$Γ$- convergence and homogenisation for a class of degenerate functionals
Nicolas Dirr Federica Dragoni Paola Mannucci Claudio Marchi









Γ- convergence and homogenisation for a class of degenerate functionals.
    Nicolas DirrCardiff School of Mathematics, Cardiff University, Cardiff, UK, e-mail: dirrnp@cardiff.ac.uk. Federica Dragoni
Cardiff School of Mathematics, Cardiff University, Cardiff, UK, e-mail: DragoniF@cardiff.ac.uk. Paola Mannucci 
Dipartimento di Matematica “Tullio Levi-Civita” ,  Università di Padova, Padova, Italy, e-mail: mannucci@math.unipd.it Claudio Marchi
Dipartimento di Ingegneria dell'Informazione,  Università di Padova, Padova, Italy, e-mail: claudio.marchi@unipd.it

    December 30, 2023
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This paper is on Γ-convergence for degenerate integral functionals related to homogenisation problems  in the Heisenberg group. Here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the Heisenberg group.  
Without using special geometric features, these functionals would be neither coercive nor periodic, so classic results do not apply.  All the results apply to the more general case of Carnot groups.



Keywords:   Gamma-convergence, homogenisation,  Carnot groups, Heisenberg group, 
Hörmander condition,
degenerate functionals.




§ INTRODUCTION
 



Γ-convergence is a notion of convergence of a family of functionals F_ε to a  functional F_∞  which goes back to E. de Giorgi (see <cit.>) and guarantees the convergence of minimisers of the functionals F_ε to minimisers of the limit (or effective) functional F_∞; for a precise definition  and properties see Section <ref>.The convergence of minimisers  implies, under suitable conditions, convergence of solutions of the Euler-Lagrange equations, and is therefore a useful tool for homogenisation problems, in particular in the random and nonlinear case, see e.g. <cit.>. If the functional is integral, i.e. of the form


    u↦
    F(u)=∫_Af(x,∇  u(x)) dx,
 where A is a (Borel) domain, ∇ u is the distributional gradient of the real-valued function u assumed to be in a suitable L^p-space,  and f:^N×^N→ with some regularity and growth assumptions, then the corresponding Euler-Lagrange equation is a nonlinear divergence form equation.





 The above functional can be generalised to degenerate functionals in the setting of 
 Carnot groups. In this paper we focus specifically  on  the n-dimensional  Heisenberg group ^n, which is a step 2 Carnot group defined on ^2n+1 (see Section <ref> for definitions and properties).
Thus the family of functionals considered here is of the form


    u↦
    F(u)=∫_Af(x,∇_  u(x)) dx

where A is a domain on ^N=^2n+1 while ∇_u is the horizontal gradient in the Heisenberg group
which belongs to a suitable m-dimensional subspace of the “space of derivatives” (tangent space),
see Definition <ref>. Since m=2n, we have m<N:
as a consequence, such functionals are typically not coercive in the classical sense, so classical results do not apply.

Working in the setting of the Heisenberg group, the scaling  needs to adapt to the underlying geometrical structure, therefore we will consider the following anisotropic scaling 
 δ_1/ε(x_1,x_2,x_3)=(ε^-1x_1,ε^-1x_2,ε^-2x_3), with (x_1,x_2,x_3)∈^N≡^n×^n
×. 
 Hence the scaling is anisotropic w.r.t. the last component. The anisotropy can be understood heuristically in another way: at each point, some directions are “forbidden”, i.e. paths of the associated control problem can move only on a 2n-dimensional subspace of a (2n+1)-dimensional space. By varying their direction often (i.e. by the use of non-trivial commutators from the Hörmander condition) they are able to reach any given point but the cost for “zig-zagging” to get in the forbidden direction is higher, so typically they move slower in these directions, which makes  a faster rescaling necessary.
 


The limit functional F_∞ is of the same form,  i.e.

    F_∞(u)=∫_A f_0(∇_ u(x))dx,

where the integral function f_0 does not depend  on x anymore
(however  the horizontal gradient still depends on x trough the vector fields).
The corresponding Euler-Lagrange equation will not be elliptic  but only subelliptic,
we refer to  <cit.> for an overview on subelliptic equations.

The study of homogenisation in subelliptic settings started with the periodic case (see e.g. <cit.>).
The first result for the stochastic case in this degenerate setting  is  <cit.>, where the authors studied the case Hamilton-Jacobi (first order) case for Hamiltonian depending on the horizontal gradient in the case of Carnot groups.





As Γ-convergence has nice compactness results, the main difficulty is in general the identification of the Γ-limit as again an integral functional. Here it is  used that the integrand can be retrieved by considering minimisation problems over small cubes with affine  boundary conditions, see <cit.>. A generalisation to the setting of the Heisenberg group requires a suitable adaption of the notion of “affine”, namely H-affine functions, see  Section <ref> for the definition and references. 
Recently some results for Γ-convergence of degenerate functionals in very general geometries have been proved in <cit.>.
Here we use that the minimal normalised energy on anisotropicaly  (Heisenberg dilations)
scaled cubes  is  subadditive  by constructing admissible functionals  on large cubes and patching together translated minimisers  on translated cubes. Here we need  to use the specific properties of translations in the Heisenberg group. Note that 
 cubes rescaled by an integer (i.e. δ_k(Q))  cannot be written as union of translations of the original cube Q, not even up to a set of measure zero. This a crucial difference with the Euclidean  case  but we overcome the issue by controlling the error term. 
 
 A closely related approach can be found in <cit.>, where the Γ-convergence in Cheeger-Sobolev spaces is considered. Our functional depending only on |∇_u| instead of
∇_ u would be a functional on a Cheeger-Sobolev space, but the natural tiling generalising periodicity in our case does not satisfy the assumptions of <cit.>.

All the results are written in the Heisenberg group for sake of simplicity but the proofs apply to general Carnot groups.

These results can be applied to functionals related to subelliptic p-Laplace equations and generalised to   stochastic functionals  with short correlations (as done in Dal Maso-Modica <cit.>).



This paper is organised as follows.



 In Section 2 we  give an overview on the Heisenberg group and its geometry, in particular the scaling, the horizontal gradient and the notion of periodicity.



In Section 3 we define precisely our functionals and we recall the Sobolev
spaces adapted to the structure of the Heisenberg group, in particular their embedding into L^p-spaces, through the embedding in fractional  Sobolev spaces, which will be crucial for the later Γ-convergence results.


Section 4 is devoted to  the Γ-convergence  results. We first recall the definition and some basic properties of Γ-convergence. We use compactness properties of the Γ-convergence and we give conditions under which the Γ-limit  is again an integral functional, thus recovering the results by Dal Maso-Modica, <cit.>, for our degenerate functionals.


