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

TL;DR

This paper investigates the sub-Finsler prescribed mean curvature equation for $t$-graphs in the Heisenberg group, establishing existence results for solutions under specific conditions using an approximation scheme.

Contribution

It introduces a novel approach to solving the sub-Finsler CMC equation in the Heisenberg group via a Finsler approximation scheme, extending previous work to a broader geometric setting.

Findings

Existence of Lipschitz solutions for the Dirichlet problem under certain conditions.

Development of a Finsler approximation scheme for the sub-Finsler CMC equation.

Extension of prescribed mean curvature theory to sub-Finsler geometries.

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.