Singularities of the solution to a certain Cauchy problem and an application to the Pompeiu problem
type: Item
instance of: scholarly article
title: Singularities of the solution to a certain Cauchy problem and an application to the Pompeiu problem
zbMATH Open document ID: 0797.35129
DOI: 10.1215/S0012-7094-93-07106-2
author: Peter Ebenfelt
published in: Duke Mathematical Journal
publication date: 1994-10-18T00:00:00Z
review text: This paper consists of two parts. The first is a report on an integral transform of B. Yu. Sternin and V. E. Shatalov and its application to the holomorphic Cauchy problem with constant coefficients. The second part is an application of the first to the Pompeiu problem. The main result implies that under certain hypotheses on a domain \(\Omega_ 0\) in \(\mathbb{R}^ 2\), symmetric about the \(x_ 1\)-axis, the rotation of \(\Omega_ 0\) around the \(x_ 1\)-axis in \(\mathbb{R}^ 3\), denoted by \(\Omega\) has the Pompeiu property; i.e. \(f \equiv 0\) is the only continuous function in \(\mathbb{R}^ 3\) such that \(\int_{\sigma (\Omega)} f = 0\) for every rigid motion \(\sigma\) of \(\mathbb{R}^ 3\). In fact, a stronger property of \(\Omega\) is shown to hold: a certain family of overdetermined Cauchy problems has no solutions.
Mathematics Subject Classification ID: 35N05
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 495439
zbMATH Keywords: holomorphic Cauchy problem
zbMATH Keywords: Pompeiu problem
zbMATH Keywords: overdetermined Cauchy problem
MaRDI profile type: Publication
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full work available at URL: s0012-7094-93-07106-2
OpenAlex ID: W1500761796
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description: scientific article; zbMATH DE number 495439