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Uniqueness and non-uniqueness in the Cauchy problem. (English) Zbl 0521.35003
Progress in Mathematics, 33. Boston-Basel-Stuttgart: Birkhäuser. XI, 168 p. SFr. 32.00; DM 36.00 (1983).
This book gives a nice introduction to an important chapter in the modern theory of partial differential equations: the Cauchy problem. It assumes no particular knowledge on the reader’s part of the Cauchy problem, begins with the simplest case of first order operations on \(\mathbb R^2\), and leads in a logical progression through the ideas of Carleman and Calderón, the Nirenberg-Trèves condition, and results on characteristics of higher multiplicity. The last chapter treats the application, due to Hörmander, of the pseudo-convexity condition to the Cauchy problem.
While the book is a lucid introduction to the subject, it is not for the neophyte in partial differential equations. The Cauchy problem is never defined. The usual language of modern p.d.e.’s, – characteristics, Poisson brackets, pseudo-differential operators, the basic machinery of microlocal analysis – are used without comment. While the proofs are complete, they require a fair amount of sophistication and effort on the reader’s part. On the other hand, these are lecture notes and I think that the author has made for the most part the right decision regarding level of presentation. The dedicated reader will find this a nice place to see the aforementioned p.d.a. ideas applied to a concrete problem; he will also find results accessible and easily identifiable. In the end, the brevity is appreciated. While sows of the recent crop of books on pseudo-differential operators (Hörmander, Taylor, Tréves) treat the Cauchy problem, none do so in anything approximating the detail given here. In 160 pages the reader is brought from the most basic method in the subject (Carleman’s method of weights) to the most sophisticated ones at the forefront of current research (e.g., geometric optics and the Weyl calculus of pseudodifferential operators).
The book will be a useful resource and a starting point for those getting involved in research in modern p.d.e.
Reviewer: S. G. Krantz

MSC:
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35G25 Initial value problems for nonlinear higher-order PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35S10 Initial value problems for PDEs with pseudodifferential operators
32T99 Pseudoconvex domains
78A05 Geometric optics