In Section 5 we prove the main result of the paper, that is the  homogenisation result for
Heisenberg-periodic functionals. In fact, we show a Akcoglu-Krengel
type result, <cit.>, for our anisotropic Heisenberg-periodic functionals, i.e. the convergence of normalised minimal energies over rescaled cubes. For this purpose, we exploit an underlying subadditive structure.

In Section 6 we mention some applications and further directions of research.  We  highlight how the results apply to more general functionals associated to Carnot group structures.
We then give some connections with homogenisation for subelliptic p-Laplacian.
 Finally we explain how our methods can be used to generalise the results to the 
stochastic case  with short correlations.








§ PRELIMINARIES: THE HEISENBERG GROUP.


Carnot groups
 are non-commutative Lie groups: thus they are endowed both with a non-commutative algebraic structure and with a manifold structure. The lack of commutativity in the algebraic structure reflects on the manifold structure as restrictions on the admissible motions. This means that the allowed curves are constrained to have their velocities in a lower dimensional subspace of the tangent space of the manifold. Then the associated manifold structure is not Riemannian but sub-Riemannian. 
 In this paper we give details for the Heisenberg group only but the results can be easily generalised to Carnot groups (see Section 6). We
  refer the reader to <cit.> for definitions and properties on Carnot groups and to
   <cit.> for an overview on
  sub-Riemannian manifolds.

  
  
  

To keep the paper easily readable  we omit the intrinsic definition of the Heisenberg group, introducing it directly as the following non-commutative group  structure on ^N. 


The n-dimensional Heisenberg group ^n, with n≥ 1, is a Carnot group of step 2
isomorphic to  ^N, where  N=2n+1,
endowed with the  following non-commutative  group operation:

    x* y:=(x_1+y_1, x_2+y_2,x_3+y_3+x_1·y_2- x_2·y_1/2)

for all x=(x_1,x_2,x_3),y=(y_1,y_2,y_3)∈^N≡^n×^n× and where by · we indicate the standard inner product in ^n.





In all Carnot groups it is possible to define  a natural scaling, induced by the Lie algebra stratification, namely dilations. The dilations replace the multiplication by scalars in the standard vector space structure of the Euclidean ^N.


The dilations in the Heisenberg group are the family of group homeomorphisms defined as, for all t>0,  
δ_t:^N→^N  with

    δ_t(x)=(t x_1,t x_2, t^2 x_3),
      ∀ 
    x=(x_1,x_2,x_3)∈^N≡^n×^n×.


Thus  the dilations in ^n coincide with the standard Euclidean scaling in the first 2n components while the last component scales as t^2.

The following properties of dilations are true in all Carnot groups and they can be easily checked in the Heiseberg group by using formulas (<ref>) and  (<ref>).


For all  t, s >0, the following properties hold true:


  (1)  δ_1 = id;

  (2) δ^-1_t = δ_t^-1;
  
  (3)
  δ_t∘δ_s =δ_t  s;

  (4) for every x, y∈^N one has δ_t(x) * δ_t(y) = δ_t(x* y).


We now recall the notion of homogenous dimension. In a general Carnot group , the homogenous dimension is the natural number  
𝒬:=∑_i=1^r i dim _i,
 where r is the step of the  stratified  associated Lie algebra 
  (see e.g. <cit.> for more details).
In ^n one can easily show that

    𝒬=2n+2.

The homogeneous dimension  is correlated to the scaling of measures since it coincides with the Hausdorff dimension w.r.t. every homogeneous metric. 
 In the paper we always indicate simply by |A| the N-dimensional Lebesgue measure of the Borel set A of ^N. Then for all t>0,
|t  A|=t^N|A| while
one can easily show that

    A^t:=δ_t(A)
      ⇒  
    |A^t|=t^𝒬 |A|.


 
 Since the Heisenberg group (as all Carnot groups) is non-abelian, translations to the right or to the left determine two different families of homeomorphism on the group. As standard in this setting, we consider the left-translations, which are defined, for all y∈^N as 
 L_y:^N→^N with
 
    L_y(x):=y* x,

where * is the group operation defined in (<ref>).


Using the left-translations it is possible to define a sub-Riemannian structure on each Carnot group by introducing a suitable family of  left-invariant vector fields spanning to the first layer of the Lie algebra stratification. We omit the general definition on Carnot groups (see e.g. <cit.>). In the specific case of the Heisenberg group, the vector fields can be found as 

    X_i(x)=dL_x(e_i),

 where e_i are the unit vectors of the standard Euclidean basis on ^N for i=1,…, 2n. 
 One can also easily show that, for all j=1,…,n
 
    X_2n+1(x)=dL_x(e_2n+1)
     =[
     X_j,X_n+j](x),

 where [·,·] are the standard Lie brackets (called also commutators) defined for vector fields. 
 In the case n=1 the vector fields are 

    X_1(x)=([      1;      0; -x_2/2 ])  and   
    X_2(x)=([     0;     1; x_1/2 ]),
      ∀  x=(x_1,x_2,x_3)∈^3.

We recall that the previous vector fields are left-invariant by definition.
For later use we introduce the following simplified notation: given any function u:^N→, the translation L_z of the function  u is simply u∘ L_z, i.e.

    L_y(u)(x):=u(y*x).

Thus X_i is a left-invariant vector field if   for all u∈ C^∞(^N) and for all fixed y∈^N

    X_i(  L_y(u))(x)=(X_iu) (y* x),   ∀ x∈^N,

(while this is in general false considering instead the right-translations).

We recall that the vector fields X_i for i=1,…, 2n span a bracket generating distribution with step 2 (see e.g. <cit.> for some details).







The previous vector fields allow us to define  derivatives of any order, just considering how a vector field acts on smooth functions.  
 Given a function u:^N→, we denote
 the horizontal gradient  of u 
  by 

    ∇_ u = (X_1 u,…, X_2nu)^T.


In the case of n=1 the horizontal gradient can be explicitly  written as

    ∇_ u=[  u_x_1-x_2/2u_x_3; u_x_2+x_1/2 u_x_3 ]∈^2.



We now recall that a differential operator ℒ on a the Heisenberg group  is called homogeneous of degree κ if for every u∈ C^∞(^N) one has

    ℒ(δ_t u) = t^κδ_t(ℒu),

where  the scaled function is defined as δ_t u(x) := u(δ_t(x)), for all x∈^N.

Then we  have the following result.


For every i = 1,...,2n, each left-invariant vector field X_i, defined  in (<ref>),
is  homogeneous of degree κ = 1, i.e., for any u∈ C^∞(^N) one has

    X_i(δ_t u) = t δ_t(X_i u).


The proof is a very simple computation in the Heisenberg group while for general Carnot groups the reader can find a proof e.g. in <cit.>.


