Extremality Conditions and Regularity of Solutions to Optimal Partition Problems Involving Laplacian Eigenvalues

Let Omega subset of R(N)be an open bounded domain and m is an element of N. Given k(1),...k(m) is an element of N, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following form inf {F(lambda(k1)(omega 1),...,lambda k(m) (omega(m))) : (omega 1,...omega(m)) is an element of P-m(Omega)}, where lambda(ki) (omega i) denotes the k (i)-th eigenvalue of (-Delta, H-0(1)(omega(i))) counting multiplicities, and P-m(Omega) is the set of all open partitions of , Omega namely P-m(Omega) = {omega(1),...,.m(m)) : omega(i) subset of Omega open, omega(i) boolean AND omega(j) = empty set for all i not equal j}. While the existence of a quasi-open optimal partition follows from a general result by Bucur, Buttazzo and Henrot [Adv Math Sci Appl 8(2):571-579, 1998], the aim of this paper is to associate with such minimal partitions and their eigenfunctions some suitable extremality conditions and to exploit them, proving as well the Lipschitz continuity of some eigenfunctions, and the regularity of the partition in the sense that the free boundary is, up to a residual set, locally a hypersurface. This last result extends the ones in the paper by Caffarelli and Lin [J Sci Comput 31(1-2):5-18, 2007] to the case of higher eigenvalues.