Deformations of balls in Schiffer's conjecture for Riemannian symmetric spaces
type: Item
instance of: scholarly article
title: Deformations of balls in Schiffer's conjecture for Riemannian symmetric spaces
zbMATH Open document ID: 0865.53043
DOI: 10.1007/BF02761034
author: Alexander M. Semenov
author: Mark L. Agranovsky
published in: Israel Journal of Mathematics
publication date: 1997-07-10T00:00:00Z
review text: Let us recall that the Schiffer problem is to decide whether in \(\mathbb{R}^n\) the balls are the only simply connected bounded domains for which the overdetermined Dirichlet boundary value problem \(\Delta u+\lambda u=0\) in \(\Omega\), \(u=1\) and \({\partial u\over\partial n}=0\) on \(\partial\Omega\), has a solution. The first author [J. Math. Anal. Appl. 178, No. 1, 269-279 (1993; Zbl 0793.35063)] in \(\mathbb{R}^2\) and \textit{T. Kobayashi} [Commun. Anal. Geom. 1, No. 4, 515-541 (1993; Zbl 0847.35090)] in \(\mathbb{R}^n\) proved (using different methods) that the balls are isolated among classes of domains \(\Omega\) that have such overdetermined eigenvalues. In the present paper, the authors give a beautiful proof, exploiting in a very clever and non-elementary way some elementary ideas from the reviewer [J. Anal. Math. 37, 128-144 (1980; Zbl 0449.35024)] that this result is still true in any non-compact symmetric space of rank 1. (The same proof also works in \(\mathbb{R}^n\), it is just simpler).
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 952558
zbMATH Keywords: Schiffer conjecture
zbMATH Keywords: symmetric spaces
Wikidata QID: Q115391687
reviewed by: Carlos A. Berenstein
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02761034
OpenAlex ID: W2075399377
description: scientific article; zbMATH DE number 952558