A STUDY OF COMPARISON, EXISTENCE AND REGULARITY OF VISCOSITY AND WEAK SOLUTIONS FOR QUASILINEAR EQUATIONS IN THE HEISENBERG GROUP

. In this manuscript, we are interested in the study of existence, uniqueness and comparison of viscosity and weak solutions for quasilinear equa- tions in the Heisenberg group. In particular, we highlight the limitation of applying the Euclidean theory of viscosity solutions to get comparison of so- lutions of sub-elliptic equations in the Heisenberg group. Moreover, we will be concerned with the equivalence of diﬀerent notions of weak solutions under appropriate assumptions for the operators under analysis. We end the paper with an application to a Rad´o property.

1. Introduction and main results. The main concern of this paper is to analyse weak and viscosity solutions of quasilinear equations in divergence form in the Heisenberg group: − div 0 (A(p, ∇ 0 u)) = f (p, u), (1) where A = A(p, ξ) : Ω × R 2 → R 2 , div 0 and ∇ 0 are the intrinsic divergence and gradient operators in the Heisenberg group. We refer the reader to Section 2 for definitions and notation. Observe that under the symmetric condition: to study viscosity solutions of (1) is equivalent to consider viscosity solutions of: where ∇ 2, * 0 u is the symmetrized Hessian of u in the Heisenberg group and where div 0 (A(·, ∇ 0 u)) is the horizontal divergence of p → A(p, ξ) with respect to p at − tr M (p, ∇ 1 u(p))∇ 2, * 0 u = F (p, u, ∇ 1 u), where M is a matrix field. As a motivation for considering differential equations with intrinsic structures in the Heisenberg group, let us consider the sub-elliptic Laplacian operator: where X 1 and X 2 are given as in (3). In terms of Euclidean derivatives, the subelliptic Laplace equation may be written as: Observe that the matrix:  is not uniformly elliptic for all (x, y, z). Hence the results from the Euclidean theory of viscosity solutions of uniformly elliptic operators are not applied directly (see the fundamental works [11,5] and the references therein).
With respect to distributional or weak solutions, we may write the sub-Laplace equation in terms of the Euclidean divergence operator as: The matrix ∂ ξ A is not uniformly elliptic for all p, and hence the comparison and uniqueness results for weak solutions in the Euclidean framework for uniformly elliptic operators are not applied (see the seminal work [36] and the reference there in).
Therefore, in order to obtain satisfactory existence, uniqueness and regularity results, we apply an intrinsic theory derived from the differential and metric structures of the Heisenberg group as a Carnot group.
The Heisenberg group is a sub-Riemannian group. Recently, different applications of the geometry of sub-Riemannian manifolds have been considered. As an example, the roto-translation group serves as a model for the structure of the visual cortex. Indeed, in [8], the authors consider the sub-Riemannian cortical model of image completion which gives rise to a diffusion driven motion by mean curvature. The roto-translation group appears as a natural model for the process of lifting of a plane image to a regular surface. We refer the reader to [9,10,8,6] and the references therein for details and more applications of sub-Riemannian structures.
The theory of viscosity solutions has been applied to differential equations in the Heisenberg group since the 80's. However, one of the main limitations to apply directly the Euclidean theory is the loss of uniform ellipticity. Moreover, the lack of a Crandall-Ishii Lemma intrinsic to sub-Riemannian structures is also a limitation.
To the best of our knowledge, the available Crandall-Ishii Lemma in sub-Riemannian groups and, in particular, Carnot groups, depends on the corresponding Euclidean result via a translation of sub-elliptic jets into Euclidean jets. We refer the reader to [4] and [2] for a derivation of the Crandall-Ishii Lemma in the Heisenberg group and Carnot groups. A related issue is the development of comparison results for viscosity solutions. We refer the reader to [28] for a survey of comparison results. Towards getting comparison principles, it is usual to assume that a given second order operator F = F(p, u, ∇ 1 u, ∇ 2, * 0 u) does not depend on the spatial variable p, or the first-derivative variable ∇ 1 u, and also assume that F has bounded away from zero derivatives ∂F/∂u (strict monotonicity in u). For examples of these results we quote [28], [31,Proposition 4.1], [30, Theorem 2.1] (here the authors remove the strictly increasing assumptions but they assume a sign condition). At this point we would like to quote the works [1] and [29] where the authors propose various form of partial nondegeneracy to weaken the uniform ellipticity assumption and apply their results to some sub-elliptic second order equations.
In our work, any viscosity solution will be an Euclidean solution (see Definition 2.3). This allows to apply comparison principles from the Euclidean theory to the sub-elliptic equations. However, as we will see in Section 3 and specifically in Remark 1, standard assumptions in the Euclidean context used to get comparison, such as (3.14) in [11], (6) and (10) in [1], or the Lipschitz hypothesis on G in [1, Theorem 3.2], are not satisfied in general by the operators treated here. To overcome this scenario, we employ intrinsic techniques from the sub-Riemannian framework.
