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

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 $C^1$.


Introduction
The aim of this paper is to study the regularity of the characteristic curves of the boundary of a set with continuous prescribed mean curvature in the first Heisenberg group H 1 with a sub-Finsler structure. Such a structure is defined by means of an asymmetric left-invariant norm || · || K in H 1 associated to a convex set K ⊂ R 2 containing 0 in its interior, see [38]. We assume in this paper that K has C 2 boundary with positive geodesic curvature.
Following De Giorgi [14], the authors of [38] defined a notion of sub-Finsler Kperimeter, see also [18]. Given a measurable set E ⊂ H 1 and an open subset Ω ⊂ H 1 , it is said that E has locally finite K-perimeter in Ω if for any relatively compact open set V ⊂ Ω we have where H 1 0 (V ) is the space of horizontal vector fields of class C 1 with compact support in V , and ||U || K,∞ = sup p∈V ||U p || K . Both the divergence and the integral are computed with respect to a fixed left-invariant Riemannian metric g on H 1 . When S = ∂E ∩ Ω is a Euclidean Lipschitz surface the K-perimeter coincides with the area functional the horizontal projection of N to the horizontal distribution in H 1 and || · || K, * is the dual norm of || · || K .
We say that a set E with Euclidean Lipschitz boundary has prescribed K-mean curvature f ∈ C 0 (Ω) if, for any bounded open subset V ⊂ Ω, E is a critical point of the functional This notion extends the classical one in Euclidean space and the one introduced in [21] for the sub-Riemannian area. We refer the reader to the introduction of [21] for a brief historical account and references.
We say that a set E has constant prescribed K-mean curvature if there exists λ ∈ R such that E has prescribed K-mean curvature λ. In Proposition 2.2 we consider a set E with Euclidean Lipschitz boundary and positive K-perimeter. We show that if E is a critical point of the K-perimeter for variations preserving the volume up to first order then E has constant prescribed K-mean curvature on any open set Ω avoiding the singular set S 0 and where |∂E| K (Ω) > 0. This result can be applied to isoperimetric regions in H 1 with Euclidean Lipschitz boundary.
The main result of this paper is Theorem 3.1, where we prove that the boundary S of a set E with prescribed continuous K-mean curvature is foliated by horizontal characteristic curves of class C 2 in its regular part. The minimal assumptions we require for the boundary S of E are to be Euclidean Lipschitz and H-regular. The result holds in particular when the boundary of E is of class C 1 . As we point out in Remark 3.5, C 2 regularity is optimal since the Pansu-Wulff shapes obtained in [38] have prescribed constant mean curvature and their boundaries are foliated by characteristic curves with the same regularity as that of ∂K, that may be just C 2 . In the proof of the Theorem 3.1 we exploit the first variation formula of the area following the arguments developed in [20,21] and make use of the biLipschitz homeomorphism considered in [35]. One of the main differences in our setting is that the area functional strongly depends on the inverse π K of the Gauss map of ∂K. Therefore the first variation of the area depends on the derivative of the map that describes the boundary ∂K. In order to use the bootstrap regularity argument in [20,21] we need to invert this map on the boundary ∂K, that is possible since the geodesic curvature of ∂K is strictly positive, see Lemma 3.2. Moreover, the C 2 regularity of the characteristic curves implies that, on characteristic curves of a boundary with prescribed continuous K-mean curvature f , the ordinary differential equation is satisfied. In this equation ν h = N h /|N h | is the classical sub-Riemannian horizontal unit normal, Z is the unit characteristic vector field tangent to the characteristic curves and D the Levi-Civita connection associated to the left-invariant Riemannian metric g on H 1 . Equation (*) was proved to hold for C 2 surfaces in [38]. For regularity assumptions below H-regular and Euclidean Lipschitz, equation (*) holds in a suitable weak sense, a result proved in [1] for the sub-Riemannian area, when K coincides with the unit disk centered at 0. Moreover, in Proposition 4.2 we stress that equation (*) is equivalent to This manuscript is a natural continuation of the many recent papers concerning sub-Riemannian area minimizers [22,11,8,7,5,13,2,1,26,27,28,41,29,17,4,10,30,32,24,23,6]. The sub-Riemannian perimeter functional is a particular case of the sub-Finsler functionals considered in this paper where the convex set is the unit disk D centered at 0. In the pioneering paper [22] N. Garofalo and D.M. Nhieu showed the existence of sets of minimal perimeter in Carnot-Carathéodory spaces satisfying the doubling property and a Poincaré inequality. In [31] Leonardi and Rigot showed the existence of isoperimetric sets in Carnot groups. However the optimal regularity of the critical points of these variational problems involving the sub-Riemannian area is not completely understood. Indeed, even in the sub-Riemannian Heisenberg group H 1 there are several examples of non-smooth area minimizers: S. D. Pauls in [37] exhibited a solution of low regularity for the Plateau problem with smooth boundary datum; on the other hand in [8,39,34] the authors provided solutions of Bernstein's problem in H 1 that are only Euclidean Lipschitz.
