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

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 $\mathrm{\Omega }$ in the Heisenberg group ${\mathbb{H}}^n$. When the prescribed datum $H$ is constant and strictly smaller that the Finsler mean curvature of $\mathrm{\partial }\mathrm{\Omega }$, 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.