On the global existence of holomorphic solutions and the semi-global existence of real analytic solutions of linear partial differential equations
type: Item
instance of: scholarly article
title: On the global existence of holomorphic solutions and the semi-global existence of real analytic solutions of linear partial differential equations
zbMATH Open document ID: 0632.35010
DOI: 10.1007/BF02790330
author: B. George
published in: Journal d'Analyse Mathématique
publication date: 1987-01-01T00:00:00Z
review text: This paper contains an erroneous claim at a key-point in the reasoning: Theorem 3.5 is correct only in the two-dimensional case. The correct statement can be found by \textit{H. Suzuki} [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 11, 253-258 (1972; Zbl 0255.35005)] for the first order equations, and, for higher order equations by the reviewer and \textit{Y. Takei} [Bicharacteristical convexity and the semi-global existence of holomorphic solutions of linear differential equations with holomorphic coefficients, RIMS-Preprint 581 (1987)]. Since another seemingly interesting part of this paper, i.e., {\S} 5, is based upon Theorem 3.5, it is not a mathematical claim. It should also be noted that, although the author uses the expression ``Theorem 3.5'', there is no proof for it. Apparently the author jumped to the erroneous conclusion that the bicharacteristical convexity entails the positive-definiteness of the generalized Levi form; simple and interesting counter-examples to this claim can be found in the above quoted papers. The reviewer hopes that the editorial board of Journal d'Analyse Mathématique announces for the benefit of the mathematical world that this paper contains serious errors.
Mathematics Subject Classification ID: 35G05
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35A20
zbMATH DE Number: 4027973
zbMATH Keywords: Bicharacteristical convexity
zbMATH Keywords: semi-global existence
zbMATH Keywords: holomorphic solutions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4027973