Bounds for the first eigenvalue of the elastically supported membrane on convex domains
type: Item
instance of: scholarly article
title: Bounds for the first eigenvalue of the elastically supported membrane on convex domains
zbMATH Open document ID: 1035.35087
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2004-01-05T00:00:00Z
full work available at URL: http://hdl.handle.net/20.500.11850/53592
review text: This paper deals with lower bounds for the first eigenvalue of the following eigenvalue problem \[ \Delta u + \lambda u = 0 \;\text{in} \;\Omega, \qquad \frac{\partial u}{\partial n} + h u = 0 \;\text{on} \;\partial \Omega, \] where \(h > 0 \) is a constant and \(\Omega\) is a plane domain. The method is based on Barta's principle and is closely related to \textit{L. E. Payne} and \textit{G. A. Philippin} [SIAM J. Math. Anal. 14, 1154--1162 (1983; Zbl 0521.73056)] and \textit{R. P. Sperb} [Z. Angew. Math. Phys. 44, 639--653 (1993; Zbl 0785.35044)]. The solution of the torsion problem is extensively used as an auxiliary function. The paper contains numerical results and a nice example showing that the eigenvalue is not a monotone function of the domain.
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 2021120
zbMATH Keywords: eigenvalues of the Laplacian
zbMATH Keywords: elastically supported membrane
author: René Sperb
reviewed by: Bodo Dittmar
MaRDI profile type: Publication
OpenAlex ID: W2071777082
DOI: 10.1007/S00033-003-3212-3
description: scientific article; zbMATH DE number 2021120