Regularity of the optimal sets for some spectral functionals

In this paper we study the regularity of the optimal sets for the shape optimization problem min{λ1(Ω)+⋯+λk(Ω) : Ω⊂Rd open, |Ω|=1}, where λ1(·) , … , λk(·) denote the eigenvalues of the Dirichlet Laplacian and | · | the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer Ωk∗ is composed of a relatively open regular part which is locally a graph of a C∞ 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∗∈ [ 5 , 7 ] 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.