claim. For every n 2 N , we have capW 1;q

n [

Ui

i D0



capW 1;q .An / 6

n X

capW 1;q .Ai /:

ŒcapW 1;q .Ui /

i D0

Proof of the claim. For every j 2 N , by the strong additivity property applied to jS1 Ui and Uj we have the sets i D0

capW 1;q

j [

i D0

The set

jS1 i D0



Ui C capW 1;q

 j[1 i D0



Ui \ Uj 6 capW 1;q

Ui \ Uj is open and contains Aj

1.

 j[1 i D0

 Ui C capW 1;q .Uj /:

Subtracting capW 1;q .Aj

1/

and

capW 1;q .Aj / from both sides, we get capW 1;q

j [

Ui

i D0

6 capW 1;q



 j[1 i D0

capW 1;q .Aj /

Ui



capW 1;q .Aj

1/

C ŒcapW 1;q .Uj /

Adding these estimates for j 2 ¹1; : : : ; nº, the claim follows.

capW 1;q .Aj /: 4

380

A. Sobolev capacity

As n ! 1 in Choquet’s estimate, by the increasing set lemma for open sets we deduce that capW 1;q

1 [

i D0

1  X Ui 6 lim capW 1;q .An / C ŒcapW 1;q .Ui / n!1

capW 1;q .Ai /:

i D0

Given  > 0, by the definition of capacity on arbitrary sets we may choose the sequence of open sets .Un /n2N such that the series in the right-hand side is smaller than 1 1 S S Ai , we deduce that Ui is an open set containing . Since i D0

i D0

capW 1;q

1 [

i D0

 Ai 6 lim capW 1;q .An / C ; n!1

and the conclusion follows.



The increasing set lemma enters in the proof of the celebrated inner regularity property of the capacity due to Choquet, see [84] and Section 5.8.3 in [164]: Proposition A.15. For every Borel set A  RN and every  > 0, there exists a compact set K  A such that capW 1;q .A/ 6 capW 1;q .K/ C : For example, if A is a closed set or a countable union of closed sets, then A is the limit of a nondecreasing sequence of compact sets .Kn /n2N . In this case, Proposition A.15 follows directly from the increasing set lemma: capW 1;q .A/ D lim capW 1;q .Kn /: n!1

The decreasing counterpart of the increasing set lemma is false. This is an important difference between measures and mere capacities (cf. Exercise 2.1). Example A.16. Let .rn /n2N be a decreasing sequence of real numbers converging to 1. The sequence .An /n2N defined by An D B.0I rn / n BŒ0I 1 is decreasing, and 1 T Ai is empty. One would then expect that the set i D0

lim capW k;q .An / D 0;

n!1

but this is not correct since, for every n 2 N, we have capW k;q .An / > capW k;q [email protected] .rn C 1/=2//; and the right-hand side is bounded from below by a positive constant (Exercise A.2).

A.4. Measures on dual Sobolev spaces

381

Exercise A.2. For every r > 0, prove that capW k;q [email protected] r// > min ¹r N ; r N

kq

º capW k;q [email protected] 1//:

We have benefited from the elegant presentation of first-order capacities by Willem, see Chapter 7 in [345]; another good source is the paper by Frehse [135]. An alternative approach based on Bessel capacities for exponents 1 < q < C1 can be found in Chapter 2 in [7].

A.4 Measures on dual Sobolev spaces We establish the existence of diffuse measures supported by compact sets of positive capacity: Proposition A.17. For every compact set S  RN , there exists a nonnegative finite 1 Borel measure , supported in S , such that .S / D .capW k;q .S // q and, for every nonnegative function ' 2 Cc1 .RN /, we have ˆ 06 ' d 6 k'kW k;q .RN / : RN

In particular, for every compact set K  RN ,

1

0 6 .K/ 6 .capW k;q .K// q ; whence capW k;q .K/ D 0 implies that .K/ D 0. Such a measure is called diffuse with respect to the W k;q capacity (cf. Definition 14.6). The main ingredients in the proof of Proposition A.17 are the Riesz representation theorem and the Hahn–Banach theorem. To explain the idea, assume that  is a nonnegative finite Borel measure, supported in S , which satisfies the conclusion of the proposition. Consider the linear functional LW C 0 .S / ! R defined for  2 C 0 .S / by ˆ L./ D

 d:

(A.4)

S N

For every nonnegative function ' 2 Cc1 .R / such that  6 ' in S , we have ˆ L./ 6 ' d 6 k'kW k;q .RN / ; RN

whence L./ 6 inf ¹k'kW k;q .RN / W ' 2 Cc1 .RN / is nonnegative, and  6 ' in S º: (A.5)

382

A. Sobolev capacity

The quantity in the right-hand side defines a sublinear function P in C 0 .S /: .a/ for every  2 C 0 .S / and every t > 0, P .t / D tP ./; .b/ for every 1 ; 2 2 C 0 .S /, P .1 C 2 / 6 P .1 / C P .2 /: Therefore, P satisfies the assumptions of the Hahn–Banach theorem, see Theorem 1.1 in [53] or Theorem 3.1 in [300]. Observe that any linear functional LW C 0 .S / ! R satisfying estimate (A.5) is nonnegative: if  6 0, then L./ 6 0, since we can take ' D 0. Lemma A.18. If LW C 0 .S / ! R is a nonnegative linear functional, then L is continuous, and there exists a unique nonnegative finite Borel measure  supported by S , such that identity (A.4) holds. Proof of Lemma A.18. For every  2 C 0 .S /, we have kkC 0 .S / S 6  6 kkC 0 .S / S ; whence, by the nonnegativity of L, we get kkC 0 .S / L.S / 6 L./ 6 kkC 0 .S / L.S /; and the continuity of the linear functional L follows. By the Riesz representation theorem (Proposition 2.9), there exists a unique finite Borel measure supported by S such that, for every  2 C 0 .S /, ˆ L./ D  d: S

To prove that  is nonnegative on compact subsets K  S , let .n /n2N be a bounded sequence of nonnegative functions in C 0 .S / converging pointwise to the characteristic function K . By nonnegativity of L, for every n 2 N we have ˆ n d D L.n / > 0: S

Letting n ! 1, it follows from the dominated convergence theorem that .K/ > 0. By the inner regularity of Borel measures (Proposition 2.5),  is nonnegative on every Borel set. 

A.4. Measures on dual Sobolev spaces

383

Proof of Proposition A.17. Consider the sublinear function P W C 0 .S / ! R defined for  2 C 0 .S / by P ./ D inf ¹k'kW k;q .RN / W ' 2 Cc1 .RN / is nonnegative, and  6 ' in S º: By the definition of the Sobolev capacity, we have 1

P .S / D .capW k;q .S // q : Thus, by the Hahn–Banach theorem, the linear functional defined in the vector subspace of constant functions by 1

RS 3 tS 7 ! t .capW k;q .S // q

has a linear extension LW C 0 .S / ! R such that, for every  2 C 0 .S /, L./ 6 P ./: By Lemma A.18, L is a continuous linear functional, and there exists a nonnegative finite Borel measure  supported by S satisfying (A.4). For every nonnegative function ' 2 Cc1 .RN /, we also have ˆ L.'/ D ' d 6 P .'/ 6 k'kW k;q .RN / ; S

and the conclusion follows.



Appendix B

Hausdorff measure “Il y avait lieu de chercher à établir une comparaison moins grossière des ensembles que celle de la puissance.”1 Maurice Fréchet

We consider the family of Hausdorff capacities that arises in the definition of the Hausdorff measure. The Hausdorff capacities have some more geometric flavor compared to the Sobolev capacity. The presentation focuses on their connection with density estimates of Borel measures.

