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 $$ J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla_{\mathbb{G}} u(x)|^p+\chi_{{u>0}}(x)\Big)\,dx$$ whenever $p>p^#:=\frac{2Q}{Q+2}$, are locally Lipschitz continuous with respect Carnot-Carath\'eodory distance on $\mathbb{G}$. This implies an H\"older continuous regularity from an Euclidean point of view.