Regularity for almost minimizers of a one-phase Bernoulli-type functional in Carnot Groups of step two

Abstract

We prove that nonnegative almost minimizers of the horizontal Bernoulli-type functional $$J(u,\hat{I}\text{c◯}):={\int }_{\hat{I}}\text{c◯}(|{\nabla }_{\mathbb{G}}u(x){|}^2+\ddot{I}{‡}_{\{u>0\}}(x))dx$$ are Lipschitz continuous in the intrinsic sense.

36 pages