B.1 Density estimate Given an exponent 0 6 s < C1 and a gauge 0 < ı 6 C1, we first introduce the Hausdorff capacity Hıs on compact sets by measuring balls as if they were objects of dimension s: Definition B.1. Given a compact set K  RN , the Hıs capacity of K is defined as Hıs .K/

D inf

` °X i D0

!s ris W K



` [

i D0

± B.xi I ri / and 0 < ri 6 ı ;

where !s is evaluated in terms of the Gamma function € by s

!s D

2 : s €. 2 C 1/

We have adopted Hausdorff’s original definition based on coverings with balls, which makes more transparent the equivalence between the N -dimensional Hausdorff measure and the Lebesgue measure on RN , see Definition 1 in [163]. When s is an integer, !s is the volume of the unit ball in Rs . The Hausdorff measure of dimension s – integer or not – is defined as the limit Hs .K/ D lim Hıs .K/: ı!0

1 “It

was necessary to seek to establish a less rough comparison of sets than that of their power.”

386

B. Hausdorff measure

s The Hausdorff capacity H1 , with gauge ı D C1, is also called the Hausdorff content of dimension s. In this case, we impose no restriction on the size of the balls used to cover the set K. For the purpose of identifying sets of zero Hausdorff measure, the Hausdorff content successfully does the job, since we have s Hs .K/ D 0 if and only if H1 .K/ D 0:

The Hausdorff dimension of a compact set K  RN is defined as s dimH .K/ D inf ¹s > 0W H1 .K/ D 0º:

For example, the Hausdorff dimension of a set of zero W k;q capacity is, at most, N kq (see Proposition 10.4 for k D 1). Exercise B.1. Let s 6 t and a compact set K  RN . .a/ Prove that H1t .K/ 6 ! t =!s H1s .K/. s t .b/ Deduce that if H1 .K/ D 0, then H1 .K/ D 0. The Hıs capacity is not a legitimate measure, since additivity fails. It is nevertheless finitely subadditive: for every collection of compact subsets K1 ; : : : ; Kn  RN , Hıs

n [

j D1

n
X Hıs .Kj /; Kj 6 j D1

which readily follows from the definition. It also has some properties that we would naively expect from the Hausdorff measure: Hıs is finite on every compact subset of RN and, for every 0 < r < ı, we have Hıs .BŒxI r/ 6 !s r s : Equality holds provided that s 6 N : Proposition B.2. For every x 2 RN and every 0 < r < ı, we have ´ !s r s if s 6 N , s Hı .BŒxI r/ D 0 if s > N . Proof. The inequality Hıs .BŒxI r/ 6 !s r s follows by covering BŒxI r with single balls B.xI rN / of radii r < rN 6 ı. We now prove the reverse inequality for 0 6 s 6 N : if .B.xi I ri //i 2¹0;:::;`º is a collection of balls covering BŒxI r, it follows from the subadditivity of the Lebesgue measure that rN 6

` X i D0

riN :

B.1. Density estimate

387

Since 0 6 s=N 6 1, s

N

r D .r /

s N

6

` X

riN

i D0

 Ns

6

` X

ris :

i D0

Minimizing the right-hand side over all coverings of B.xI r/, we deduce the reverse inequality. We now consider the case s > N . Given 0 <  6 ı and balls .B.xi I ri //i 2¹0;:::;`º , with 0 6 ri 6 , which cover BŒxI r, we have Hıs .BŒxI r/

6

` X

!s ris

i D0

We can choose the balls so that

` P

i D0

D

` X

!s ris N riN

6 !s 

s N

i D0

` X

riN :

i D0

riN 6 r N C 1. Thus,

Hıs .BŒxI r/ 6 !s  s

N

.r N C 1/:

Letting  ! 0, we have the conclusion.



The Hıs capacity characterizes density estimates of the type .B.xI r// 6 C r s , where  is a nonnegative measure, see the proof of Theorem 2 in §47 in [139]: Proposition B.3. Let ˛ > 0, and let  be a nonnegative Borel measure on RN . Then, we have  6 ˛Hıs on every compact subset of RN if and only if, for every x 2 RN and every 0 < r 6 ı, .B.xI r// 6 ˛!s r s : Proof. It suffices to establish the equivalence for ˛ D 1. For the direct implication, if  6 Hıs , then, by the monotonicity of , we have .B.xI r// 6 .BŒxI r/ 6 Hıs .BŒxI r/; for every x 2 RN and every r > 0. If in addition r < ı, then we get .B.xI r// 6 Hıs .BŒxI r/ 6 !s r s : The case r D ı is obtained from the increasing set property of  (Exercise 2.1), by letting r ! ı from below.

388

B. Hausdorff measure

Conversely, given a compact set K  RN , let .B.xi I ri //i 2¹0;:::;`º be a collection of balls covering K. Since the measure  is subadditive, we have .K/ 6

` X i D0

.B.xi I ri //:

If the radii satisfy ri 6 ı, then, by our assumption on , we get .K/ 6

` X

!s ris :

i D0

Taking the infimum over all collections of balls covering K, the conclusion follows. 

B.2 Frostman’s lemma s The lack of additivity of the Hausdorff content H1 can be compensated by Frostman’s lemma, see §47 in [139]. This important result ensures the existence of positive density measures supported in compact sets of positive Hausdorff measure.

Proposition B.4. For every compact set S  RN , there exists a nonnegative finite s s Borel measure , supported by S , such that .S / > 3 s H1 .S / and  6 H1 on N every compact subset of R . We present a proof due to Howroyd based on the Hahn–Banach theorem and weighted coverings, see Theorem 2 in [170] or Chapter 8 in [228]. We first explain the main idea. Assume that  is a nonnegative measure, supported by S , that satisfies s  6 H1 . Consider the linear functional LW C 0 .S / ! R defined for  2 C 0 .S / by ˆ L./ D  d: (B.1) S

If .˛i /i 2¹0;:::;`º are positive numbers such that  6

` P

i D0

˛i B.xi Iri / in S , then,

by the monotonicity of  and the density assumption, we have L./ 6

` X i D0

˛i .BŒxi I ri / 6

` X i D0

s ˛i H1 .BŒxi I ri / 6

` X i D0

˛i !s ris ;

B.2. Frostman’s lemma

389

whence, for every  2 C 0 .S /, we deduce that L./ 6 inf

` °X i D0

˛i !s ris W  6

` X i D0

± ˛i B.xi Iri / and ˛i > 0 :

(B.2)

The quantity in the right-hand side defines a sublinear function in C 0 .S /. Taking  D S to be the characteristic function of S , the pointwise condition S 6

` X i D0

˛i B.xi Iri /

may be interpreted as a weighted covering of S by balls .B.xi I ri //i 2¹0;:::;`º with weights .˛i /i 2¹0;:::;`º . In particular, the right-hand side of (B.2) is bounded from s above by H1 .S/. Interestingly, both quantities are comparable, see Lemma 8.16 in [228]: Lemma B.5. Let S  RN be a compact set. Then, for every weighted covering of S with balls .B.xi I ri //i 2¹0;:::;`º and weights .˛i /i 2¹0;:::;`º , we have s H1 .S /

63

s

` X

˛i !s ris :

i D0

Proof of Lemma B.5. We first assume that the weights .˛i /i 2¹0;:::;`º are positive rational numbers, and write ˛i D pi =q for some common denominator q 2 N . Repeating pi times each ball B.xi I ri /, without loss of generality we may suppose that pi D 1 for every i 2 ¹0; : : : ; `º. Thus, qS 6

` X i D0

B.xi Iri / :

