Regularity of Lipschitz boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group H1

For a strictly convex set K ⊂ R 2 of class C 2 we consider its associated sub-Finsler K-perimeter | ∂ E | K in H 1 and the prescribed mean curvature functional | ∂ E | K − ∫ 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 .