This in particular implies that
the horizontal 
gradient is homogeneous of degree one with respect to the dilations δ_λ, i.e., for every u∈ C^∞(^N) we have

    ∇_ (δ_t u)
    = t
    δ_t( ∇_ u).

 

For later use,  
it is very useful to  introduce the N× 2n-matrix associated to the vector fields, that is
 
    σ:=(X_1,…,X_2n),

where 
 X_i are the left-invariant vector fields defined in (<ref>)
  and   the extended matrix of vector fields, which is the N× N-matrix

    σ_Ext:=(X_1,…,X_2n, X_2n+1),

where X_2n+1(x)=dL_x(e_2n+1), and e_2n+1 is the unit vector spanning the  2n+1-direction (and associated to the second layer of the stratification for the Lie algebra).

In the 1-dimensional Heisenberg group ^1, the matrix σ is the 2× 3-matrix given by 

    σ(x_1,x_2,x_3)=
    [      1      0;      0      1; -x_2/2  x_1/2 ],

while
σ_Ext is the 3× 3-matrix given by 

    σ_Ext(x_1,x_2,x_3)=
    [      1      0      0;      0      1      0; -x_2/2  x_1/2      1 ].


A trivial computation shows the following property, which will be very useful later:
given the quadratic matrix defined in (<ref>), then 

    det(σ_Ext(x))=1,
      ∀  x∈^N.

The property above means that the left-translations are an  isometry for the associated L^p-spaces, i.e. informally setting for all fixed z∈^N y:=L_z(x)=z* x, we have dy=dx.

Property (<ref>) can be generalised to all Carnot groups in exponential coordinates or to more general Carnot-type groups (see e.g. <cit.> for properties and definitions of Carnot-type groups). 


 
Trivially ∇_u=σ^T ∇ u 
where ∇ u 
denotes   
the standard (Euclidean) gradient  
of u.






 §.§ Periodicity in the Heisenberg group.


Being the Heisenberg group a Lie group,
a very natural notion of periodicity can be introduced by left-translations,
see 
<cit.>.
We refer also to the Phd thesis <cit.> where periodicity in the Heisenberg group (but also in more general structures as Grushin spaces) is studied in  details with many properties and examples.
Given any Ω⊂^N, we say that Ω is H-periodic with period T>0,
 whenever L_Tk(Ω)=(Tk)* Ω = Ω for all k∈^N.

 For later use in the paper we  fix the period T=2. In fact, recalling that L_y∘  L_z(x)=
 L_y* z(x),  the composition of two left-translations with period T=1  is not anymore a integer left-translation since k* h ∉^N, because the third component becomes
 k_3+h_3+
 k_1h_2-k_2h_1/2, which is in general not anymore an integer.

 Instead the composition of two left-translations with period 2 is still a translation of the same type since,
  for all k, h∈^N,
 2k* 2h=2z 

 with z=(k_1+h_1,k_2+h_2,
 k_3+h_3+k_2·h_1
 -k_1·h_2
 )∈^N.
 (Note that z≠ k*h since the third component is different by a factor 1/2 in the mixed term.)
 One could very simply  adapt everything to period T=1 by choosing a different representation of the Heisenberg group, where the group law is expressed by polynomials with integer coefficients; in that case the unit cell needs to be rescaled   to a unit cube (e.g. [-1/2,1/2]^N), see e.g. <cit.>.
 
 We introduce the following simplified notation for the left-translations with period 2, that is

    τ_k(x):=2k* x,
      ∀ k∈^N,   x∈^N.

We recall that, for all k,h∈^N, the following properties hold true:

    τ_k∘τ_h
    =τ_k* h  and  τ_k^-1=τ_-k.


A definition of periodicity adapted to the Heisenberg group structure can be given for functions as follows.


We say that the function
f:^N→  is H-periodic  whenever

    f(τ_k(x))=f(2k* x)=f(x),
      ∀  x∈^N, k∈^N.


To construct a large class of periodic functions we need to introduce a H-periodic tiling of ^N.
Thus we consider the semiopen cube Q=[-1,1)^N. We call Q unit cell and consider
τ_k(Q)
=2k*Q.
Then one can easily show that the family
{τ_k(Q)}_k∈^N
fullfills

    ⋃_k∈^Nτ_k(Q)=^N
      and  τ_k(Q)∩τ_h(Q)=∅,
     ∀  k≠ h.

see Figure <ref>  and   <cit.>.







We next want to highlight a few facts about the scaling of tilings since this will be crucial later when we will study our homogenisation problem.
First recall that, in the Heisenberg group, if we scale the unit cell, then we do not get anymore hypercubes but hyper-rectangles since the scaling is anisotropic. In Figure <ref> we show how the cube Q scales for t>1 and for t<1.
Then if we want to build a tiling of ^N  starting by a rescaled cell, we need to be very careful and adapt the translations to the Heisenberg  scaling.

Given the unit cell  Q=[-1,1)^N and a t>0,
the scaled unit cell as 
    Q^t=δ_t(Q)  and   Q^t_k:=τ_δ_t(k)(Q^t),

then the family
{Q^t_k}_k∈^N
is  a tiling of ^N in the sense that

    ⋃_k∈^NQ^t_k
    =^N
      and  
    Q^t_k∩
    Q^t_h=∅,
     ∀  k≠ h.



The result follows easily from the properties for t=1 and  from the fact that
δ_t(k)*δ_t(Q)=δ_t(k*Q).







§ A CLASS OF DEGENERATE FUNCTIONALS.





Affine functions can be introduced in different ways in the Heisenberg group setting and they have been  studied in <cit.>.
For the purpose of the paper, we say that a function u:^N→ is H-affine (in the Heisenberg group) if

    u(x)=q·π_m (x)+a,

 for  m=2n and for some q∈^m and a∈,  where π_m :^N→^m is the projection on the first m components and · is the standard inner product on ^m.   The following lemma is an immediate  property of H-affine functions in all Carnot-type groups and it will be key  for  our later results.
 
  For all fixed q∈^m, we have
 
    ∇_ u=q
       ⇔   
     u(x)=q·π_m (x)+a,

 for some a∈ and for all x∈^N.
 
 
 One implication (from the right to the left) 
 follows trivially from the fact that  q·π_m (x)+a does not depend on the last coordinate and the structure of the horizontal gradient.
 

 The other implication follows from the fact that ∇_ u=q means X_iu(x)=q_i=constant for all i=1,…,m, then 
 
    u_2n+1=[X_1,X_n+1]u =X_1(X_n+1u)-X_n+1(X_1u)=X_1(q_n+1)-X_n+1(q_1)=0,

 where we indicate by u_i the partial derivative of u w.r.t. the variable x_i, for i=1,…,N.
Using u_2n+1(x)=0, for all x∈^N, X_iu(x)=q_i implies u_i(x)=q_i for all i=1,…, 2n, which gives
 u(x)=q·π_m (x)+a  for some a∈.
 