In [36] P. Pansu conjectured that the boundaries of isoperimetric sets in H 1 are given by the surfaces now called Pansu's spheres, union of all sub-Riemannian geodesics of a fixed curvature joining two point in the same vertical line. This conjecture has been solved only assuming a priori some regularity of the minimizers of the area with constant prescribed mean curvature. In [41] the authors solved the conjecture assuming that the minimizers of the area are of class C 2 , using the description of the singular set, the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces developed in [7]. Hence the a priori regularity hypothesis are central to study the sub-Riemannian isoperimetric problem. Motivated by this issue, it was shown in [9] that a C 1 boundary of a set with continuous prescribed mean curvature is foliated by C 2 characteristic curves. Regularity results for Lipschitz viscosity solutions of the minimal surface equation were obtained in [4]. Furthermore, in [21] the authors generalized the previous result when the boundary S is immersed in a three-dimensional contact sub-Riemannian manifold. Finally M. Galli in [20] improved the result in [21] only assuming that the boundary S is Euclidean Lipschitz and H-regular in the sense of [19]. The Bernstein problem in H 1 with Euclidean Lipschitz regularity was treated by S. Nicolussi and F. Serra-Cassano [35]. Partial solutions of the sub-Riemannian isoperimetric problem have been obtained assuming Euclidean convexity [33], or symmetry properties [12,40,32,17]. An analogous sub-Finsler isoperimetric problem might be considered. Candidate solutions would be the Pansu-Wulff shapes considered in [38]. See [38,18] for partial results in the sub-Finsler isoperimetric problem and [42] for earlier work.
We have organized this paper into several sections. In Section 2 we introduce sub-Finsler norms in the first Heisenberg group H 1 and their associated sub-Finsler perimeter, the notion of H-regular surfaces, intrinsic Euclidean Lipschitz graphs and the definition of sets with prescribed mean curvature. Moreover, at the end of this section we prove Proposition 2.2. Section 3 is dedicated to the proof of the main Theorem 3.1, that ensures that the characteristic curves are C 2 . Finally in Section 4 we deal with the K-mean curvature equation, see Proposition 4.1 and Proposition 4.2.
A basis of left invariant vector fields is given by For p ∈ H 1 , the left translation by p is the diffeomorphism L p (q) = p * q. The horizontal distribution H is the planar distribution generated by X and Y , that coincides with the kernel of the (contact) one-form ω = dt − ydx + xdy. We shall consider on H 1 the left invariant Riemannian metric g = ·, · , so that {X, Y, T } is an orthonormal basis at every point, and let D be the Levi-Civita connection associated to the Riemannian metric g. The following relations can be easily computed Setting J(U ) = D U T for any vector field U in H 1 we get J(X) = Y , J(Y ) = −X and J(T ) = 0. Therefore −J 2 coincides with the identity when restricted to the horizontal distribution. The Riemannian volume of a set E is, up to a constant, the Haar measure of the group and is denoted by |E|. The integral of a function f with respect to the Riemannian measure by f dH 1 .

2.2.
The pseudo-hermitian connection. The pseudo-hermitian connection ∇ is the only affine connection satisfying the following properties: 1. ∇ is a metric connection, 2. Tor(U, V ) = 2 J(U ), V T for all vector fields U, V .
