This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) 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 within Sobolev spaces; 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 summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between functional analytic insights and calculus-type estimates within the context of Sobolev spaces. 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 PDEs.

Readership

Graduate students and research mathematicians interested in PDEs.

Reviews

"In this textbook on partial differential equations (PDE) the author gives a broad survey of many important topics in this area. Generally the approach of the author is to explain the fundamental ideas of a subject in a clearest possible setting, and to emphasize the importance of nonlinear concepts and of generalized solutions. Highly recommend the book for students as well as for lectures in PDE."

-- Monatshefte für Mathematik

"The community has been rewarded with several new, outstanding graduate texts in PDEs. The most recent of these, published by the American Mathematical Society in their "Graduate Studies" series, is by Lawrence Evans. The text spawned from the author's UC Berkeley notes, which had limited availability; many instructors were eager to see the formal publication of those notes. The text is very attractively designed and packaged. Its exposition is laid out in large print with ample margins and separation between sections, formulas, and the main results. The format in Evans's text is inviting and unintimidating; students will appreciate the space to make marginal notes. The ideas are clearly explained and the principal definitions and theorems are carefully stated. It is entirely possible that the Evans PDE text could eventually become the benchmark. It is a standard treatment with good notation. It is extremely well written, with a very attractive format."

-- SIAM Review

"Well written; proofs are given in full detail and pictures are inserted when needed; moreover, the exposition is perfectly self-contained, the development within each part and section being rigorous and complete. The natural audience consists of Ph.D. students, willing to begin research activity on calculus of variations, viscosity solutions for Hamilton-Jacobi or systems of conservation laws; here they may find the necessary rudiments, expounded in an attractive form."

-- Mathematical Reviews

"For a student wishing to specialise in the theory of PDEs it provides a very solid foundation."

-- The Mathematical Gazette

"This excellent textbook is meant as an introduction to mathematical analysis of partial differential equations. Throughout the book the reader is acquainted with various approaches and techniques to initial and boundary-value problems. [This book is recommended] as the first textbook for anyone who wants to learn the theory of partial differential equations."

-- European Mathematical Society Newsletter

Table of Contents

• Introduction

• Four important linear partial differential equations
• Nonlinear first-order PDE
• Other ways to represent solutions

• Sobolev spaces
• Second-order elliptic equations
• Second-order elliptic equations
• Linear evolution equations

• The calculus of variations
• Nonvariational techniques
• Hamilton-Jacobi equations
• Systems of conservation laws
• Appendices
• Bibliography
• Index