*Preprint*

**Inserted:** 27 jul 2022

**Last Updated:** 27 jul 2022

**Year:** 2022

**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.

**Keywords:**
Heisenberg group, prescribed mean curvature, Constant Mean Curvature, Sub-Finsler Structure

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