Maximization of Neumann eigenvalues

TL;DR

This paper investigates the maximization of Neumann eigenvalues for domains and densities, proving existence of optimal densities, characterizing solutions in one dimension, and providing numerical approximations in two dimensions.

Contribution

It introduces a relaxed formulation for the eigenvalue maximization problem, proves existence of optimal densities, and offers numerical methods for approximations in two dimensions.

Findings

Optimal densities exist for the relaxed problem.

In one dimension, maximizers are unions of equal segments.

Numerical approximations for optimal densities in 2D are provided.

Abstract

This paper is motivated by the maximization of the $k$-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of ${\mathbb R}^N$ with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in ${\mathbb R}^N$ with prescribed mass and prove the existence of an optimal density. For $k=1,2$ the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For $k \ge 3$ this question remains open, except in one dimension of the space where we prove that the maximal densities correspond to a union of $k$ equal segments. This result provides sharp upper bounds for Sturm-Liouville eigenvalues and proves the validity of the P\'olya conjecture in the class of densities in ${\mathbb R}$. Based on the relaxed formulation, we provide numerical approximations of optimal densities for $k=1, \dots, 8$ in ${\mathbb R}^2$.