By David Gilbarg, Neil S. Trudinger
Bankruptcy 1. advent half I: Linear Equations bankruptcy 2. Laplace's Equation 2.1 The suggest price Inequalities 2.2 greatest and minimal precept 2.3 The Harnack Inequality 2.4 Green's illustration 2.5 The Poisson vital 2.6 Convergence Theorems 2.7 inside Estimates of Derivatives 2.8 The Dirichlet challenge; the tactic of Subharmonic capabilities 2.9 skill difficulties bankruptcy three. The Classical greatest precept 3.1 The susceptible greatest precept 3.2 The powerful greatest precept 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence shape Notes difficulties bankruptcy four. Poisson's Equation and Newtonian strength 4.1 Holder Continuity 4.2 The Dirichlet challenge for Poisson's Equation 4.3 Holder Estimates for the second one Derivatives 4.4 Estimates on the Boundary 4.5 Holder Estimates for the 1st Derivatives Notes difficulties bankruptcy five. Banach and Hilbert areas 5.1 The Contraction Mapping 5.2 the strategy of Cintinuity 5.3 The Fredholm substitute 5.4 twin areas and Adjoints 5.5 Hilbert areas 5.6 The Projection Theorem 5.7 The Riesz illustration Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm substitute in Hilbert areas 5.10 susceptible Compactness Notes difficulties bankruptcy 6. Classical options; the Schauder process 6.1 The Schauder inside Estimates 6.2 Boundary and worldwide Estimates 6.3 The Dirichlet challenge 6.4 inside and Boundary Regularity 6.5 an alternate technique 6.6 Non-Uniformly Elliptic Equations 6.7 different Boundary stipulations; the Obliue by-product challenge 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes difficulties bankruptcy 7. Sobolev areas 7.1 L^p areas 7.2 Regularization and Approximation by way of tender services 7.3 susceptible Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) areas 7.6 Density Theorems 7.7 Imbedding Theorems 7.8 power Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness effects 7.11 distinction Quotients 7.12 Extension and Interpolation Notes difficulties bankruptcy eight Generalized strategies and Regularity 8.1 The susceptible greatest precept 8.2 Solvability of the Dirichlet challenge 8.3 Diferentiability of susceptible strategies 8.4 international Regularity 8.5 international Boundedness of susceptible recommendations 8.6 neighborhood homes of susceptible strategies 8.7 The robust greatest precept 8.8 The Harnack Inequality 8.9 Holder Continuity 8.10 neighborhood Estimates on the Boundary 8.11 Holder Estimates for the 1st Derivatives 8.12 The Eigenvalue challenge Notes difficulties bankruptcy nine. powerful recommendations 9.1 greatest Princiles for robust ideas 9.2 L^p Estimates: initial research 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet challenge 9.7 an area greatest precept 9.8 Holder and Harnack Estimates 9.9 neighborhood Estimates on the Boundary Notes difficulties half II: Quasilinear Equations bankruptcy 10. greatest and comparability ideas 10.1 The comparability precept 10.2 greatest rules 10.3 A Counterexample 10.4 comparability rules for Divergence shape Operators 10.5 greatest ideas for Divergence shape Operators Notes difficulties bankruptcy eleven. Topological mounted aspect Theorems and Their software 11.1 The Schauder Fixes aspect Theorem 11.2 The Leray-Schauder Theorem: a unique Case 11.3 An software 11.4 The Leray-Schauder mounted element Theorem 11.5 Variational difficulties Notes bankruptcy 12. Equations in Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet challenge for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes difficulties bankruptcy thirteen. Holder Estimates for the Gradient 13.1 Equations of Divergence shape 13.2 Equations in Variables 13.3 Equations of common shape; the internal Estimate 13.4 Equations of common shape; the Boundary Estimate 13.5 software to the Dirichlet challenge Notes bankruptcy 14. Boundary Gradient Estimates 14.1 normal domain names 14.2 Convex domain names 14.3 Boundary Curvature stipulations 14.4 Non-Existence effects 14.5 Continuity Estimates 14.6 Appendix: Boundary Curvature and the space functionality Notes difficulties bankruptcy 15. worldwide and inside Gradient Bounds 15.1 A greatest precept for the Gradient 15.2 the overall Case 15.3 inside Gradient Bounds 15.4 Equations in Divergence shape 15.5 chosen life Theorems 15.6 lifestyles Theorems for non-stop Boundary Values Notes difficulties bankruptcy sixteen. Equations of suggest Curvature kind 16.1 Hypersurfaces in R^(n+1) 16.2 inside Gradient Bounds 16.3 software to the Dirichlet challenge 16.4 Equations in autonomous Variables 16.5 Quasiconformal Mappings 16.6 Graphs with Quasiconformal Gauss Map 16.7 functions to Equations of suggest Curvature kind 16.8 Appendix Elliptic Parametric Functionals Notes difficulties bankruptcy 17. totally Nonlinear Equations 17.1 greatest and comparability ideas 17.2 the strategy of Continuity 17.3 Equations in Variables 17.4 Holder Estimates for moment Derivatives 17.5 Dirichlet challenge for Uniformly Elliptic Equations 17.6 moment by-product Estimates for Equations of Monge-Ampere style 17.7 Dirichlet challenge for Equations of Monge-Amperere sort 17.8 worldwide moment by-product Holder Estimates 17.9 Nonlinear Boundary price difficulties Notes difficulties Bibliography Epilogue topic Index Notation Index
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