The prescribed mean curvature equation for $t$-graphs in the sub-Finsler Heisenberg group ${\mathbb{H}}^n$

Abstract

We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body $K_0\subseteq {\mathbb{R}}^{2n}$, for $t$-graphs on a bounded domain $\Omega$ in the Heisenberg group ${\mathbb{H}}^n$. When the prescribed datum $H$ is constant and strictly smaller that the Finsler mean curvature of $\partial \Omega$, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme.

We realized that our approximating Finsler equation had a degeneracy problem close to the singular set.Then, we regularized it perturbing it with a Euclidean curvature term, depending on a new parameter $\eta >0$, that anyway tends to disappear when $\eta \to 0$. In order to provide estimates for $\sup |u|$, we need to assume condition (1.4) in order to exploit Proposition 4.6