Existence and regularity of solutions to optimal partition problems involving Laplacian eigenvalues

In this talk we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, with monotone cost functions. We prove the existence of an open optimal partition proving as well its regularity in the sense that the common boundary is, up to a residual set, locally a regular hypersurface. The proof involves a careful study of an associate Schrodinger system with competition terms, as well as several free boundary techniques (joint work with M. Ramos and S. Terracini).