Introduction to Partial Differential Equations

by Gerald B. Folland

The second edition of Introduction to Partial Differential Equations, which originally appeared in the Princeton series Mathematical Notes, serves as a text for mathematics students at the intermediate graduate level. The goal is to acquaint readers with the fundamental classical results of partial differential equations and to guide them into some aspects of the modern theory to the point where they will be equipped to read advanced treatises and research papers. This book includes many more exercises than the first edition, offers a new chapter on pseudodifferential operators, and contains additional material throughout.

The first five chapters of the book deal with classical theory: first-order equations, local existence theorems, and an extensive discussion of the fundamental differential equations of mathematical physics. The techniques of modern analysis, such as distributions and Hilbert spaces, are used wherever appropriate to illuminate these long-studied topics. The last three chapters introduce the modern theory: Sobolev spaces, elliptic boundary value problems, and pseudodifferential operators.

Genres: Technical, Reference

352 pages, Hardcover

First published December 1, 1976

Reviews

[Tomáš Ševček · Rating 5 out of 5]

This book is still used in graduate-level classes on PDEs at the University of Washington (where it first originated based on Folland's notes from 1975). At first, I was surprised that the courses did not follow Evans as it is more widely used these days, but I soon realized why. While Evans aims to cover a broad range of PDE topics (ranging from the most introductory techniques for solving certain types of PDEs, through the general theory of linear second-order PDEs, to numerous techniques for non-linear PDEs), Folland's main objective is to cover the theory of linear second-order partial differential operators (with the exception of the first chapter in which some existence and uniqueness results for non-linear equations are covered too). The approaches to studying the theory differ significantly between the two books. Evans seems to emphasize individual function spaces, while Folland provides a treatment through the lens of distribution and (in part) potential theory, both of which are omitted in Evans. This approach (together with the fact that the last chapter briefly introduces pseudodifferential operators) gives the students a very solid foundation for continuing with microlocal analysis, which is a fairly active area of research in contemporary PDEs and is also one of the areas of research at UW.

As for difficulty, reading Folland might be a bit more challenging than Evans at first (especially if you have no background in PDEs whatsoever, in which case I do not recommend this book despite the title saying it is an introduction to PDEs, which really means the book is an introduction to the abstract theory of PDEs). You have to be familiar with vector calculus and functional analysis in order to read both, but in Folland, some knowledge of distribution theory is assumed (one should definitely be familiar with Schwartz functions, tempered and compactly supported distributions and the corresponding topologies on those spaces as well as with basic operations on distributions such as differentiation, convolution and the Fourier transform of distributions). Furthermore, more of complex analysis is needed for Folland as Evans usually assumes the functions are real-valued.

All in all, I recommend Folland if you aim to study distribution-heavy areas of PDEs in the future. If you are, however, looking for breadth or if you have gaps in less advanced PDE techniques, I think Evans still remains the better choice.