On the Ashbaugh–Benguria conjecture about lower-order Dirichlet eigenvalues of the Laplacian

Wang, Qiaoling; Xia, Changyu

doi:10.2140/apde.2021.14.2069

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On the Ashbaugh–Benguria conjecture about lower-order Dirichlet eigenvalues of the Laplacian

Qiaoling Wang and Changyu Xia

Vol. 14 (2021), No. 7, 2069–2078

DOI: 10.2140/apde.2021.14.2069

Abstract

We prove an isoperimetric inequality for lower-order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the celebrated Ashbaugh–Benguria inequality conjectured by Payne, Pólya and Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh and Benguria.

Keywords

Ashbaugh–Benguria conjecture, isoperimetric inequality, eigenvalues, Dirichlet problem, Dirichlet eigenvalues, Payne–Pólya–Weinberger conjecture

Mathematical Subject Classification 2010

Primary: 35P15

Secondary: 58C40

Milestones

Received: 7 December 2018

Accepted: 15 June 2020

Published: 10 November 2021

Authors

Qiaoling Wang
Departamento de Matemática Universidade de Brasília Brasília Brazil

Changyu Xia
Departamento de Matemática Universidade de Brasília Brasília Brazil

Vol. 14, No. 7, 2021