 We will later often use the following notation  for H-linear functions:

    l_q(x)=q·π_m (x).

 


We next recall that the definition of Sobolev spaces in the setting of Hörmander vector fields, which applies in particular to the Heisenberg group. We refer to <cit.> 

 for more details on these spaces.

Let k>1 be an integer, 1≤α≤ + ∞ and A a domain on ^N. We define the space 

    W_^k,α(A) = 
    {  
    u ∈ L^α(A)  |
      ^J u ∈ L^α(A) ,  ∀ J ∈^m,   |J| ≤ k  }

where ^Ju=X_1^J_1…  X_m^J_mu for J=(J_1,…,J_m).
Endowed with the norm

    u_ W_^k,α (A)
    = ( ∑_|J|≤  k ∫_A  
    |^J u |^α  dx  )^1/α,

W_^k,α(A) is a Banach space, and is an Hilbert space in the case α=2.


Moreover, for any 1≤α<+ ∞, the embeddings

    W_^k, α(A) ↪ W^k/r, α(A)  ,

 hold true, where r is the step of the  stratified  associated Lie algebra, thus in Heisenberg group r=2
 (see e.g. <cit.>). 
Later we will also need the following compact embedding. 

 W_^k, α(A)
is compactly embedded into L^α(A). 


This follows from the previous embedding and the fact that the 
fractional Sobolev space  W^k/r, α(A) is compactly embedded into L^p(A) (see e.g. <cit.>).  



For each domain A⊂^N, we indicate by

    W_,0^k,α(A)

the closure of C^∞_0(A) w.r.t. the Sobolev norm
·_ W_^k,α (A).

 This means that, whenever the  boundary ∂ A is regular enough, the trace of u vanishes on the boundary of the set.

We will use this notation to express the Dirichlet boundary conditions:
 more precisely

    u-u_0∈ W_,0^k,α(A)

are all the functions 
u∈ W_^k,α(A) 
   which coincide  on ∂ A
   (in the sense of Sobolev space) 
   with  some u_0∈ W_^k,α(A).   
 
 Next we recall the following Poincaré inequality, which is key for later results.
 
 
 
 Given a bounded domain A⊂^N,
 then there exists a constant C>0 such that
 
    ∫_A |u|^α dx≤ C
      ∫_A |∇_ u|^α dx,
        ∀   u∈ W^1,α_,0(A).

 
 
 This follows from the results in <cit.>.
 
 
 

 Consider now a function  f:^N×^m→ with N=2n+1 and m=2n, we introduce the integral functional defined, for all  domain A⊂^N, as

    F(u,A):=
    {∫_A f(x,∇_ u)  d  x,     
    u∈  W_^1,α (A),
    
    +∞,     else.
    .

We introduce  the following properties for the integrand function

f:^N×^m→ (with N=2n+1 and m=2n)

    f(x,q)  is measurable in  x   and convex in  q;



    ∃ 
    C_1,C_2>0 and α>1  such that
       C_1|q|^α≤
    f(x,q)
    ≤ C_2(|q|^α+1
    ),   q∈^m,  x∈^N.


Moreover for the later homogenisation problem we will assume H-periodicity for the functional in the sense of Definition <ref>; more precisely

    f(τ_k(x),q)=f(2k* x,q)=f(x,q),
      ∀  x∈^N, q∈^m, k∈^N.
 



The main example is f(x,q)=a(x)|q|^α, which trivially satisfies (<ref>) and (<ref>) whenever a:^N→ is bounded (with a strictly positive lower bound) and measurable,
while assumption (<ref>) is equivalent to requiring that  a(·) is H-periodic.



We want to study the minimisation problem for F(u,A) with H-affine boundary condition, i.e.

    m(F,u_0,A):=min{F(u,A) |  u-u_0∈ W_,0^1,α(A)
    },

with u_0(x)=q·π_m(x)+a for some  q∈^m and a∈.


 

Note that  under assumptions (<ref>) and (<ref>),  the infimum of F(u,A) on the set of  functions u such that  
u-u_0∈ W_,0^1,α(A) is indeed a minimum by standard arguments, using the convexity, the embedding in Lemma <ref> and the  Poincaré inequality (see Lemma <ref>).





§ A Γ-CONVERGENCE RESULT FOR DEGENERATE FUNCTIONALS.


To keep the paper self contained we next  recall briefly the definition of Γ-convergence.



 §.§ Very brief introduction to Γ-convergence.

In homogenisation theory, we consider a family of solutions to equations with rapidly oscillating coefficients and investigate if they converge to a solution of a homogenised equation with slowly oscillating or constant coefficients. If these equations are the Euler-Lagrange equations of a suitable family of functionals with rapidly oscillating coefficients, and if both minimisers and solutions of the Euler-Lagrange equation are unique, then we can study convergence of the family of functionals instead. 

We need a notion of convergence of functionals which guarantees that minimisers of the approximating functionals converge to minimisers of the limit functional.


A suitable mathematical setup to make this rigorous is
the notion of Γ-convergence.
Let us  briefly recall the definition of Γ-convergence (see <cit.> for more details on this subject).




Let U be a metric space and for ε >0 let F_ε: U→
be a family of functionals on U. 
We say that F_ε 
Γ-converge to F: U→ 
if the following conditions are verified:


  * for all u∈ U and for all u_ε→ u, there holds 
lim inf_ε→ 0F_ε(u_ε)≥ F(u) 
 (Γ-liminf inequality);

  * for all u∈ U there exist u_ε→ u, such that 
lim_ε→ 0F_ε(u_ε)= F(u) 
 (Γ-limsup inequality).



The convergence of minimisers to minimisers is formalised in the following way, 
which can be easily derived from Definition <ref>.


If F_ε Γ-converge to F in U, also the corresponding minimal values (or infima) 
converge. Moreover, if u_ε is a minimiser of F_ε and u_ε→ u∈ U, 
then u is a minimiser of F. 


Hence, the asymptotic behaviour of minimisers of F_ε (and therefore solutions of the Euler-Lagrange equations, see Section 6) can be partly understood by considering 
the Γ-limit of F_ε. 

Moreover, Γ-convergence has nice compactness properties, i.e. in general it is easy to show that a Γ-limit along subsequences exists. The problem is then to identify this limit (see Section 5) and to show properties of this limit, in particular that it is again an integral functional.




 §.§ Γ-convergence limit.



We say that a family
(A_ρ)_ρ>0 of open subsets of ^N with Lipschitz boundary  is a substantial family (around x) as ρ→0^+ if, for every positive ρ, there hold

    A_ρ⊂ B(x,ρ) := {y∈^N |  |y-x|<ρ}  and  
    |A_ρ|≥ c |B(x,ρ)|,

where c is a constant independent of ρ (see the monograph <cit.> for other properties). 