On the other hand, the theory of weak solutions has also been developed in sub-Riemannian groups. For completeness, we provide, in the appendix, a full proof of existence of weak solutions for the equations treated here and in John domains. Most of the literature on existence is developed in non-characteristic domains [46,44,43,17,45,47,42,39,41,40]. However, following [34, Theorem 3.2.1], the class of non-characteristic domains is a subclass of John domains, and hence our results are slightly more general.
We make now some comments on the relation between weak and viscosity solutions. In the Euclidean setting, the relation is well understood from the works [27,22,25] and [23]. In the Heisenberg scenario, the basic reference is [3].
Having taking into account the above comments, we now summarize our main contributions and the organization of the manuscript.
• Comparison result for semicontinuous viscosity solutions of (2). In order to obtain this result, we do not assume that the differential operator has partial derivatives with respect to u bounded away from zero. Also, we will not assume any sign condition. We refer the reader to Section 3. • Equivalence of weak and viscosity solutions. Based on the above results and regularity theory, we derive the equivalence of the notions of solutions considered in this work. We refer to Section 4.1. • Radó type property. As an application of the above results, we provide a Radó type theorem for horizontal critical points. See Section 5. • Existence of weak (or distributional) solutions to (1). This will achieve by standard methods by introducing obstacle problems. We prove existence in John domains. See the Appendix.
2. Preliminaries. Basic notation. We shall use the following standard notation in the work. The Euclidean interior product is denoted by ·, · . If γ, β ∈ R n , the vector γ ⊕ β is defined as the vector in R 2n whose first n entries are those of γ, followed by the components of β. The Euclidean norm of a vector in R n will be denoted by · . For matrices, by A < B and A ≤ B we mean that B − A is positive definite and positive semi-definite, respectively. Also, we denote by S n (R) the set of n × n symmetric and positive semi-definite matrices with real coefficients. The trace of a matrix A is denoted by tr(A). By Ω we shall always denote an open and bounded domain in R 3 where ∂Ω has Lebesgue measure zero. We also introduce the following functional spaces: U SC(Ω) = {u : Ω → R : u is upper semicontinuous in Ω}.
2.1. Heisenberg group. We denote by H the first-order Heisenberg group whose underlying manifold is R 3 , and whose group operation is given by: The group H is a Lie group with Lie algebra h generated by the left-invariant basis: where p = (x, y, z) ∈ R 3 . We equip h with an interior product (a Riemann structure) so that the frame (3) is orthonormal. We recall that the exponential mapping is a global diffeomorphism that takes the vector xX 1 + yX 2 + zX 3 in the Lie algebra h to the point (x, y, z) in the Lie group H. This allows us to identify vectors in h with points in H. For t > 0, we denote by δ t the dilation of the group given as δ t (p) = (tx, ty, t 2 z) The two dimensional linear space generated by the vectors X 1 (p) and X 2 (p) is denoted by H 0,p . The distribution H 0 is called the horizontal distribution.
2.1.1. Metric structure and Calculus in H. The metric structure on H is given by the Carnot-Carathéodory distance (CC distance in brief) which is defined as following: an absolutely continuous curve γ ∈ W 1,2 ((0, 1), R 3 ) is said to be horizontal if there is a control v ∈ L 2 ((0, 1), R 3 ) such that: for a. e. t in (0, 1). For any p, q ∈ H, the CC distance between p and q is defined as: where the infimum is taken over all horizontal curves γ, with associated control v, so that γ(0) = p and γ(1) = q.
For computational purposes, we shall also use a smooth gauge out of the diagonal defined as follows: This gauge is comparable to the CC distance. The corresponding distance is: Also, for any p ∈ H and δ > 0, we write: to denote the ball in the Heisenberg group with center at p and radius δ.
Given a smooth function u in R 3 , and a multi-index I = (i 1 , i 2 , i 3 ), the derivative X I u is defined by: In general, the class of C k H functions is larger than the class of Euclidean C k (Ω) (see [16,Remark 5.9] for examples).
For an Euclidean smooth function u : R 3 → R, Taylor expansion around 0 implies: for p → 0, and where ∇u and ∇ 2 u stand for the Euclidean gradient and Hessian of u, respectively. The horizontal Taylor expansion of u at 0 is: where the gradient of u with respect to the frame (3) is: and the symmetrized horizontal second derivative matrix, denoted by ∇ 2 0 u, is given by: In addition, the horizontal gradient of u is : Finally, for a vector field V = (V 1 , V 2 ), its horizontal divergence operator is defined by

