Regularity of Lipschitz boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group \(\mathbb{H}^1\) (Q2232191)

For a strictly convex set \(K\subset \mathbb{R}^2\) of class \(C^2\), the authors consider its associated sub-Finsler \(K\)-perimeter \(|\partial E|_K\) in \(H^1\) and the prescribed mean curvature functional \(|\partial E|_K-\int_E f\) associated to a continuous function \(f\). Given a critical set for this functional with Euclidean Lipschitz and intrinsic regular boundary, the authors 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 \(C^1\).