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Monotonicity formula and regularity for general free discontinuity problems. (English) Zbl 1283.49056
Summary: 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 us to prove partial regularity results (that is 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.

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 49N60 Regularity of solutions in optimal control
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##### References:
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