Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 13, pp. 1–13.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

H-CONVERGENCE FOR EQUATIONS DEPENDING ON

MONOTONE OPERATORS IN CARNOT GROUPS

ALBERTO MAIONE

Abstract. This article presents some results related to the convergence of

solutions and momenta of Dirichlet problems for sequences of monotone oper-

ators in the sub-Riemannian framework of Carnot groups.

1. 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

A(u) = −div(A(x,∇u)),

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.

In recent years, this theory found numerous applications in literature, such as
homogenization. We refer the interested reader to [3, 4, 5, 6, 7, 9, 11, 12, 13, 15, 16,
20, 28, 30] for details. In particular, De Arcangelis and Serra Cassano [14] extended
into the setting of Banach spaces the original Murat and Tartar H-compactness the-
orem, working with weights. A linear counterpart of this study, in Carnot groups,
was faced up by Baldi, Franchi, Tchou and Tesi [1, 2, 21]. This environment has
become of particular interest for analysis and PDEs over the previous decades, see
e.g. [10, 17, 18, 25, 27].

The class of linear operators considered in [1, 2, 21] is made of matrix-valued
measurable functions, that is, operators of the form

A(u) = −divG(A(x)∇Gu), (1.1)

where A is a (m × m)-matrix-valued measurable function and ∇G and divG are,
respectively, the intrinsic gradient and the intrinsic divergence (see Definition 2.2
for details). We remind that a definition of intrinsic curl, curlG, 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].

2010 Mathematics Subject Classification. 35B40, 35J66, 35R03, 47H05.

Key words and phrases. H-convergence; Carnot groups; monotone operators; div-curl lemma.
c©2021 Texas State University.

Submitted June 18, 2020. Published March 11, 2021.

1



2 A. MAIONE EJDE-2021/13

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

A(u) = −divG(A(x,∇Gu)) (1.2)

for a given A ∈M(α, β; Ω). The class M(α, β; Ω) is defined as follows.

Definition 1.1. Let Ω ⊂ G be open, 2 ≤ p <∞ and α ≤ β be positive constants.
We define M(α, β; Ω) the class of Carathéodory functions A : Ω× Rm → Rm such
that

(i) A(x, 0) = 0;
(ii) 〈A(x, ξ)−A(x, η), ξ − η〉 ≥ α|ξ − η|p;
(iii) |A(x, ξ)−A(x, η)| ≤ β [1 + |ξ|p + |η|p]

p−2
p |ξ − η|

for every ξ, η ∈ Rm and a.e. x ∈ Ω.

The main result of this article is the following theorem.

Theorem 1.2. Let Ω ⊂ G be open, connected and bounded, 2 ≤ p < ∞, α ≤ β
positive constants and let (An)n ⊂ M(α, β; Ω). Then, up to subsequences, there
exists Aeff ∈M(α, β; Ω) such that

(An)n H-converges to Aeff .

We would like to stress that, for p = 2, Theorem 1.2 generalizes several previous
results. For instance, if the Carnot groups G is the Euclidean space Rn, then The-
orem 1.2 immediately gives [32, Theorem 11.2]. Moreover, in the sub-Riemannian
framework of Carnot groups, if we restrict to operators (1.1), then Theorem 1.2
generalizes both [21, Theorem 4.4], if G is the first Heisenberg group, [1, Theorem
6.4], if G is a general Heisenberg group and [2, Theorem 5.4], in any Carnot group.

The structure of this article is the following one: in Section 2, we give the defi-
nitions 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.

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 V1, . . . , Vk linear subspaces of g, usually called layers, such that

(i) g = V1 ⊕ · · · ⊕ Vk ;
(ii) [V1, Vi] = Vi+1 for any i < k, where [V1, Vi] is the sub-algebra of g generated

by the commutation [X,Y ], with X ∈ V1, Y ∈ Vi;
(iii) Vk 6= {0} and Vi = {0} for any i > k, where 0 is the identity element of g.

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.



EJDE-2021/13 H-CONVERGENCE FOR MONOTONE OPERATORS IN CARNOT GROUPS 3

It is clear from Definition 2.1, that the first layer V1 plays the role of generator
of the algebra g, by commutation. For this reason, we refer to V1 as the horizontal
layer, while the other layers Vi, 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.,

dim(G) = dim(g) =

k∑
i=1

mi,

where mi := dim(Vi) for any i, and the homogeneous dimension, defined by

Q :=

k∑
i=1

i mi.

