Elliptic Partial Differential Equations of Second Order

1. Front Matter

   Pages i-x

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2. Introduction

   1. Introduction

      • David Gilbarg, Neil S. Trudinger

      Pages 1-9

3. Linear Equations

   1. Front Matter

      Pages 11-11

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   2. Laplace’s Equation

      • David Gilbarg, Neil S. Trudinger

      Pages 13-29

   3. The Classical Maximum Principle

      • David Gilbarg, Neil S. Trudinger

      Pages 30-49

   4. Poisson’s Equation and the Newtonian Potential

      • David Gilbarg, Neil S. Trudinger

      Pages 50-67

   5. Banach and Hilbert Spaces

      • David Gilbarg, Neil S. Trudinger

      Pages 68-81

   6. Classical Solutions; the Schauder Approach

      • David Gilbarg, Neil S. Trudinger

      Pages 82-136

   7. Sobolev Spaces

      • David Gilbarg, Neil S. Trudinger

      Pages 137-165

   8. Generalized Solutions and Regularity

      • David Gilbarg, Neil S. Trudinger

      Pages 166-200

4. Quasilinear Equations

   1. Front Matter

      Pages 201-201

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   2. Maximum and Comparison Principles

      • David Gilbarg, Neil S. Trudinger

      Pages 203-220

   3. Topological Fixed Point Theorems and Their Application

      • David Gilbarg, Neil S. Trudinger

      Pages 221-238

   4. Equations in Two Variables

      • David Gilbarg, Neil S. Trudinger

      Pages 239-263

   5. Hölder Estimates for the Gradient

      • David Gilbarg, Neil S. Trudinger

      Pages 264-277

   6. Boundary Gradient Estimates

      • David Gilbarg, Neil S. Trudinger

      Pages 278-299

   7. Global and Interior Gradient Bounds

      • David Gilbarg, Neil S. Trudinger

      Pages 300-327

   8. Equations of Mean Curvature Type

      • David Gilbarg, Neil S. Trudinger

      Pages 328-380

5. Back Matter

   Pages 381-404

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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 of G. 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 oflsolde 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.

• differential equation
• functional analysis
• partial differential equation
• potential theory

• Department of Mathematics, Stanford University, Stanford, USA

  David Gilbarg

• Department of Pure Mathematics, Australian National University, Canberra, Australia

  Neil S. Trudinger

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