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