Let us notice that, when G is not Rn, the homogeneous dimension of G is always
bigger than the topological one. In the sequel, we denote m := m1, for simplicity.

2.2. Functional setting. Through the paper, (X1, . . . , Xm) denotes a basis of the
horizontal layer V1, |Ω| the Lebesgue measure of any set Ω ⊂ G and, if ξ, η ∈ Rm, we
denote by |ξ| and 〈ξ, η〉 the Euclidean norm and the scalar product, respectively.
The subbundle of the tangent bundle TG, which is spanned by the vector fields
X1, . . . , Xm, 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 Φ = (Φ1, . . . ,Φm) : G→ Rm.

Definition 2.2. Let u ∈ L1
loc(G), let Xiu exist in sense of distributions, and

assume XiΦi ∈ L1
loc(G) for i = 1, . . . ,m. We define the intrinsic gradient of u and

the intrinsic divergence of Φ, respectively, as

∇Gu :=

m∑
j=1

(Xju)Xj = (X1u, . . . ,Xmu), divG(Φ) :=

m∑
i=1

XiΦi.

Definition 2.3. For 1 ≤ p <∞ we define

W 1,p
G (Ω) := t{u ∈ Lp(Ω) : Xju ∈ Lp(Ω) for j = 1, . . . ,m},

endowed with its natural norm, W 1,p
G,0(Ω) the closure of C∞c (Ω)∩W 1,p

G (Ω) inW 1,p
G (Ω)

and W−1,p′

G (Ω) the dual space of W 1,p
G,0(Ω). Notice that, if Ω is bounded, then

‖u‖p
W 1,p

G,0 (Ω)
:=

∫
Ω

|∇Gu|p dx

defines an equivalent norm on W 1,p
G,0(Ω) (see [24, Section 2] and, for more details,

[23, 26]). Finally, we denote Lp(Ω, HG) the set of measurable sections Φ ∈ Lp(Ω)m.

Proposition 2.4 ([19, Corollary 4.14]). If 1 < p <∞, then W 1,p
G (Ω) is independent

of the choice of the basis (X1, . . . , Xm).

2.3. Monotone operators. Let us recall the definition of monotone operators.
See, for instance, [22] for more details.

Definition 2.5 ([22, Definitions 1.1–1.3, Chapter III]). Let V be a reflexive Banach
space, V ∗ its dual space and let A : V → V ∗ be a mapping. We say that

• A is monotone, if

〈A(u)−A(v), u− v〉V ∗×V ≥ 0 for all u, v ∈ V ;



4 A. MAIONE EJDE-2021/13

• A is strictly-monotone, if it is monotone and

〈A(u)−A(v), u− v〉V ∗×V = 0 implies u = v ;

• A is coercive, if there exists an element v ∈ V such that

〈A(u)−A(v), u− v〉V ∗×V
‖u− v‖V

→∞ as ‖u‖V →∞ ;

• A is continuous on finite dimensional subspaces of V if, for any finite di-
mensional subspace M of V , the restriction of A to M is weakly continuous,
namely, if A : M → V ∗ is weakly continuous.

Operator (1.2) is strictly-monotone, in sense of Definition 1.1. The following
result will be crucial later on.

Theorem 2.6 ([22, Corollary 1.8, Chapter III]). 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
u ∈ K such that

〈A(u), v − u〉X∗×K ≥ 0

for any v ∈ K.

3. Existence results for equations driven by monotone operators

Let Ω ⊂ G be open, connected and bounded, 2 ≤ p < ∞, V = W 1,p
G,0(Ω) and

V ∗ = W−1,p′

G (Ω). Moreover, let A : V → V ∗ be as in (1.2).

Proposition 3.1. Let A ∈ M(α, β; Ω). Then, for every f ∈ V ∗ there exists a
unique (weak) solution u ∈ V of

− divG(A(·,∇Gu)) = f in Ω , (3.1)

i.e., ∫
Ω

〈A(x,∇Gu),∇Gϕ〉dx =

∫
Ω

f ϕ dx ∀ϕ ∈ C∞c (Ω). (3.2)

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

〈B(u), v〉V ∗×V :=

∫
Ω

(
〈A(x,∇Gu),∇Gv〉 − f v

)
dx ∀u, v ∈ V.