In the previous line the torsion tensor Tor(U, V ) is given by From the above definition and the Koszul formula it follows easily that ∇X = ∇Y = 0 and ∇J = 0. For a general discussion about the pseudo-hermitian connection see for instance [15, § 1.2]. Given a curve γ : I → H 1 we denote by ∇/ds the covariant derivatives induced by the pseudo-hermitian connection along γ.

2.3.
Sub-Finsler norms. Given a convex set K ⊂ R 2 with 0 ∈ int(K) an associated asymmetric norm || · || in R 2 , we define on H 1 a left-invariant norm || · || K on the horizontal distribution by means of the equality for any p ∈ H 1 . The dual norm is denoted by || · || K, * . If the boundary of K is of class C ℓ , ℓ 2, and the geodesic curvature of ∂K is strictly positive, we say that K is of class C ℓ + . When K is of class C 2 + , the outer Gauss map N K is a diffeomorphism from ∂K to S 1 and the map defined for non-vanishing horizontal vector fields U = f X + gY , satisfies 2.4. Sub-Finsler perimeter. Here we summarize some of the results contained in subsection 2.4 in [38]. Given a convex set K ⊂ R 2 with 0 ∈ int(K), the norm || · || K defines a perimeter functional: given a measurable set E ⊂ H 1 and an open subset Ω ⊂ H 1 , we say that E has locally finite K-perimeter in Ω if for any relatively compact open set V ⊂ Ω we have where H 1 0 (V ) is the space of horizontal vector fields of class C 1 with compact support in V , and ||U || K,∞ = sup p∈V ||U p || K . The integral is computed with respect to the Riemannian measure dH 1 of the left-invariant Riemannian metric g. When K = D, the closed unit disk centered at the origin of R 2 , the K-perimeter coincides with classical sub-Riemannian perimeter.
If K, K ′ are bounded convex bodies containing 0 in its interior, there exist constants α, β > 0 such that and it is not difficult to prove that As a consequence, E has locally finite K-perimeter if and only if it has locally finite K ′ -perimeter. In particular, any set with locally finite K-perimeter has locally finite sub-Riemannian perimeter.
Riesz Representation Theorem implies the existence of a |∂E| K measurable vector field ν K so that for any horizontal vector field U with compact support of class In addition, ν K satisfies |∂E| K -a.e. the equality ||ν K || K, * = 1, where || · || K, * is the dual norm of || · || K . Given two convex sets K, K ′ ⊂ R 2 containing 0 in their interiors, we have the following representation formula for the sub-Finsler perimeter measure |∂E| K and the vector field ν K Indeed, for the closed unit disk D ⊂ R 2 centered at 0 we know that in the Euclidean Here |∂E| D is the standard sub-Riemannian measure. Moreover, where dS is the standard Riemannian measure on S. Hence we get, for a set E with Euclidean Lipschitz boundary S where dS is the Riemannian measure on S, obtained from the area formula using a local Lipschitz parameterization of S, see Proposition 2.14 in [19]. It coincides with the 2-dimensional Hausdorff measure associated to the Riemannian distance induced by g. We stress that here N is the outer unit normal. This choice is important because of the lack of symmetry of || · || K and || · || K, * .
We say that S ⊂ H 1 is an H-regular surface if for each p ∈ H 1 there exist a neighborhood U and a function f ∈ C 1 H (U ) such that ∇ H f = 0 and S ∩U = {f = 0}. Then the continuous horizontal unit normal is given by Given an oriented Euclidean Lipschitz surface S immersed in H 1 , its unit normal N is defined H 2 -a.e. in S, where H 2 is the 2-dimensional Hausdorff measure associated to the Riemannian distance induced by g. In case S is the boundary of a set E ⊂ H 1 , we always choose the outer unit normal. We say that a point p belongs to the singular set S 0 of S if p ∈ S is a differentiable point and the tangent space T p S coincides with the horizontal distribution H p . Therefore the horizontal projection of the normal N h at singular points vanishes. In S S 0 the horizontal unit normal ν h is defined H 2 -a.e. by where N h is the horizontal projection of the normal N . The vector field Z is defined H 2 -a.e. on S S ′ 0 by Z = J(ν h ), and it is tangent to S and horizontal. H-regularity plays an important role in the regularity theory of sets of finite sub-Riemannian perimeter. In [19], B. Franchi, R. Serapioni and F. Serra-Cassano proved that the boundary of such a set is composed of H-regular surfaces and a singular set of small measure.
