$G$-convergence of elliptic and parabolic operators depending on vector fields

We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness theory, in a setting in which the existence of affine functions is not always guaranteed, due to the nature of the family of vector fields.


Introduction
The asymptotic behaviour, as h → ∞, from the point of view of Gconvergence of a sequence of equations like (1) E h u = f in bounded domains Ω of R n has been widely studied, in particular when E h is an elliptic or parabolic operator in divergence form, i.e., where ∇ denotes the Euclidean gradient, x ∈ Ω and t ∈ (0, T ), with T > 0. G-convergence was introduced for linear elliptic operators E h (for symmetric matrices a h with positive eigenvalues) by Spagnolo in a series of papers at the end of '60s (see [25][26][27]) and studied later by the same author in many other papers. Surely worthy to be recalled are [6], in the case of homogenization, where for the first time an explicit expression of the limit operator is given, [3] in which also a comparison between elliptic and parabolic G-convergence is made, and [24] among the first papers about linear parabolic operators. Regarding linear operators E h with non-symmetric matrices a h , or E h nonlinear, this study involves a further difficulty: the lack of uniqueness of a representative for the limit operator, see e.g. [23]. This problem is bypassed in the '70s by Murat and Tartar, who extended the notion of G-convergence, and call it H-convergence, to general linear and monotone elliptic operators E h , see e.g. [29,30]. We want to give a brief, and certainly not exhaustive, account regarding the case of nonlinear operators: we refer to [2] for elliptic operators and to [28] for parabolic ones. Finally, we mention [33], where linear operator also of degree greater than two are considered, and the book [21] where nonlinear both elliptic and parabolic operators, homogenization and random operators are presented.
The aim of this paper is to extend the classical results for sequences of monotone operators to the more general setting of operators modeled on vector fields (i.e. replacing the Euclidean gradient ∇ in (1) with a family of vector fields X), continuing along the path traced in the recent papers [16][17][18]. Before entering into the details, we want to recall that the literature concerning homogenization, G-convergence and integral representation of abstract functionals depending on vector fields is pretty vast, ranging from more rigid structures like Carnot groups, see e.g. [1, 10-12, 16, 19] and the references therein, up to the more general setting considered in [7,8,17,18]. The setting we will take into account embraces the huge family of vector fields satisfying the Hörmander condition. We stress however that our results still apply to families of vector fields not satisfying the Hörmander condition, provided that the following hypotheses are fulfilled: we consider a bounded domain Ω of R n and a family of m ≤ n vector fields X = (X 1 , . . . , X m ), defined and Lipschitz continuous on an open neighborhood Ω 0 of Ω, such that the following conditions hold (H1) let d : R n × R n → [0, ∞] be the so-called Carnot-Carathéodory distance function induced by X, see e.g. [14]. Then, d(x, y) < ∞ for any x, y ∈ Ω 0 , so that d is a standard distance in Ω 0 , and d is continuous with respect to the usual topology of R n ; (H2) for any compact set K ⊂ Ω 0 there exist r K , C K > 0, depending on K, such that for any x ∈ K and r < r K . B d (x, r) denotes the open metric ball with respect to d, that is, B d (x, r) := {y ∈ Ω 0 : d(x, y) < r}; (H3) there exist geometric constants c, C > 0 such that for every ball B = B d (x, r) with cB := B d (x, cr) ⊆ Ω 0 , for every u ∈ Lip(cB) and for every (LIC) the n-dimensional vectors X 1 (x), . . . , X m (x) are linearly independent for any x ∈ Ω \ Z X , where Z X is a Lebesgue measure zero subset of Ω.
As already mentioned, the main goal of the paper is to extend to the monotone, and possibly parabolic, case the result contained in [18], where the authors dealt only with the elliptic and linear case. We recall that G-convergence when the equations (1) represent the Euler-Lagrange equations of a family of functionals, may be connected with Γ-convergence, see e.g. [4,5]. This is the approach used in the linear elliptic case in [18] and it cannot be followed in the case of parabolic problems driven by monotone operators, because the corresponding PDEs cannot be seen as Euler-Lagrange equations of appropriate functionals. To mark once more the difference with respect to [18] and to better stress the novelty of the paper, let us review the classical approach introduced by De Giorgi and Spagnolo [6,[24][25][26][27], which is based on the compensated compactness and the existence of affine functions.

