Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions
type: Item
instance of: scholarly article
title: Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions
zbMATH Open document ID: 1055.35080
author: Antoine Henrot
author: Edouard Oudet
published in: Archive for Rational Mechanics and Analysis
publication date: 2004-01-05T00:00:00Z
review text: Consider the family \(\mathcal F\) of all bounded open subsets of the plane having fixed area (say equal to \(1\)), and let \(\lambda_k(\Omega)\) be the \(k-\)th eigenvalue of \(\Omega\in\mathcal F\) for the Dirichlet boundary conditions. The well-known Faber-Krahn inequality asserts that \(\lambda_1(\Omega)\) is minimized by the disk, and Szegő proved that \(\lambda_2(\Omega)\) is minimized by the union of two identical disks (a disconnected set). Looking for minimizers of \(\lambda_2\) among (arbitrary) connected domains has no solution, and so it is natural to ask the following question: What is the convex plane domain \(\Omega^*\) of unit area which minimizes \(\lambda_2\)? (The minimizer \(\Omega^*\) is actually shown to exist). Numerical experiments lead \textit{A. B. Troesch} [Math. Comput. 27, 767--772 (1973; Zbl 0271.35018)] to conjecture that \(\Omega^*\) is a stadium, i.e. the convex hull of two identical tangent disks. The main result of the paper under review is to show that the above conjecture is false; more generally it is shown that the minimizer \(\Omega^*\) has no arc of circle in its boundary. In the course of the proof, the authors also prove some interesting facts about \(\Omega^*\); namely that it is at least \(C^1\) and at most \(C^2\) and that, if it is assumed to be \(C^{1,1}\) then it has two (and only two) straight lines in its boundary, and these lines are parallel; finally, under the \(C^{1,1}\) assumption the second eigenvalue is simple.
reviewed by: Alessandro Savo
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 2021063
zbMATH Keywords: convex planar domains
zbMATH Keywords: second eigenvalue
zbMATH Keywords: minimizers
MaRDI profile type: Publication
full work available at URL: s00205-003-0259-4
OpenAlex ID: W2074576683
DOI: 10.1007/S00205-003-0259-4
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description: scientific article; zbMATH DE number 2021063