On the lack of directional quasiconcavity of the fundamental mode in the clamped membrane problem
type: Item
instance of: scholarly article
title: On the lack of directional quasiconcavity of the fundamental mode in the clamped membrane problem
zbMATH Open document ID: 1233.35149
author: Tobias Weth
published in: Archiv der Mathematik
publication date: 2011-12-13T00:00:00Z
review text: It is well known that the first eigenfunction of the Laplacian in a bounded convex region under Dirichlet boundary condition has convex level sets. In 1994 the reviewer asked if a corresponding statement holds if the domain is merely convex in one direction [in: Partial differential equations of elliptic type. Cambridge: University Press. Symp. Math. 35, 165--183 (1994; Zbl 0827.35083)]. In this note the author provides a negative answer in the form of a well designed two-dimensional counterexample.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 74K15
zbMATH DE Number: 5988054
zbMATH Keywords: quasiconcavity
zbMATH Keywords: superlevel set
zbMATH Keywords: principle Dirichlet eigenfunction
zbMATH Keywords: counterexample
MaRDI profile type: Publication
full work available at URL: s00013-011-0308-4
OpenAlex ID: W1971957527
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DOI: 10.1007/S00013-011-0308-4
description: scientific article; zbMATH DE number 5988054