Regularity result for a shape optimization problem under perimeter constraint

We study the problem of optimizing the eigenvalues of the Dirichlet Laplace operator under perimeter constraint. We prove that optimal sets are analytic outside a closed singular set of dimension at most $d-8$ by writing a general optimality condition in the case the optimal eigenvalue is multiple. As a consequence we find that the optimal $k$-th eigenvalue is strictly smaller than the optimal $(k+1)$-th eigenvalue. We also provide an elliptic regularity result for sets with positive and bounded weak curvature.