Multiphase free discontinuity problems: monotonicity formula and regularity results

The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding state functions. Our main results state that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. The proof relies on a multiphase monotonicity formula and on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.