The book under review can be used as textbook for a first PDE course either for the duration of one semester, or with additional material, for a two-semester course at a more leisurely pace. The material is suitable for both undergraduate and beginning graduate courses. It covers first order PDEs and the theory of linear second order equations, classified into elliptic, parabolic, and hyperbolic equations.
There are eight chapters in the book, but only chapters 2 to 7 contain material useful in the classroom. These chapters go over the standard topics a first PDEs course is expected to cover. At the end of each of Chapters 2–7 there are exercises of varying difficulty. Chapter 1 serves as introduction and overview, while Chapter 8 is an epilogue; it discusses further directions and problems in PDEs. The pages for both of these first and last chapters could have been better used for expanding on concepts already mentioned or for additional topics, (e.g. shocks and conservations laws, weak solutions, eigenvalues/eigenfunctions, hints/solutions for the exercises…). My guess is that the role of Chapter 8 is motivational, to encourage students into pursuing the further study of PDEs; I am unsure how well it succeeds, as most of the terms and problems listed there are not explained.
One of the points at which the book succeeds is that it presents the material in a clear manner and it covers it without heavy reference to the theory of ODEs. The main drawback is that at a comparable price, there are many books out there (e.g. Evans’ Partial Differential Equations in the same series), which not only cover the same material but go well beyond.
Florin Catrina is Assistant Professor of Mathematics at St. John's University in Queens, New York.