Maximization of the second non-trivial Neumann eigenvalue

Abstract

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of $\mathbb{R}^N$ with prescribed measure $m$ attains its maximum on the union of two disjoint balls of measure $m/2$. As a consequence, the Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.