The stability of some eigenvalue estimates
type: Item
instance of: scholarly article
title: The stability of some eigenvalue estimates
zbMATH Open document ID: 0770.35049
author: Antonios D. Melas
published in: Journal of Differential Geometry
publication date: 1993-04-01T00:00:00Z
review text: Let \(\Omega\) be a bounded convex domain in \(\mathbb{R}^ n\), and consider the Laplacian on \(L^ 2(\Omega)\) with Dirichlet boundary condition. Let also \(D\) denote a ball of \(\mathbb{R}^ n\) with same volume as \(\Omega\). The author first shows that if the first eigenvalue associated to \(\Omega\) is close enough to the one associated to \(D\), then \(\Omega\) is close to a ball in the sense of Hausdorff distance. Rather precise estimates are given, and the same result is also obtained if the quotient of the first two eigenvalues associated to \(\Omega\) is close enough to the corresponding quantity associated to \(D\). Finally, the first result is extended to nonconvex domains in two dimensions. The proofs are based on isoperimetric inequalities.
reviewed by: André Martinez
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 147455
zbMATH Keywords: Dirichlet eigenvalues
zbMATH Keywords: isoperimetric inequalities
zbMATH Keywords: bounded convex domain
zbMATH Keywords: nonconvex domains
Wikidata QID: Q115180721
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214448441
OpenAlex ID: W1588533103
DOI: 10.4310/JDG/1214448441
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description: scientific article; zbMATH DE number 147455