Lipschitz Regularity of the Eigenfunctions on Optimal Domains

We study the optimal sets for spectral functionals depending on the eigenvalues of the Dirichlet-Laplacian, which are bi-Lipschitz with respect to each variable, a prototype being the sum of the first p eigenvalues. We prove the Lipschitz continuity of the eigenfunctions on an optimal set and, as a corollary, we deduce that this optimal set is open. For functionals depending only on a generic subset of the spectrum, as for example the k-th eigenvalue, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.