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

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