2.6. Sets with prescribed mean curvature. Consider an open set Ω ⊂ M , and an integrable function f ∈ L 1 loc (Ω). We say that a set of locally finite K-perimeter E ⊂ Ω has prescribed K-mean curvature f in Ω if, for any bounded open set B ⊂ Ω, E is a critical point of the functional If S = ∂E ∩ Ω is a Euclidean Lipschitz surface then S has prescribed K-mean curvature f if it is a critical point of the functional for any bounded open set B ⊂ Ω. If E has boundary S = ∂E ∩ Ω of class C 2 , standard arguments imply that E has prescribed K-mean curvature f in Ω if and only if and ν h is the outer horizontal unit normal, see [38]. Since by [38, Lemma 2.1] the Levi-Civita connection D and the pseudo-hermitian connection ∇ coincide for horizontal vector fields, we obtain that It is important to remark that the mean curvature H K strongly depends on the choice of ν h . When K is centrally symmetric, π K (−u) = −π K (u) and so the mean curvature changes its sign when we take −ν h instead of ν h . When K is not centrally symmetric, there is no relation between the mean curvatures associated to ν h and −ν h . A set E ⊂ H 1 with Euclidean Lischiptz boundary has locally finite K-perimeter: we know that it has locally bounded sub-Riemannian perimeter by Proposition 2.14 in [19] and we can apply the perimeter estimates in § 2.3. Letting H 2 be the Riemannian 2-dimensional Hausdorff measure, the Riemannian outer unit normal N is defined H 2 -a.e. in ∂E, and it can be proven that We say that a set E of locally finite K-perimeter in an open set Ω ⊂ H 1 has constant prescribed K-mean curvature if there exists λ ∈ R such that E has prescribed K-mean curvature λ. This means that E is a critical point of the functional E → |∂E| K (B) − λ|E ∩ B| for any bounded open set B ⊂ Ω.
Our next result implies that Euclidean Lipschitz isoperimetric boundaries (for the K-perimeter) have constant prescribed K-mean curvature. Proof. Since the K−perimeter of E in Ω is positive there exists a horizontal vector field U 0 with compact support in Ω so that E div U 0 dH 1 > 0. Let {ψ s } s∈R be the flow associated to U 0 and define Let W any vector field with compact support in Ω and associated flow {ϕ s } s∈R .
This means that the flow of W − λU 0 preserves the volume of E up to first order. By our assumption on E we get where Q is defined in (2.7). Now Lemma 2.3 implies Q(W ) = λQ(U 0 ) and, from the definition of H 0 , we get This implies that E is a critical point of the functional E → |∂E| K − H 0 |E| and so it has prescribed K-mean curvature equal to the constant H 0 . Writing W (s) = f (s)X σ(s) + g(s)Y σ(s) we have ||W (s)|| K, * = ||(f (s), g(s))||, where || · || is the planar asymmetric norm associated to the convex set K. We have for a constant C > 0 that only depends on K. The derivates of f and g can be estimated in terms of the covariant derivative D ds W = D ds (N s ) h along σ. Since D ds we get an uniform estimate on the derivatives of f and g independent of p. So the quotient (2.8) is uniformly bounded above by a constant independent of p.
To compute the derivative of A K (ϕ s (S)) at s = 0 we write The uniform estimate of the quotient (2.8) allows us to apply Lebesgue's dominated convergence theorem and Leibniz's rule to compute the derivative of A K (ϕ s (S)), given by Given a point p ∈ (S B) ∩ supp(U ), since supp(U ) ⊂ Ω and Ω ∩ S 0 = ∅ we get (N h ) p = 0 and so where e i is an orthonormal basis of T p (∂E), we get that Using Euclidean rotations about the vertical axis x = y = 0, that are isometries of the Riemannian metric g, we may assume that D is contained in the plane y = 0. Since the vector field Y is a unit normal to this plane, the intrinsic graph Gr(u) is given by {exp p (u(p)Y p ) : p ∈ D}, where exp is the exponential map of g, and can be parameterized by the map Φ u (x, t) = (x, u(x, t), t − xu(x, t)).
