Introduction / Chapter 1: |
Linear Equations / Part I: |
Laplace''s Equation / Chapter 2: |
The Mean Value Inequalities / 2.1: |
Maximum and Minimum Principle / 2.2: |
The Harnack Inequality / 2.3: |
Green''s Representation / 2.4: |
The Poisson Integral / 2.5: |
Convergence Theorems / 2.6: |
Interior Estimates of Derivatives / 2.7: |
The Dirichlet Problem; the Method of Subharmonic Functions / 2.8: |
CapacityProblems / 2.9: |
The Classical Maximum Principle / Chapter 3: |
The Weak Maximum Principle / 3.1: |
The Strong Maximum Principle / 3.2: |
Apriori Bounds / 3.3: |
Gradient Estimates for Poisson''s Equation / 3.4: |
A Harnack Inequality / 3.5: |
Operators in Divergence FormNotesProblems / 3.6: |
Poisson''s Equation and Newtonian Potential / Chapter 4: |
H+ lder Continuity / 4.1: |
The Dirichlet Problem for Poisson''s Equation / 4.2: |
H+ lder Estimates for the Second Derivatives / 4.3: |
Estimates at the Boundary / 4.4: |
H+ lder Estimates for the First DerivativesNotes Problems / 4.5: |
Banach and Hilbert Spaces / Chapter 5: |
The Contraction Mapping / 5.1: |
The Method of Cintinuity / 5.2: |
The Fredholm Alternative / 5.3: |
Dual Spaces and Adjoints / 5.4: |
Hilbert Spaces / 5.5: |
The Projection Theorem / 5.6: |
The Riesz Representation Theorem / 5.7: |
The Lax-Milgram Theorem / 5.8: |
The Fredholm Alternative in Hilbert Spaces / 5.9: |
Weak CompactnessNotesProblems / 5.10: |
Classical Solutions; the Schauder Approach / Chapter 6: |
The Schauder Interior Estimates / 6.1: |
Boundary and Global Estimates / 6.2: |
The Dirichlet Problem / 6.3: |
Interior and Boundary Regularity / 6.4: |
An Alternative Approach / 6.5: |
Non-Uniformly Elliptic Equations / 6.6: |
Other Boundary Conditions; the Obliue Derivative Problem / 6.7: |
Appendix 1: Interpolation Inequalities / 6.8: |
Appendix 2: Extension LemmasNotesProblems / 6.9: |
Sobolev Spaces / Chapter 7: |
L^p spaces / 7.1: |
Regularization and Approximation by Smooth Functions / 7.2: |
Weak Derivatives / 7.3: |
The Chain Rule / 7.4: |
The W^(k,p) Spaces / 7.5: |
Density Theorems / 7.6: |
Imbedding Theorems / 7.7: |
Potential Estimates and Imbedding Theorems / 7.8: |
The Morrey and John-Nirenberg Estimes / 7.9: |
Compactness Results / 7.10: |
Difference Quotients / 7.11: |
Extension and InterpolationNotesProblems / 7.12: |
Generalized Solutions and Regularity / Chapter 8: |
Solvability of the Dirichlet Problem / 8.1: |
Diferentiability of Weak Solutions / 8.3: |
Global Regularity / 8.4: |
Global Boundedness of Weak Solutions / 8.5: |
Local Properties of Weak Solutions / 8.6: |
Local Estimates at the Boundary / 8.7: |
H+ lder Estimates for the First Derivatives / 8.11: |
The Eigenvalue ProblemNotesProblems / 8.12: |
Strong Solutions / Chapter 9: |
Maximum Princiles for Strong Solutions / 9.1: |
L^p Estimates: Preliminary Analysis / 9.2: |
The Marcinkiewicz Interpolation Theorem / 9.3: |
The Calderon-Zygmund Inequality / 9.4: |
L^p Estimates / 9.5: |
A Local Maximum Principle / 9.6: |
H+ lder and Harnack Estimates / 9.8: |
Local Estimates at the BoundaryNotesProblems / 9.9: |
Quasilinear Equations / Part II: |
Maximum and Comparison Principles / Chapter 10: |
The Comparison Principle / 10.1: |
Maximum Principles / 10.2: |
A Counterexample / 10.3: |
Comparison Principles for Divergence Form Operators / 10.4: |
Maximum Principles for Divergence Form Operators Notes Problems / 10.5: |
Topological Fixed Point Theorems and Their Application / Chapter 11: |
The Schauder Fixes Point Theorem / 11.1: |
The Leray-Schauder Theorem: a Special Case / 11.2: |
An Application / 11.3: |
The Leray-Schauder Fixed Point Theorem / 11.4: |
Variational ProblemsNotes / 11.5: |
Equations in Two Variables / Chapter 12: |
Quasiconformal Mappings / 12.1: |
h+ lder Gradient Estimates for Linear Equations / 12.2: |
The Dirichlet Problem for Uniformly Elliptic Equations / 12.3: |
Non-Uniformly Elliptic EquationsNotesProblems / 12.4: |
H+ lder Estimates for the Gradient / Chapter 13: |
Equations of Divergence Form / 13.1: |
Equations of General Form; the Interior Estimate / 13.2: |
Equations of General Form; the Boundary Estimate / 13.4: |
Application to the Dirichlet ProblemNotes / 13.5: |
Boundary Gradient Estimates / Chapter 14: |
General Domains / 14.1: |
Convex Domains / 14.2: |
Boundary Curvature Conditions / 14.3: |
Non-Existence Results / 14.4: |
Continuity Estimates / 14.5: |
Appendix: Boundary Curvature and the Distance FunctionNotesProblems / 14.6: |
Introduction / Chapter 1: |
Linear Equations / Part I: |
Laplace''s Equation / Chapter 2: |