Monotonicity formula and regularity for general free discontinuity problems

We give a general monotonicity formula for local minimizers of free discontinuity problems which have a critical deviation from minimality, of order d − 1. This result allows to prove partial regularity results (i.e. closure and density estimates for the jump set) for a large class of free discontinuity problems involving general energies associated to the jump set, as for example free boundary problems with Robin conditions. In particular, we give a short proof to the De Giorgi-Carriero-Leaci result for the Mumford-Shah functional.