Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19 (Graduate Studies in Mathematics)

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 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.

This is by far the best book on PDE. The text is extremely clear, and most of the rather technical proofs are prefaced with "heuristic" calculations to help the reader understand what is going on. The chapter on the calculus of variations is the best exposition I have found of the subject, and Evans completely dispenses with the awful "delta" notation which never made any sense.

The text doesn't make much use of the Fourier transform and doesn't even mention distributions, and this gives his book a definite nonlinear flavor (which is a good thing). This should become the standard introduction to PDE on the graduate level.


Best textbook for a modern one-year course in PDE available.   September 29, 1998
16 out of 18 found this review helpful

I have taught a one-year course in PDE based on Evans' book and found it extremely cogent and stimulating both for myself and for the students. The treatment is up-to-date, with a definite nonlinear flavor. Beyond that, the exercises are very good, and the treatment is sufficiently detailed to make class preparation fairly fast. It does demand mathematical dexterity and maturity of the students right from the start, though.


This book opened a new world for me   February 24, 2006
Matthias Heymann (NYC)
9 out of 11 found this review helpful

After several bad lectures I had already almost given up on PDE. But when I got this book into my hands, I was immediately drawn into the subject and couldn't put it down until I finished it completely. After less than a month I felt how a new world had opened up to me, and I can since feel its effect when I go to lectures.

An important feature of this book is also its perfect layout which is very easy on the eye: I have seen so many mathbooks before that try to put as much information as possible on every square inch of paper, which is hard and slow to read - not this one: Evans' book is a pleasure to just look at.

Some experience in functional analysis is very helpful.


A Fine Treatise on the Subject   May 2, 2007
ikantspel (South Carolina)
2 out of 2 found this review helpful

This is a superb exposition of a difficult, yet enriching subject. This book is intended only as a beginning text (in a relative sense) and is by no means an attempt to give an exhaustive view of many topics discussed therein.

The first few chapters discuss classical solution techniques to frequently encountered PDEs such as the heat and Laplace equation. Methods of solution are discussed including Fourier transform methods and other classical methods to obtain strong solutions and/or representation formulas. The author, from this point, focuses on weak solution techniques for second order PDEs and systems in addition to conservation laws and other nonlinear PDEs. There is also a self-contained chapter on Sobolev spaces that proves to be fairly useful.

There is a necessary mathematical maturity needed to fully benefit from this text. The reader should be relatively comfortable with standard topics from classical analysis. It would help if the reader has seen Lebesgue spaces and is familiar with basic functional analysis and operator theory although many of these topics are reviewed in the apendices.

While this book is dense and difficult at times, it has a prominent place on my bookshelf.


This will become the standard text in PDE   November 19, 1999
14 out of 17 found this review helpful

This is a very well written textbook for graduate-level students as well as an excellent reference for researchers. The outlook of the author, a leader in his field, is non-linear and very broad and includes maechanics and geometry. Any department library needs this book.