Geometrically, each point of S belongs to at least q balls. Applying q-times Wiener’s covering lemma (Lemma 9.2) q times, we find disjoint sets of indices J1 ; : : : ; Jq in ¹0; : : : ; `º such that, for each j 2 ¹1; : : : ; qº, the balls .B.xi I ri //i 2Jj are disjoint and .B.xi I 3ri //i 2Jj covers S . In particular, by the definition of the Hausdorff content, X X s !s ris : !s .3ri /s D 3s H1 .S / 6 i 2Jj

i 2Jj

Summing this inequality with respect to j , we get s qH1 .S /

63

s

q X X

j D1 i 2Jj

!s ris

s

63

` X i D1

!s ris :

390

B. Hausdorff measure

We have thus proved that s H1 .S / 6 3s

` X

˛i !s ris ;

i D0

when all the weights are positive rational numbers. The case of real positive weights follows by taking rational numbers ˛Q i > ˛i , for which the estimate above holds, and then taking the infimum over all rational upper bounds of ˛i .  Proof of Proposition B.4. Consider the sublinear function P W C 0 .S / ! R defined for  2 C 0 .S / by P ./ D inf

` °X i D0

˛i !s ris W  6

` X i D0

± ˛i B.xi Iri / and ˛i > 0 :

By the previous lemma, we have s 3 s H1 .S / 6 P .S /:

By the Hahn–Banach theorem, the functional s RS 3 tS 7 ! 3 s H1 .S / t;

thus has a linear extension LW C 0 .S / ! R such that, for every  2 C 0 .S /, L./ 6 P ./:

(B.3)

In particular, L is a nonnegative functional. By Lemma A.18, L is a continuous linear functional represented in terms of a nonnegative Borel measure  supported by S . s We show that  6 H1 using the characterization of density measures on balls (Proposition B.3). If W S ! R is a continuous function supported in a ball B.xI r/, then  6 kkC 0 .S / B.xIr / ; whence, by Property (B.3), we have ˆ  d D L./ 6 kkC 0 .S / !s r s : S

Take a nondecreasing sequence of nonnegative continuous functions in C 0 .RN / converging pointwise to the characteristic function B.xIr / . Using the monotone convergence theorem and the previous inequality, we deduce that .B.xI r/ \ S / 6 !s r s : Since  is supported by S , the conclusion follows.



B.3. Regularity and uniform approximation

391

B.3 Regularity and uniform approximation We extend the definition of the Hıs capacity to every subset of RN , first by inner regularity on open sets, and then by outer regularity, as for the Sobolev capacity. Definition B.6. Given an open set U  RN , the Hıs capacity of U is defined as Hıs .U / D sup ¹Hıs .K/W K  RN is compact and K  U º: This approach avoids technical issues concerning the inner regularity of Hıs , see pp. 9–10 in [79] and Theorem 2.10.22 in [128]. In our case, the inner regularity on open sets holds by definition. The Hıs capacity is also outer regular on compact sets: Proposition B.7. For every compact set K  RN and every  > 0, there exists an open set U  RN such that Hıs .U / 6 Hıs .K/ C : Proof. Let .B.xi I ri //i 2¹0;:::;`º be a collection of balls covering K such that ri 6 ı and ` X !s ris 6 Hıs .K/ C : i D0

Taking U D

` S

i D0

B.xi I ri /, it follows that, for every compact set L  U , the same

collection of balls covers L, whence Hıs .L/ 6

` X i D0

!s ris 6 Hıs .K/ C :

Taking the supremum in the left-hand side over all compact subsets of U , the conclusion follows.  In view of the outer regularity of Hıs on compact sets, there is no ambiguity in defining Hıs on every subset of RN via outer regularity: Definition B.8. For every A  RN , the Hıs capacity of A is defined as Hıs .A/ D sup ¹Hıs .U /W U  RN is open and A  U º: The monotonicity of the Hıs capacity on compact sets is inherited by all subsets: Proposition B.9. For every A; B  RN such that A  B, we have Hıs .A/ 6 Hıs .B/:

392

B. Hausdorff measure

The subadditivity property is also satisfied by the Hıs capacity: Proposition B.10. For every sequence .An /n2N of subsets of RN , we have 1 1 [  X Hıs An 6 Hıs .An /: nD0

nD0

The proof follows the same steps as for the Sobolev capacity (Proposition A.9), but is shorter since the finite subadditivity on compact sets is straightforward. One then shows the countable subadditivity on open sets, and finally on arbitrary subsets. We define the Hausdorff measure of dimension s of a set A  RN as the limit Hs .A/ D lim Hıs .A/: ı!0

The Hausdorff measure H is not  -finite when 0 6 s < N , but induces finite Borel measures by restriction to Borel subsets of finite measure: s

Proposition B.11. Let 0 6 s < C1 and let A  RN be a Borel set. If we have Hs .A/ < C1, then A is Hs measurable and Hs bA is a finite Borel measure on RN . This proposition is a consequence of Carathéodory’s theorem on measurable sets (see Proposition 11.16 in [134]), since the Hausdorff measure Hs satisfies the property of metric additivity, see Theorem 16 in [299] or the proof of Proposition 11.17 in [134]: Lemma B.12. For any disjoint sets A1 ; A2  RN such that d.A1 ; A2 / > 2ı, we have Hıs .A1 [ A2 / D Hıs .A1 / C Hıs .A2 /: The proof of the lemma relies on the fact that an open ball of diameter at most d.A1 ; A2 / cannot intersect simultaneously A1 and A2 . Letting ı ! 0 in the identity above, we deduce the metric additivity: Corollary B.13. For every disjoint sets A1 ; A2  RN , if d.A1 ; A2 / > 0, then Hs .A1 [ A2 / D Hs .A1 / C Hs .A2 /:

The Hausdorff capacities Hıs converge uniformly to the Hausdorff measure Hs on sets of finite Hausdorff measure, see Lemma 1.7 in [127]. Such a property can be deduced from the strong approximation of measures on terms of densities (Proposition 14.3), but there is a simpler proof based on the subadditivity of the Hıs capacity: Proposition B.14. Let A  RN be a Borel set. If Hs .A/ < C1, then, for every ı > 0 and every subset B  A, we have 0 6 Hs .B/

Hıs .B/ 6 Hs .A/

Hıs .A/:

B.3. Regularity and uniform approximation

393

Proof. Since A is measurable and Hs .A/ < C1, for every B  A we have Hs .B/ D Hs .A/

Hs .A n B/:

Since Hıs 6 Hs , for every ı > 0 we then get 0 6 Hs .B/

Hıs .B/ 6 Hs .A/

ŒHıs .B/ C Hıs .A n B/:

On the other hand, by the subadditivity of the Hıs capacity, Hıs .A/ 6 Hıs .B/ C Hıs .A n B/: Combining the two estimates, the conclusion follows.



Appendix C

Solutions and hints to the exercises 1.1 (p. 8). If vW .0; C1/ ! R is such that v.r/ D u.x/ where r D jxj, then N

v 00 .r/ C

1 r

v 0 .r/ D u.x/ D 0;

and now rewrite the ODE as d N .r dr

1 0

v .r// D 0:

1.2 (p. 10). .a/ Write the integral as ˆ x ˆ .x y/00 .y/ dy C 1

C1

x/00 .y/ dy;

.y

x

and integrate by parts. .b/ Proceed as in the proof of Proposition 1.2. 1.3 (p. 14). .a/ From the mean value identity (1.4), deduce that ˆ 1 ju.x/j 6 juj; !N r N RN for every x 2 RN and every r > 0, and then let r ! 1.

