Maximization of the second non-trivial Neumann eigenvalue

TL;DR

This paper proves that the second non-trivial Neumann eigenvalue of the Laplace operator is maximized by two disjoint equal-measure balls, confirming the Pólya conjecture for this case and extending results to non-smooth domains.

Contribution

It establishes the maximizer for the second Neumann eigenvalue as two disjoint equal balls and confirms the Pólya conjecture for this eigenvalue, including in non-smooth settings.

Findings

Maximum of the second Neumann eigenvalue achieved by two disjoint equal balls

Pólya conjecture holds for the second Neumann eigenvalue

Relaxed inequality valid for non-smooth domains and densities

Abstract

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{\'o}lya 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.