Spectral optimization problems with internal constraint

International audienceWe consider spectral optimization problems with internal inclusion constraints, of the form min {lambda(k)(Omega): D subset of Omega subset of R-d, vertical bar Omega vertical bar = m}, where the set D is fixed, possibly unbounded, and lambda(k) is the k-th eigenvalue of the Dirichlet Laplacian on Omega. We analyze the existence of a solution and its qualitative properties, and rise some open questions