A Basic Course in Partial Differential Equations

✍ Scribed by Qing Han

• Publisher: American Mathematical Society
• Year: 2011
• Tongue: English
• Leaves: 305
• Series: Graduate Studies in Mathematics 120
• Category: Library

✦ Synopsis

This is a textbook for an introductory graduate course on partial differential equations. Han focuses on linear equations of first and second order. An important feature of his treatment is that the majority of the techniques are applicable more generally. In particular, Han emphasizes a priori estimates throughout the text, even for those equations that can be solved explicitly. Such estimates are indispensable tools for proving the existence and uniqueness of solutions to PDEs, being especially important for nonlinear equations. The estimates are also crucial to establishing properties of the solutions, such as the continuous dependence on parameters.

Han's book is suitable for students interested in the mathematical theory of partial differential equations, either as an overview of the subject or as an introduction leading to further study.

Readership: Advanced undergraduate and graduate students interested in PDEs.

✦ Table of Contents

Preface

Chapter 1 Introduction
1.1. Notation
1.2. Well-Posed Problems
1.3. Overview

Chapter 2 First-Order Differential Equations
2.1. Noncharacteristic Hypersurfaces
2.2. The Method of Characteristics
2.2.1. Linear Homogeneous Equations
2.2.2. Quasilinear Equations
2.2.3. General Nonlinear Equations.
2.3. A Priori Estimates
2.3.1. L°°-Estimates
2.3.2. L2-Estimates
2.3.3. Weak Solutions
2.4. Exercises

Chapter 3 An Overview of Second-Order PDEs
3.1. Classifications
3.2. Energy Estimates
3.3. Separation of Variables
3.3.1. Dirichlet Problems
3.3.2. Initial/Boundary-Value Problems
3.4. Exercises

Chapter 4 Laplace Equations
4.1. Fundamental Solutions
4.1.1. Green's Identities.
4.1.2. Green's Functions
4.1.3. Poisson Integral Formula.
4.1.4. Regularity of Harmonic Functions.
4.2. Mean-Value Properties
4.3. The Maximum Principle
4.3.1. The Weak Maximum Principle
4.3.2. The Strong Maximum Principle.
4.3.3. A Priori Estimates.
4.3.4. Gradient Estimates
4.3.5. Removable Singularity.
4.3.6. Perron's Method.
4.4. Poisson Equations
4.4.1. Classical Solutions
4.4.2. Weak Solutions.
4.5. Exercises

Chapter 5 Heat Equations
5.1. Fourier Transforms
5.1.1. Basic Properties
5.1.2. Examples
5.2. Fundamental Solutions
5.2.1. Initial-Value Problems
5.2.2. Regularity of Solutions
5.2.3. Nonhomogeneous Problems
5.3. The Maximum Principle
5.3.1. The Weak Maximum Principle
5.3.2. The Strong Maximum Principle
5.3.3. A Priori Estimates.
5.3.4. Interior Gradient Estimates.
5.3.5. Harnack Inequalities
5.4. Exercises

Chapter 6 Wave Equations
6.1. One-Dimensional Wave Equations
6.1.1. Initial-Value Problems
6.1.2. Mixed Problems
6.2. Higher-Dimensional Wave Equations
6.2.1. The Method of Spherical Averages
6.2.2. Dimension Three
6.2.3. Dimension Two
6.2.4. Properties of Solutions
6.2.5. Arbitrary Odd Dimensions
6.2.6. Arbitrary Even Dimensions
6.2.7. Global Properties
6.2.8. Duhamel's Principle
6.3. Energy Estimates
6.4. Exercises

Chapter 7 First-Order Differential Systems
7.1. Noncharacteristic Hypersurfaces
7.1.1. Linear Partial Differential Equations
7.1.2. Linear Partial Differential Systems
7.2. Analytic Solutions
7.2.1. Real Analytic Functions.
7.2.2. Cauchy-Kovalevskaya Theorem
7.2.3. The Uniqueness Theorem of Holmgren
7.3. Nonexistence of Smooth Solutions
7.4. Exercises

Chapter 8 Epilogue
8.1. Basic Linear Differential Equations
8.1.1. Linear Elliptic Differential Equations
8.1.2. Linear Parabolic Differential Equations
8.1.3. Linear Hyperbolic Differential Equations
8.1.4. Linear Symmetric Hyperbolic Differential System
8.2. Examples of Nonlinear Differential Equations
8.2.1. Nonlinear Differential Equations
8.2.2. Nonlinear Differential Systems
8.2.3. Variational Problems

Bibliography

Index