Two extremum problems for Neumann eigenvalues

Neumann eigenvalues being non-decreasing with respect to domain inclusion, it makes sense to study the two shape optimization problems $\min\{\mu_k(\Omega):\Omega \mbox{ convex},\Omega \subset D, \}$ (for a given box $D$) and $\max\{\mu_k(\Omega):\Omega \mbox{ convex},\omega \subset \Omega, \}$ (for a given obstacle $\omega$). In this paper, we study existence of a solution for these two problems in two dimensions and we give some qualitative properties. We also introduce the notion of {\it self-domains} that are domains solutions of these extremal problems for themselves and give examples of the disk and the square. A few numerical simulations are also presented.