Singularities of solutions of equations with noninvolutive characteristics. I: The case of second order Fuchsian equations
type: Item
instance of: scholarly article
title: Singularities of solutions of equations with noninvolutive characteristics. I: The case of second order Fuchsian equations
zbMATH Open document ID: 0780.35002
author: Sunao Ōuchi
published in: Journal of the Mathematical Society of Japan
publication date: 1994-02-06T00:00:00Z
review text: Let \(L_ i(t,z,\partial_ t, \partial_ z)\) (\(i=1,2\)), \((t,z)\in\mathbb{C}^ 1\times \mathbb{C}^ n\), be linear partial differential operators with the order \(i\): \[ \begin{aligned} L_ 2(t,z,\partial_ t,\partial_ z) &= \partial^ 2_ t+A_ 1 (t,z,\partial_ z) \partial_ t+ A_ 2(t,z,\partial_ z),\\ L_ 1(t,z,\partial_ t, \partial_ z) &= a_ 0(t,z)\partial_ t+ \sum^ n_{i=1} a_ i(t,z)\partial_{z_ i}+ d(t,z), \end{aligned} \] where their coefficients are holomorphic in a neighbourhood \(\Omega\) of the origin. Let \(K_ i\) (\(i=1,2\)) be the characteristic surfaces of \(L_ 2(t,z,\partial_ t,\partial_ z)\) through \(t=z_ 1=0\). Put \[ L(t,z,\partial_ t, \partial_ z)= t^ 2 L_ 2(t,z,\partial_ t, \partial_ z)+ tL_ 1(t,z,\partial_ t, \partial_ z)+c(t,z), \] which is a Fuchsian operator with respect to \(K_ 0=\{t=0\}\). Consider the following equation \(L(t,z,\partial_ t, \partial_ z)=u(t,z)=f(t,z)\), where \(f(t,z)\) is holomorphic on \(\Omega-K_ 1\) and \(u(t,z)\) is holomorphic in a neighbourhood of \((\Omega-K_ 1)\cap \{t=0\}\). In this paper an integral representation of \(u(t,z)\) is obtained under some conditions on the characteristic exponent, and it is shown that \(u(t,z)\) is holomorphic on the universal covering space of \(\Omega'- \bigcup^ 2_{i=0} K_ i\) for a small neighbourhood \(\Omega'\) of the origin. The following Fuchsian operator is also treated: \(L(t,z,\partial_ t, \partial_ z)= tL_ 2(t,z,\partial_ t, \partial_ z)+ L_ 1(t,z,\partial_ t, \partial_ z)\), for which Cauchy problem \(L(t,z,\partial_ t, \partial_ z) u(t,z)=f(t,z)\), \(u(0,z)=u_ 0(z)\), is considered and similar results to the preceding are obtained.
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35G05
zbMATH DE Number: 469059
zbMATH Keywords: singularity
zbMATH Keywords: second order Fuchsian equations
zbMATH Keywords: characteristic surfaces
zbMATH Keywords: integral representation
zbMATH Keywords: Cauchy problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04520215
OpenAlex ID: W2162424710
DOI: 10.2969/JMSJ/04520215
description: scientific article; zbMATH DE number 469059