Compensated compactness: for any pair
. Roughly speaking, the compensated compactness theory ensures that the Euclidean inner product remains continuous with respect to weak convergence, even thought neither sequence is assumed to be relatively compact in L 2 (Ω; R n ). Here the lack of compactness is compensated by the boundedness of some combinations of partial derivatives. Murat and Tartar in [29,30] extended the compensated compactness with a result involving the notion of curl (notice that curl∇v = 0). This theory is known as div-curl lemma and its extension to the setting of Sobolev spaces depending on vector fields is a pretty delicate task, mainly because a proper notion of intrinsic curl, curl X , ensuring that curl X Xv = 0, is not always available. We recall here that a possible notion of curl in the setting of Carnot groups has been given in [11], using the intrinsic complex of differential forms of Rumin. In Theorem 3.1 and Theorem 4.2, we show that in fact the classical technique due to Spagnolo is sufficient to get compensated compactness even without a proper generalization of the div-curl lemma, and in particular of the curl. The compensated compactness is usually used to prove the closure of the class of operators in divergence form, meaning that if A h := −div(a h (x, ∇)) G-converges to A , then the limit operator A is in divergence form, i.e., A = −div(a(x, ∇)) , for some function a. In the Euclidean setting, the definition of a goes through the existence of suitable affine functions.
Existence of affine functions: for any fixed ξ ∈ R n , there exists a unique smooth enough function u (at least C 2 ) such that ∇u = ξ. In the purely Euclidean framework, affine functions exist and can be represented by the Euclidean scalar product of the fixed vector ξ with x ∈ R n . Another example in which one can prove the existence of such functions is provided by the Heisenberg group. On the other hand, if we consider the case of the Grushin plane, we easily get an example of a family of vector fields satisfying our assumptions but for which affine functions may not exist. To be more precise, let n = 2 and consider the Grushin gradient X = (X 1 , X 2 ) Then, for any fixed ξ = (ξ 1 , ξ 2 ) ∈ R 2 with ξ 2 = 0, one can easily show that there exists no function u ∈ C 2 (Ω X ) such that Xu(x) = ξ. Despite the possible non-existence of X-affine functions, we are however able to prove G-compactness by using the classical Euclidean affine functions and exploiting either the linear independence condition (LIC) on the X-gradient and the algebraic structure of the family X. See the proofs of Theorem 3.3 and Theorem 4.4 for the details. We finally stress that our proof drastically simplifies the one in [18] for merely linear elliptic operators.
The paper is organized as follows: in Section 2, we provide the functional setting of Sobolev spaces depending on vector fields and we state the main properties of the classes of monotone operators we are interested in. In Section 3, we state and prove the main result in the elliptic framework and, in Section 4, in the parabolic setting. Finally, in Lemma 4.5, we show that the parabolic limit and the elliptic one coincide when the parabolic sequence of monotone operators is independent of time.

Notations and Preliminaries
2.1. Functional setting. Let X(x) := (X 1 (x), . . . , X m (x)) be a given family of first order linear differential operators with Lipschitz coefficients on a bounded domain Ω of R n , that is, with c ji (x) ∈ Lip(Ω) for j = 1, . . . , m, i = 1, . . . , n. In the following, we will refer to X as X-gradient. As usual, we identify each X j with the vector field (c j1 (x), . . . , c jn (x)) ∈ Lip(Ω; R n ) and we call for any ϕ ∈ C ∞ c (Ω) and j = 1, . . . , m, then (2) becomes and its domain is the set W 1,p X,0 (Ω) defined as follows.
Definition 2.3. We define the anisotropic Sobolev spaces in the sense of Folland and Stein [9] as . . , m} . These spaces, endowed with the norm . Since vector fields X j have Lipschitz continuous coefficients, then, by definition, Here W 1,p (Ω) denotes the classical Sobolev space, or, equivalently, the space W 1,p X (Ω) associated to the family X = (∂ 1 , . . . , ∂ n ). Inclusion (3) can be strict, in particular when the number of vector fields is strictly less than the dimension of the space, and turns out to be continuous. By the Lipschitz regularity assumption, the validity of the classical result 'H = W ' of Meyers and Serrin [20] is still guaranteed as proved, independently, in [13] and [15].
As a consequence of conditions (H1) -(H3), it has been proved in [14,18] the validity of a Rellich-type theorem and a Poincaré inequality in W 1,p X,0 (Ω). . Under the hypotheses of the previous theorem and also assuming that Ω is connected, there exists a positive constant c p,Ω , depending only on p and Ω, such that where V ′ is the dual space of V , and we identify the dual space of H, H ′ , with H itself, in such a way that where the embeddings are dense and continuous. In a similar way, we define and, by (4), we have V ⊂ H ⊂ V ′ with continuous and dense embeddings. We endow V and H, respectively, with the following norms: and V ′ with the natural norm [31,Chapter 23] or [22,Chapter 3]).

