Regularity of Lipschitz boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group H^1

Abstract

Given a strictly convex set $K\subset\rr^2$ of class $C^2$ we consider its associated sub-Finsler $K$-perimeter $|\ptl E|_K$ in $\hh^1$ and the prescribed mean curvature functional $|\ptl E|_K-\int_E f$ associated to a function $f$. Given a critical set for this functional, we prove that where the boundary of $E$ is Euclidean lipschitz and intrinsic regular, the characteristic curves are of class $C^2$. Moreover, this regularity is optimal. The result holds in particular when the boundary of $E$ is of class $C^1$