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.


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