The following result states that the integral function f can be obtained from the minima of the Dirichlet problem for 
F with affine boundary data. To this purpose, for any domain
A with Lipschitz boundary and for every u_0 H-affine data, we introduce the following regularised variational problem

    m_reg(F,u_0,A):=inf{F(u,A) |  u∈ C^∞(^N),  u=u_0 on ∂ A
    }.

Since the functional depends on u only through its horizontal gradient, therefore the constant a in the definition of H-affine function does not affect the results; we now consider directly the
H-linear functions defined in (<ref>) as boundary data.
We state now a useful property, key for the later results.



Let A be a N-dimensional domain with Lipschitz boundary.
For q∈^m and
l_q(y)=q·π_m(y),  and for every smooth function u such that u=l_q on ∂ A,
we have

    ∫_A∇_ u  dy = q  |A|.



We write y=(y_1,y_2,y_3)∈^N≡^n×^n×.
First
we show that

    ∫_A∇_ u  dy =  ∫_∂ A(q·π_m) ν_0 d ^N-1∈^m,

where ν_0 is the horizontal normal, i.e. ν_0=σ^T ν with σ matrix of the vector fields  defined in (<ref>) and ν the outward unit normal to ∂ A, while d ^N-1 is the Hausdorff measure defined on ∂ A. 
To prove the claim (<ref>)
we use that the vector fields  in the Heisenberg group are divergence free
  and we combine a simply integration by parts  with Remark <ref>, which gives
  
    ∫_AX_i u  dy=
    ∫_∂ Au  ν^i_0 
    d ^N-1
    = ∫_∂ A(q·π_m) ν^i_0 d ^N-1,

where ν^i_0 is the i-component of ν_0  for i=1,…, m=2n.

Then we can use the divergence theorem again (together with the fact that the vector fields in Carnot groups are divergence free) to conclude:

    ∫_∂ A(q·π_m) ν^i_0 d ^N-1=
     ∫_A  X_i (q·π_m)  d  x= 
    q_i  ∫_A d  x=
     q_i |A|,

for all i=1,…, m.


 We now use the previous lemma to show that, whenever the integrand function f does not depend on x, then  H-affine functions are minimisers for 
 problem (<ref>) with H-affine boundary condition.
 
 
 Given a domain with Lipschitz boundary A⊂^N, consider the problem (<ref>) with u_0(x)=q·π_m(x)+a for some q∈^m and a∈,
and  F defined in (<ref>) with f(x,q)=f(q) convex, then
 
    m_reg(F,u_0,A)=∫_Af(∇_u_0)  d x=f(q)|A|.

 

The result follows by combining Lemmas <ref> and  <ref> with Jensen's inequality.


Now following the arguments of Dal Maso-Modica <cit.> for the standard non degenerate case, we prove that the integrand function f can be retrieved in term of m_reg.


Under assumptions (<ref>)-(<ref>),
there exists a measurable subset N of ^N with | N|=0 such that

    f(x,q)=lim_ρ→0^+m_reg(F,l_q,A_ρ)/|A_ρ|,

for every q∈^m and with l_q H-linear as in (<ref>), x∈^N∖ N and every substantial family (A_ρ)_ρ>0 around x.


We will use the same arguments as in <cit.> for the non-degenerate case.
We just sketch the main steps.

 Let us at the moment  assume  that there exists some R>0 such that f does not depend on x for |q|>R.
Then by Jensen's inequality and by Lemma <ref>, we obtain

    inf{∫_A_ρf(x,∇_ u)  dy |  u∈ C^∞(^N),  u=l_q on ∂ A_ρ}   =   ∫_A_ρf(x,q)  dy 
       =     |A_ρ|f(x,q),

for every q∈^m, ρ>0, x∈^N and l_q defined by (<ref>).
Exactly as in <cit.>, we can deduce

    |f(x,q)-m_reg(F, l_q, A_ρ)/|A_ρ||≤1/|A_ρ|∫_A_ρsup_q∈^m|f(x,q)-f(y,q)|  dy.

It remains to prove that there exists a measurable subset N of ^N with | N|=0 such that

    lim_ρ→0^+1/|A_ρ|∫_A_ρϕ(x,y)  dy=0,

for every x∈^N∖ N and every substantial family {A_ρ} around x,
where

    ϕ(x,y):= sup_q∈^mω(x,y,q),  ω(x,y,q):=|f(x,q)-f(y,q)|,  ∀ x,y∈^N,q∈^m.

We observe that ω(x,y,q)=0 for every q∈^m with |q|≥ R and that, arguing as in <cit.> (recall that f is convex w.r.t. q), there exists a positive constant K such that: |ω(x,y,q_1)-ω(x,y,q_2)|≤ K|q_1-q_2| for every x,y∈^N and q_1,q_2∈^m.

Fix a dense subset D of ^m. The Lebesgue's Differentiation Theorem ensures that there exists a measurable subset N of ^N with | N|=0 such that

    lim_ρ→0^+1/|A_ρ|∫_A_ρω(x,y,p)  dy=0,    ∀ x∈^N∖ N, p∈ D,

for every substantial family {A_ρ}  around x. Moreover, as in <cit.>, for every ε>0, there exists a finite set {p_1,…,p_k}⊂ D such that

    ϕ(x,y)≤∑_i=1^kω(x,y,p_i)+Kε,     ∀ x,y∈^N.

Therefore, we infer

    lim sup_ρ→0^+1/|A_ρ|∫_A_ρϕ(x,y)  dy≤ Kε,
        ∀ x∈^N∖ N,

for every substantial family {A_ρ}  around x.
By the arbitrariness of ε, we accomplish the proof.


Let us now remove the additional assumption of Step 1. Taking into account the convexity and the coercivity of f w.r.t. q, by the same arguments as in <cit.>, we obtain that there exists a measurable set N'⊂^N, with | N'|=0, such that

    f(x,q)≥lim sup_ρ→ 0^+m_reg(F,l_q,A_ρ)/|A_ρ|,    ∀ x∈^N∖ N',  q∈^m,

for every family (A_ρ)_ρ>0 as in the statement.
In order to obtain the reverse inequality, we first observe that the same arguments of <cit.> ensure that there exists an increasing sequence {f_i}_i∈ of functions such that f=sup_if_i and each f_i satisfies the assumptions of step 1. For each i∈, we denote F_i and N_i respectively the corresponding functional and the negligible set given by step 1. We set N”:=∪_i=1^∞ N_i.
Step 1 for f_i and the inequality f≥ f_i entail

    f_i(x,q)=lim_ρ→0^+m_reg(F_i,l_q,A_ρ)/|A_ρ|≤lim inf_ρ→0^+m_reg(F,l_q,A_ρ)/|A_ρ|,  ∀ x∈^N∖ N”, q∈^m,

for every family (A_ρ)_ρ>0 as in the statement. Passing to the limit as i→+∞, one deduces

    f(x,q)≤lim inf_ρ→0^+m_reg(F,l_q,A_ρ)/|A_ρ|,    ∀ x∈^N∖ N”,  q∈^m,

for every family (A_ρ)_ρ>0 as in the statement. Finally, we accomplish the proof by choosing N= N'∪ N”.