Horizontal Sobolev spaces.
In order to treat variational problem in H, we consider horizontal Sobolev spaces: equipped with the norm: Also, we define the Sobolev space S 1,t 0 (Ω) as the closure of C ∞ 0 (Ω) with respect to the above norm in S 1,t (Ω). Usually, we shall write · t for · L t (Ω) .
The following is the Sobolev-Poincaré inequality in the Heisenberg group in John domains (see [18, Corollary 1.6, Theorem 1.30] and also [12] for global results). We recall that a bounded open set Ω in a metric space (M, d) is a John domain if there exist a point p 0 ∈ Ω and C > 0 such that for every p ∈ Ω there exists a continuous rectifiable curve parametrized by arclength γ : [0, T ] → Ω, T ≥ 0, such that γ(0) = p, γ(T ) = p 0 and: for all t ∈ [0, T ]. We refer the reader to [6] and the references therein for further properties of John domains. As an illustration, we point out that in view of [33, Theorem 1.3], any C 1,1 domain in the Heisenberg group is a John domain.

Convolution and approximation to identity in H.
Let ρ ∈ C ∞ 0 (Ω) such that: For any h > 0, we let: ).
For f ∈ L 1 (Ω), we let: Then the following happens (see for instance [15,16] and [34]): We have: For a more complete discussion of the Heisenberg group and more general Carnot groups, we refer the reader to [6,15] and [28]. [11] in the Euclidean setting and [4] in the Heisenberg group scenario. Let u ∈ U SC(Ω). The second-order superjet of u at p is defined as follows:

Viscosity solutions. The basic reference in what follows is
It is well-known (see [4] and [2, Lemma 2.2]) that subelliptic jets may be seen as appropriate derivatives of test functions touching the given function by above or below. More precisely, if u is upper semicontinuous, let us consider: for all q close to p and similarly define K −,2 (v, p) for test functions touching the lower semicontinuous function v from below around p. Hence, by the results in [4] and [2], it follows that: and: Finally, we shall also consider the theoretic closure of the sets defined above. We define J 2,+ (u, p) as the set of (η, X ) in R 3 × S 2 (R) so that there exists a sequence (p n , u(p n ), η n , X n ) converging to (p, u(p), η, X ) satisfying (η n , X n ) ∈ J 2,+ (u, p n ) for all n. In a similar way, we define J 2,− (v, p).

2.3.
A sub-elliptic Maximum Principle. The following result is a standard tool towards proving comparison principle for viscosity solutions in sub-Riemannian structures. For a complete discussion see [2].
Theorem 3.1. Assume hypotheses (H1)-(H3). Let u ∈ U SC(Ω) be a subsolution of (11) and v ∈ LSC(Ω) be a supersolution of (11) so that u ≤ v on ∂Ω. Then: Proof. Reasoning by contradiction, suppose that: Now we apply the subelliptic maximum principle to u − v. In order to do so, we shall check that if p τ , q τ ∈ Ω are so that: then we indeed have that p τ and q τ belong to Ω. To prove the statement, observe that in view of the compactness of Ω, there exist p 0 , q 0 ∈ Ω so that, up to a subsequence that we do not re-label: p τ → p 0 and q τ → q 0 as τ → ∞.

Examples.
• Hamilton-Jacobi equations. The comparison result may be applied to Hamilton-Jacobi equations perturbed by a viscosity term: where ν is a positive constant and ∆ 0 denotes the sub-elliptic Laplacian operator in H. • Degenerate p-Laplacian type equations. Consider the p-Laplacian operator for p > 2: where: for η = 0 and we let M (0) = 0. Observe that we may write: Hence, in view of Theorem 3.1, the comparison principle holds.
Remark 1. The following assumption, together with strict monotonicity in u, is used in [11,Theorem 3.3] to derive a comparison principle for fully non-linear operators F (p, u, ∇u, ∇ 2 u) = 0 in the Euclidean framework: there is a modulus of for all p, q ∈ Ω, r ∈ R and X, Y in the space of (3 × 3)-symmetric matrices so that the next structural condition holds: where I is the (3 × 3)-identity matrix. We claim that, as a function depending on Euclidean quantities, the operator (11) does not meet, in general, the estimate (18). Indeed, denote by σ(p) and σ * (p) the following matrices: and let: where: and: ω(η) = η α (1, 1) .
Consider the matrices: where: By [21,Theorem 7.7.6], the matrices X τ and Y τ satisfy (19). Next, we define: and Observe that: as τ → +∞, for all γ > 0. The left hand side of (18) is given by: Note that From (25) and the definitions of p τ and q τ , we obtain: Note that: where:

