By David Gilbarg, Neil S. Trudinger
Bankruptcy 1. creation 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 necessary 2.6 Convergence Theorems 2.7 inside Estimates of Derivatives 2.8 The Dirichlet challenge; the strategy of Subharmonic services 2.9 ability difficulties bankruptcy three. The Classical greatest precept 3.1 The vulnerable greatest precept 3.2 The robust 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 tactic 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 vulnerable Compactness Notes difficulties bankruptcy 6. Classical strategies; the Schauder process 6.1 The Schauder inside Estimates 6.2 Boundary and international Estimates 6.3 The Dirichlet challenge 6.4 inside and Boundary Regularity 6.5 an alternate procedure 6.6 Non-Uniformly Elliptic Equations 6.7 different Boundary stipulations; the Obliue spinoff 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 via soft 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 recommendations and Regularity 8.1 The susceptible greatest precept 8.2 Solvability of the Dirichlet challenge 8.3 Diferentiability of vulnerable strategies 8.4 worldwide Regularity 8.5 international Boundedness of susceptible ideas 8.6 neighborhood homes of susceptible options 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 suggestions 9.1 greatest Princiles for robust strategies 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 a neighborhood 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 fastened aspect Theorems and Their program 11.1 The Schauder Fixes aspect Theorem 11.2 The Leray-Schauder Theorem: a different Case 11.3 An software 11.4 The Leray-Schauder fastened 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 inner Estimate 13.4 Equations of normal shape; the Boundary Estimate 13.5 software to the Dirichlet challenge Notes bankruptcy 14. Boundary Gradient Estimates 14.1 basic 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 lifestyles Theorems 15.6 life Theorems for non-stop Boundary Values Notes difficulties bankruptcy sixteen. Equations of suggest Curvature sort 16.1 Hypersurfaces in R^(n+1) 16.2 inside Gradient Bounds 16.3 program to the Dirichlet challenge 16.4 Equations in self sustaining Variables 16.5 Quasiconformal Mappings 16.6 Graphs with Quasiconformal Gauss Map 16.7 purposes to Equations of suggest Curvature sort 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 spinoff Estimates for Equations of Monge-Ampere variety 17.7 Dirichlet challenge for Equations of Monge-Amperere variety 17.8 international moment by-product Holder Estimates 17.9 Nonlinear Boundary worth difficulties Notes difficulties Bibliography Epilogue topic Index Notation Index
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Elliptic Partial Differential Equations of Second Order, 2nd edition by David Gilbarg, Neil S. Trudinger