Maximization of Neumann Eigenvalues

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