H-CONVERGENCE FOR EQUATIONS DEPENDING ON MONOTONE OPERATORS IN CARNOT GROUPS

. The aim of this paper is to present some results, re-lated to the convergence of solutions of Dirichlet problems for sequences of monotone operators, that generalize well-known results of Murat-Tartar, De Arcangelis-Serra Cassano and Baldi-Franchi-Tchou-Tesi to the sub-Riemannian framework of Carnot groups.


Introduction
The term H-convergence was coined by François Murat and Luc Tartar in the 70's and it is addressed to differential operators. Tartar [31,32] reported applications of the H-convergence to many different frameworks covering, among other things, the case involving monotone operators (see Definition 2.5) of the form where A is a Carathéodory function satisfying uniformly ellipticity and continuous conditions, in the setting of Hilbert spaces. See [32,Chapter 11] for details and [29,Chapter 2.3] for a general discussion about this topic.
The class of linear operators considered in [1,2,21] is made of matrix-valued measurable functions, that is, operators of the form where A is a (m × m)-matrix-valued measurable function and ∇ G and div G are, respectively, the intrinsic gradient and the intrinsic divergence (see Definition 2.2 for details). We remind that a definition of intrinsic curl, curl G , can be found in [2,Section 5]. The key tool in [1,2,21] was an extension to Carnot groups of Murat and Tartar' Div-curl lemma [32,Lemma 7.2], namely [2,Theorem 5.1].
Motivated by the previous results, in this paper we look for extensions to Carnot groups, in the general setting of Banach spaces, of the original result of Murat and Tartar [32,Theorem 11.2] and we provide a H-compactness theorem for (nonlinear) monotone operators, working with operators of the form for a given A ∈ M(α, β; Ω). The class M(α, β; Ω) is defined as follows.
We define M(α, β; Ω) the class of Carathéodory functions A : Ω × R m → R m such that for every ξ, η ∈ R m and a.e. x ∈ Ω.
The main result of this article is the following theorem. The structure of this article is the following one: in Section 2, we give the definitions of Carnot groups and the functional setting required throughout the paper. In Section 3, we study the main properties of the class of monotone operators we are interested in and, in Section 4, after defining a proper notion of H-convergence (see Definition 4.1), we prove Theorem 1.2.

Preliminaries
2.1. Carnot groups. Let us recall just few definitions concerning Carnot groups. We refer the interested reader to [8].
Definition 2.1. A Carnot group G of step k is a connected, simply connected and nilpotent Lie group, whose Lie algebra g admits a step k stratification, that is, there exist V 1 , . . . , V k linear subspaces of g, usually called layers, such that Typical examples of Carnot groups are the Euclidean space, the only Abelian Carnot group of step 1 and the Heisenberg group, a Carnot group of step 2.
It is clear from Definition 2.1, that the first layer V 1 plays the role of generator of the algebra g, by commutation. For this reason, we refer to V 1 as the horizontal layer, while the other layers V i , 1 < i ≤ k, are called vertical layers.
We can define two different dimensions on G: the topological dimension, which is its dimension as Lie group, i.e., where m i := dim(V i ) for any i, and the homogeneous dimension, defined by Let us notice that, when G is not R n , the homogeneous dimension of G is always bigger than the topological one. In the sequel, we denote m := m 1 , for simplicity.

2.2.
Functional setting. Through the paper, (X 1 , . . . , X m ) denotes a basis of the horizontal layer V 1 , |Ω| the Lebesgue measure of any set Ω ⊂ G and, if ξ, η ∈ R m , we denote by |ξ| and ξ, η the Euclidean norm and the scalar product, respectively. The subbundle of the tangent bundle T G, which is spanned by the vector fields X 1 , . . . , X m , is called the horizontal bundle and is denoted by HG. Each section Φ of HG is called horizontal sections and is identified with canonical coordinates with respect to the moving frame, by a function We define the intrinsic gradient of u and the intrinsic divergence of Φ, respectively, as defines an equivalent norm on W 1,p G,0 (Ω) (see [24,Section 2] and, for more details, [23,26]). Finally, we denote L p (Ω, HG) the set of measurable sections Φ ∈ L p (Ω) m .
is independent of the choice of the basis (X 1 , . . . , X m ).

Monotone operators.
Let us recall the definition of monotone operators. See, for instance, [22] for more details. . Let V be a reflexive Banach space, V * its dual space and let A : V → V * be a mapping. We say that • A is strictly-monotone, if it is monotone and Operator (1.2) is strictly-monotone, in sense of Definition 1.1. The following result will be crucial later on. . Let X be a Banach space, let K be a closed, nonempty and convex subset of X and let A : K → X * be monotone, coercive and continuous on finite dimensional subspaces of K. Then, there exists