.b/ For every x; y 2 RN and every r > 0, we have ˇ ˇ ˆ ˇ ˇ 1 juj; ju.x/ u.y/j D ˇˇ u uˇˇ 6 !N r N A.x;yIr / B.xIr / B.yIr /

where A.x; yI r/ is the symmetric difference between balls:

A.x; yI r/ D ŒB.xI r/ n B.yI r/ [ ŒB.yI r/ n B.xI r/: For every r >

jx yj , 2

we have

A.x; yI r/  B

x C y 2

Ir C

jx

2

yj 

nB

x C y 2

Ir

jx

2

yj 

Evaluating the measure of the annulus in the right-hand side, deduce that jA.x; yI r/j 6 C1 jx

yjr N

1

:

:

396

C. Solutions and hints to the exercises

Apply this estimate and the boundedness of u to get ju.x/

u.y/j 6

C2 jx r

yjkukL1 .RN / ;

and let r ! 1. 2.1 (p. 22). .a/ Since

1 S

Ak is the union of the disjoint sets

kD0

AnC1 n An ;

An ;

AnC2 n AnC1 ;

:::;

by the additivity of  we have 

1 [

kD0

1  X Ak D .An / C .Ak n Ak

1 /;

kDnC1

and then let n ! 1.

.b/ We have that An is the union of the disjoint sets An n AnC1 ;

whence .An / D Let n ! 1 to conclude.

AnC1 n AnC2 ;

1 X

kDn

1 \

:::;

.Ak n AkC1 / C 

Ak ;

kD0 1 \

kD0

 Ak :

2.2 (p. 22). .a/ Write B D A [ .B n A/ [ ; [ ; [ : : : Then, by the additivity and nonnegativity of , .B/ D .A/ C .B n A/ > .A/: .b/ Since

1 S

Ak is the union of the disjoint sets

kD0

A0 ;

A1 n A0 ;

A2 n .A0 [ A1 /;

:::;

by the additivity and monotonicity of , we have 

1 [

kD0

1 k[1 1   X X
Ak D .A0 / C  Ak n Aj 6 .A0 / C .Ak /: kD1

j D0

kD1

397

2.3 (p. 22). Taking A D B D ;, we have kkM./ > 0. Assuming kkM./ D 0 and taking B D ;, we deduce that .A/ 6 0 for every Borel set A; a similar argument yields .B/ 6 0 for every Borel set B. Therefore  D 0. The identity ktkM./ D jt jkkM./ ; holds for t > 0; for t > 0 one interchanges the role of A and B. Finally, for any ;  2 M./ and any Borel sets A; B  , we have . C /.A/

. C /.B/ 6 kkM./ C kkM./ I

maximizing the left-hand side with respect to A and B, we obtain the triangle inequality. 2.4 (p. 24). Given a Borel set E   satisfying the conclusion of the Jordan decomposition theorem, by the additivity of the measure we have .A/ 6 .A \ E/

and .B/ > .B n E/;

for every Borel sets A; B  . Consequently, .A/

.B/ 6 .A \ E/

.B n E/ D C .A/ C  .B/;

By the monotonicity of the measures  and C , this implies that kkM./ 6 C ./ C  ./: To obtain the reverse inequality, it suffices to estimate the norm kkM./ by evaluating the measure  on the sets E and  n E. 2.5 (p. 24). Since, for every m; n 2 N, kfm

fn kL1 ./ D kfm

fn kM./ ;

we deduce that .fn /n2N is a Cauchy sequence in L1 ./. By the completeness of L1 ./, this sequence converges to some function f 2 L1 ./. Thus, for every Borel set A  , we have ˆ ˆ fn D .A/ D lim f: n!1 A

A

2.6 (p. 25). .a/ By additivity of , for every Borel subsets A; B   we have .A/ D .A n B/ C .A \ B/; .B/ D .B n A/ C .A \ B/:

398

C. Solutions and hints to the exercises

Hence, we get .B/ D .A n B/

.A/

.B n A/;

where the sets A n B and B n A are disjoint. Conclude using the inner regularity of . .b/ Given disjoint compact sets K; L  , let .n /n2N be a sequence of functions in Cc0 ./ converging pointwise to the difference of characteristic functions K L , and such that, for every n 2 N, we have jn j 6 1 in . By the dominated convergence theorem, we have ˆ ˆ lim n d D .K L / d D .K/ .L/: n!1





Therefore, .K/

.L/ 6 sup

²ˆ



³  dW  2 Cc0 ./ and jj 6 1 in  ;

and we then take the supremum in the left-hand side with respect to K and L. To get the reverse inequality, it suffices to observe that, for every  2 Cc0 ./ such that jj 6 1 in , we have ˆ ˆ  d 6 jj djj 6 jj./; 



and the conclusion follows from the characterization (2.1) of the total variation norm in terms of the measure jj. 2.7 (p. 32). Proceed as in the proof of Lemma 2.16. 2.8 (p. 33). For every ' 2 Cc1 ./, we have ˇ ˇˆ ˆ ˇ ˇ ˇ 6 C kun ˇ u ' u ' n ˇ ˇ 



ukL1 ./ :

Hence,

ˆ



u ' D lim

n!1

ˆ

un ':



3.1 (p. 41). Given n 2 N, let gn W X ! R be the bounded function defined by ´ jf jp 2 f in ¹jf j 6 nº; gn D 0 otherwise: Applying the functional estimate with test function gn , we get ˆ p 0 jf jp d 6 C kf kLpp .¹jf j6nºI/ : ¹jf j6nº

Thus, kf kLp .¹jf j6nºI/ 6 C . Conclude using Fatou’s lemma.

399

x \ W 1;q ./. For every ˆ 2 C 1 .I x RN /, by the diver4.1 (p. 50). Let u 2 C 1 ./ 0 gence theorem we have ˆ



G
ˆD

ˆ



u div ˆ D

ˆ



ru
ˆ C

ˆ



ˆ

@

u .ˆ
n/ d:

Thus, ˆ

@

u .ˆ
n/ d D

(C.1)

.G C ru/
ˆ:

x RN / converging Given  2 C 1 .@/, take a bounded sequence .ˆk /k2N in C 1 .I pointwise to 0 in , and such that ˆk
n D  on @. Applying identity (C.1) with test function ˆk , and letting k ! 1, it follows from the dominated convergence theorem that ˆ u d D 0: @

Conclude that u D 0 on @. 4.2 (p. 51). Given a sequence of positive numbers .n /n2N converging to zero, let un W BŒ0I 1 ! R be the function defined by 1

un .x/ D .1 C n2 / 2

1

.jxj2 C n2 / 2 :

Then, un 2 C01 .BŒ0I 1/ and the sequence .un /n2N converges uniformly to u. Moreover, x run .x/ D : 1 .jxj2 C n2 / 2 Thus, the sequence .run /n2N is uniformly bounded and converges pointwise to x=jxj in B.0I 1/ n ¹0º. Conclude using the stability property. Alternatively, for every 0 < r < 1 and every ˆ 2 C 1 .BŒ0I 1I RN /, by the divergence theorem we have ˆ

B.0I1/nBŒ0Ir 

u div ˆ D

ˆ

B.0I1/nBŒ0Ir 

ru
ˆ

ˆ

@B.0Ir /

u.ˆ
n/ d;

where n is the outer normal vector on @B.0I r/. The last integral converges to zero as r ! 0.

