Lipschitz regularity of the eigenfunctions on optimal domains

We study the optimal sets Ω∗ ⊂ R for spectral functionals F ( λ1(Ω), . . . , λp(Ω) ) , which are bi-Lipschitz with respect to each of the eigenvalues λ1(Ω), . . . , λp(Ω) of the Dirichlet Laplacian on Ω, a prototype being the problem min { λ1(Ω) + · · ·+ λp(Ω) : Ω ⊂ R, |Ω| = 1 } . We prove the Lipschitz regularity of the eigenfunctions u1, . . . , up of the Dirichlet Laplacian on the optimal set Ω∗ and, as a corollary, we deduce that Ω∗ is open. For functionals depending only on a generic subset of the spectrum, as for example λk(Ω) or λk1(Ω) + · · · + λkp(Ω) , our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.