Regularity of Lipschitz boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group $\mathbb{H}^1$

TL;DR

This paper proves that sets with Lipschitz boundaries and prescribed sub-Finsler mean curvature in the Heisenberg group have characteristic curves of class C^2, establishing optimal regularity results under certain conditions.

Contribution

It establishes the optimal regularity of characteristic curves for critical sets with Lipschitz boundaries in the sub-Finsler Heisenberg setting, extending prior regularity results.

Findings

Characteristic curves are of class C^2.

Regularity is proven to be optimal.

Results hold for C^1 boundaries.

Abstract

For a strictly convex set $K\subset \mathbb{R}^2$ of class $C^2$ we consider its associated sub-Finsler $K$-perimeter $|\partial E|_K$ in $\mathbb{H}^1$ and the prescribed mean curvature functional $|\partial E|_K-\int_E f$ associated to a function $f$. Given a critical set for this functional with Euclidean Lipschitz and intrinsic regular boundary, we prove that their characteristic curves are of class $C^2$ and that this regularity is optimal. The result holds in particular when the boundary of $E$ is of class $C^1$.