The tangent plane to any point in S = Gr(u) is generated by the vectors A unit normal to S is given by N =Ñ /|Ñ | wherẽ Therefore the horizontal projection of the unit normal to S is given by We also assume that S = Gr(u) is an H-regular surface, meaning thatÑ h and Z in (2.9) and are continuous. Hence also (u x + 2uu t ) is continuous.
In particular horizontal curves in Gr(u) satisfy the ordinary differential equation From (2.2), the sub-Finsler K-area for a Euclidean Lipschitz surface S is where N h K, * = N h , π(N h ) with π = (π 1 , π 2 ) = π K and dS is the Riemannian area measure. Therefore when we consider the intrinsic graph S = Gr(u) we obtain Observe that the K-perimeter of a set was defined in terms of the outer unit normal. Hence we are assuming that S is the boundary of the epigraph of u.
Given v ∈ C ∞ 0 (D), a straightforward computation shows that and F is the function Since (u x + 2uu t ) is continuous and π is at least C 1 the function M is continuous.

Characteristic curves are C 2
Here we prove our main result, that characteristic curves in an intrinsic Euclidean Lipschitz H-regular surface with continuous prescribed K-mean curvature are of class C 2 . The reader is referred to Theorem 4.1 in [21] for a proof of the the sub-Riemannian case. The proof of Theorem 3.1 depends on Lemmas 3.2 and 3.3.
Theorem 3.1. Let K be a C 2 + convex set in R 2 with 0 ∈ int(K) and || · || K the associated left-invariant norm in H 1 . Let Ω ⊂ H 1 be an open set and E ⊂ Ω a set of prescribed Kmean curvature f ∈ C 0 (Ω) with an Euclidean Lipschitz and H-regular boundary S. Then the characteristic curves of S ∩ Ω are of class C 2 .
Proof. By the Implicit Function Theorem for H-regular surfaces, see Theorem 6.5 in [19], given a point p ∈ S, after a rotation about the vertical axis, there exists an open neighborhood B ⊂ H 1 of p such that B ∩ S is the intrinsic graph Gr(u) of a function u : D → R, where D is a domain in the vertical plane y = 0, and B ∩ E is the epigraph of u. The function u is Euclidean Lipschitz by our assumption. .Since Gr(u) has prescribed continuous mean curvature f , from equation (2.11) we get The function M is defined in (2.12). By Remark 4.3 in [21] implies that (3.1) holds for each v ∈ C 0 0 (D) for which v x + 2uv t exists and is continuous.
Let Γ(s) be a characteristic horizontal curve passing through p whose velocity is the vector fieldZ defined in (2.9), that only depends on u x + 2uu t . Since S is H-regular the function u x + 2uu t is continuous and Γ(s) is of class C 1 . Let us consider the function F defined in (2.13) and define Hence F (g(s)) = M (s). The function F is C 1 for any convex set K of class C 2 + and, from Lemma 3.2, we obtain that F ′ (x) > 0 for each x ∈ R. Therefore F −1 is also C 1 and g(s) = F −1 (M (s)). Thanks to Lemma 3.3 we obtain that M is C 1 along Γ and we conclude that also g is C 1 along Γ. SoZ is C 1 and the curve Γ is C 2 .
Lemma 3.2. Let K ⊂ R 2 be a convex body of class C 2 + such that 0 ∈ int(K). Then the function F defined in (2.13) is C 1 and F ′ (x) > 0 for each x ∈ R.
Proof. Parameterize the lower part of the boundary of the convex body K by a function φ defined on a closed interval I ⊂ R. The function φ is of class C 2 inI and the graph becomes vertical at the endpoints of I. As K is of class C 2 + we have φ ′′ (x) > 0 for each x ∈ R. Take x ∈ R, then we have where N K is the outer unit normal to ∂K. Let ϕ(x) ∈I be the point where (ϕ(x), φ(ϕ(x))) = π(x, −1).