4.
Weak solutions for equations in divergence form. Throughout this section, we shall study second-order equations in divergence form as follows: where Ω is a John domain, A : Ω × R 2 → R 2 and f : Ω × R → R. We refer the reader to the Appendix for the proofs of existence and uniqueness of weak solutions of equations like (30). We highlight the parallelism between the assumptions on A and f in (30) to obtain the mentioned results to the corresponding well-known conditions in the Euclidean scenario. Moreover, even do one can not apply directly the Euclidean results in the Heisenberg framework, the use of slightly modifications of Euclidean techniques works in the sub-Riemannian setting.
The notion of weak solutions is provided in the next definition.
Definition 4.1. A function u ∈ S 1,t (Ω) is a weak solution to equation (30) if: for all v ∈ S 1,t 0 (Ω). 4.1. Equivalence of viscosity and weak solutions. In this section, we shall prove the equivalence of the notions of viscosity and weak solutions to equations in divergence form. We first state the following regularity result for viscosity solutions.
where A is smooth and satisfies the polynomial growth rate (Hw1). Moreover, we assume that the matrix:    is positive semi-definite, that the following symmetric property holds: and finally that f satisfies (Hw2). Then u ∈ S 1,t loc (Ω). Proof. For each > 0, consider the inf-convolution in the Heisenberg group: Hence, there exists r( ) → 0 such that: By the results in [37], the sequence {u } is increasing, (Euclidean) semi-concave and converges uniformly to u in Ω. Hence: is concave. As in [32, Lemma 2.1], we derive that u is a viscosity supersolution of: f (q, r).
Moreover since A is smooth and u is almost everywhere twice differentiable, we have: a. e. in Ω (indeed, by Aleksandrov's Theorem, (∇ 0 u (p), ∇ 2, * 0 u (p)) ∈ J 2,− (u , p) for a. e. p in Ω ). Let ϕ ∈ C ∞ 0 (Ω) and take small enough so that K := supp ϕ ⊂ Ω . Hence, (32) implies: Choose {ψ j } as a sequence of mollifications of ψ. Then, ψ j is smooth and concave in Ω, ψ j → ψ in S 1,q loc (Ω) for all q ∈ [1, ∞), ∇ 0 ψ j L ∞ (K) is uniformly bounded (since the Euclidean gradients are locally uniformly bounded [20]), and by [13, pag. 242]: We let: Integration by parts and Dominated Convergence Theorem give: In the sequel we shall prove that: From ∇ 2 u ,j 1 I, the decomposition (see [2, Lemma 3.1]): where M = 0 in the Heisenberg scenario, and by the uniform (in j) local boundedness of ∇ 0 u ,j , we obtain up to a multiplicative constant: By assumption, the matrix:    is positive semidefinite. Hence by the local boundedness of ∇ 0 u ,j (with a bound independent of j) and recalling (31): for some c ∈ R and where div 0 (A(·, ∇ 0 u ,j )) is the horizontal divergence of p → A(p, ξ) with respect to p at ξ = ∇ 0 u ,j . By Fatou's Lemma: Hence, combining (35) and the pointwise convergence of −div 0 A(p, ∇ 0 u ,j ) to −div 0 (A(p, ∇ 0 u )) (which follows from the pointwise convergence of ∇ 2, * 0 ψ j to ∇ 2, * 0 ψ, (34) and the smoothness of A) we derive (36). So far, we have obtained: for any ϕ ∈ C ∞ 0 (Ω). As in [32, Lemma 2.3], we have the following Caccioppoli's estimate: for all ζ ∈ C ∞ 0 (Ω) and where K := supp ζ. Since u is increasing in and converges uniformly to u in Ω, we have: for all < 0 . Hence, by (37), the sequence {∇ 0 u } is uniformly bounded in L t on compact sets. The previous fact together with the uniform convergence of u to u in Ω and the boundedness of u in Ω imply that {u } is a bounded sequence in S 1,t (K) for each compact K ⊂ Ω. Hence, up to a subsequence, {u } converges weakly to v ∈ S 1,t loc (Ω) in S 1,t (Ω) on compact set. Since the embedding S 1,t in L t is compact for all t (see [34,Corollary 4.1.13]), we derive that u = v a. e. and hence u ∈ S 1,t loc (Ω).
In the remaining of this section, we shall prove the equivalence of the notions of viscosity and weak solutions. To do so, we shall assume the following: (H) The solution from Theorem 6.1 is continuous up to the boundary. Moreover, we assume that A and f satisfy the structure assumptions (Hw1)-(Hw5), where in (Hw4) we require that A is strictly monotone , that A has continuous partial derivatives satisfying (31), that: verify (H1)-(H3) from Section 3 and that M ≥ 0.
If u ∈ C(Ω) is a weak solution to (38), then u is a viscosity solution as well. Conversely, if u ∈ C(Ω) is a viscosity solution to (38), then u is a weak solution.