We denote by F= F(α,C_1,C_2) the set of all functional F which satisfy assumptions (<ref>)-(<ref>) with the same constants α, C_1 and C_2. In the next result, we obtain a characterization of Γ-convergence in terms of the convergence of the minima of problems with Dirichlet boundary conditions.
We like also to mention that very recently some results in this direction have been proved in <cit.> in much more general geometries but with quite different techniques.


Let {F_n}_n∈ be a sequence of functionals in F. Let D be a dense subset of ^m. Let B be a family of open bounded subsets of ^N which contains a substantial family around every point x∈^N. Assume that for each q∈ D and for each B∈ B there exists lim_n m(F_n,q,B).
Then, there exists a functional F_∞∈ F such that the sequence {F_n}_n∈ Γ-converge to F_∞ and

    lim_n→+∞m(F_n,l_q,A)=m(F_∞,l_q,A),

for every q∈^m and for every A  bounded domain of ^N with Lipschitz boundary.
 

The proof follows exactly the same arguments of the proof of <cit.> so we just sketch the main issues.
We first claim that the space F can be endowed with a metric d such that ( F,d) is a compact metric space and a sequence {F_n}_n of functionals in F is convergent w.r.t. to d to some F∈ F if and only if it Γ-converges to F.
Indeed, this property can be obtained following the same arguments of <cit.> and taking advantage of the properties of W_,0^k,α and of W_^k,α for the Heisenberg group, in particular the Rellich compact injection  and the Poincaré inequality respectively in Lemma <ref> and in Lemma  <ref>. Hence, we shall omit it.

Even if the rest of the proof follows the arguments in <cit.>, for the sake of completeness, let us recall the role of Theorem <ref>. Let F_k_1(n) and F_k_2(n) two subsequences of F_n which Γ-converge respectively to some F_∞' and to some F_∞”. We claim: F_∞'=F_∞”.
Actually, we have

    m_reg(F_∞',l_q,B)=m_reg(F_∞”,l_q,B),     ∀ q∈^m,  B∈ B.

Theorem <ref> ensures that there exists a measurable set N⊂^N, with | N|=0, such that

    f_∞'(x,q)=f_∞”(x,q),    ∀ x∈^N∖ N,  q∈ D,

where f_∞' and f_∞” are the integrands of F_∞' and respectively of F_∞”. Finally, the convexity of f_∞' and of f_∞” permits to extend the previous equality to every q∈^m.







§ PERIODIC HOMOGENISATION FOR DEGENERATE FUNCTIONALS WITH H-AFFINE DATA.


Given the functional F(u,A) defined in (<ref>), we now introduce for all ε>0 the following rescaled  functionals:

    F_ε(u,A)=(ρ^H_εF)(u,A)
    :=
    {∫_A
    f((x), ∇_u(x)
    ) d  x,      u∈ W^1,k_(A) 
    
    +∞,     else,
    .

and for all
z∈^N, the following translated  functionals:

    (τ^H_zF)(u,A):=
    {∫_A f(z*
    x,∇_ u(x))  d  x,     
    u∈  W_^k,α (A)
    
    +∞,     else.
    .

Following the idea in <cit.>, for all fixed q∈^m, for all bounded domain A⊂^N, and with N=2n+1 and m=2n,
 we introduce the following notation

    μ_q(A):= m(F,l_q,A)=min{∫_Af(x,∇_ u(x))  d x  |
    u-l_q∈ W^1,α_,0 (A)
    },

where we recall that l_q(x)=q·π_m(x) is a H-affine boundary data.


We next define  

    τ_z^Hμ_q (A):=μ_q(τ_z^H(A))=μ_q(z* A).



Given a bounded domain A of ^N, there holds


    τ_z^Hμ_q (A)=min{
    (τ^H_zF) (w,A) |
    w-l_q∈ W^1,α_,0 (A)
    }.



Note that since the functional F depends only on the gradient of the function, F(w,A)=F(w+l_q(z),A). Thus to prove (<ref>) is the same of proving

    τ_z^Hμ_q (A)=
    min{(τ^H_zF)(w+l_q(z),A)
      |
    w-l_q∈ W^1,α_,0 (A)
    }.

In order to prove (<ref>)
we start looking at the right-hand side and defining
 v(x):=w(x)+l_q(z). Since w and v differ only by a constant, obviously

    min{(τ^H_zF)(w+l_q(z),A)
      |
    w-l_q∈ W^1,α_,0 (A)
    }
    
    =   min{∫_A
    f(z* x,∇_[w(x)+l_q(z)
    ])  d x 
      |
    w-l_q∈ W^1,α_,0 (A)
    }
    
    =   min{∫_A
    f(z* x,∇_ v(x)
    )  d x 
      |
    v-L_z(l_q)∈ W^1,α_,0 (A)
    },

where we recall that L_z(l_q)(x):=l_q(z* x) by definition of translated function.

Now we consider the following change of variables y=z* x (equivalently x=z^-1* y where z^-1 is the inverse element w.r.t. the group law *).

An easy computation shows that the Jacobian of the change of variables is exactly the matrix σ_Ext defined in (<ref>). 
Then property (<ref>) tells that | J|=1. Since X_i are defined as left-invariant vector fields 
for all i=1,… m
(see (<ref>)) we also know  that

    ∇_ v (x)=∇_(L_z^-1 (v))(x)=∇_ v(z^-1* x).

Moreover x∈ A if and only if y∈ z* A and 

    v-L_z(l_q)∈ W^1,α_,0 (A) 
       if and only if   
    L_z^-1(v)-l_q∈  W^1,α_,0 (z* A),
 
where 
L_z^-1(v)(x):=v(z^-1* x):
in fact  on ∂ (z* A) we have v(z^-1* y)=v(x)=l_q(z* x)=l_q(z* z^-1* y)=l_q(y).
Then in the new variables y=z* x we have

    min{∫_A
    f(z* x,∇_ v(x)
    )  d x 
      |
    v-L_z(l_q)∈ W^1,α_,0 (A)
    }
    
    =   min{∫_z* A
    f(y,∇_ v(z^-1* y)
    )  d y
      |
     L_z^-1(v)-l_q
     ∈ W^1,α_,0 (z* A)
    }.