Therefore, if we consider the normal N K of the previous equality we obtain Hence φ ′ (ϕ(x)) = x and so ϕ is the inverse of φ ′ , that is invertible since φ ′′ (x) > 0 for each x ∈ R. Notice that for each x ∈ R. is satisfied along any characteristic curve γ.
Proof. Let Γ(s) be a characteristic curve passing through p in Gr(u). Let γ(s) be the projection of Γ(s) onto the xt-plane, and (a, b) ∈ D the projection of p to the xtplane. We parameterize γ by s → (s, t(s)). By Remark 2.5 the curve s → (s, t(s)) satisfies the ordinary differential equation t ′ = 2u. For ε small enough, Picard-Lindelöf's theorem implies the existence of r > 0 and a solution t ε :]a − r, a + r[→ R of the Cauchy problem We define γ ε (s) = (s, t ε (s)) so that γ 0 = γ. Here we exploit an argument similar to the one developed in [35]. By Theorem 2.8 in [44] we gain that t ε is Lipschitz with respect to ε with Lipschitz constant less than or equal to e Lr . Fix s ∈]a − r, a + r[, the inverse of the function ε → t ε (s) is given byχ t (−s) = χ t (−s) − b where χ t is the unique solution of the following Cauchy problem Again by Theorem 2.8 in [44] we have thatχ t is Lipschitz continuous with respect to t, thus the function ε → t ε is a locally biLipschitz homeomorphisms.
We consider the following Lipschitz coordinates (3.4) G(ξ, ε) = (ξ, t ε (ξ)) = (s, t) around the characteristic curve passing through (a, b). Notice that, by the uniqueness result for (3.2), G is injective. Given (s, t) in the image of G using the inverse functionχ t defined in (3.3) we find ε such that t ε (s) = t, therefore G is surjective. By the Invariance of Domain Theorem [3], is a homeomorphism. The Jacobian of G is defined by almost everywhere in ε. Any function ϕ defined on D can be considered as a function of the variables (ξ, ε) by makingφ(ξ, ε) = ϕ(ξ, t ε (ξ)). Since the function G is C 1 with respect to ξ we have Furthermore, by [ when h goes to 0. Puttingṽh/(t ε+h − t ε ) in (3.6) instead ofṽ we gain Using Lebesgue's dominated convergence theorem and letting h → 0 we have Let η : R → R be a positive function compactly supported in I and for ρ > 0 we consider the family η ρ (x) = ρ −1 η(x/ρ), that weakly converge to the Dirac delta distribution. Putting the test functions η ρ (ε)ψ(ξ) in (3.7) and letting ρ → 0 we get for each ψ ∈ C ∞ 0 ((a − r, a + r)). Since u x + 2uu t is continuous, M in (2.12) is continuous, thus alsoM . Hence thanks to Lemma 3.4 we conclude that M is C 1 along γ, thus by Remark 2.5 is also C 1 along Γ.
Remark 3.5. Let K be a convex body of class C 2 + such that 0 ∈ K. Following [38] we consider a clockwise-oriented P -periodic parameterization γ : R → R 2 of ∂K. For a fixed v ∈ R we take the translated curve s → γ(s + v) − γ(v) = (x(s), y(s)) and we consider its horizontal lifting Γ v (s) to H 1 starting at (0, 0, 0) ∈ H 1 for s = 0, given by The Pansu-Wulff shape associated to K is defined by In [38,Theorem 3.14] it is shown that the horizontal liftings Γ v , for each v ∈ [0, P ), are solutions for H K = 1, therefore S K has constant prescribed K-mean curvature equal to 1. Since the curves Γ v have the same regularity as ∂K, the C 2 regularity result for horizontal curves obtained in Theorem 3.1 is optimal.
Corollary 3.6. Let K be a C 2 + convex set in R 2 with 0 ∈ int(K) and || · || K the associated left-invariant norm in H 1 . Let Ω ⊂ H 1 be an open set and E ⊂ Ω a set of prescribed K-mean curvature f ∈ C 0 (Ω) with C 1 boundary S. Then the characteristic curves in S S 0 are of class C 2 .
