Regularity results for some 1-homogeneous functionals

Abstract

We consider local minimizers for a class of 1-homogeneous integral functionals defined on BV loc (Ω) , with Ω⊂ R 2 . Under general assumptions on the functional, we prove that the boundary of the subgraph of such minimizers is (locally) a lipschitz graph in a suitable direction. The proof of this statement relies on a regularity result holding for boundaries in R 2 which minimize an anisotropic perimeter. This result is applied to the boundary of sublevel sets of a minimizer u∈BV loc (Ω) . We also provide an example which shows that such regularity result is optimal.