  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 ${\mathbb
R}^2$.

Analysis of PDEs;Optimization and Control