Regularity of all minimizers of a class of spectral partition problems

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs [Please download the PDF to view the mathematical expression] where ([[omega].sub.1], ..., [[omega].sub.m]) are the sets of the partition and [[lambda].sub.j]([[omega].sub.i]) is the j-th Laplace eigenvalue of the set [w.sub.i] with zero Dirichlet boundary conditions. Keywords: elliptic competitive systems; optimal partition problems; Laplacian eigenvalues; segregation phenomena; extremality conditions; regularity of free boundary problems; blowup techniques