On the monodromy of the ramified Cauchy problem
type: Item
instance of: scholarly article
title: On the monodromy of the ramified Cauchy problem
zbMATH Open document ID: 1040.35002
DOI: 10.1016/S0021-7824(01)01240-5
author: Renaud Camalès
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-09-07T00:00:00Z
review text: Let \(a(x,D)=\sum_{\mid\alpha\mid\leq m}a_\alpha(x)D^\alpha\) be a linear differential operator of order \(m\) with coefficients \(a_\alpha(x)\) which are holomorphic in a neighbourhood of the origin of \(\mathbb C^{n+1}\). For a \((n+1)\)-tuple \(x=(x_0,\ldots,x_n)\) the corresponding \(n\)-tuple \((x_1,\ldots,x_n)\) will be denoted by \(x'\). Suppose that \(a_{(m,0,\ldots,0)}=1\) and that \(a(x,D)\) is an operator with multiple characteristics of constant multiplicity, meaning that there exist holomorphic function germs \(\lambda_i(x;\xi')\) at \((0;1,0,\ldots,0)\) and integers \(m_i \geq 1\) such that we can write the principal symbol \(g(x,\xi)\) of \(a(x,D)\) as \(g(x,\xi)=\prod_{i=1}^d(\xi_0 - \lambda_i(x,\xi'))^{m_i}\) with \(\lambda_i(0;1,0,\ldots,0) \neq \lambda_j(0;1,0,\ldots,0)\) if \(i\neq j\). Then the following result is due to \textit{E. Leichtnam} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, 369--443 (1990; Zbl 0717.35018)]. Denote \(S=\{x_0=0\}\) and \(T=\{x_0=x_1=0\}\). Let \(\Omega\) be an open connected neigbourhood of the origin of \(\mathbb C^{n+1}\) such that \(\Omega \cap S\) is connected and let \(v\) and \(w_h\), \(0\leq h < m\), be holomorphic functions on the universal covering of \(\Omega \setminus K\) and \(\Omega \cap S \setminus T\), respectively. Then there exists an open connected neighbourhood \(\Omega' \subseteq \Omega\) of the origin of \(\mathbb C^{n+1}\) and a unique solution \(u\) of the ramified Cauchy problem \(a(x,D)u(x)=v(x)\), \(D_0^hu(x)=w_h(x)\) for \(x_0=0\), \(0\leq h < m\), that is holomorphic on the universal covering of \(\Omega' \setminus K\). Here \(K=\bigcup_{i=1}^dK_i\) with \(K_i\) the characteristic hypersurfaces issuing from \(T\). Leichtnam also proved that, if \(v\) and \(w_h\) are of finite determination, then also the solution \(u\) is of finite determination; and if the monodromy of \(v\) is solvable, then the monodromy of \(u\) is solvable. In this paper the author studies the eigenvalues of the monodromy…
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 1975946
zbMATH Keywords: ramified Cauchy problem
zbMATH Keywords: monodromy
zbMATH Keywords: functions of finite determination
MaRDI profile type: Publication
cites work: Systemes d'équations aux dérivées partielles à caractéristiques multiples: problème de Cauchy ramifie, hyperbolicite partielle
cites work: Q4740878
cites work: Le problème de Cauchy ramifié
cites work: Non abelian ramification
full work available at URL: s0021-7824(01)01240-5
OpenAlex ID: W1982114112
description: scientific article; zbMATH DE number 1975946