Notice that the integral in the previous definition is the Bochner integral and that
The natural space of solutions to parabolic PDEs is the Banach space u ′ is to be intended as the generalized derivative of u (see Definition 2.6). In the following proposition we recall some results regarding the space W, see, e.g., Proposition 1.2, Corollary 1.1 and Proposition 1.3 in [22].
Proposition 2.7. The space W continuously embeds in C 0 ([0, T ]; H) and, for every u, v ∈ W and for every t, s ∈ [0, T ], the following generalized integration by parts formula holds: Moreover, the space W compactly embeds in L p (0, T ; H).

Now consider the operators
Definition 2.8. Let g ∈ V ′ . We define u ∈ V a solution to the problem Moreover, if f ∈ V ′ and ϕ ∈ H, we define u ∈ W a solution to the problem We conclude this part by recalling a result useful for the sequel. . . ϑ m be non-negative numbers such that ϑ 1 + · · · + ϑ m ≤ 1, and assume that a.e. in U .

2.3.
Position of the problems.
Remark 2.18. Consider the sequence of problems (E h ) and denote by u h (g) their solutions. Suppose that A h G-converges to A. Then, it holds that u h (g h ) → u(g) strongly in L p (Ω) , a h (·, Xu h (g h )) → a(·, Xu(g)) weakly in L p ′ (Ω; R m ) , provided that (g h ) h ⊂ V ′ strongly converges to g in V ′ . The analogous holds for problems (P h ) where one considers two sequences of data (f h ) h and (ϕ h ) h , provided that f h → f strongly in V ′ and ϕ h → ϕ strongly in H.

Elliptic G-convergence
In this section we state and prove a G-compactness result in the elliptic case, namely Theorem 3.3. In all this section we always assume that Ω is a bounded domain of R n , that 2 ≤ p < ∞ and that X satisfies conditions (H1), (H2), (H3) and (LIC), given in the Introduction.
and assume that Then, Proof. Fix ϕ ∈ C ∞ c (Ω) and consider the quantities div X M h , ϕ V ′ ×V and div X M h , v h ϕ V ′ ×V . By (9) and (10), we get div X M = g in D ′ (Ω) . Moreover, Consider the right hand side terms. By assumptions we have that Then, the thesis follows by (11). such that, up to subsequences, the following convergences hold u h (g) → B(g) strongly in L p (Ω) , a h (·, Xu h (g)) → M (g) weakly in L p ′ (Ω; R m ) .

