Extension of an inequality of Pólya-Szegö to wedge-like domains
type: Item
instance of: scholarly article
title: Extension of an inequality of Pólya-Szegö to wedge-like domains
zbMATH Open document ID: 0772.35043
DOI: 10.1007/BF00914359
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1993-05-16T00:00:00Z
review text: This is an interesting note which establishes an isoperimetric inequality for the first eigenvalue of the clamped membrane problem: \[ \Delta u+\lambda u=0\quad\text{in } D,\quad u=0\quad\text{on }\partial D, \] where \(D\) is a domain contained in a wedge of angle \(\pi/\alpha\). The precise result proved is: The first eigenvalue \(\lambda_ 1\) satisfies for any \(\alpha\geq 1/2\): \[ \lambda_ 1\leq{j^ 2_ \alpha\over 2A_ \alpha}\left[1+\left({\pi\over 2B_ \alpha}- \alpha\right){{\partial J_ \alpha\over\partial\alpha}(j_ \alpha)\over j_ \alpha J_{\alpha+1}(j_ \alpha)}\right] \] where \(j_ \alpha\) is the first positive root of \(J_ \alpha\), and \(A_ \alpha\), \(B_ \alpha\) are integrals depending on the domain and its boundary. It is important to observe that this inequality is best possible in the sense that equality holds when \(D\) is a circular sector of angle \(\pi/\alpha\).
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 168469
zbMATH Keywords: isoperimentric inequality for the first eigenvalue
zbMATH Keywords: clamped membrane problem
author: René Sperb
reviewed by: Brian Straughan
MaRDI profile type: Publication
cites work: A Faber-Krahn Inequality for Wedge-Like Membranes
cites work: Isoperimetric Inequalities in Mathematical Physics. (AM-27)
full work available at URL: https://doi.org/10.1007/bf00914359
OpenAlex ID: W1971656635
description: scientific article; zbMATH DE number 168469