The hyperbolic Cauchy problem
type: Item
instance of: scholarly article
title: The hyperbolic Cauchy problem
zbMATH Open document ID: 0762.35002
author: Kunihiko Kajitani
author: Tatsuo Nishitani
published in: Lecture Notes in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: This is a collection of highly technical lectures given by the authors on the question of existence and uniqueness of solution of the Cauchy problem for hyperbolic operators with multiple characteristics. Part I considers the Cauchy problem for hyperbolic operators in Gevrey classes while Part II treats the hyperbolic operators in \(C^ \infty\) classes. To overcome the difficulty which arises from the loss of derivatives of solutions in the theory of partial differential operators, the authors introduce the so-called Fourier integral operators with complex-valued phase function \(\varphi(x,\xi)=x\cdot\xi-i\Lambda(x,\xi)\), where \(\Lambda(x,\xi)\) is a real-valued function in a suitable class. For the Cauchy problem in Gevrey class \(E^{\{\kappa\}}(\kappa\geq 1)\), \(\Lambda(x,\xi)\) is chosen to satisfy \(\Lambda(x,\xi)=O(|\xi|^{1/\kappa})\) while in the \(C^ \infty\) class it is chosen to satisfy \(\Lambda(x,\xi)=O(\log|\xi|)\), as \(|\xi|\to\infty\). For \(\varphi=x\cdot\xi-i\Lambda(x,\xi)\), the authors define the Fourier integral operator \(I_ \varphi(x,D)u(x)=\int e^{i\varphi(x,\xi)}\hat u\hat d\xi\) for \(u\in C_ 0^ \infty(\mathbb{R}^ n)\times E^{\{\kappa\}}(\mathbb{R}^ n)\) (resp. \(C_ 0^ \infty(\mathbb{R}^ n))\), where \(u\) stands for the Fourier transform of \(u\) and \(d\xi=(2\pi)^{-n}d\xi\). If \(\exp\Lambda(x,\xi)\) is chosen to compensate for the loss of derivatives of solutions, then it can be expected that operators with loss of infinite (resp. finite) derivatives in Gevrey classes (resp. \(C^ \infty\) classes) will be transformed to ``good'' operators by conjugation with such a Fourier integral operator. For a pseudo-differential operator \(p(x,D)\), the authors define \(p_ \Lambda(x,D)=I^ R_{-\varphi}(x,D)p(x,D)I_ \varphi(x,D)\), where \(I^ R_{-\varphi}(x,D)\) denotes the reverse operator of \(I_{- \varphi}(x,D)\). The symbol \(p_ \Lambda(x,\xi)\) is thus given by \(p_ \Lambda(x,\xi)\sim p(x+i\Lambda_ \xi,\xi-i\Lambda_ x)+\cdots\), where \(p(x+iy,…
reviewed by: Eutiquio C. Young
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35S10
Mathematics Subject Classification ID: 35L99
Mathematics Subject Classification ID: 35S30
zbMATH DE Number: 53222
zbMATH Keywords: hyperbolic operators in \(C(infinity)\) classes
zbMATH Keywords: hyperbolic operators in Gevrey classes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0090882
OpenAlex ID: W2506827042
DOI: 10.1007/BFB0090882
description: scientific article; zbMATH DE number 53222