400

C. Solutions and hints to the exercises

4.3 (p. 51). Take un W BŒ0I 1 ! R defined by un .x/ D

.jxj2

1 C n2 /˛=2

1 ; .1 C n2 /˛=2

where .n /n2N is a sequence of positive numbers converging to zero, and proceed as in Exercise 4.2. 4.4 (p. 52). Given a sequence of mollifiers .n /n2N in Cc1 .RN /, we have r.n  u/ D 0 in RN , whence n  u D 0 in RN . Since the sequence .n  u/n2N converges to u in Lq ./, we have u D 0 almost everywhere in . 4.5 (p. 52). Given a sequence of mollifiers .n /n2N in Cc1 .RN /, by the classical Leibniz formula and the approximation property of Sobolev functions (Proposition 4.3) we have rŒ.n  u/.n  v/ D .n  ru/ .n  v/ C .n  u/ .n  rv/ in RN . This sequence converges in Lq .RN I RN / to the function ´ .ru/v C u.rv/ in , F D 0 in RN n . The conclusion follows from the stability property. 4.6 (p. 52). By the mean value theorem, for every t 2 R we have jH.t /j 6 C jt j, whence H.u/ 2 Lq ./. Let .n /n2N be a sequence of mollifiers in Cc1 .RN /. Since H.0/ D 0, the function H.n  u/ has compact support in RN . By the divergence theorem, we thus have ˆ ˆ H.n  u/ div ˆ D H 0 .n  u/ .n  ru/
ˆ; RN

RN

for every ˆ 2 Cc1 .RN I RN /. The sequence .H.n  u//n2N converges in Lq .RN / to ´ H.u/ in , vD 0 in RN n .

Since H 0 is bounded and continuous, the sequence .H 0 .n  u/ .n  ru//n2N converges in Lq .RN I RN / to ´ H 0 .u/ru in , F D 0 in RN n .

401

The chain rule still holds for Lipschitz-continuous functions H W R ! R such that H.0/ D 0, but the proof has to be substantially modified, see Theorem 4.2 in [35] and Theorem 2.1 in [12]. The operator u 2 W01;q ./ 7! H.u/ 2 W01;q ./ is also continuous, see Theorem 1 in [220]. 4.7 (p. 52). Take a sequence .rn /n2N in the interval .0; 1/ converging to 1. For every n 2 N, the function un W B.0I 1/ ! R defined for x 2 B.0I 1/ by ´ u.x=rn / if jxj < rn , un .x/ D 0 otherwise, belongs to W01;q .B.0I 1//, and the sequence .un /n2N converges to u in W01;q .B.0I 1//. To conclude, take a sequence of mollifiers .n /n2N such that supp n C B.0I rn / b B.0I 1/. The sequence .n  un /n2N has the required properties. 4.8 (p. 54). Let u 2 W01;2 ./ be the solution of the Dirichlet problem. Take a nondecreasing function H 2 C 1 .R/ such that H.t / D 0 for every t > 0 and H 0 is bounded in R. By the chain rule for Sobolev functions (Exercise 4.6), we have H.u/ 2 W01;2 ./. Thus, H.u/ is an admissible test function, and by nonnegativity of , we have ˆ ˆ 

H 0 .u/jruj2 D

H.u/ 6 0:



Since the integrand in the left-hand side is nonnegative, we deduce that rH.u/ D H 0 .u/ru D 0; whence H.u/ D 0 almost everywhere in . To conclude, choose H satisfying H.t / < 0 for every t < 0. 4.9 (p. 55). By the partial converse of the dominated convergence theorem, for every subsequence .fnk /k2N there exist a further subsequence .fnki /i 2N converging almost everywhere to f , and h 2 Lq .XI / such that jfnki j 6 h almost everywhere. By the dominated convergence theorem, it follows that .H.fnki //i 2N converges strongly to H.f / in L1 .XI /. Since the limit H.f / does not depend on the subsequence .H.fnk //k2N , we deduce the convergence of the entire sequence .H.fn //n2N . 4.10 (p. 56). Given an increasing sequence of positive numbers .rn /n2N converging to 1, take un W B.0I 1/ ! R defined for x 2 B.0I 1/ by 8 rn . 1 rn

402

C. Solutions and hints to the exercises

4.11 (p. 58). Prove that, for x ¤ 0, ru.x/ D

 1 ˛ ˛ log jxj

and observe that ru 2 LN .B.0I 1/I RN / for ˛
0.

4.15 (p. 69). The case p D 1 follows from (4.12) as  ! 0. For 1 < p < C1, adapt the identity  ˆ ˆ ˆ 1 p p 2 jg.u/j D .p 1/ t jg.u/j dt; 

0

¹jg.u/j>tº

as in the proof of Proposition 4.24 using estimate (4.12). 4.16 (p. 73). Since   R2 and r > 1, for every  2 Lr ./ we have  2 .W01;2 .//0 , see Exercise 4.12 for the case r < 2. Hence, the Dirichlet problem with density  has a solution in W01;2 ./. Compared to the case N > 3, the argument requires some minor modification due to the lack of the critical Sobolev embedding W01;2 ./ 6 L1 ./.

403

5.1 (p. 80). Applying the differential inequality, we get .H 1

ˇ 0

/ 6 .1

ˇ/. B/

almost everywhere in Œ0; s, whence the function t 2 Œ0; s 7 ! .1

ˇ/Bt C H.t /1

ˇ

is non-increasing. This gives the estimate of H.t /. If H.t / > 0, it then follows that .1 ˇ/Bt < H.0/1 ˇ . 5.2 (p. 83). Let 0 < r < 1 be such that ' D 1 in the open ball B.0I r/. By an explicit computation, for every x 2 B.0I r/ with x ¤ 0 we have 1  1  ˛ 1 ju.x/j D ˛.N 2/ N log .1 C o.1//; jxj jxj and, for every e 2 RN n ¹0º, 2

jD u.x/Œe; ej D .N

1  1  2/ N log jxj jxj

ˇ ˇjej2 ˇ

˛ˇ

ˇ x 2 ˇ N
e C o.1/ˇˇ: jxj

Therefore, u is a function in L1 .B.0I 1// for any ˛ > 0, but u 62 W 2;1 .B.0I 1// if ˛ 6 1. The distributional and the pointwise Laplacians coincide, since, by the dominated convergence theorem and the divergence theorem, we have ˆ ˆ ˆ u ' D ' u; u ' D lim B.0I1/

for every ' 2

!0 B.0I1/nB.0I/

B.0I1/

Cc1 ./.

5.3 (p. 88). Take a sequence of smooth functions Hn W R ! R such that .1/ Hn .0/ D 0 and .Hn /n2N converges pointwise to T , .2/ .Hn0 /n2N is uniformly bounded and converges pointwise to Œ ; . By the chain rule for Sobolev functions, we have Hn .u/ 2 W 1;q ./ and rHn .u/ D Hn0 .u/ru. Conclude using the stability property. 7.1 (p. 117). .a/ Proceed as in Example 6.2; see also Lemma 17.6. .b/ Let u be the solution of the linear Dirichlet problem with density b! in !. By .a/, we have .v u/ > 0 in the sense of .C01 .!// x 0 . It follows from the weak maximum principle (Proposition 6.1) that R! v D v u > 0 almost everywhere in !.

404

C. Solutions and hints to the exercises

8.1 (p. 126). .a/ It suffices to observe that, for every x 2 Lu \ Lv , B.xIr /

ju C v

Œu.x/ O C v.x/j O 6

B.xIr /

ju

u.x/j O C

B.xIr /

jv

v.x/j: O

.b/ For every x 2 Lu , we have B.xIr /

j˛u

˛ u.x/j O D˛

B.xIr /

ju

u.x/j: O

8.2 (p. 126). By the Lipschitz continuity of H , for every x 2 Lu we have B.xIr /

jH.u/

H.u.x//j O 6C

B.xIr /

ju

u.x/j: O

8.3 (p. 133). By the mean value property of superharmonic functions (Proposition 2.18), the function r7 !

u; B.xIr /

is non-increasing. As r ! 0, we then get lim

r !0 B.xIr /

uD

sup

u:

0 0, for every x 2 RN we have B.0I r/  B.xI jxj C r/, and then ˆ 1 Mu.x/ > juj > juj: !N .jxj C r/N B.0Ir / B.xIjxjCr /

Thus,

1 lim inf jxj Mu.x/ > jxj!1 !N N

ˆ

B.0Ir /

juj:

To conclude, let r ! 1.

