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

For a strictly convex set K subset of R^2 of class C^2 we consider its associated sub-Finsler K-perimeter vertical bar partial derivative |\partial E|_K in H^1 and the prescribed mean curvature functional vertical bar partial derivative |\partial E|_K - \int_E f associated to a continuous 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.