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{µk(Ω) : Ω convex, Ω ⊂ D, } (for a given box D) and max{µk(Ω) : Ω convex, ω ⊂ Ω, } (for a given obstacle ω). 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 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.