Ebook: Elliptic Partial Differential Equations of Second Order, 2nd edition

Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem; the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 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 4.5 Holder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 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 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions; the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 Density Theorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 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 Diferentiability 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 Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p 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 Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Holder Estimates for the Gradient 13.1 Equations of Divergence Form 13.2 Equations in Two Variables 13.3 Equations of General Form; the Interior Estimate 13.4 Equations of General Form; the Boundary Estimate 13.5 Application to the Dirichlet Problem Notes Chapter 14. Boundary Gradient Estimates 14.1 General Domains 14.2 Convex Domains 14.3 Boundary Curvature Conditions 14.4 Non-Existence Results 14.5 Continuity Estimates 14.6 Appendix: Boundary Curvature and the Distance Function Notes Problems Chapter 15. Global and Interior Gradient Bounds 15.1 A Maximum Principle for the Gradient 15.2 The General Case 15.3 Interior Gradient Bounds 15.4 Equations in Divergence Form 15.5 Selected Existence Theorems 15.6 Existence Theorems for Continuous Boundary Values Notes Problems Chapter 16. Equations of Mean Curvature Type 16.1 Hypersurfaces in R^(n+1) 16.2 Interior Gradient Bounds 16.3 Application to the Dirichlet Problem 16.4 Equations in Two Independent Variables 16.5 Quasiconformal Mappings 16.6 Graphs with Quasiconformal Gauss Map 16.7 Applications to Equations of mean Curvature Type 16.8 Appendix Elliptic Parametric Functionals Notes Problems Chapter 17. Fully Nonlinear Equations 17.1 Maximum and Comparison Principles 17.2 The Method of Continuity 17.3 Equations in Two Variables 17.4 Holder Estimates for Second Derivatives 17.5 Dirichlet Problem for Uniformly Elliptic Equations 17.6 Second Derivative Estimates for Equations of Monge-Ampere Type 17.7 Dirichlet Problem for Equations of Monge-Amperere Type 17.8 Global Second Derivative Holder Estimates 17.9 Nonlinear Boundary Value Problems Notes Problems Bibliography Epilogue Subject Index Notation Index