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Partial Differential Equations of Mathematical Physics

S. L. Sobolev
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Florin Catrina
, on

Everything about this book calls for a single word review, “classic”: the publisher (Dover Publications) and their traditional selection of titles, the topic of the book, the style of exposition, the author himself.

Probably everyone reading this review has a sense of the type of books (at least in Mathematics) that Dover chooses for publication. This book is no exception. It is a re-publication of a translation from the Russian by Commander E. R. Dawson (edited by T. A. A. Broadbent) of the third edition of the lecture notes for a course on “Equations of Mathematical Physics”, originally published by Pergamon Press in 1964. The 30 sections of the book are called “lectures” and they correspond to the division of the material chosen by the author when teaching this course.

As is stated in Lecture 1, the book deals with “what are usually called the classical equations of mathematical physics, namely, the wave equation, Laplace’s equation, and the equation of heat conduction.” The presentation of the topics is clear and it flows naturally. It does not take much effort to almost visualize the professor expounding the material at the blackboard.

There are no exercises, neither at the end of the lectures nor dispersed throughout the text. This has the disadvantage of making the book a difficult choice for instructors teaching a PDE course; but it pays off in fluency. The presentation is gradual and the necessary background is built up thoughtfully, so that the reader is tempted to try to do the proofs before reading them in the book.

“Sobolev Spaces” has become a household name in Analysis: modern (even first) courses in PDEs always mention Sobolev spaces. It may come as a disappointment, then, that this book does not contain anything about them. It has a little about the regularity of weak solutions, but this is not its scope. The real goal is to give an in-depth understanding of the three types of second order differential equations (elliptic, parabolic, hyperbolic), with just enough generality to catch the physical motivations and applications and to justify the use of the machinery of Functional Analysis. For example, all applications are in one, two, or three dimensions.

I would not consider this book a reference book in the sense of Gilbarg and Trudinger’s Elliptic PDEs of Second Order or Adams’ Sobolev Spaces; neither do I consider the book to be the ideal textbook (such as Evans) for a first year PDE course. But I believe that it belongs on the mandatory reading list right after that.

I guess that the person who would benefit most from this book is the graduate student at the end of the first year. During the summer after the first PDE course, if he or she has the time and the discipline to cover two lectures a week, it may just make for a very productive summer. Actively reading it will seriously reinforce the material learned in class, it will rigorously justify and prove results which are not covered in a first PDE course, and it will invite the reader to learn about further applications of Functional Analysis. Last but not least, the book will give the young reader a sense of “how it used to be done”.

Florin Catrina is Assistant Professor of Mathematics at St. John's University in Queens, New York.

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