Classics in Mathematics David Gilbarg· Neil S.Trudinger Elliptic Partial Differential Equations

of Second Order

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David Gilbarg was born in New York in 1918, and was educated there through

undergraduate school. He received his Ph.D. degree at Indiana University in 1941. His work in fluid dynamics during the war years motivated much of his later research on flows with free boundaries. He was on the Mathematics Faculty at Indiana Uni­versity from 1946 to 1957 and at Stanford University from 1957 on. His principal interests and contri­butions have been in mathematical fluid dynamics and the theory of elliptic partial differential equations.

Neil S. Trudinger was born in Ballarat, Australia in 1942. After schooling and undergraduate education in Australia, he completed his Ph.D. at Stanford University, USA in 1966.

He has been a Professor of Mathematics at the Australian National University, Canberra since 1973. His research contributions, while largely focused on non-linear elliptic partial differentipl equations, have also spread into geometry, functional analysis and computational mathematics. Among honours received are Fellow­ships of the Australian Academy of Science and of the Royal Society of London.

David Gilbarg • Neil S. Trudinger

Elliptic Partial Differential Equations

of Second Order

Reprint of the 1998 Edition

Springer

David Gilbarg Stanford University Department of Mathematics Stanford. CA 94305-2125 USA e-mail: gilbarg@math.stanford.edu

Neil S. Trudinger The Australian National University School of Mathematical Sciences Canberra ACT 0200 Australia e-mail: neiltrudinger@anu.edu.au

Originally published as Vol. 224 of the Grundlehren der mathematischen Wissenschaften

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Die Deutsche Bibliothek· CIP-Einbeitsaufnahme

Gilbarg. David: Elliptic partial differential equations of second order I David Gilbarg; Neil S. Trudinger. -Reprint of the 1998 ed. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; 1bkyo: Springer.lOO1 (Classics in mathematics)

Mathematics Subject Classification (2000): 3SJxx

ISSN 1431-0821 ISBN-I3: 978-3-540-41160-4 e-ISBN-13: 978-3-642-61798-0 001: 10.1007/978-3-642-61798-0

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Preface to the Revised Third Printing

This revision of the 1983 second edition of "Elliptic Partial Differential Equations of Second Order" corresponds to the Russian edition, published in 1989, in which we essentially updated the previous version to 1984. The additional text relates to the boundary Holder derivative estimates of Nikolai Krylov, which provided a fundamental component of the further development of the classical theory of elliptic (and parabolic), fully nonlinear equations in higher dimensions. In our presentation we adapted a simplification of Krylov's approach due to Luis Caffarelli.

The theory of nonlinear second order elliptic equations has continued to flourish during the last fifteen years and, in a brief epilogue to this volume, we signal some of the major advances. Although a proper treatment would necessi­tate at least another monograph, it is our hope that this book, most of whose text is now more than twenty years old, can continue to serve as background for these and future developments.

Since our first edition we have become indebted to numerous colleagues, all over the globe. It was particularly pleasant in recent years to make and renew friendships with our Russian colleagues, Olga Ladyzhenskaya, Nina Ural'tseva, Nina Ivochkina, Nikolai Krylov and Mikhail Safonov, who have contributed so much to this area. Sadly, we mourn the passing away in 1996 of Ennico De Giorgi, whose brilliant discovery forty years ago opened the door to the higher-dimen­sional nonlinear theory.

October 1997 David Gilbarg . Neil S. Trudinger

Preface to the First Edition

This volume is intended as an essentially self-contained exposition of portions of the theory of second order quasilinear elliptic partial differential equations, with emphasis on the Dirichlet problem in bounded domains. It grew out of lecture notes for graduate courses by the authors at Stanford University, the final material extending well beyond the scope of these courses. By including preparatory chapters on topics such as potential theory and functional analysis, we have attempted to make the work accessible to a broad spectrum of readers. Above all, we hope the readers of this book will gain an appreciation of the multitude of ingenious barehanded techniques that have been developed in the study of elliptic equations and have become part of the repertoire of analysis.

Many individuals have assisted us during the evolution of this work over the past several years. In particular, we are grateful for the valuable discussions with L. M. Simon and his contributions in Sections 15.4 to 15.8; for the helpful comments and corrections of J. M. Cross, A. S. Geue, J. Nash, P. Trudinger and B. Turkington; for the contributions ofG. Williams in Section 10.5 and of A. S. Geue in Section 10.6; and for the impeccably typed manuscript which resulted from the dedicated efforts ofIsolde Field at Stanford and Anna Zalucki at Canberra. The research of the authors connected with this volume was supported in part by the National Science Foundation.

