Uniqueness and non-uniqueness in the Cauchy problem
type: Item
instance of: scholarly article
title: Uniqueness and non-uniqueness in the Cauchy problem
zbMATH Open document ID: 0521.35003
author: Claude Zuily
published in: Progress in Mathematics
publication date: 1983-01-01T00:00:00Z
review text: 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.
reviewed by: Steven G. Krantz
Mathematics Subject Classification ID: 35-01
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35S10
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 78A05
zbMATH DE Number: 3824310
zbMATH Keywords: Cauchy problem
zbMATH Keywords: uniqueness
zbMATH Keywords: non-uniqueness
zbMATH Keywords: Weyl calculus
zbMATH Keywords: pseudoconvex
zbMATH Keywords: geometric optics
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3824310