Spectral optimization problems with internal constraint

We consider spectral optimization problems with internal inclusion constraints, of the formmin{λk(Ω):D⊂Ω⊂Rd,|Ω|=m}, where the set D is fixed, possibly unbounded, and λk is the k-th eigenvalue of the Dirichlet Laplacian on Ω. We analyze the existence of a solution and its qualitative properties, and rise some open questions.