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

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 $$J_p(u,\mathrm{\Omega }):={\int }_{\mathrm{\Omega }}(\mathrm{\mid }{\mathrm{\nabla }}_{\mathbb{G}}u(x){\mathrm{\mid }}^p+{\chi }_{\{u>0\}}(x))dx$$ whenever $p>p^{\mathrm{\#}}:=\frac{2Q}{Q+2}$, are locally Lipschitz continuous with respect Carnot-Carathéodory distance on $\mathbb{G}$. This implies an Hölder continuous regularity from an Euclidean point of view.