August 1977 David Gilbarg Stanford

Neil S. Trudinger Canberra

Note: The Second Edition includes a new, additional Chapter 9. Consequently Chapters 10 and 15 referred to above have become Chapters 11 and 16.

Table of Contents

Chapter I. Introduction.

Part I. Linear Equations

Chapter 2. Laplace's Equation 13

2.1. The Mean Value Inequalities. . . 13 2.2. Maximum and Minimum Principle 15 2.3. The Harnack Inequality. 16 2.4. Green's Representation 17 2.5. The Poisson Integral . . 19 2.6. Convergence Theorems . 21 2.7. Interior Estimates of Derivatives 22 2.8. The Dirichlet Problem; the Method of Subharmonic Functions 23 2.9. Capacity 27 Problems. . . . . . . . . . . . . . . . . . . . . . . . .. 28

Chapter 3. The Classical Maximum Principle. 31

3.1. The Weak Maximum Principle . 32 3.2. The Strong Maximum Principle. . . 33 3.3. Apriori Bounds . . . . . . . . . 36 3.4. Gradient Estimates for Poisson's Equation. 37 3.5. A Harnack Inequality. . . . 41 3.6. Operators in Divergence Form 45 Notes . . 46 Problems. . . . . . . . . . . . 47

Chapter 4. Poisson's Equation and the Newtonian Potential 51

4.1. Holder Continuity . . . . . . . . . . . . 4.2. The Dirichlet Problem for Poisson's Equation 4.3. Holder Estimates for the Second Derivatives 4.4. Estimates at the Boundary. . . . . . . . .

51 54 56 64

x

4.5. Holder Estimates for the First Derivatives Notes .. Problems ................ .

Table of Contents

67 70 70

Chapter 5. Banach and Hilbert Spaces. . . .

5.1. The Contraction Mapping Principle. 5.2. The Method of Continuity.

73

74 74 75 79 80 81 82 83 83 85 85 86

5.3. The Fredholm Alternative . 5.4. Dual Spaces and Adjoints 5.5. Hilbert Spaces . . . . . . 5.6. The Projection Theorem .. 5.7. The Riesz Representation Theorem 5.8. The Lax-Milgram Theorem . . . 5.9. The Fredholm Alternative in Hilbert Spaces 5.10. Weak Compactness. Notes .. Problems ...... .

Chapter 6. Classical Solutions; the Schauder Approach 87

6.1. The Schauder Interior Estimates 89 6.2. Boundary and Global Estimates . 94 6.3. The Dirichlet Problem. . . . . . 100 6.4. Interior and Boundary Regularity. 109 6.5 .. An Alternative Approach . . . . 112 6.6. Non-Uniformly Elliptic Equations 116 6.7. Other Boundary Conditions; the Oblique Derivative Problem. 120 6.8. Appendix I : Interpolation Inequalities. 130 6.9. Appendix 2: Extension Lemmas 136 Notes . . 138 Problems ............ 141

Chapter 7. Sobolev Spaces. . . . . . . . . . . . . . . . . . 144

7.1. U Spaces . . . . . . . . . . . . . . . . . . . . . 145 7.2. Regularization and Approximation by Smooth Functions 147 7.3. Weak Derivatives. 149 7.4. The Chain Rule . 151 7.5. The W"·P Spaces . 153 7.6. Density Theorems 154 7.7. Imbedding Theorems 155 7.8. Potential Estimates and Imbedding Theorems. 159 7.9. The Morrey and John-Nirenberg Estimates. 164 7.10. Compactness Results . . . . . . . . . . . 167

Table of Contents

7.11. Difference Quotients 7.12. Extension and Interpolation Notes .. Problems .......... .

Chapter 8. Generalized Solutions and Regularity

8.1. The Weak Maximum Principle ... 8.2. Solvability of the Dirichlet Problem. 8.3. Differentiability of Weak Solutions . 8.4. Global Regularity . . . . . . . . 8.5. Global Boundedness of Weak Solutions 8.6. Local Properties of Weak Solutions. 8.7. The Strong Maximum Principle. 8.8. The Harnack Inequality. . . . 8.9. Holder Continuity . . . . . . 8.10. Local Estimates at the Boundary 8.11. Holder Estimates for the First Derivatives 8.12. The Eigenvalue Problem. Notes .. Problems ...... .