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
space V .

Fix u, v ∈ V . Then, by Definition 1.1 (ii)

〈B(u)− B(v), u− v〉V ∗×V ≥ α‖u− v‖pV ≥ 0 ,

〈B(u)− B(v), u− v〉V ∗×V
‖u− v‖V

≥ α‖u− v‖p−1
V .

Let (un)n be strongly convergent to u in V . By Hölder’s inequality, we have

〈B(un)− B(u), un − u〉V ∗×V ≤ ‖A(·,∇Gun)−A(·,∇Gu)‖Lp′ (Ω,HG)‖un − u‖V .



EJDE-2021/13 H-CONVERGENCE FOR MONOTONE OPERATORS IN CARNOT GROUPS 5

Notice that (A(·,∇Gun))n strongly converges to A(·,∇Gu) in Lp
′
(Ω, HG) since, by

Definition 1.1 (iii) and Hölder’s inequality

‖A(·,∇Gun)−A(·,∇Gu)‖p
′

Lp′ (Ω,HG)

≤ βp
′
∫

Ω

[1 + |∇Gun|p + |∇Gu|p]
p−2
p−1 |∇Gun −∇Gu|p

′
dx

≤ βp
′
(∫

Ω

[1 + |∇Gun|p + |∇Gu|p] dx
) p−2

p−1
(∫

Ω

|∇Gun −∇Gu|p dx
) p′

p

= βp
′
[|Ω|+ ‖un‖pV + ‖u‖pV ]

p−2
p p′ ‖un − u‖p

′

V .

Moreover, by Theorem 2.6, there exists u ∈ V such that

〈B(u), v − u〉V ∗×V ≥ 0 ∀v ∈ V (3.3)

and, choosing v1 := u+ ϕ and v2 := u− ϕ, we obtain

〈B(u), ϕ〉V ∗×V = 0 ∀ϕ ∈ V.
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)

0 =

∫
Ω

〈A(x,∇Gu)−A(x,∇Gv),∇Gu−∇Gv〉dx ≥ α‖u− v‖pV ≥ 0 ,

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.

Proposition 3.3. Let A ∈ M(α, β; Ω), let A be as in (1.2) and let A−1 be its
inverse operator. Then

(a) 〈A(u)−A(v), u− v〉V ∗×V ≥ α‖u− v‖pV ;

(b) ‖A−1(f)−A−1(g)‖pV ≤ ( 1
α )p

′‖f − g‖p
′

V ∗ ;

(c) ‖A(u)−A(v)‖V ∗ ≤ β[|Ω|+ ‖u‖pV + ‖v‖pV ]
p−2
p ‖u− v‖V

for any u, v ∈ V and for any f, g ∈ V ∗.

Proof. Fix u, v ∈ V and f, g ∈ V ∗ such that

A(u) = f and A(v) = g in Ω .

Notice that (a) directly follows from Definition 1.1 (ii). Moreover, recalling that

〈A(u)−A(v), u− v〉V ∗×V ≤ ‖A(u)−A(v)‖V ∗‖u− v‖V ∀u, v ∈ V,
and applying (a), with u = A−1(f) and v = A−1(g), we obtain

α‖A−1(f)−A−1(g)‖pV ≤ ‖f − g‖V ∗‖A
−1(f)−A−1(g)‖V ,

which implies (b).
Finally, by Definition 1.1 (iii),

‖A(·,∇Gu)−A(·,∇Gv)‖Lp′ (Ω,HG) ≤ β [|Ω|+ ‖u‖pV + ‖v‖pV ]
p−2
p ‖u− v‖V ,

i.e.,

〈A(u)−A(v), u− v〉V ∗×V ≤ ‖A(·,∇Gu)−A(·,∇Gv)‖Lp′ (Ω,HG)‖u− v‖V

≤ β[|Ω|+ ‖u‖pV + ‖v‖pV ]
p−2
p ‖u− v‖2V .



6 A. MAIONE EJDE-2021/13

Then, (c) follows by the definition of ‖ · ‖V ∗ . �

4. 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.

Definition 4.1. Let An ∈M(α, β; Ω) and let Aeff ∈M(α′, β′; Ω), for some α ≤ β
and α′ ≤ β′ positive constants. Fix f ∈ W−1,p′

G (Ω) and let un, u∞ ∈ W 1,p
G,0(Ω) be,

respectively, weak solutions of

−divG(An(·,∇Gu)) = f in Ω

−divG(Aeff(·,∇Gu)) = f in Ω .

