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

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.