.b/ Observe that if Mu 2 L1 .RN /, then lim inf jxjN Mu.x/ D 0: jxj!1

9.2 (p. 140). .a/ Compute

´

u using the integration formula in polar coordinates:

B.0I 12 /

ˆ

B.0I 12 /

u D N

ˆ

1 2

0

1 dr: rj log rj˛

.b/ Since B.0I jxj/  B.xI 2jxj/, we have 1 Mu.x/ > u> N ! .2jxj/ N B.xI2jxj/

ˆ

u:

B.0Ijxj/

Compute the integral in the right-hand side and conclude. 9.3 (p. 142). Use Cavalieri’s principle (1.5) and the improved maximal inequality (9.3) to get ˆ ˆ 1  ˆ q q 2 juj dt: .Mu/ 6 C q t RN

0

¹juj> 2t º

406

C. Solutions and hints to the exercises

Conclude using Fubini’s theorem. 9.4 (p. 142). For every  > 0, by Cavalieri’s principle and the improved maximal inequality (9.3) we have ˆ ˆ 1 Mu D jB.0I R/ \ ¹Mu > t ºj dt B.0IR/

0

6 jB.0I R/j  C

ˆ

1 

Applying Fubini’s theorem, we deduce that ˆ ˆ Mu 6 jB.0I R/j  C C B.0IR/

To conclude, take  D 2

ffl

B.0IR/ juj

C t

RN

ˆ

¹u> 2t º

juj logC

 juj dt:

 2juj  

:

(assuming that the average integral is positive).

9.5 (p. 150). Apply the triangle inequality and Jensen’s inequality. 10.1 (p. 163). Note that ˇ ˇ ˇ ˇ ˇ ˇ ˇu.y/ u.z/ dz ˇˇ D ˇˇ ˇ B.xIr /

Œu.y/ B.xIr /

ˇ ˇ u.z/ dz ˇˇ 6

B.xIr /

ju.y/

u.z/j dz:

Integrating with respect to y over B.xI r/, we obtain the first inequality. By the triangle inequality, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ju.y/ u.z/j 6 ˇu.y/ u./ d ˇ C ˇu.z/ u./ d ˇˇ: B.xIr /

B.xIr /

Integrating with respect to y and z over B.xI r/, we get the second inequality. 10.2 (p. 169). .a/ By the Minkowski inequality, we have ˆ 1 k.B.
I r//kLp.RN / 1 dr: kNkLp .RN / 6 N 0 rN 1

(C.2)

Using the density assumption for small values of r and the boundedness of  for large values of r, estimate the Lp norm of the integrand as in the proof of Lemma 10.12: 8 N p 1 N 0, we have and krH. 't /kLq .RN / 6

¹j'j > t º  ¹H. 't / > 2º

C1 kr'kLq .¹ t 6j'j6tº/ : 2 t

Applying the weak trace inequality to the function H. 't /, we then get C 0 C1 kr'kLq .¹ t 6'6tº/ : 2 t

1

.¹j'j > t º/ p 6 Thus,

C2 p q q kr'kL q .RN / kr'kLq .¹ t 6'6tº/ : t 2 Integrate over .0; 1/, and conclude using Cavalieri’s principle in the left-hand side and Fubini’s theorem in the right-hand side. tp

1

.¹j'j > t º/ 6

16.2 (p. 270). .a/ By Jensen’s inequality, for every ball B.xI r/  RN we have B.xIr /

jruj 6



q

B.xIr /

jruj

 q1

6

C r

N q

krukLq .RN / :

Conclude using the Poincaré–Wirtinger inequality (10.5) and Proposition 16.9. .b/ Since uO 2 Lq .RN / is uniformly continuous, we have lim u.x/ O D 0: jxj!1

17.1 (p. 278). .a/ We may assume that w.0/ D 0. By the mean value identity (1.4) for harmonic functions, for every x 2 RN and every r > 0 we have w.x/ D w.x/

w.0/ D

Œw.y/ B.xIr /

w.z/ dy dz:

B.0Ir /

Thus, jw.x/j 6

B.xIr /

B.0Ir /

jw.y/

w.z/j dy dz:

Since the balls B.xI r/ and B.0I R/ are contained in B.0I r C jxj/, we have jw.x/j 6

.r C jxj/2N r 2N

B.0Ir Cjxj/

B.0Ir Cjxj/

jw.y/

w.z/j dy dz:

Taking r D jxj, we deduce that jw.x/j 6 22N ŒwBMO : Thus, w is a bounded harmonic function in RN . By Liouville’s theorem (see Exercise 1.3), we deduce that w D w.0/ D 0.

411

.b/ The function u N is weakly harmonic. By Proposition 17.4, we also N 2 have that  6 C 00 H1 . It thus follows from Proposition 17.3 that u N has bounded mean oscillation. Conclude using Weyl’s lemma (Proposition 2.14) and Assertion .a/. 17.2 (p. 291). The direct implication follows along the lines of the proof of Proposition 17.13. To prove the converse, observe that the linear functional ˆ V 3  7 !  d 

x is well defined in the vector subspace V D ¹W  2 C01 ./º, and has a contin1 uous linear extension to L ./. Conclude using the Riesz representation theorem (Proposition 3.3). 17.3 (p. 297). .a/ Use the scaling invariance of the capacity in dimension two. .b/ Observe that, for every ' 2 Cc1 .R2 /, ˆ x1 ˆ x2 @2 ' '.x1 ; x2 / D .s; t / ds dt: 1 1 @x1 @x2 .c/ From .a/, we deduce that, for every non-empty compact set K  R2 , we have 0 capD 2;1 .K/ D capD 2;1 .¹0º/. From .b/, we have capD 2;1 .¹0º/ > 0. Since H1 D1 on non-empty compact sets, the conclusion follows. 19.1 (p. 315). Take a measurable set E  A such that .fn /n2N converges uniformly to f in E. Since ˆ ˆ ˆ ˆ jfn f j d 6 jfn f j d C jfn j d C jf j d X

E

XnE

and E has finite measure, we have ˆ ˆ lim sup jfn f j d 6 lim sup n!1

n!1

X

XnE

jfn j d C

XnE

ˆ

XnE

jf j d:

For every ı > 0, by Egorov’s theorem we may choose E such that .X n E/ 6 ı. Conclude using the equi-integrability of the sequence .fn /n2N . 21.1 (p. 341). For almost every 0 < r <  < d.a; @/, identity (21.4) becomes  ˆ  ˆ ds 1 : (C.3) u d D u d u 2 s r @B.aIr / @B.aI/ B.aIs/ Since lim

ˆ

s!0 B.aIs/

u D u.¹aº/;

412

we have

C. Solutions and hints to the exercises

1 lim r !0 log 1=r

ˆ

 r

 ds D u.¹aº/: u s B.aIs/

ˆ

Dividing identity (C.3) by log 1=r, and letting r ! 0, the conclusion follows. 21.2 (p. 341). Proceed as in the proof of Proposition 21.2, and apply the counterpart of Lemma 21.4 in dimension two (Exercise 21.1). 22.1 (p. 359). The function t 7! .t C 1/ log .t C 1/ t is the Legendre transform of the function s 7! es .1 C s/. Thus, for every s; t > 0 and C > 0, by Young’s inequality we have: st 6 es=C C .C t C 1/ log .C t C 1/: N 2 Under the stronger assumption 0 6  6 C H1 , for some ˛ > 0 and ı > 0, it follows from the exponential integrability of the Newtonian potential (Proposition 17.8) that N=C is a nonnegative supersolution of the Dirichlet problem (Lemma 17.6). Hence, by the method of sub- and supersolutions (Proposition 22.7) we have a solution of the Dirichlet problem with density . A nonnegative measure that is diffuse with respect to HN 2 can be strongly approximated in M./ by a sequence of measures .n /n2N satisfying the previous density property (Proposition 14.3). Using the absorption estimate (22.5) and the linear elliptic L1 estimate (Proposition 3.2), we deduce that the sequence of solutions .zn /n2N converges to the solution of the Dirichlet problem (22.3) with density .

