Necessary and sufficient conditions for local solvability of nonsolvable partial differential equations
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for local solvability of nonsolvable partial differential equations
zbMATH Open document ID: 0568.35010
DOI: 10.1007/BF02790181
author: B. George
published in: Journal d'Analyse Mathématique
publication date: 1984-01-01T00:00:00Z
review text: The principal result of this paper is as follows. Let P be a partial differential operator with analytic coefficients, and let p be the principal symbol of P. Let Char p be the characteristic variety of P. Assume there is a \(\lambda \in {\mathbb{R}}^ N\setminus \{0\}\) such that \(p(0,...,0,\lambda)\neq 0\). Let \(g\in {\mathcal E}'({\mathbb{R}}^ N)\) be such that \(WF_ A(g)\cap Char p\neq \emptyset\) in some neighborhood of \(0\in {\mathbb{R}}^ N\). Then there is a neighborhood U of \(0\in {\mathbb{R}}^ N\) and a distribution \(v\in {\mathcal D}'(U)\) such that \(Pv=g\) on U. Other results of this type are obtained. These generalize results of Greiner-Kohn-Stein for the Hans Lewy operator. The principal technique is to relate the problem to the Boutet de Monvel/Guillemin theory of Toeplitz operators. The facilitates passing from microlocal theory to local theory.
Mathematics Subject Classification ID: 35F05
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 3906943
zbMATH Keywords: local solvability
zbMATH Keywords: non-solvable equations
zbMATH Keywords: analytic coefficients
zbMATH Keywords: principal symbol
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: microlocal theory
reviewed by: Steven G. Krantz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3906943