Lipschitz regularity for almost minimizers of a one-phase $p$-Bernoulli-type functional in Carnot Groups of step two

Abstract

In this paper, in a Carnot group $\mathbb{G}$ of step $2$ and homogeneous dimension $Q$, we prove that almost minimizers of the (horizontal) one-phase $p$-Bernoulli-type functional $$J_p(u,\Omega ):={\int }_{\Omega }(|{\nabla }_{\mathbb{G}}u(x){|}^p+{\chi }_{\{u>0\}}(x))dx$$ whenever $p>p^{\#}:=\frac{2Q}{Q+2}$, are locally Lipschitz continuous with respect Carnot-Carathéodory distance on $\mathbb{G}$. This implies an Hölder continuous regularity from an Euclidean point of view.

60 pages. arXiv admin note: substantial text overlap with arXiv:2311.02975