Orient BlackSwan

About the BookAbout the AuthorTable of Contents

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.

Qing Han, University of Notre Dame, IN

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
2.3. A Priori Estimates
2.4. Exercises

Chapter 3. An Overview of Second-Order PDEs
3.1. Classifications
3.2. Energy Estimates
3.3. Separation of Variables
3.4. Exercises

Chapter 4. Laplace Equations
4.1. Fundamental Solutions
4.2. Mean-Value Properties
4.3. The Maximum Principle
4.4. Poisson Equations
4.5. Exercises

Chapter 5. Heat Equations
5.1. Fourier Transforms
5.2. Fundamental Solutions
5.3. The Maximum Principle
5.4. Exercises

Chapter 6. Wave Equations
6.1. One-Dimensional Wave Equations
6.2. Higher-Dimensional Wave Equations
6.3. Energy Estimates
6.4. Exercises

Chapter 7. First-Order Differential Systems
7.1. Noncharacteristic Hypersurfaces
7.2. Analytic Solutions
7.3. Nonexistence of Smooth Solutions
7.4. Exercises

Chapter 8. Epilogue
8.1. Basic Linear Differential Equations
8.2. Examples of Nonlinear Differential Equations
Bibliography
Index