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 $\min \{{\mu }_k(\mathrm{\Omega }):\mathrm{\Omega }convex,\mathrm{\Omega }\subset D,\}$ (for a given box $D$) and $\max \{{\mu }_k(\mathrm{\Omega }):\mathrm{\Omega }convex,\omega \subset \mathrm{\Omega },\}$ (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.