Cauchy problem for hyperbolic \(\mathcal{D}\)-modules with regular singularities
type: Item
instance of: scholarly article
title: Cauchy problem for hyperbolic \(\mathcal{D}\)-modules with regular singularities
zbMATH Open document ID: 0934.58021
author: Andrea D'Agnolo
author: Francesco Tonin
published in: Pacific Journal of Mathematics
publication date: 2000-04-09T00:00:00Z
review text: The aim of the paper is to generalize classical results concerning the hyperbolic Cauchy problem for a linear partial differential operator with analytic coefficients [see \textit{M. Kashiwara} and \textit{T. Oshima}, Ann. Math., II. Ser. 106, 145-200 (1977; Zbl 0358.35073)] to the case of general systems of linear partial differential equations. More precisely, let \(\mathcal M\) be a coherent \(\mathcal D\)-module on the complexification of a real analytic manifold \(M.\) Assume that the characteristic variety of \(\mathcal M\) is hyperbolic with respect to a submanifold \(N \subset M\) and \(\mathcal M\) has regular singularities along a regular involutive submanifold of real type. The authors prove that under the above assumptions the Cauchy problem is well posed in the space of distributions. Thus, they generalize results of \textit{M. Kashiwara} and \textit{P. Schapira} [`Sheaves on manifolds', Springer, Berlin (1990; Zbl 0709.18001)], and others obtained in the case of Sato's hyperfunctions. In addition, they recover a classical result in the case when \(\mathcal M\) is induced by a single differential operator or by a normal square system whose characteristics have constant multiplicities since the above hypotheses correspond to Levi conditions [\textit{L. Hörmander}, `The analysis of linear partial differential operators. II', Springer (1983; Zbl 0521.35002)].
Mathematics Subject Classification ID: 58J15
Mathematics Subject Classification ID: 35S15
Mathematics Subject Classification ID: 35R05
Mathematics Subject Classification ID: 35F15
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 55N30
Mathematics Subject Classification ID: 35G05
Mathematics Subject Classification ID: 35E05
zbMATH DE Number: 1186194
zbMATH Keywords: hyperbolic Cauchy problem
zbMATH Keywords: regular singularities
zbMATH Keywords: characteristic variety
zbMATH Keywords: distributions
zbMATH Keywords: coherent \(\mathcal D\)-modules
zbMATH Keywords: involutive submanifold
zbMATH Keywords: boundary value problem
zbMATH Keywords: propagation
zbMATH Keywords: geometric optics
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: pjm.1998.184.1
OpenAlex ID: W2024950629
DOI: 10.2140/PJM.1998.184.1
description: scientific article; zbMATH DE number 1186194