×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI
References:
[1] Alt, H. W.; Caffarelli, L. A., Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325, 105-144, (1981) · Zbl 0449.35105
[2] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000 · Zbl 0957.49001
[3] Bucur, D.; Giacomini, A., A variational approach to the isoperimetric inequality for the Robin eigenvalue problem. arch, Ration. Mech. Anal., 198, 927-961, (2010) · Zbl 1228.49049
[4] Dal Maso, G.; Morel, J.-M.; Solimini, S., A variational method in image segmentation: existence and approximation results, Acta Math., 168, 89-151, (1992) · Zbl 0772.49006
[5] De Giorgi, E.; Carriero, M.; Leaci, A., Existence theorem for a minimum problem with free discontinuity set. arch, Ration. Mech. Anal., 108, 195-218, (1989) · Zbl 0682.49002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.