22.2 (p. 359). Use min ¹z; 0º as a test function in the Euler–Lagrange equation. Deduce that r min ¹z; 0º D 0 almost everywhere in . 22.3 (p. 359). Take a sequence of measures .n /n2N in .W01;2 .//0 converging strongly to  in M./ (see Propositions 14.1 and 14.2). For every n 2 N, let zn 2 W01;2 ./ be the solution of the Dirichlet problem (22.3) with density n . By the contraction estimate (22.5) and the linear elliptic L1 estimate (Proposition 3.2), the sequence .zn /n2N converges to the solution of the Dirichlet problem (22.3) with density . 22.4 (p. 363). Since the W 2;1 capacity and the HN 2 Hausdorff measure vanish on the same sets (cf. Proposition 17.15), there exists a compact set K  ¹u D 0º such that HN 2 .K/ > 0. By Frostman’s lemma (Proposition B.4), there exists a positive N 2 measure  supported in K such that  6 H1 . The Dirichlet problem (22.3) with density  has a solution (Exercise 22.1), and we may then proceed as in the proof of Proposition 22.5.

413

A.1 (p. 373). Given a nonnegative function ' 2 Cc1 .RN / such that ' > 1 on @B.0I r/, let 'r .x/ D '.rx/. The function 'r is admissible to estimate the capacity of @B.0I 1/. By a change of variables, we have capW k;q [email protected] 1// 6 k'r kqW k;q .RN / 6 max ¹r

N

; r kq

N

ºk'kqW k;q .RN / :

A.2 (p. 381). First consider the case ` D 2. The compact sets L1 D K n A2 and L2 D K n A1 are disjoint. Thus, there exist disjoint open sets B1 and B2 such that Li  Bi for i 2 ¹1; 2º. The conclusion then follows with K1 D K n B2 and K 2 D K n B1 . For ` > 3, proceed by induction using the case ` D 2. s B.1 (p. 386). .b/ If H1 .K/ D 0, then H1s .K/ 6 Hs .K/ D 0: By the estimate in (a), t we then have H1 .K/ 6 H1t .K/ D 0:

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Index —A— absolutely continuous measure, 219, 233 absorption estimate, 4, 312, 341 —B— balayage, 115 Banach–Saks property, 159 Besicovitch covering lemma, 143 Bessel kernel, 369 Bôcher–Koebe lemma, 14 Borel set, 22 bounded mean oscillation, 276 boundedness principle, 205 boxing inequality, 162 Brezis–Merle inequality, 279 —C— Calderón–Zygmund regularity, 82 Campanato’s lemma, 265 capacity capD 2;1 (homogeneous), 292 cap1 (Laplacian), 181, 363 capNewt (Newtonian), 181 capW k;q (Sobolev), 367 Hıs (Hausdorff), 385 comparison, 155 equivalence, 181, 271, 292 non-equivalence (W 1;2 and W 2;1 ), 292 quasi-everywhere, 128 quasicontinuous, 304 Cavalieri’s principle, 14 classical solution, 17 closure lemma (Sobolev spaces), 55 co-area formula, 326 integration in polar coordinates, 13 coincidence set, 176 composition in Sobolev spaces first order, 52, 88 second order (Maz0 ya), 186 concentrated measure, 224 conjugate exponent, 41

continuity principle Evans–Vasilesco, 306 Lewi–Stampacchia (obstacle), 191 contraction estimate, 46 convolution, 9, 26 pointwise convergence, 127 covering lemma Besicovitch, 143 Wiener, 140 —D— d@ (distance to boundary), 321 density measure, 387 existence (Frostman), 388 diffuse measure, 219 decomposition in dual spaces, 225, 307 existence, 381, 388 Frostman’s lemma, 388 functional characterization (W 1;2 ), 307 strong approximation, 216, 218, 223 Dirac mass, 24 Dirichlet problem classical solution, 17 linear, 40 nonlinear, 44 Schrödinger operator, 356 weak solution, 40 Dirichlet problem (linear), 17, 40 boundary condition, 321 Calderón–Zygmund estimate, 82 classes of solutions bounded, 291 continuous, 299 variational (W01;2 ), 53 compactness, 91 continuity principle, 306 equivalent formulations, 97, 98, 324 existence, 41 exponential integrability, 281, 287 extension, 115 localization, 104 maximum principle, 17, 95 precise representative, 131

450

Index

regularity with measure data, 77, 87 Stampacchia estimate (L1 ), 78 uniqueness, 43 Dirichlet problem (nonlinear), 44 absorption estimate, 4, 312, 341 contraction estimate, 46 existence, 44, 64, 311, 315 method of sub- and supersolutions, 328 non-existence, 47 Perron–Remak method, 332 stability, 342 subcritical, 337 subsolution, 327 supercritical, 344, 347 Dirichlet problem (Schrödinger), 356 existence, 357, 359 method of sub- and supersolutions, 356 non-existence, 359 divergence (fractional), 250 duality map, 160 —E— energy functional, 62 Dirichlet, 1, 200 Thomas–Fermi, 70 equi-integrable, 47 Euler–Lagrange equation, 64 Evans–Vasilesco continuity principle, 306 exceptional set, 127, 128, 134 —F— fractional divergence, 250 gradient, 246 fractional Sobolev space, 241 connection with W 1;q , 244 interpolation inequality, 243 Sobolev inequality, 241 Frostman’s lemma, 273, 388 function balayage, 116 bounded mean oscillation, 276 harmonic, 7, 30 maximal, 124 quasicontinuous, 304 superharmonic, 12, 33 truncation, 87 weak Lp (Marcinkiewicz), 84

fundamental solution, 9 fundamental theorem of the calculus of variations, 50 —G— Gagliardo seminorm, 241 gauge, 385 gradient (fractional), 246 Green function, 44 Green’s identity, 325 Grishin’s property, 104 —H— Hıs , 385 s (content), 386 H1 harmonic function, 7 Hausdorff capacity, 385 comparison with Sobolev cap, 155 diffuse measure (existence), 388 equivalence with Sobolev cap, 271, 292 gauge, 385 metric additivity, 392 monotonicity, 391 outer regularity, 391 subadditivity, 392 uniform convergence, 392 Hausdorff content, 386 Hausdorff dimension, 386 Hausdorff measure, 385 Hölder continuity integral characterization, 267 Morrey’s condition, 270 —I— increasing set lemma capacity, 378 measure, 22 inequality boxing, 162 Brezis–Merle, 279 capacitary strong, 259 weak, 374 Gagliardo–Nirenberg interpolation, 89 Hardy, 239, 290 Hardy–Littlewood, 142 interpolation (W 1;2 ), 88 interpolation (weak W 1;q ), 91 Kato, 99, 103, 109 maximal, 139, 143, 150

Index Minkowski, 167 Poincaré, 52, 353 Poincaré–Wirtinger, 146, 163 Sobolev, 56 fractional, 241 trace, 235 Adams, 167 Gagliardo, 238 Maz0 ya, 216, 258 Trudinger, 90 Young, 317 inner regularity capacity, 380, 391 measure, 25 integrability condition, 328, 343 integration in polar coordinates, 13 interpolation inequality (W 1;2 ), 88 inverse maximum principle, 107 —J— Jordan decomposition theorem, 23 —K— Kato measure, 303 Kato’s inequality L1 , 99 finite charge, 117 measures, 103, 109 up to boundary, 329 —L— Laplace equation, 7 Laplacian, 1 fundamental solution, 9 Lebesgue decomposition theorem, 224 Lebesgue point, 126 L1 function, 123 Cauchy property, 128 superharmonic function, 131 Lebesgue set, 126 Lebesgue’s differentiation theorem, 123 Legendre transform, 317 lower semicontinuity, 25