We say that (An)n H-converges to Aeff if, as n→∞,

un → u∞ weakly in W 1,p
G,0(Ω) (convergence of solutions)

and

An(·,∇Gun)→ Aeff(·,∇Gu∞) weakly in Lp
′
(Ω, HG) (convergence of momenta).

Before proving Theorem 1.2, we need two preliminary results.

Lemma 4.2. Let An ∈M(α, β; Ω) and define An : W 1,p
G,0(Ω)→W−1,p′

G (Ω) as

An(u) := − divG(An(·,∇Gu)) in Ω .

Then, there exist a continuous and invertible operator A∞ : W 1,p
G,0(Ω)→W−1,p′

G (Ω)

and a subsequence (Am)m of (An)n, such that

A−1
m (f)→ A−1

∞ (f) weakly in W 1,p
G,0(Ω)

for every f ∈W−1,p′

G (Ω).

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

An(u) = f in Ω (4.1)

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 un ∈ V , weak solution of
(4.1), that is, un = A−1

n (f) for any n ∈ N. Moreover, by Proposition 3.3 (b)

‖un‖V ≤
( 1

α

) 1
p−1 ‖f‖

1
p−1

V ∗ ,

i.e., (un)n is bounded in V, reflexive Banach space and, therefore, there exist
u∞(f) ∈ V and (um)m, diagonal subsequence of (un)n, such that

um → u∞(f) weakly in V.

Notice that, by the lower semicontinuity of the norm and by Proposition 3.3(a),

〈f, u∞〉V ∗×V = lim
m→∞

〈Am(um), um〉V ∗×V ≥ α lim inf
m→∞

‖um‖pV ≥ α‖u∞‖
p
V



EJDE-2021/13 H-CONVERGENCE FOR MONOTONE OPERATORS IN CARNOT GROUPS 7

and, since

〈f, u∞〉V ∗×V ≤ ‖f‖V ∗‖u∞‖V ,

it follows that

‖u∞‖V ≤
( 1

α

) 1
p−1 ‖f‖

1
p−1

V ∗ .

Step 2. Define S : X → V as

S(f) := lim
m→∞

A−1
m (f) for any f ∈ X.

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),

‖A−1
m (f)−A−1

m (g)‖V ≤
( 1

α

) 1
p−1 ‖f − g‖

1
p−1

V ∗ ∀m ∈ N

and, passing to the limit, by the lower semicontinuity of the norm, we obtain

‖S(f)− S(g)‖V ≤ lim inf
m→∞

‖A−1
m (f)−A−1

m (g)‖V ≤
( 1

α

) 1
p−1 ‖f − g‖

1
p−1

V ∗ .

For the sake of completeness, the extension of S to V ∗ \X is defined as

S(f) := lim
n→∞

S(fn)

for any f ∈ V ∗ and (fn)n ⊂ X such that fn → 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),

〈S(f)− S(g), f − g〉V×V ∗ = lim
m→∞

〈A−1
m (f)−A−1

m (g), f − g〉V×V ∗

= lim
m→∞

〈Am(um)−Am(vm), um − vm〉V ∗×V

≥ α lim
m→∞

‖um − vm‖pV ≥ 0 .

Moreover,

‖Am(um)−Am(vm)‖pV ∗
≤ βp [|Ω|+ ‖um‖pV + ‖vm‖pV ]

p−2 ‖um − vm‖pV

≤ βp

α
[|Ω|+ ‖um‖pV + ‖vm‖pV ]

p−2 〈Am(um)−Am(vm), um − vm〉V ∗×V

≤ βp

α

[
|Ω|+

( 1

α

)p′‖f‖p′V ∗ +
( 1

α

)p′‖g‖p′V ∗]p−2〈A−1
m (f)−A−1

m (g), f − g〉V×V ∗ .

Passing to the limit,

‖f − g‖pV ∗

≤ βp

α

[
|Ω|+

( 1

α

)p′‖f‖p′V ∗ +
( 1

α

)p′‖g‖p′V ∗]p−2〈S(f)− S(g), f − g〉V×V ∗ .

