Vol. 14, No. 7, 2021

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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
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