Partial Differential Equations

This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, a significantly expanded bibliography. About the First Edition: I have used this book for both regular PDE and topics courses. It has a wonderful combination of insight and technical detail. … Evans’ book is evidence of his mastering of the field and the clarity of presentation. ―Luis Caffarelli, University of Texas It is fun to teach from Evans’ book. It explains many of the essential ideas and techniques of partial differential equations … Every graduate student in analysis should read it. ―David Jerison, MIT I usePartial Differential Equationsto prepare my students for their Topic exam, which is a requirement before starting working on their dissertation. The book provides an excellent account of PDE’s … I am very happy with the preparation it provides my students. ―Carlos Kenig, University of Chicago Evans’ book has already attained the status of a classic. It is a clear choice for students just learning the subject, as well as for experts who wish to broaden their knowledge … An outstanding reference for many aspects of the field. ―Rafe Mazzeo, Stanford University

Table of Contents

Chapter 1. Introduction

Part I. Representation Formulas for Solutions

Chapter 2. Four Important linear Partial Differential Equations

Chapter 3. Nonlinear First-Order PDE

Chapter 4. Other Ways to Represent Solutions

Part II. Theory for linear Partial Differential Equations

Chapter 5. Sobolev Spaces

Chapter 6. Second-Order Elliptic Equations

Chapter 7. linear Evolution Equations

Part Ill. Theory for Nonlinear Partial Differential Equations

Chapter 8. The Calculus of Variations

Chapter 9. Nonvariational Techniques

Chapter 10. Hamilton-Jacobi Equations

Chapter 11. Systems of Conservation Laws

Chapter 12. Nonlinear Wave Equations

Appendices

Appendix A. Notation

Appendix B. Inequalities

Appendix C. Calculus

Appendix D. Functional Analysis

Appendix E. Measure Theory