Regularity of the optimal sets for the second Dirichlet eigenvalue

This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set center dot minimizes the functional FA(center dot) = A2(center dot) + AI center dot I, among all subsets of a smooth bounded open set D C Rd, where A2(center dot) is the second eigenvalue of the Dirichlet Laplacian on center dot and A > 0 is a fixed constant, then center dot is equivalent to the union of two disjoint open sets center dot+ and center dot_, which are C1;-regular up to a (possibly empty) closed set of Hausdorff dimension at most d - 5, contained in the one-phase free boundaries D pi 8 center dot+ \ 8 center dot_and D pi 8 center dot_ \ 8 center dot+.