We obtain the conclusion, defining A∞ := S−1 : V → V ∗. �



8 A. MAIONE EJDE-2021/13

Lemma 4.3. Let An be as in the Lemma 4.2. Then, for any f ∈ W−1,p′

G (Ω),

there exists a continuous operator M : W−1,p′

G (Ω) → Lp
′
(Ω, HG) such that, up to

subsequences

An(·,∇GA−1
n (f))→M(f) weakly in Lp

′
(Ω, HG).

Proof. Let X be a countable and dense subspace of Lp
′
(Ω, HG) and let f ∈ X.

Then, by Definition 1.1(iii) and Hölder’s inequality∫
Ω

|An(x,∇GA−1
n (f))|p

′
dx ≤ βp

′
∫

Ω

[1 + |∇GA−1
n (f)|p]

p−2
p−1 |∇GA−1

n (f)|p
′
dx

≤ βp
′
[|Ω|+ ‖A−1

n (f)‖pV ]
p−2
p p′‖A−1

n (f)‖p
′

V ,

i.e.,

‖An(·,∇GA−1
n (f))‖Lp′ (Ω,HG) ≤ β[|Ω|+ ‖A−1

n (f)‖pV ]
p−2
p ‖A−1

n (f)‖V
and, by Proposition 3.3,

‖An(·,∇GA−1
n (f))‖Lp′ (Ω,HG) ≤

β

α
1

p−1

[
|Ω|+

( 1

α

)p′‖f‖p′V ∗] p−2
p ‖f‖

1
p−1

V ∗ .

Therefore, (An(·,∇GA−1
n (f)))n is bounded in Lp

′
(Ω, HG) and, by the countability

of X, there exists a diagonal subsequence of (An(·,∇GA−1
n (f)))n weakly convergent

to M = M(f) in Lp
′
(Ω, HG).

We define M : X → Lp
′
(Ω, HG) as

M(f) := lim
m→∞

Am(·,∇GA−1
m (f)) for any f ∈ X.

If f, g ∈ X, then, by Proposition 3.3,

‖Am(·,∇GA−1
m (f))−Am(·,∇GA−1

m (g))‖Lp′ (Ω,HG)

≤ β

α
1

p−1

[
|Ω|+

( 1

α

)p′‖f‖p′V ∗ +
( 1

α

)p′‖g‖p′V ∗] p−2
p ‖f − g‖

1
p−1

V ∗ .

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].

Theorem 4.4 ([2, Theorem 5.1]). 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 ∈ I2

0 , let
a(σ) > 1 and b > 1 be such that

a(σ) >
Qp

Q+ (σ − 1)p
and b >

Qq

Q+ q
.

Finally, let En, E ∈ Lploc(Ω, HG) and Dn, D ∈ Lqloc(Ω, HG) be such that

(i) En → E weakly in Lploc(Ω, HG);
(ii) Dn → D weakly in Lqloc(Ω, HG);

(iii) the components of (curlGE
n)n of weight σ are bounded in L

a(σ)
loc (Ω, HG);

(iv) (divGD
n)n is bounded in Lbloc(Ω, HG).

Then 〈Dn, En〉 → 〈D,E〉 in D′(Ω), i.e.,∫
Ω

〈Dn(x), En(x)〉ϕ(x) dx→
∫

Ω

〈D(x), E(x)〉ϕ(x) dx for any ϕ ∈ D(Ω).



EJDE-2021/13 H-CONVERGENCE FOR MONOTONE OPERATORS IN CARNOT GROUPS 9

Proof of Theorem 1.2. We denote A∞ and M the operators defined in Lemma 4.2
and Lemma 4.3, and define

C := M ◦ A∞ : W 1,p
G,0(Ω)→ Lp

′
(Ω, HG) .

Let us show the existence of Aeff ∈M(α, β; Ω) such that

C(u) = Aeff(x,∇GA−1
∞ (f))

for any f ∈W−1,p′

G (Ω) and for any u ∈W 1,p
G,0(Ω) such that

A∞(u) = f a.e. x ∈ Ω . (4.2)

Fix f ∈W−1,p′

G (Ω) 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 Aeff : Ω× Rm → Rm as

Aeff(x, ξ) := C(v) if ∇Gv(x) = ξ a.e. x in ω .

