Partial Differential Equations is a large book (over 700 pages) on an even larger subject. Part I of the book summarizes the classical theory of PDEs, starting with four canonical examples of partial differential equations: the transport equation, Laplace’s equation, the heat equation, and the wave equation. Part II covers the theory of linear PDEs in a modern setting, and Part III is devoted to nonlinear PDEs.
The book Partial Differential Equations is more unified than the subject of partial differential equations. There is no grand unified theory of PDEs, though there are unifying themes and common techniques. PDEs have been classified into three broad categories — elliptic, parabolic, and hyperbolic — though this classification breaks down much as the original classification of life into plants and animals broke down on closer inspection. First you discover that things like bacteria are not plants or animals. Then later you discover bacteria-like creatures (archaea) that are not so bacteria-like after all. Evans recognizes this problem with PDEs and mentions “the false impression that that there is some kind of general and useful classification scheme available in general.”
This book would be invaluable for a graduate student preparing to do research in PDEs; I wish I had had a copy in graduate school. (A couple years after I graduated I ran across a set of Evans’ lecture notes and greatly appreciated them. These notes have been subsumed into the present book.) Although the content is advanced, the book is a textbook, replete with numerous exercises. Evans has simplified his presentation slightly for the sake of pedagogy, not always seeking the weakest hypotheses or the strongest conclusions. As large as the book is, it would have been far larger had Evans not shown restraint. As he says in the preface
I have made a huge number of editorial decisions about what to keep and what to toss out, and can only claim that this selection seems to me about right.
Evans’ book is well written. It is a bit intimidating, despite the author’s approachable style, due to the vast subject area it surveys. However, Evans may be correct that the selection of material for the book is about right. The book is written for an audience that will value the book’s thorough presentation.
John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.
BLL — The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.