Partial differential equations / Jeffrey Rauch

This book is meant to provide graduate students of mathematics or physics with samples of problems, results and methods in PDEs ranging from the classical to the fairly recent. Only rudiments of functional analysis are assumed as background. The arrangement of the text seems to have achieved an optimal balance between the formal presentation of results and their motivations, and the lively style enhances the rigor rather than vice versa. The emphasis is on evolution equations, with the elliptic problems appearing as limits of such, accordingly treated at the end of the book. The basic examples are the Schrödinger equation, the heat and the wave equations, and the Cauchy-Riemann equation. Exercises form the backbone of the text: many of them, with adequate hints, contain results which would be likely to appear in the main body of other texts. The table of contents is as follows. Chapter 1: Power series methods—after some preliminary examples, the Cauchy-Kovaleskaya theorem is explained and exploited, but not proved. The last three sections are devoted to the Holmgren theorem, to its global version by F. John and to a nice discussion of shocks. Chapter 2: Some harmonic analysis—Fourier transform and Sobolev spaces. Chapter 3: Solution of initial value problems by Fourier synthesis—Fourier transform approach mainly to the Schrödinger equation but also to the heat, the wave and the Cauchy-Riemann equations. This chapter also includes a discussion of the Hadamard-Petrovskiĭ condition. Chapter 4: Propagators and x-space methods. The solutions calculated in the previous chapter in terms of their Fourier transform are transformed back and studied in terms of the original space variable. We also find here a nice discussion and examples of radiation problems. Chapter 5: The Dirichlet problem. This chapter deals with second order equations and, as indicated in the title, with the first boundary problem; it begins with the variational approach which is later generalized to obtain the Fredholm alternative for problems not in divergence form. The method of separation of variables, elliptic regularity results and the maximum principle are also presented in this chapter. The book contains an appendix entitled "A crash course in distribution theory'', and a bibliography of some 40 items including most of the important books on the subject.
To sum up, in the reviewer's opinion, this is an outstanding text presenting a healthy challenge not only to students but also to teachers used to more traditional or more pedestrian developments of the subject. (MathSciNet)

Bibliogr. p. [259]-260. Index