Moreover, B is invertible and
Proof. By Theorem 2.14, the sequence (u h (g)) h is bounded in V and then, up to a subsequence, (u h (g)) h weakly converges in V . By Theorem 2.4, and up to a further subsequence, it strongly converges in L p (Ω). Fix g ∈ V ′ and define B(g) := lim h→∞ u h (g) .
Since, by Definition 2.10 (iii), that is, the sequence (a h (·, Xu h (g))) h turns out to be bounded in L p ′ (Ω; R m ). Therefore, for every g ∈ V ′ and up to a subsequence, there exists a limit M (g) ∈ L p ′ (Ω; R m ) such that (a h (·, Xu h (g))) h weakly converges to M (g) in L p ′ (Ω; R m ). Moreover, since that is, (12) holds. Fix f, g ∈ V ′ . By Definition 2.10, it holds that a.e. x ∈ Ω and, letting h → ∞, we get 1/p R m , by means of Lemma 2.9 and Theorem 3.1. Thus, (13) follows by Cauchy-Schwarz inequality.
We conclude by showing the invertibility of B. Fix f, g ∈ V ′ and assume that B(f ) = B(g). By (13), it holds that M (f ) = M (g) and, by (12), we conclude that f = g. Moreover, B(V ′ ) is dense in V . In fact, if g o ∈ V ′ satisfies g o , B(g) V ′ ×V = 0 for every g ∈ V ′ then, in particular, g o , B(g o ) V ′ ×V = 0 and, by Definition 2.10 (ii) and the lower semicontinuity of the norm · V , we get   (7).
where A : V → V ′ is the operator in X-divergence form associated to a and defined by A(u) = div X (a(·, Xu)) for any u ∈ V .
Proof. Let us divide the proof of Theorem 3.3 in three steps.
Step 1. Construction of the limit operator and useful estimates. Let B and M be the operators introduced in Lemma 3.2 and define By (12), div X M (g) = g = A(B(g)) for every g ∈ V ′ .
As a consequence of Theorem 3.1 and Lemma 3.2, it easy to show that N satisfies for every v, w ∈ V . (To get an idea of how to prove (14) and (15), we remind the interested reader to the equivalent estimates in the parabolic case, namely (19) and (20), provided in the proof of Theorem 4.4.) Step 2. Let us show that A in an operator in X-divergence form, by providing the existence of a function a(x, ·) : R m → R m satisfying N (u) = a(x, Xu) for any u ∈ V a.e. x ∈ Ω .
Fix an open set ω such that ω ⊂ Ω, let φ ∈ C 1 0 (Ω) be such that φ ≡ 1 in ω and, for any ξ ∈ R n , define where (ξ, x) R n = ξ 1 x 1 + . . . + ξ n x n . Then (17) Xw where C(x) is the coefficient matrix of the X-gradient (see Section 2). Notice that, if ξ,ξ ∈ R n are such that ξ −ξ ∈ Ker C(x), then Indeed, by (15), it holds that and, by (17), the right hand side, and then the left hand side, is zero.
By (18),ã is well-defined and, by condition (LIC), we can defineã(x, ·) in the whole space R m . Consider now a sequence of open sets (ω j ) j such that ω j ⊂ Ω for every j ∈ N and such that ∪ ∞ j=1 ω j = Ω. Moreover, let (φ j ) j ⊂ C 1 0 (Ω) be such that φ j ≡ 1 on ω j for every j ∈ N, and, for any fixed ξ ∈ R n , define the maps where C(x)ξ = η. Then, by previous considerations onã, (16) follows.
Step 3. We conclude by showing that a ∈M Ω (α, β ′ , p). By construction, a(x, ·) : R m → R m turns out to be continuous for a.e. x ∈ Ω and a(·, η) : Ω → R m turns out to be measurable for every η ∈ R m . Then, a : Ω × R m → R m is a Carathéodory function. Moreover, by (14), a satisfies condition (ii) of Definition 2.10 and, by (15), condition (iii) ′ of Remark 2.11. Let us prove that a(·, 0) = 0. Let w h be the solution to (E h ), with g = 0. Since, by Definition 2.10 (i), a h (x, 0) = 0 a.e. in Ω then, by the uniqueness of the solution to problem (E h ), we get that w h ≡ 0 a.e. in Ω for every h ∈ N and, by Lemma 3.2 and the definition of M , we get (up to subsequences) that is, N (0) = 0. Fixξ = 0 ∈ R n . Then, N w j ξ (x) = 0 in ω j for any j ∈ N and, by the definition of a, we finally get a(x, 0) = 0 a.e. x ∈ Ω.

