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

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 $C^1$.

In the second version of the manuscript we add a new relation between the $K$-mean curvature $H_K$ and the sub-Riemannian mean curvature

References in corpus (2)

• Variational formulas for submanifolds of fixed degree
• Sub-Finsler Heisenberg Perimeter Measures