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

TL;DR

This paper proves that almost minimizers of a one-phase p-Bernoulli functional in Carnot groups of step two are locally Lipschitz continuous with respect to the Carnot-Carathéodory distance, ensuring Euclidean H"older regularity.

Contribution

It establishes Lipschitz regularity for almost minimizers of a p-Bernoulli functional in Carnot groups of step two, extending regularity results to a sub-Riemannian setting.

Findings

Almost minimizers are locally Lipschitz continuous in Carnot groups.

Regularity holds for p > 2Q/(Q+2), where Q is the homogeneous dimension.

Ensures Euclidean H"older continuity from Lipschitz regularity.

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.