On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions
type: Item
instance of: scholarly article
title: On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions
zbMATH Open document ID: 0833.35035
DOI: 10.1215/S0012-7094-95-07801-6
author: Mark S. Ashbaugh
author: Rafael D. Benguria
published in: Duke Mathematical Journal
publication date: 1996-03-18T00:00:00Z
review text: The Rayleigh's conjecture for the clamped plate -- when the edges are clamped, the form of gravest pitch is doubtless the circle -- is considered. Namely, the eigenvalue problem \(\Delta^2 u= \lambda u\) in \(\Omega\subset \mathbb{R}^n\), \(u= |\nabla u|= 0\) on \(\partial\Omega\), where \(\Omega\) is any bounded domain in \(\mathbb{R}^n\) \((n= 2,3)\) with smooth boundary is investigated. Let \(\lambda(\Omega)\) be the principal eigenvalue. The main result is the following. For the eigenvalue problem mentioned above one has \(\lambda(\Omega)\geq \lambda(\Omega^*)\), where \(\Omega^*\) is the ball in \(\mathbb{R}^n\) having the same volume as the domain \(\Omega\) and equality holds if and only if \(\Omega\) is a ball.
Mathematics Subject Classification ID: 35J35
Mathematics Subject Classification ID: 31A30
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J40
zbMATH DE Number: 784722
zbMATH Keywords: biharmonic operator
zbMATH Keywords: Rayleigh's conjecture
zbMATH Keywords: principal eigenvalue
reviewed by: Valery V. Karachik
MaRDI profile type: Publication
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full work available at URL: s0012-7094-95-07801-6
OpenAlex ID: W1514532039
Wikidata QID: Q59158264
description: scientific article; zbMATH DE number 784722