To conclude we now define 
u:=L_z^-1
(v). 
Using again the property of left-invariant vector fields, we have 
∇_
v (z^-1* y)
=∇_ u(z* z^-1* y)= ∇_ u(y),
 then

    min{∫_z* A
    f(y,∇_ v(z^-1* y)
    )  d y
      |
     L_z^-1(v)-l_q
     ∈ W^1,α_,0 (z* A)
    }
    
    =   min{∫_z* A
    f(y,∇_ u(y)
    )  d y
      |
    u-l_q
     ∈ W^1,α_,0 (z* A)
    }
    =τ_z^Hμ_q (A).

The chains of identities in (<ref>), (<ref>) and (<ref>) give identity (<ref>) and conclude the proof.


The following result is an immediate consequence of the previous lemma in the case of H-periodic functionals.


Assume (<ref>), then, for all bounded  domains A⊂^N  and 
for all q∈^m and z∈^N

    τ_z^Hμ_q (A)=μ_q (A).



In the following lemma we show how the assumptions on the integrand f(x,q) are inherited by μ_q(A).


Let A be a bounded domain in ^N with Lipschitz boundary, q∈^m and μ_q(A) defined in (<ref>) and let  f:^N×^m→ be measurable. 


  (i) If f satisfies assumption (<ref>), we have

    C_1|q|^α|A|≤μ_q(A)≤ C_2(|q|^α+1) |A|,

where C_1,C_2 and α are the same constants given in  (<ref>).

  (ii) If f satisfies  assumption (<ref>), we have
 for all q_1,q_2∈^m and for all λ∈ (0,1)

    μ_λ q_1+(1-λ)q_2(A)≤λμ_q_1+(1-λ)μ_q_2.




Using that l_q= q·π_m is admissible for the minimum defining μ_q(A) and that 
∇_
l_q(x)=q, we get

    μ_q(A)≤∫_Af(x, ∇_ l_q(x))  d x
    =
    ∫_Af(x,  q)  d x
    ≤ 
    C_2(|q|^α+1)|A|.

Moreover

    μ_q(A)   =
    min{∫_Af(x,∇_ u(x))  d x  |
    u-l_q∈ W^1,α_,0 (A)
    }
       ≥ C_1
    min{∫_A|∇_ u(x)|^α  d x  |
    u-l_q∈ W^1,α_,0 (A)
    }
       =C_1∫_A|∇_ l_q(x)|^α  d x=
    C_1|q|^α|A|,

where for the last identity we  use Lemma <ref> for the convex function f(x,q)=|q|^α, which tells that the minimisers are the H-affine functions, whenever f does not depend on x. 

It remains to prove (ii). To this end it is enough to remark that for all functions u_1 and u_2 which are admissible respectively  for μ_q_1 and μ_q_2, then u:=λ u_1+(1-λ)u_2 is admissible for μ_λ q_1+(1-λ)q_2, which implies

    μ_λ q_1+(1-λ)q_2(A)   ≤∫_A f(x,∇_u(x))d x
       
    =∫_A f(x,
    λ∇_u_1(x)+(1-λ)∇_u_2(x)
    )
     d x
       ≤λ∫_A f(x,
     ∇_u_1(x)) d x
     +
    (1-λ)
    ∫_A f(x,
     ∇_u_2(x)) d x.

Taking the minimum over all admissible u_1 and u_2, we get property (ii).


To prove the convergence of the functional F_ε(u,A) as ε→ 0^+, we need to show now a sort of  Akcoglu-Krengel  type result (see <cit.>) for periodic functionals, adapted to the anisotropic structure of the Heisenberg group. 
In <cit.>  the authors prove a very interesting Akcoglu-Krengel  type result for general metric measure spaces.  We need to mention that unfortunately the result therein does not apply to our case. In fact it is quite easy to show that the Heisenberg group endowed with the Carnot-Carathéodory metric (or also with the homogeneous metric) and the Lebesgue measure is a (G,{δ_t}_t>0)-metric measure space 
where G is the subgroup of homeomorphisms on the Heisenberg group defined by the left-translations w.r.t. an element in ^N. Nevertheless one can also show that in general that space is not “meashable”  according to the definition introduced in <cit.>.
  We give a self-contained proof which can be later adapted to the stochastic case (which will be a topic in a forthcoming paper, see Section 6).
  
  We now recall that, defining for all t>0, 
 Q^t=δ_t(Q),
  we know that |δ_t(Q)|=t^𝒬|Q| (see (<ref>)), where 𝒬 is the homogeneous dimension, then in ^n in particular 𝒬=2n+2=N+1. 




The next lemma tells that, as t→+∞, we can reduce to take the limits only over integer subsequences.


Assume that the limit exists for integer sequences, i.e.  for h∈,

    lim_h→∞μ_q(Q^h)/|Q^h|=:C.
 
 Then for all sequences {t_k}⊂ with t_k→∞ 
it holds

    lim_k→∞μ_q(Q^t_k)/|Q^t_k|=C.



For t>0 we define

    e_t:= μ_q(Q^t)/|Q^t|.

Fix ε>0 and choose N  large enough that
|C-e_h|<ε for h≥ N.

Denote by C^+:=lim sup_k→∞e_t_k, and 
C^-:=lim inf_k→∞e_t_k, which are both finite by Lemma <ref>.
We can find k such that 

    e_t_k≥ C^+-ε  and   t_k>N.

Define N_k:=[ t_k]≥ N,  (where by [·] we indicate the integer part of a real number)
and let u_k be a function with H-affine boundary conditions on Q^N_k such that F(u_k, Q^N_k)=μ_q(Q^N_k). We extend
u_k to Q^t_k by letting it equal to the boundary condition on Q^t_k∖ Q^N_k,
i.e. u_k: ^N→ given by

    u_k(x):=
    {[      u_k(x), if x∈ Q^N_k,;      l_q(x),        else, ].

whose restriction to Q^t_k is an admissible function for  μ_q(Q^t_k).
Note that

    f(x,∇_u_k)=f(x,∇_l_q)=f(x,q)≤ C_2(|q|^α+1)  on  
    Q^t_k∖ Q^N_k,

 hence

    F(u_k,Q^t_k)= ∫_Q^N_kf(x,∇_u_k)dx+∫_Q^t_k∖ Q^N_k
    f(x,q)dx≤  F(u_k,Q^N_k) +C| Q^t_k∖ Q^N_k|,
 where the constant depends on q and α.
 Since u_k is admissible for Q^t_k, so μ_q(Q^t_k)≤ F(u_k,Q^t_k),
we estimate

    C^+   ≤    e_t_k+ε≤F(u_k,Q^t_k)/|Q^t_k|+ε≤F(u_k,Q^N_k)/|Q^t_k| +ε+ C| Q^t_k∖ Q^N_k| /|Q^t_k|
       =   ε+μ_q(Q^N_k)/|Q^t_k|+C| Q^t_k∖ Q^N_k| /|Q^t_k|=e_N_k|Q^N_k|/|Q^t_k|+ε+ C| Q^t_k∖ Q^N_k| /|Q^t_k|.

