A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions
type: Item
instance of: scholarly article
title: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions
zbMATH Open document ID: 0757.35052
DOI: 10.2307/2946578
author: Mark S. Ashbaugh
author: Rafael D. Benguria
published in: Annals of Mathematics. Second Series
publication date: 1993-01-16T00:00:00Z
review text: The authors present a proof for the Payne-Polya-Weinberger conjecture. It says that the ratio \(\lambda_ 2/\lambda_ 1\) of the membrane eigenvalues \[ \Delta\varphi+\lambda\varphi=0\quad\text{in }D, \qquad \varphi|_{\partial D}=0 \] is maximal for the ball. The proof relies on the Rayleigh principle and uses rearrangement techniques together with specific properties of the Bessel functions and their zeros. Extensions to problems with more general operators are given.
reviewed by: Catherine Bandle
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 91444
zbMATH Keywords: Payne-Polya-Weinberger conjecture
zbMATH Keywords: membrane eigenvalues
zbMATH Keywords: Rayleigh principle
Wikidata QID: Q59158304
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946578
OpenAlex ID: W2000523938
description: scientific article; zbMATH DE number 91444