# Monotonicity formula and regularity for general free discontinuity problems

created by bucur on 07 Mar 2012
modified on 18 Jul 2013

[BibTeX]

Accepted Paper

Inserted: 7 mar 2012
Last Updated: 18 jul 2013

Journal: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
Year: 2013

Abstract:

We give a general monotonicity formula for local minimizers of free discontinuity problems which have a {\it critical} deviation from minimality, of order $d-1$. This result allows to prove partial regularity results (i.e. closedness 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.