Parabolic G-convergence
The last section of this paper is devoted to the study of the G-convergence of sequences of parabolic operators depending on vector fields. As done in the elliptic case (Section 3), we provide in Theorem 4.4 a G-compactness theorem, Theorem 4.4, which follows from preliminary results, Theorem 4.1, Theorem 4.2 and Lemma 4.3. We conclude this section by showing in Lemma 4.5 that, whenever the sequence of Carathéodory functions (a h ) h , that defines the monotone parabolic operators (∂ t + div X (a h (x, X))) h , does not depend on t for every h ∈ N, then the parabolic G-limit is the operator where div X (a(x, X)) is the elliptic G-limit of the sequence of operators (div X (a h (x, X))) h . In all this section we always assume that Ω is a bounded domain of R n , that 2 ≤ p < ∞ and that X satisfies conditions (H1), (H2), (H3) and (LIC), given in the Introduction.
The first result of this section, which is proved in [32,Lemma 3], shows that the sequence of solutions (u h ) h to problems (P h ), that are naturally compact in L p (0, T ; H) as stated in Proposition 2.7, converges to its limit in the space C 0 ([0, T ]; H). Lemma 3]). Let u h ∈ W be the solution to problem (P h ) and let u be the limit, up to subsequences, of (u h ) h in L p (0, T ; H). Then, for some f, g ∈ V ′ . Then, Proof. The proof can be obtained in a way similar to the analogous one showed in the elliptic case, namely Theorem 3.1. For reader's convenience we provide in the following the main calculations.
Fix ϕ ∈ C ∞ c (Ω × (0, T )) and consider the quantity Consider the right hand side terms. By assumptions, As regards the second term, by Proposition 2.7 and Theorem 4.1, we get For the third term one can proceed as in the proof of Theorem 3.1 and conclude.
The proof of the next result is classical and can be obtained following and adapting (since this is for p = 2) the analogous one contained in [24] or, for p ≥ 2, the proof contained in [28].
Fix sequences of open sets ω j and I j such that ω j ⊂ Ω and I j ⊂ (0, T ) for every j ∈ N and satisfying ∪ j∈N ω j × I j = Ω × (0, T ), and a sequence of cut-off functions Φ j ∈ C 1 0 (Ω × (0, T )) such that Φ j (x, t) ≡ 1 in ω j × I j for any j ∈ N. Then, for any fixed ξ ∈ R n , denoted by η = η(x) the vector such that C(x)ξ = η ∈ R m , we define a(x, t, η(x)) := lim j→+∞ N w j ξ (x) for any x ∈ Ω \ Z X , where, set (ξ, x) R n = ξ 1 x 1 + . . . + ξ n x n , the map w j ξ is defined by w j ξ (x, t) := ξ, x R n Φ j (x, t) for any (x, t) ∈ ω j × I j . Notice that a is well-defined by condition (LIC) on the X-gradient, which ensures that (X 1 (x), . . . , X m (x)) are linearly independent outside Z X . The last part of the proof follows verbatim as in Step 2 and Step 3 of Theorem 3.3, where one uses (19) and (20), instead of (14) and (15), and Lemma 4.3, instead of Lemma 3.2.
Proof. Fix g ∈ V ′ and denote by u h (g) ∈ V the solution to problem div X (a h (x, Xw)) = g in V ′ .
In a similar way, for any fixed f ∈ V ′ and ϕ ∈ H, denote by v h (f, ϕ) ∈ W the solution to problem (21) w ′ + div X (a h (x, Xw)) = f in V ′ w(0) = ϕ in H .
Let f (x, t) := g(x) for any t ∈ [0, T ] and define ϕ h := u h (g) for any h ∈ N.

(23)
Moreover, since u h (g) is also a solution to problem (21), with fixed data f (x, t) := g(x) for any t ∈ [0, T ] and ϕ h := u h (g) and, since v h (0) = u h (g), then, by the uniqueness of the solution to problem (21), it holds that v h (x, t) = u h (x) for any t ∈ [0, T ] and, by (22), that v(g, u(0)) = u(g), that is, v turns out to be independent of t. By (23), since the left hand side term is independent of t, we have that b(x, t, ξ) = b(x, ξ) . Therefore, div X (a(x, Xu(g))) = g in V ′ , div X (b(x, t, Xv(g, u(0)))) = div X (b(x, Xu(g))) = g in V ′ , that is, where A : V → V ′ and B : V → V ′ are, respectively, defined by Aw := div X (a(x, Xw)) and Bw := div X (b(x, Xw)) , w ∈ V .
Then, A = B and, by the convergence of momenta, we finally get a = b.