Regularity of the optimal sets for some spectral functionals

In this paper we study the regularity of the optimal sets for the sum of the first k eigenvalues of the Dirichlet Laplacian among sets of finite measure. We prove that the topological boundary of a minimizer is composed of a relatively open regular part which is locally a graph of a C^{1,s} function and a closed singular part, which is empty if d<d*, contains at most a finite number of isolated points if d=d* and has Hausdorff dimension smaller than (d- d*) if d> d*, where the natural number d* is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.