On Ashbaugh-Benguria's Conjecture about Lower Order Dirichlet Eigenvalues of the Laplacian

TL;DR

This paper proves a new isoperimetric inequality for lower order Dirichlet Laplacian eigenvalues, advancing the understanding of eigenvalue ratios and supporting the Ashbaugh-Benguria conjecture.

Contribution

It establishes a strengthened inequality for eigenvalue ratios, contributing a significant step toward proving the Ashbaugh-Benguria conjecture.

Findings

Proved an isoperimetric inequality for lower order eigenvalues.

Strengthened the Ashbaugh-Benguria inequality.

Made progress toward the Ashbaugh-Benguria conjecture.

Abstract

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality conjectured by Payne-P\'olya-Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh-Benguria.