Chapter 9. Strong Solutions

9.1. Maximum Principles for Strong Solutions 9.2. LP Estimates: Preliminary Analysis ... 9.3. The Marcinkiewicz Interpolation Theorem. 9.4. The Calderon-Zygmund Inequality 9.5. LP Estimates. . . . . . . . 9.6. The Dirichlet Problem. . . . . 9.7. A Local Maximum Principle .. 9.8. Holder and Harnack Estimates. 9.9. Local Estimates at the Boundary Notes .. Problems. . . . . . . . . . . . .

Part II. Quasilinear Equations

Chapter 10. Maximum and Comparison Principles

10.1. The Comparison Principle 10.2. Maximum Principles .............. . 10.3. A Counterexample. . . . . . . . . . . . . . . . 10.4. Comparison Principles for Divergence Form Operators 10.5. Maximum Principles for Divergence Form Operators. Notes Problems ...................... .

xi

168 169 173 173

177

179 181 183 186 188 194 198 199 200 202 209 212 214 216

219

220 225 227 230 235 241 244 246 250 254 255

259

263 264 267 268 271 277 277

XII Table of Contents

Chapter II. Topological Fixed Point Theorems and Their Application . 279

11.1. The Schauder Fixed Point Theorem. . . . . 279 11.2. The Leray-Schauder Theorem: a Special Case 280 11.3. An Application. . . . . . . . . . . . 282 11.4. The Leray-Schauder Fixed Point Theorem 286 11.5. Variational Problems 288 Notes. . . . . . . . . . . . . . . 293

Chapter 12. Equations in Two Variables. 294

12.1. Quasiconformal Mappings. . . 294 12.2. Holder Gradient Estimates for Linear Equations 300 12.3. The Dirichlet Problem for Uniformly Elliptic Equations 304 12.4. Non-Uniformly Elliptic Equations 309 Notes. . 315 Problems . . . . . . . . . . . . . . 317

Chapter 13. Holder Estimates for the Gradient 319

13.1. Equations of Divergence Form. . . 319 13.2. Equations in Two Variables . . . . 323 13.3. Equations of General Form; the Interior Estimate. 324 13.4. Equations of General Form; the Boundary Estimate. 328 13.5. Application to the Dirichlet Problem 331 Notes. . . . . . . . . . . . . . . . . . . . . . . . 332

Chapter 14. Boundary Gradient Estimates . 333

14.1. General Domains. . . . . . . 335 14.2. Convex Domains. . . . . . . 337 14.3. Boundary Curvature Conditions 341 14.4. Non-Existence Results . . . . 347 14.5. Continuity Estimates . . . . . 353 14.6. Appendix: Boundary Curvatures and the Distance Function 354 Notes. . 357 Problems . . . . . . . . . . . . . . . . . . . . . . . .. 358

Chapter 15. Global and Interior Gradient Bounds. 359

15.1. A Maximum Principle for the Gradient 359 15.2. The General Case. . . . . . 362 15.3. Interior Gradient Bounds . . 369 15.4. Equations in Divergence Form 373 15.5. Selected Existence Theorems. 380 15.6. Existence Theorems for Continuous Boundary Values 384 Notes. . 385 Problems . . . . . . . . . . . . . . . . . . . . . . 386

Table of Contents xiii

Chapter 16. Equations of Mean Curvature Type 388

16.1. Hypersurfaces in IR"+ 1. . . . . . . 388 16.2. Interior Gradient Bounds . . . . . . 401 16.3. Application to the Dirichlet Problem . 407 16.4. Equations in Two Independent Variables. 410 16.5. Quasiconformal Mappings. . . . . . . 413 16.6. Graphs with Quasiconformal Gauss Map 423 16.7. Applications to Equations of Mean Curvature Type . 429 16.8. Appendix: Elliptic Parametric Functionals . 434 Notes. . 437 Problems . . . . . . . . . . . . 438

Chapter 17. Fully Nonlinear Equations 441

17.1. Maximum and Comparison Principles. 443 17.2. The Method of Continuity. . . . . . 446 17.3. Equations in Two Variables . . . . . 450 17.4. Holder Estimates for Second Derivatives. 453 17.5. Dirichlet Problem for Uniformly Elliptic Equations 463 17.6. Second Derivative Estimates for Equations of

Monge-Ampere Type. . . . . . . . . . . . . 467 17.7. Dirichlet Problem for Equations of Monge-Ampere Type 471 17.8. Global Second Derivative Holder Estimates 476 17.9. Nonlinear Boundary Value Problems 481 Notes. . 486 Problems . . . . . . . . . . . 488

Bibliography 491

Epilogue 507

Subject Index 511

Notation Index 516