Elliptic Partial Differential Equations of Second Order

✍ Scribed by David Gilbarg, Neil S. Trudinger

• Publisher: Springer
• Year: 2001
• Tongue: English
• Leaves: 532
• Edition: 3
• Category: Library

✦ Synopsis

From the reviews:
"This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student. Although the material has been developed from lectures at Stanford, it has developed into an almost systematic coverage that is much longer than could be covered in a year's lectures". Newsletter, New Zealand Mathematical Society, 1985
" ... as should be clear from the previous discussion, this book is a bibliographical monument to the theory of both theoretical and applied PDEs that has not acquired any flaws due to its age. On the contrary, it remains a crucial and essential tool for the active research in the field. In a few words, in my modest opinion, “. . . this book contains the essential background that a researcher in elliptic PDEs should possess the day s/he gets a permanent academic position. . . .” SIAM Newsletter

✦ Table of Contents

Cover
Title: Elliptic Partial Differential Equations of Second Order
Copyright
Preface to the Revised Third Printing
Preface to the First Edition
Table of Contents
Chapter 1. Introduction
Part I. Linear Equations
Chapter 2. Laplace's Equation
2.1. The Mean Value Inequalities
2.2. Maximum and Minimum Principle
2.3. The Harnack Inequality
2.4. Green's Representation
2.5. The Poisson Integral
2.6. Convergence Theorems
2.7. Interior Estimates of Derivatives
2.8. The Dirichlet Problem; the Method of Subharmonic Functions
2.9. Capacity
Problems
Chapter 3. The Classical Maximum Principle
3.1. The Weak Maximum Principle
3.2. The Strong Maximum Principle
3.3. Apriori Bounds
3.4. Gradient Estimates for Poisson's Equation
3.5. A Harnack Inequality
3.6. Operators in Divergence Form
Notes
Problems
Chapter 4. Poisson's Equation and the Newtonian Potential
4.1. Hölder Continuity
4.2. The Dirichlet Problem for Poisson's Equation
4.3. Hölder Estimates for the Second Derivatives
4.4. Estimates at the Boundary
4.5. Hölder Estimates for the First Derivatives
Notes
Problems
Chapter 5. Banach and Hilbert Spaces
5.1. The Contraction Mapping Principle
5.2. The Method of Continuity
5.3. The Fredholm Alternative
5.4. Dual Spaces and Adjoints
5.5. Hilbert Spaces
5.6. The Projection Theorem
5.7. The Riesz Representation Theorem
5.8. The Lax-Milgram Theorem
5.9. The Fredholm Alternative in Hilbert Spaces
5.10. Weak Compactness
Notes
Problems
Chapter 6. Classical Solutions; the Schauder Approach
6.1. The Schauder Interior Estimates
6.2. Boundary and Global Estimates
6.3. The Dirichlet Problem
6.4. Interior and Boundary Regularity
6.5. An Alternative Approach
6.6. Non-Uniformly Elliptic Equations
6.7. Other Boundary Conditions; the Oblique Derivative Problem
6.8. Appendix 1: Interpolation Inequalities
6.9. Appendix 2: Extension Lemmas
Notes
Problems
Chapter 7. Sobolev Spaces
7.1. L^p Spaces
7.2. Regularization and Approximation by Smooth Functions
7.3. Weak Derivatives
7.4. The Chain Rule
7.5. The W^{k,p} Spaces
7.6. Density Theorems
7.7. Imbedding Theorems
7.8. Potential Estimates and Imbedding Theorems
7.9. The Morrey and John-Nirenberg Estimates
7.10. Compactness Results
7.11. Difference Quotients
7.12. Extension and Interpolation
Notes
Problems
Chapter 8. Generalized Solutions and Regularity
8.1. The Weak Maximum Principle
8.2. Solvability of the Dirichlet Problem
8.3. Differentiability of Weak Solutions
8.4. Global Regularity
8.5. Global Boundedness of Weak Solutions
8.6. Local Properties of Weak Solutions
8.7. The Strong Maximum Principle
8.8. The Harnack Inequality
8.9. Hölder Continuity
8.10. Local Estimates at the Boundary
8.11. Hölder Estimates for the First Derivatives
8.12. The Eigenvalue Problem
Notes
Problems
Chapter 9. Strong Solutions
9.1. Maximum Principles for Strong Solutions
9.2. L^p Estimates: Preliminary Analysis
9.3. The Marcinkiewicz Interpolation Theorem
9.4. The Calderon-Zygmund Inequality
9.5. L^p Estimates
9.6. The Dirichlet Problem
9.7. A Local Maximum Principle
9.8. Hölder and Harnack Estimates
9.9. Local Estimates at the Boundary
Notes
Problems
Part II. Quasilinear Equations
Chapter 10. Maximum and Comparison Principles
10.1. The Comparison Principle
10.2. Maximum Principles
10.3. A Counterexample
10.4. Comparison Principles for Divergence Form Operators
10.5. Maximum Principles for Divergence Form Operators
Notes
Problems
Chapter 11. Topological Fixed Point Theorems and Their Application
11.1. The Schauder Fixed Point Theorem
11.2. The Leray-Schauder Theorem: a Special Case
11.3. An Application
11.4. The Leray-Schauder Fixed Point Theorem
11.5. Variational Problems
Notes
Chapter 12. Equations in Two Variables
12.1. Quasiconformal Mappings
12.2. Hölder Gradient Estimates for Linear Equations
12.3. The Dirichlet Problem for Uniformly Elliptic Equations
12.4. Non-Uniformly Elliptic Equations
Notes
Problems
Chapter 13. Hölder Estimates for the Gradient
13.1. Equations of Divergence Form
13.2. Equations in Two Variables
13.3. Equations of General Form; the Interior Estimate
13.4. Equations of General Form; the Boundary Estimate
13.5. Application to the Dirichlet Problem
Notes
Chapter 14. Boundary Gradient Estimates
14.1. General Domains
14.2. Convex Domains
14.3. Boundary Curvature Conditions
14.4. Non-Existence Results
14.5. Continuity Estimates
14.6. Appendix: Boundary Curvatures and the Distance Function
Notes
Problems
Chapter 15. Global and Interior Gradient Bounds
15.1. A Maximum Principle for the Gradient
15.2. The General Case
15.3. Interior Gradient Bounds
15.4. Equations in Divergence Form
15.5. Selected Existence Theorems
15.6. Existence Theorems for Continuous Boundary Values
Notes
Problems
Chapter 16. Equations of Mean Curvature Type
16.1. Hypersurfaces in R^{n+1}
16.2. Interior Gradient Bounds
16.3. Application to the Dirichlet Problem
16.4. Equations in Two Independent Variables
16.5. Quasiconformal Mappings
16.6. Graphs with Quasiconformal Gauss Map
16.7. Applications to Equations of Mean Curvature Type
16.8. Appendix: Elliptic Parametric Functionals
Notes
Problems
Chapter 17. Fully Nonlinear Equations
17.1. Maximum and Comparison Principles
17.2. The Method of Continuity
17.3. Equations in Two Variables
17.4. Hölder Estimates for Second Derivatives
17.5. Dirichlet Problem for Uniformly Elliptic Equations
17.6. Second Derivative Estimates for Equations of Monge-Ampère Type
17.7. Dirichlet Problem for Equations of Monge-Ampere Type
17.8. Global Second Derivative Hölder Estimates
17.9. Nonlinear Boundary Value Problems
Notes
Problems
Bibliography
Epilogue
Subject Index
Notation Index