Existence results for equations driven by monotone operators
Let Ω ⊂ G be open, connected and bounded, 2 Remark 3.2. By standard approximation arguments, (3.2) holds for every ϕ ∈ V .
Proof of Proposition 3.1. Let f ∈ V * and let B : V → V * be defined by Let us show that B is strictly-monotone, coercive and continuous on any finite dimensional subspace of V . To obtain the weak continuity on finite dimensional Banach spaces, it is enough to prove that B is strongly continuous in the whole Let (u n ) n be strongly convergent to u in V . By Hölder's inequality, we have Notice that (A(·, ∇ G u n )) n strongly converges to A(·, ∇ G u) in L p (Ω, HG) since, by Definition 1.1 (iii) and Hölder's inequality and, choosing v 1 := u + ϕ and v 2 := u − ϕ, we obtain Then, u satisfies (3.2). Finally, if u, v ∈ V are weak solutions of (3.1) then, by Remark 3.2 (choosing ϕ = u − v ∈ V ) and by Definition 1.1 (ii) that is, the solution of (3.1) is unique.
As a direct consequence of Proposition 3.1, A is continuous and invertible in V . We conclude this section providing useful estimates.
Proof. Fix u, v ∈ V and f, g ∈ V * such that Notice that (a) directly follows from Definition 1.1 (ii). Moreover, recalling that and applying (a), with u = A −1 (f ) and v = A −1 (g), we obtain Finally, by Definition 1.1 (iii), i.e., Then, (c) follows by the definition of · V * .

H-convergence and Div-curl lemma
The following statement of H-convergence is a natural adaptation of the original definition of Murat and Tartar in our context. (Ω) and let u n , u ∞ ∈ W 1,p G,0 (Ω) be, respectively, weak solutions of We say that (A n ) n H-converges to A eff if, as n → ∞,
Before proving Theorem 1.2, we need two preliminary results.
Then, there exist a continuous and invertible operator A ∞ : W 1,p G,0 (Ω) → W −1,p G (Ω) and a subsequence (A m ) m of (A n ) n , such that Proof. For the sake of simplicity, let us denote V = W 1,p G,0 (Ω) and V * = W −1,p G (Ω). We divide the proof of the lemma into three steps.
Step 1. Let X be a fixed countable and dense subset of V * . We show that, for any fixed f ∈ X, the sequence of solutions of weakly converges, up to subsequences, in V . Moreover, we provide an upper-bound for its limit, in terms of f . Fix f ∈ X. Then, by Proposition 3.1, there exists u n ∈ V , weak solution of (4.1), that is, u n = A −1 n (f ) for any n ∈ N. Moreover, by Proposition 3.3 (b) i.e., (u n ) n is bounded in V, reflexive Banach space and, therefore, there exist u ∞ (f ) ∈ V and (u m ) m , diagonal subsequence of (u n ) n , such that Notice that, by the lower semicontinuity of the norm and by Proposition 3.3(a), Step 2. Define S : X → V as Let us show that S can be extended to the whole space V * . Since X is countable and dense in V * , it is sufficient to show that S is continuous in (X, · V * ). Fix f, g ∈ X. Then, by Proposition 3.3(b), and, passing to the limit, by the lower semicontinuity of the norm, we obtain For the sake of completeness, the extension of S to V * \ X is defined as for any f ∈ V * and (f n ) n ⊂ X such that f n → f in V * .
Step 3. Let us finally prove that, as a consequence of Theorem 2.6, S is invertible in V * . To this aim, we show that S is monotone and coercive in V * . Fix f, g ∈ V * . Then, by Proposition 3.3(a),

Moreover,
Passing to the limit, We obtain the conclusion, defining A ∞ := S −1 : V → V * . (Ω) → L p (Ω, HG) such that, up to subsequences HG). Proof. Let X be a countable and dense subspace of L p (Ω, HG) and let f ∈ X. Then, by Definition 1.1(iii) and Hölder's inequality Therefore, (A n (·, ∇ G A −1 n (f ))) n is bounded in L p (Ω, HG) and, by the countability of X, there exists a diagonal subsequence of (A n (·, ∇ G A −1 n (f ))) n weakly convergent to M = M (f ) in L p (Ω, HG).
We define M : X → L p (Ω, HG) as If f, g ∈ X, then, by Proposition 3.3, Therefore, by the lower semicontinuity of the norm, M can be extended to the whole space V * , and the thesis follows.
We recall now the statement of Div-curl lemma, in the framework of Carnot groups, given by Baldi, Franchi, Tchou and Tesi [2]. Let Ω ⊂ G be an open set and let p, q > 1 be a Hölder's conjugate pair. Moreover, following the notations of [2], if σ ∈ I 2 0 , let a(σ) > 1 and b > 1 be such that Finally, let E n , E ∈ L p loc (Ω, HG) and D n , D ∈ L q loc (Ω, HG) be such that (i) E n → E weakly in L p loc (Ω, HG); (ii) D n → D weakly in L q loc (Ω, HG); (iii) the components of (curl G E n ) n of weight σ are bounded in L (Ω) and for any u ∈ W 1,p G,0 (Ω) such that and ω open set such that ω ⊂ Ω. For any v ∈ W 1,p G,0 (Ω), weak solution of (3.1), we define the Carathéodory function A eff : Ω × R m → R m as
To conclude the proof of the theorem, we show that C(u ∞ ) = A eff (x, ∇ G u ∞ ) a.e. x ∈ Ω . (4.11) Let u ∞ ∈ W 1,p G,0 (Ω) be the (unique) weak solution of (4.2), let (u m ) m be weakly convergent to u ∞ in W 1,p G,0 (Ω) and define D m 2 = A m (x, ∇ G u m ) and E m 2 = ∇ G u m . Then, by Theorem 4.4, for any ϕ ∈ D(ω 1 ) and, following the same techniques of the first part of the proof, Finally, varying ϕ ∈ D(ω 1 ) and ξ 1 ∈ R m , we obtain (4.11).