A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues

Abstract

We prove the Ashbaugh--Benguria conjecture for bounded domains with smooth boundary in ${\mathbb{R}}^m$. More precisely, among all smooth bounded domains of fixed volume, the ball minimizes the sum of the reciprocals of the first $m$ nonzero Neumann eigenvalues. Equality is attained precisely by balls.

15 pages. Comments and suggestions are welcome!