451 —M— maximal function, 124 maximal inequality Lebesgue functions, 139 potentials, 150 Sobolev functions, 143 maximum principle inverse, 107 quantitative, 15, 36 strong, 17, 351 weak, 17, 95 mean value property harmonic functions, 14 superharmonic functions, 12, 34 measure, 21 absolutely continuous (diffuse), 219 approximation, 26 compactness, 27 concentrated, 224 decomposition (Lebesgue), 224 density, 387 diffuse, 219 Hausdorff, 385 lower semicontinuity of norm, 25 monotonicity, 22 norm, 22 positive and negative parts, 23 regularity, 25 strong convergence, 24 subadditivity, 22 total variation, 23 weak convergence, 24 method of sub- and supersolutions, 328 integrability condition, 328 Schrödinger operator, 356 metric additivity, 392 minimizing sequence, 63 mollifier, 26 monotone set lemma, 22 —N— Newtonian potential, 8, 29, 32 bounded mean oscillation, 276 Cavalieri’s representation, 32 exponential integrability, 279, 282 Poisson equation, 8, 29

452

Index

norm Gagliardo seminorm, 241 Morrey (density measures), 86 Sobolev, 50, 367 total variation (measure), 22 normal derivative, 111, 112 comparison, 197 —O— !N (measure of B.0I 1/), 11 obstacle problem barrier (Lebesgue), 194 capacitor problem, 204 continuity principle, 191 energy, 200 equivalence, 202 total charge, 195 outer normal vector (n), 6 outer regularity Hausdorff capacity, 391 measure, 25 Sobolev capacity, 372 —P— Perron–Remak method, 188, 332 Poincaré inequality, 52, 146, 163, 353 Poisson equation, 8 approximation by convolution, 36 weak solution, 28 positive distributions (Schwartz), 35 potential, 131 bounded, 205 bounded mean oscillation, 277 continuous, 306 exceptional set, 134 Hölder-continuous, 209 Lp , 207 quasicontinuous, 304 precise representative, 126 —Q— quantitative maximum principle, 15 quasi-everywhere, 128 quasicontinuity, 304 Lusin property, 303 potentials, 304 Sobolev, 304, 376

—R— removable singularity bounded, 172 Hölder-continuous, 176 Lp , 175 Schwarz’s principle, 171 representation formula fractional, 246 order one, 58 order two, 10 Riesz potential Cavalieri’s representation, 167 fractional gradient, 247 trace inequality (Adams), 167 Riesz representation theorem, 27, 41 —S—  (surface measure), 7 N (measure of @B.0I 1/), 8 Schrödinger operator, 351 Dirichlet problem, 356 strong maximum principle L1 potential, 352 Lp potential, 355 L1 potential, 351 Schwarz’s principle, 171 semi-additivity, 372 semimeasure, 219 set exceptional, 128 Lebesgue, 126 sign condition, 62 smooth function, 6 smooth open set, 6 normal vector (n), 6 Sobolev capacity, 367 capacitary inequality, 259, 374 comparison with Hausdorff cap, 155 diffuse measure (existence), 381 equivalence with Hausdorff cap, 271, 292 increasing set lemma, 378 monotonicity, 372 pointwise convergence, 374 regularity, 372, 380 semi-additivity, 372 strong additivity, 376 truncation, 185

Index Sobolev space, 49, 60 characterization (fractional), 244 completeness, 50 composition, 52 embedding, 57, 60 fractional, 241 norm, 50, 367 precise representative, 128 Sobolev exponent, 57 Sobolev inequality, 56 stability property, 50 trace inequality, 235 trace operator, 236, 251, 253 space Cc1 ./, 8 x 39 C01 ./, Mp .RN /, 86 M./, 22 W s;q ./, 241 1;q W0 ./, 49 W 1;q ./, 60 X./, 92 strong additivity, 376 strong approximation (measure) by capacitary measures, 216 by density measures, 218 precise density bound, 218 with compact support, 114 strong capacitary inequality, 259 strong convergence (measures), 24 strong maximum principle, 17, 351 superharmonic, 12, 33 mean value property, 12, 34 minimization (total charge), 198 minimum (two functions), 189 modulus of continuity, 177 Poisson equation, 34 Riesz’s characterization, 29 —T— T (truncation), 87 theorem Banach–Saks property, 159 Besicovitch covering lemma, 143 Bôcher–Koebe lemma, 14 Campanato’s lemma, 265 Cavalieri’s principle, 14 closure lemma, 55 dominated convergence (converse), 55, 374

453 Fréchet–Riesz representation, 53 Frostman’s lemma, 273, 388 Hahn–Banach theorem, 42, 382 increasing set lemma, 378 Jordan decomposition theorem, 23 Lebesgue’s decomposition theorem, 224 Lebesgue’s differentiation theorem, 123 Liouville’s theorem, 14, 278 Mazur’s lemma, 158 mean value property, 12 method of sub- and supersolutions, 328 Rellich–Kondrashov compactness, 54 Riesz representation theorem, 27, 41 Schauder’s fixed point theorem, 330 Schwarz’s principle, 171 Vitali’s theorem, 315 Weyl’s lemma, 30 Wiener’s covering lemma, 140 total variation norm, 22 trace inequality, 235, 258 Adams, 167 Gagliardo, 238 Maz0 ya, 216, 258 trace operator, 236 range, 251, 253 truncation function, 87 method (Stampacchia), 78 —V— vanishing average condition, 322 —W— weak approximation (measure), 26 weak capacitary inequality, 374 weak convergence (measures), 24 weak gradient, 50 approximation by convolution, 51 weak harmonic function, 30 weak maximum principle, 17, 95 weak normal derivative, 112 weak solution (Poisson equation), 28 weighted covering, 389 Weyl’s lemma, 30 Wiener’s covering lemma, 140

Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet, one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PDEs and GMT in a simple and elegant way by investigating properties like existence and regularity of solutions of linear and nonlinear elliptic PDEs. Inspired by a variety of sources, from the pioneer balayage scheme of Poincaré to more recent results related to the Thomas–Fermi and the Chern–Simons models, the problems covered in this book follow an original presentation, intended to emphasize the main ideas in the proofs. Classical techniques like regularity theory, maximum principles and the method of sub- and supersolutions are adapted to the setting where merely integrability or density assumptions on the data are available. The distinguished role played by capacities and precise representatives is also explained.

ISBN 978-3-03719-140-8

www.ems-ph.org

Ponce | Tracts in Mathematics 23 | Fonts Nuri /Helvetica Neue | Farben Pantone 116 / Pantone 287 | RB 40 mm

N

A

OGR H

This book invites the reader to a trip through modern techniques in the frontier of elliptic PDEs and GMT, and is addressed to graduate students and researchers having some deep interest in analysis. Most of the chapters can be read independently, and only basic knowledge of measure theory, functional analysis and Sobolev spaces is required.

From the Poisson Equation to Nonlinear Thomas–Fermi Problems

AP

Other special features are: • the remarkable equivalence between Sobolev capacities and Hausdorff contents in terms of trace inequalities; • the strong approximation of measures in terms of capacities or densities, normally absent from GMT books; • the rescue of the strong maximum principle for the Schrödinger operator involving singular potentials.

Elliptic PDEs, Measures and Capacities

Augusto C. Ponce

MO

Elliptic PDEs, Measures and Capacities

Tr a c ts i n M a t h e m a t i c s 2 3

Elliptic PDEs, Measures and Capacities

Augusto C. Ponce

Augusto C. Ponce

Tr a c ts i n M a t h e m a t i c s 2 3

WARD