Maximization of Neumann Eigenvalues

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 Pólya 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 $$\mathbb {R}^2$$ .