Functional Analysis, Sobolev Spaces and Partial Differential Equations

The author of this book, Haim Brezis, is one of the world’s top researchers in the area of Partial Differential Equations. He is an excellent presenter, and, as the readers of this book will quickly realize, he is also a master communicator in writing.

The book is the English translation of an 1983 book published in French: Analyse fonctionnelle :théorie et applications, (Dunod, Paris, reprinted in 2005). It has seen translations into numerous languages and the Springer edition was especially anticipated, as it announced a number of practice exercises following each chapter. I can honestly say that it was well worth the wait.

The structure of the book is as follows: the first six chapters deal with Functional Analysis; chapters seven through ten introduce and then focus mostly on Partial Differential Equations; chapter eleven is a collection of various facts, mainly related to the first seven chapters. At the end of each chapter there are exercises, many of which contain hints, and are broken down into parts that lead progressively to the main result.

Following Chapter 11, there are solutions for selected exercises (a significant number of complete solutions for the exercises following each chapter are included). Next, follows a list of 51 Problems. These are mostly theorems, or otherwise theoretical results, the proofs of which could be treated separately from the main text. Finally, there is a section of hints and partial solutions for the set of 51 problems.

The text is a pleasure to read. There are numerous interesting remarks and comments spread throughout, and at the end of each chapter there is a subsection of comments. The comments point to interesting areas of research related to the topics discussed and to a wealth of references to current literature.

I wholeheartedly recommend this book as both a textbook as well as for independent study.

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