| Introduction | |
| Linear Equations | |
| 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 | |
| CapacityProblems | |
| 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 FormNotesProblems | |
| Poisson''s Equation and Newtonian Potential | |
| H+ lder Continuity | |
| The Dirichlet Problem for Poisson''s Equation | |
| H+ lder Estimates for the Second Derivatives | |
| Estimates at the Boundary | |
| H+ lder Estimates for the First DerivativesNotes Problems | |
| Banach and Hilbert Spaces | |
| The Contraction Mapping | |
| The Method of Cintinuity | |
| 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 CompactnessNotesProblems | |
| 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 Obliue Derivative Problem | |
| Appendix 1: Interpolation Inequalities | |
| Appendix 2: Extension LemmasNotesProblems | |
| Sobolev Spaces | |
| L^p spaces | |
| Regularization and Approximation by Smooth Functions | |
| Weak Derivatives | |
| The Chain Rule | |
| The W^(k,p) Spaces | |
| Density Theorems | |
| Imbedding Theorems | |
| Potential Estimates and Imbedding Theorems | |
| The Morrey and John-Nirenberg Estimes | |
| Compactness Results | |
| Difference Quotients | |
| Extension and InterpolationNotesProblems | |
| Generalized Solutions and Regularity | |
| The Weak Maximum Principle | |
| Solvability of the Dirichlet Problem | |
| Diferentiability of Weak Solutions | |
| Global Regularity | |
| Global Boundedness of Weak Solutions | |
| Local Properties of Weak Solutions | |
| The Strong Maximum Principle | |
| The Harnack Inequality | |
| H+ lder Continuity | |
| Local Estimates at the Boundary | |
| H+ lder Estimates for the First Derivatives | |
| The Eigenvalue ProblemNotesProblems | |
| Strong Solutions | |
| Maximum Princiles for Strong Solutions | |
| L^p Estimates: Preliminary Analysis | |
| The Marcinkiewicz Interpolation Theorem | |
| The Calderon-Zygmund Inequality | |
| L^p Estimates | |
| The Dirichlet Problem | |
| A Local Maximum Principle | |
| H+ lder and Harnack Estimates | |
| Local Estimates at the BoundaryNotesProblems | |
| Quasilinear Equations | |
| Maximum and Comparison Principles | |
| The Comparison Principle | |
| Maximum Principles | |
| A Counterexample | |
| Comparison Principles for Divergence Form Operators | |
| Maximum Principles for Divergence Form Operators Notes Problems | |
| Topological Fixed Point Theorems and Their Application | |
| The Schauder Fixes Point Theorem | |
| The Leray-Schauder Theorem: a Special Case | |
| An Application | |
| The Leray-Schauder Fixed Point Theorem | |
| Variational ProblemsNotes | |
| Equations in Two Variables | |
| Quasiconformal Mappings | |
| h+ lder Gradient Estimates for Linear Equations | |
| The Dirichlet Problem for Uniformly Elliptic Equations | |
| Non-Uniformly Elliptic EquationsNotesProblems | |
| H+ lder 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 ProblemNotes | |
| Boundary Gradient Estimates | |
| General Domains | |
| Convex Domains | |
| Boundary Curvature Conditions | |
| Non-Existence Results | |
| Continuity Estimates | |
| Appendix: Boundary Curvature and the Distance FunctionNotesProblems | |
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