Proof. Since S is of class C 1 , in the regular part S S 0 the horizontal normal ν h is a nowhere-vanishing continuous vector fields, thus S S 0 is an H-regular surface. In particular a C 1 surface is Lipschitz, thus S S 0 verifies the hypotheses of Theorem 3.1 and the characteristic curves in S S 0 are of class C 2 .
Since the Jacobian J G defined in (3.5) is equal to ∂t ε /∂ε > 0 the change of variables G(ξ, ε) is invertible. Hence the rest of the proof of Lemma 3.3 goes in the same way as before.

The sub-Finsler mean curvature equation
Given an Euclidean Lipschitz boundary S whose characteristic curves in S S 0 are of class C 2 , for each point p ∈ S S 0 we can define the K-mean curvature H K of S by (4.1) where ν h is the outer horizontal unit normal to S. This definition was given in [38] for surfaces of class C 2 . Proof. By the Implicit Function Theorem for H-regular surfaces, Theorem 6.5 in [19], given a point p ∈ S, after a rotation about the t-axis, there exists an open neighborhood B ⊂ H 1 of p such that B ∩ S is the intrinsic graph of a function u : D → R where D is a domain in the vertical plane y = 0. The function u is Euclidean Lipschitz by our assumption. We set B ∩ S = Gr(u). We assume that E is locally the epigraph of u. Let Γ(s) be a characteristic curve passing through p in Gr(u) and γ(s) its projection on the xt-plane. The characteristic vector Z defined in (2.9) is given by Since S is H-regular, Z and the horizontal unit normal are continuous vector fields. By Lemma 3.3 we have that M = F (u x +2uu t ) defined in (2.12) satisfies the differential equation along the characteristic curves. Therefore we obtain (u x + 2uu y )(γ(s)) , As in proof of Lemma 3.2, we parametrize the lower part of the boundary of the convex body K by a function φ defined on a closed interval I ⊂ R. Again by Lemma 3.2 we have where ϕ is the inverse function of φ ′ . Furthermore the K-mean curvature defined (4.1) is equivalent to Hence we obtain H K = d ds M (γ(s)) and so H K (p) = f (p) for each p ∈ S S 0 . The following result allows us to express the K-mean curvature H K in terms of the sub-Riemannian mean curvature H D .
Proposition 4.2. Let K ⊂ R 2 be a convex body of class C 2 + such that 0 ∈ int(K) and π K = N −1 K . Let κ be the strictly positive curvature of the boundary ∂K.
Let Ω ⊂ H 1 be an open set and E ⊂ Ω a set of prescribed K-mean curvature f ∈ C 0 (Ω) with Euclidean Lipschitz and H-regular boundary S. Then, we have H D (p) = κ(π K (ν h ))f (p) for each p ∈ S S 0 , where H D (p) = D Z ν h , Z is the sub-Riemannian mean curvature, ν h be the horizontal unit normal at p to S S 0 and Z = J(ν h ) be the characteristic vector field.
Setting ν h = aX + bY we obtain

Finally, by Lemma 4.3 we get
Hence we obtain D Z ν h , Z = κ(π K (ν h )), since D Z ν h = ∇ Z ν h . Lemma 4.3. Let K ⊂ R 2 be a convex body of class C 2 + such that 0 ∈ int(K) and N K be the Gauss map of ∂K. Let κ be the strictly positive curvature of the boundary ∂K. Let S be an H-regular surface with horizontal unit normal ν h and characteristic vector field Z = J(ν h ). Then we have where (dπ) ν h is the differential of π K = N −1 K . Proof. Let α(t) = (x(t), y(t)) be an arc-length parametrization of ∂K such thaṫ x 2 (t) +ẏ 2 (t) = 1. Let ν h = aX + bY be the horizontal unit normal to S, with a = cos(θ) and b = sin(θ) and θ ∈ (− π 2 , π 2 ). Notice that θ = arctan( b a ). Then we have π K (a, b) = N −1 K ((a, b)).