Partial Differential Equations, Second edition
About this Title
Lawrence C. Evans, University of California, Berkeley, Berkeley, CA
Publication: Graduate Studies in Mathematics
Publication Year: 1998; Volume 19
ISBNs: 978-0-8218-4974-3 (print); 978-1-4704-1144-2 (online)
MathSciNet review: MR2597943
MSC: Primary 35-01
This volume is not part of this online collection.
This text presents a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDE), with particular emphasis on nonlinear equations. The exposition is divided into three parts: 1) representation formulas for solutions, 2) theory for linear partial differential equations, and 3) theory for nonlinear partial differential equations.
Included are complete treatments of the method of characteristics, energy methods, regularity for second-order elliptic, parabolic, and hyperbolic equations, maximum principles, the multidimensional calculus of variations, viscosity solutions of Hamilton–Jacobi equations, shock waves and entropy criteria for conservation laws, and much more. The author also summarizes in appendices the relevant mathematical background required.
While he has reworked and simplified much of the classical theory, the author primarily emphasizes the modern interplay between functional analytic insights and calculus-type estimates within the context of Sobolev spaces. The treatment of all topics is complete and self-contained. The book's wide scope and clear exposition make it a suitable text for a graduate course in PDE.
Graduate students and research mathematicians interested in PDEs.
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 III: 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
- Appendix A. Notation
- Appendix B. Inequalities
- Appendix C. Calculus
- Appendix D. Functional Analysis
- Appendix E. Measure Theory