Elliptic Partial Differential Equations of Second Order

Transcription

1 David Gilbarg Neil S.Trudinger Elliptic Partial Differential Equations of Second Order Reprint of the 1998 Edition Springer

2 Chapter 1. Introduction 1 Part I. Linear Equations Chapter 2. Laplace's Equation The Mean Value Inequalities Maximum and Minimum Principle The Harnack Inequality Green's Representation The Poisson Integral Convergence Theorems Interior Estimates of Derivatives The Dirichlet Problem; the Method of Subharmonic Functions Capacity 27 Problems 28 Chapter 3. The Classical Maximum Principle The Weak Maximum Principle The Strong Maximum Principle Apriori Bounds Gradient Estimates for Poisson's Equation A Harnack Inequality Operators in Divergence Form 45 Notes 46 Problems 47 Chapter 4. Poisson's Equation and the Newtonian Potential Holder Continuity The Dirichlet Problem for Poisson's Equation Holder Estimates for the Second Derivatives Estimates at the Boundary 64

3 4.5. Holder Estimates for the First Derivatives 67 Notes 70 Problems 70 Chapter 5. Banach and Hilbert Spaces The Contraction Mapping Principle The Method of Continuity The Fredholm Alternative Dual Spaces and Adjoints Hilbert Spaces The Projection Theorem The Riesz Representation Theorem The Lax-Milgram Theorem The Fredholm Alternative in Hilbert Spaces Weak Compactness 85 Notes 85 Problems 86 Chapter 6. Classical Solutions; the Schauder Approach The Schauder Interior Estimates Boundary and Global Estimates The Dirichlet Problem Interior and Boundary Regularity An Alternative Approach Non-Uniformly Elliptic Equations Other Boundary Conditions; the Oblique Derivative Problem Appendix 1: Interpolation Inequalities Appendix 2: Extension Lemmas 136 Notes 138 Problems 141 Chapter 7. Sobolev Spaces U Spaces Regularization and Approximation by Smooth Functions Weak Derivatives The Chain Rule The W k -» Spaces Density Theorems Imbedding Theorems Potential Estimates and Imbedding Theorems The Morrey and John-Nirenberg Estimates Compactness Results 167

4 XI Difference Quotients Extension and Interpolation 169 Notes 173 Problems 173 Chapter 8. Generalized Solutions and Regularity The Weak Maximum Principle Solvability of the Dirichlet Problem Differentiability of Weak Solutions Global Regularity Global Boundedness of Weak Solutions Local Properties of Weak Solutions The Strong Maximum Principle The Harnack Inequality Holder Continuity Local Estimates at the Boundary Holder Estimates for the First Derivatives The Eigenvalue Problem 212 Notes 214 Problems 216 Chapter 9. Strong Solutions Maximum Principles for Strong Solutions L" Estimates: Preliminary Analysis The Marcinkiewicz Interpolation Theorem The Calderon-Zygmund Inequality L" Estimates The Dirichlet Problem A Local Maximum Principle Holder and Harnack Estimates Local Estimates at the Boundary 250 Notes 254 Problems 255 Part II. Quasilinear Equations Chapter 10. Maximum and Comparison Principles The Comparison Principle Maximum Principles A Counterexample Comparison Principles for Divergence Form Operators Maximum Principles for Divergence Form Operators 271 Notes 277 Problems 277

5 XI1 Table of Contents Chapter 11. Topological Fixed Point Theorems and Their Application The Schauder Fixed Point Theorem The Leray-Schauder Theorem: a Special Case An Application The Leray-Schauder Fixed Point Theorem Variational Problems 288 Notes 293 Chapter 12. Equations in Two Variables Quasiconformal Mappings Holder Gradient Estimates for Linear Equations The Dirichlet Problem for Uniformly Elliptic Equations Non-Uniformly Elliptic Equations 309 Notes 315 Problems 317 Chapter 13. Holder Estimates for the Gradient Equations of Divergence Form Equations in Two Variables Equations of General Form; the Interior Estimate Equations of General Form; the Boundary Estimate Application to the Dirichlet Problem 331 Notes 332 Chapter 14. Boundary Gradient Estimates General Domains Convex Domains Boundary Curvature Conditions Non-Existence Results Continuity Estimates Appendix: Boundary Curvatures and the Distance Function Notes 357 Problems 358 Chapter 15. Global and Interior Gradient Bounds A Maximum Principle for the Gradient The General Case Interior Gradient Bounds Equations in Divergence Form Selected Existence Theorems Existence Theorems for Continuous Boundary Values 384 Notes 385 Problems 386

6 XI11 Chapter 16. Equations of Mean Curvature Type Hypersurfacesin R" Interior Gradient Bounds Application to the Dirichlet Problem Equations in Two Independent Variables Quasiconformal Mappings Graphs with Quasiconformal Gauss Map Applications to Equations of Mean Curvature Type Appendix: Elliptic Parametric Functionals 434 Notes 437 Problems 438 Chapter 17. Fully Nonlinear Equations Maximum and Comparison Principles The Method of Continuity Equations in Two Variables Holder Estimates for Second Derivatives Dirichlet Problem for Uniformly Elliptic Equations Second Derivative Estimates for Equations of Monge-Ampere Type Dirichlet Problem for Equations of Monge-Ampere Type Global Second Derivative Holder Estimates Nonlinear Boundary Value Problems 481 Notes 486 Problems 488 Bibliography 491 Epilogue 507 Subject Index 511 Notation Index 516