Let us show that

Aeff(x, ξ1) = Aeff(x, ξ2) a.e. x in ω1 ∩ ω2 (4.3)

for any ξ1 = ξ2 ∈ Rm and for any ω1, ω2 open sets such that ω1, ω2 ⊂ Ω.
We fix ϕ1, ϕ2 ∈ C1

c(Ω) such that ϕi|ωi
= 1 for i = 1, 2, and let (v1,n)n ⊂W 1,p

G,0(Ω)

and (v2,n)n ⊂W 1,p
G,0(Ω) be, respectively, weakly convergent, up to subsequences, to

v1,∞(x) = ϕ1(x) 〈ξ1, π(x)〉
v2,∞(x) = ϕ2(x) 〈ξ2, π(x)〉,

(4.4)

where π(x) := (x1, . . . , xm) for every x = (x1, . . . , xn) ∈ Ω. Moreover, define

Dn
i := An(·,∇Gvi,n) ∈ Lp

′
(Ω, HG)

Eni := ∇Gvi,n ∈ Lp(Ω, HG)

and fix f1, f2 ∈W−1,p′

G (Ω) such that

f1 = − divG(C(v1,∞)), f2 = −divG(C(v2,∞)) in Ω .

By (4.4), it holds that
∇Gv1,∞ = ξ1 in ω1

∇Gv2,∞ = ξ2 in ω2 .
(4.5)

Notice that curlG(Eni ) = 0, for any n ∈ N and i = 1, 2. Moreover, there exist

(Dm
i )m, (Emi )m, diagonal subsequences of (Dn

i )n and (Eni )n and Di ∈ Lp
′
(Ω, HG)

and Ei ∈ Lp(Ω, HG), i = 1, 2, such that

Dm
i → Di weakly in Lp

′
(Ω, HG)

Emi → Ei weakly in Lp(Ω, HG).

Therefore, by (4.5), by Lemma 4.2, Lemma 4.3, and by Theorem 4.4 (where a is
each value grater than 1, which satisfies the hypotheses of the theorem, and b = p′),
it follows that∫

Ω

〈Am(x,∇Gv2,m)−Am(x,∇Gv1,m),∇Gv2,m −∇Gv1,m〉ϕ(x) dx

→
∫

Ω

〈Aeff(x, ξ2)−Aeff(x, ξ1), ξ2 − ξ1〉ϕ(x) dx

(4.6)

for any ϕ ∈ D(ω1 ∩ ω2).



10 A. MAIONE EJDE-2021/13

Fix ϕ ≥ 0 and notice that, by Definition 1.1(ii), it holds that∫
Ω

〈Am(x,∇Gv2,m)−Am(x,∇Gv1,m),∇Gv2,m −∇Gv1,m〉ϕ(x) dx

≥ α
∫

Ω

|∇Gv2,m −∇Gv1,m|p ϕ(x) dx .

(4.7)

Then, by (4.5), (4.6) and (4.7) and Fatou’s lemma,∫
Ω

〈Aeff(x, ξ2)−Aeff(x, ξ1), ξ2 − ξ1〉ϕ(x)dx

≥ lim inf
m→∞

α

∫
Ω

|∇Gv2,m −∇Gv1,m|pϕ(x)dx

≥ α
∫

Ω

|∇Gv2,∞ −∇Gv1,∞|pϕ(x) dx

= α

∫
Ω

|ξ2 − ξ1|pϕ(x) dx .

(4.8)

Moreover, since by Definition 1.1(iii)∫
Ω

|∇Gv2,m −∇Gv1,m|pϕ(x) dx

≥ 1

βp

∫
Ω

[
1 + |∇Gv2,m|p + |∇Gv1,m|p

]2−p
× |Am(x,∇Gv2,m)−Am(x,∇Gv1,m)|pϕ(x)dx ,

(4.9)

then, by (4.5), (4.6), (4.7) and (4.9), and Fatou’s lemma,∫
Ω

〈Aeff(x, ξ2)−Aeff(x, ξ1), ξ2 − ξ1〉ϕ(x) dx

≥ α

βp

∫
Ω

[1 + |ξ2|p + |ξ1|p]2−p|Aeff(x, ξ2)−Aeff(x, ξ1)|pϕ(x) dx .

