On the solvability of linear partial differential equations
type: Item
instance of: scholarly article
title: On the solvability of linear partial differential equations
zbMATH Open document ID: 0869.35007
author: Hiroshi Koshimizu
author: Kiyoshi Takeuchi
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-07-27T00:00:00Z
review text: The authors introduce a purely algebraic method for solvability of partial differential equations, suitable also for treating \({\mathcal D}_X\)-modules, which are not hyperbolic, nor simple-characteristic. The main result they prove is as follows. Let \(N=\{x_1=0\}\) be a closed submanifold of the real analytic manifold \(M,\) and denote by \(Y\subset X\) a complexification of \(N\subset M.\) If \({\mathcal M}\) is a coherent \({\mathcal D}_X\)-module for which \(Y\) is non-characteristic, and if \({\mathcal M}\) is micro-hyperbolic in the directions \(\pm dx_1\in T_N^*M\setminus 0\) on \(N\times_M(T_M^*X\setminus 0),\) then \(\text{Ext}_{{\mathcal D}_X}^j({\mathcal M},{\mathcal B}_M)\bigl|_N\simeq 0\) for \(j>\text{proj.dim}{\mathcal M}_Y,\) where \({\mathcal B}_M\) denotes the sheaf of Sato's hyperfunctions on \(M\) and \({\mathcal M}_Y\) the system induced by \({\mathcal M}\) on \(Y.\) Some variants of the main theorem and applications, especially to the case \(P=EQ+(\)lower-order-terms), where \(E\) is elliptic and \(Q\) is semihyperbolic (resp. hyperbolic) in the direction \(dx_1\) (resp. \(\pm dx_1),\) are then given.
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 32S40
Mathematics Subject Classification ID: 58J15
zbMATH DE Number: 952360
zbMATH Keywords: \(\mathcal D_ X\)-modules
reviewed by: Alberto Parmeggiani
MaRDI profile type: Publication
full work available at URL: pjaa.72.121
OpenAlex ID: W2065899106
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DOI: 10.3792/PJAA.72.121
description: scientific article; zbMATH DE number 952360