The Ashbaugh--Benguria reciprocal-gap conjecture for Dirichlet eigenvalues

Abstract

We prove the Ashbaugh--Benguria reciprocal-gap conjecture for the Dirichlet Laplacian in every dimension $N\ge 2$. Specifically, if $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ is a bounded domain and $$0<\hat{I}{»}_1(\hat{I}\text{c◯})<\hat{I}{»}_2(\hat{I}\text{c◯})\le \hat{I}{»}_3(\hat{I}\text{c◯})\le \cdots$$ are its Dirichlet eigenvalues, then $$\sum _{i=1}^N\frac{\hat{I}{»}_1(\hat{I}\text{c◯})}{\hat{I}{»}_{i+1}(\hat{I}\text{c◯})-\hat{I}{»}_1(\hat{I}\text{c◯})}\ge \frac{N}{j_{N/2,1}^2/j_{N/2-1,1}^2-1},$$ where $j_{\hat{I}¼,1}$ denotes the first positive zero of the Bessel function $J_{\hat{I}}¼$ of the first kind of order $\hat{I}¼$. We also characterize the equality case: equality holds precisely when $\hat{I}\text{c◯}$ agrees with a Euclidean ball up to a set of Sobolev $H^1$-capacity zero. In particular, among bounded Lipschitz domains, equality holds if and only if $\hat{I}\text{c◯}$ is a Euclidean ball.

31 pages. Comments and suggestions are welcome!