(4.10)

Varying ϕ in D(ω1 ∩ ω2), (4.8) and (4.10) give

〈Aeff(x, ξ2)−Aeff(x, ξ1), ξ2 − ξ1〉 ≥ α |ξ2 − ξ1|p ,

〈Aeff(x, ξ2)−Aeff(x, ξ1), ξ2 − ξ1〉

≥ α

βp
[1 + |ξ2|p + |ξ1|p]2−p|Aeff(x, ξ2)−Aeff(x, ξ1)|p

a.e. x ∈ ω1 ∩ ω2.
If ξ1 = ξ2, we obtain (4.3), and if ξ1 6= ξ2, then Aeff satisfies Definition 1.1(ii).
Moreover, by Definition (1.1)(iii), by (4.5) and Fatou’s lemma,∫

Ω

|ξ2 − ξ1|p ϕ(x)dx

≥ lim inf
m→∞

∫
Ω

|∇Gv2,m −∇Gv1,m|pϕ(x) dx

≥ lim inf
m→∞

1

βp

∫
Ω

[
1 + |∇Gv2,m|p + |∇Gv1,m|p

]2−p
× |Am(x,∇Gv2,m)−Am(x,∇Gv1,m)|p ϕ(x) dx

≥ 1

βp

∫
Ω

[1 + |ξ2|p + |ξ1|p]2−p|Aeff(x, ξ2)−Aeff(x, ξ1)|p ϕ(x)dx



EJDE-2021/13 H-CONVERGENCE FOR MONOTONE OPERATORS IN CARNOT GROUPS 11

and, varying ϕ in D(ω1 ∩ ω2), Aeff satisfies Definition 1.1(iii).

Let un ∈W 1,p
G,0(Ω) be the (unique) weak solution of (4.1), relative to f = 0. Since

An(·, 0) = 0 a.e. in Ω by Definition 1.1(i), then un = 0 a.e. in Ω and, by Lemma
4.2 and Lemma 4.3, up to subsequences

0 = An(x,∇Gun)→ Aeff(x, 0) weakly in Lp
′
(Ω, HG).

Then, Aeff satisfies also Definition 1.1 (i) and, therefore

Aeff ∈M(α, β; Ω) .

To conclude the proof of the theorem, we show that

C(u∞) = Aeff(x,∇Gu∞) a.e. x ∈ Ω . (4.11)

Let u∞ ∈ W 1,p
G,0(Ω) be the (unique) weak solution of (4.2), let (um)m be weakly

convergent to u∞ in W 1,p
G,0(Ω) and define Dm

2 = Am(x,∇Gum) and Em2 = ∇Gum.
Then, by Theorem 4.4,∫

Ω

〈Am(x,∇Gum)−Am(x,∇Gv1,m),∇Gum −∇Gv1,m〉ϕ(x) dx

→
∫

Ω

〈C(u∞)−Aeff(x, ξ1),∇Gu∞ − ξ1〉ϕ(x) dx

for any ϕ ∈ D(ω1) and, following the same techniques of the first part of the proof,

〈C(u∞)−Aeff(x, ξ1),∇Gu∞ − ξ1〉 ≥ α|∇Gu∞ − ξ1|p ,

〈C(u∞)−Aeff(x, ξ1),∇Gu∞ − ξ1〉

≥ α

βp
[1 + |∇Gu∞|p + |ξ1|p]2−p|C(u∞)−Aeff(x, ξ1)|p;

that is,

|C(u∞)−Aeff(x, ξ1)| ≤ β[1 + |∇Gu∞|p + |ξ1|p]
p−2
p |∇Gu∞ − ξ1| a.e. x ∈ ω1.

Finally, varying ϕ ∈ D(ω1) and ξ1 ∈ Rm, we obtain (4.11). �

Acknowledgments. The author would like to thank Francesco Serra Cassano and
Andrea Pinamonti for their support and help. This research was partially supported
by the Indam-GNAMPA project 2020 “Convergenze variazionali per funzionali e
operatori dipendenti da campi vettoriali”, by MIUR, the University of Trento (Italy)
and the University of Freiburg (Germany).

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Alberto Maione

Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, 79104,
Hermann-Herder-Straße 10, Freiburg im Breisgau, Germany

Email address: alberto.maione@mathematik.uni-freiburg.de


	1. Introduction
	2. Preliminaries
	2.1. Carnot groups
	2.2. Functional setting
	2.3. Monotone operators

	3. Existence results for equations driven by monotone operators
	4. H-convergence and Div-curl lemma
	Acknowledgments

	References