The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective
type: Item
instance of: scholarly article
title: The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective
zbMATH Open document ID: 1252.35119
published in: Bulletin of the London Mathematical Society
publication date: 2012-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.4805
Mathematics Subject Classification ID: 35E10
Mathematics Subject Classification ID: 46A63
Mathematics Subject Classification ID: 35E20
zbMATH DE Number: 6048045
zbMATH Keywords: constant coefficient partial differential equation
zbMATH Keywords: augmented operator
zbMATH Keywords: surjectivity
zbMATH Keywords: counterexample
author: Thomas Kalmes
MaRDI profile type: Publication
OpenAlex ID: W2095005611
arXiv ID: 1105.4805
DOI: 10.1112/BLMS/BDR121
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review text: In their deep paper on solvability of partial differential equations in the space of vector valued distributions [J. Funct. Anal. 230, No. 2, 329--381 (2006; Zbl 1094.46006)], \textit{J. Bonet} and \textit{P. Domański} were led to the following natural question. Let \(P(D)\) be a constant coefficient partial differential operator in d variables, which is surjective on the space of distributions \(D'(X)\) on some open set \(X\subset \mathbb{R}^d\). Is the augmented operator \(P^+(D_1,\dotsc,D_{d+1}):= P(D_1,\dotsc,D_d)\) necessarily surjective on \(D'(X\times \mathbb{R})\)?NEWLINENEWLINEIn the present interesting paper, this question is answered in the negative if \(d\geq 3\). In fact, a counterexample can be chosen such that \(P(D)\) is even hypoelliptic. This implies that the kernel of \(P(D)\) in \(C^\infty(X)\) does not have property \((\Omega)\) of Vogt providing the first example of this kind. The open set \(X\) is chosen as the complement of a suitable closed cone. The proof is then based on a characterization of the surjectivity of \(P^+(D)\) on \(D'(X\times \mathbb{R})\) by a variant of Hörmanders \(\sigma_P\)-function see [\textit{T. Kalmes}, J. Math. Anal. Appl. 386, No. 1, 125--134 (2012; Zbl 1248.35044)], where also positive solutions to the question of Bonet and Domanski [loc. cit.] are provided; in particular, the answer is always positive if \(d=2\)).
reviewed by: Michael Langenbruch
description: scientific article; zbMATH DE number 6048045