Note that 
    lim_k→∞|Q^N_k|/|Q^t_k|=1  and  lim_k→∞| Q^t_k∖ Q^N_k| /|Q^t_k|=0,

so, by choosing, if necessary, N larger, we can make the right hand side ≤ C+3ε, thus, as ε was arbitrary,  we have shown C^+≤ C.

For the opposite inequality, we use estimates similar to what we did before:  we can find infinitely many k such that 

    e_t_k≤ C^-+ε  and  
     t_k> N.

Therefore we take N_k=[t_k]+1 and let u_k be a function with H-affine boundary condition l_q  on Q^t_k such that F(u_k, Q^t_k)=μ_q(Q^t_k). We extend
u_k to a function  u_k on Q^N_k which is admissible for μ_q(Q^N_k) and equals l_q on
Q^N_k∖ Q^t_k. 
Arguing as before we get

    F(u_k,Q^N_k)≤ F(u_k,Q^t_k)+C| Q^N_k∖ Q^t_k|.
 Then

    C    ≤     e_N_k+ε≤F(u_k,Q^N_k)/|Q^N_k|+ε≤F(u_k,Q^t_k)/|Q^N_k| +ε+ C| Q^N_k∖ Q^t_k| /|Q^N_k|
       =   ε+μ_q(Q^t_k)/|Q^N_k|+C| Q^N_k∖ Q^t_k| /|Q^N_k|=e_t_k|Q^t_k|/|Q^N_k|+ε+ C| Q^N_k∖ Q^t_k| /|Q^N_k|
       ≤   
    (C^-+ε)+(C^-+ε)(|Q^t_k|/|Q^N_k|-1 )+ε +C| Q^N_k∖ Q^t_k| /|Q^N_k|.

By choosing, if necessary, N larger, we can make the right hand side smaller than C^-+(2C^-+3)ε, thus we have shown C≤ C^-, but as C^-≤ C^+, we have C^-=C=C^+.










 We denote ^* the set of natural numbers excluding 0. We next prove an Akcoglu-Krengel  type result.



Let consider the (semiopen) unit cell Q=[-1,1)^N and let q∈^m and μ_q be defined in (<ref>). Assume  that f is measurable and satisfies (<ref>) and (<ref>), then 

    lim_k→ +∞μ_q(Q^k)/|Q^k|=C_q,

where C_q is the non-negative constant given by

    C_q=inf_k∈^*μ_q(Q^k)/|Q^k|
    .



Note that, by Lemma <ref> - (i),

    C_1|q|^α≤μ_q(Q^k)/|Q^k|≤ C_2(|q|^α+1),

so in particular C_q≥ 0.


Step 1. Since C_q is defined as  infumum over ^*, trivially 
μ_q(Q^k)/|Q^k|≥ C_q for all k∈^*,
which implies

    lim inf_k→ +∞μ_q(Q^k)/|Q^k|≥ C_q.


Step 2. We next show the limsup estimate.
Using the definition of infimum for C_q, for all ρ>0, and the definition of μ_q,
 then there exists k_ρ∈^*  and u_ρ∈ W^1,α_(Q^k_ρ)
 ∩ C^∞  (Q^k_ρ) 
 and such that 

    μ_q(Q^k_ρ)/|Q^k_ρ|≤ C_q+ρ/2,
       F(u_ρ,Q^k_ρ)/|Q^k_ρ|≤μ_q(Q^k_ρ)/|Q^k_ρ|+ρ/2,

where F is the functional defined in (<ref>). This sums up as follows

    F(u_ρ,Q^k_ρ)/|Q^k_ρ|≤
    C_q+ρ.

Recall the definition of 
τ_k given in
(<ref>);
we use such translations  to extend u_ρ to the whole ^N by translating periodically the gradient. 
More precisely, 
let us introduce 

    j_ρ:=
    δ_k_ρ(j)
      and  
    Q^ρ_j:= τ_j_ρ(Q^k_ρ),
      ∀  j∈^N.

Using that ^N=⋃_j∈^N Q^ρ_j, we can define

    U_ρ(x):=∑_j∈^N(
    q·π_m(j_ρ  )+ u_ρ(τ_- j_ρ(x))
    ) _ Q^ρ_j,

where by _A we indicate the 
characteristic function of the set A;
recall also that τ_k^-1=τ_-k.
The function U_ρ is well-defined since 
Q^ρ_j are all disjoint. 
We can easily check that U_ρ is continuous on ^N:  
in fact, for x∈  Q^ρ_j, then
U_ρ (x)=q·π_m(j_ρ)+
u_ρ(τ_-j_ρ(x)) and, whenever x∈∂ Q^ρ_j, we have  τ_-j_ρ(x)∈∂ Q^k_ρ which implies

    U_ρ (x)=q·π_m(j_ρ ) +q·π_m(τ_-j_ρ(x))=
    q·π_m(j_ρ ) +q·(π_m(-j_ρ)+π_m(x))=
    q·π_m(x),

which does not anymore depend on j.
The continuity of U_ρ on ^N, together with the  fact that U_ρ∈ W^1,α_(Q^ρ_j), imply that
U_ρ∈ W^1,α_,loc(^N). 


We next introduce the following two objects:

    S^ρ_k:={
    j∈^N|  Q_j^ρ⊂ Q^k=δ_k(Q)
    },
       S^ρ_k:=⋃_j∈ S^ρ_kQ_j^ρ,

and we construct a new function v_ρ, which is admissible for μ_q(Q^k), as

    v_ρ(x):=
    {   U_ρ(x),
           x∈S^ρ_k
       q·π_m(x),
        x∈  Q^k\S^ρ_k.
    .

By definition v_ρ-l_q∈ W^1,α_,0(Q^k), and

    F(v_ρ, Q^k)   =
    ∫_Q^k
    f(x,∇_v_ρ(x)) d x
    
       =
    ∫_S^ρ_k
    f(x,∇_v_ρ(x)) d x+
    ∫_Q^k\S^ρ_k
    f(x,∇_v_ρ(x)) d x

First we compute

    ∫_S^ρ_k
    f(x,∇_v_ρ(x)) d x
    =
    ∫_S^ρ_k
    f(x,∇_U_ρ(x)) d x=
    ∑_j∈ S^ρ_k∫_ Q_j^ρ
    f(x,∇_U_ρ(x)) d x,

where we have used that Q_j^ρ are disjoint. If x∈ Q_j^ρ, then
U_ρ(x)
=q·π_m(j_ρ)+u_ρ(τ_-j_ρ(x)). By using that the vector fields are left invariant, we get

    ∇_U_ρ(x)=
    ∇_(u_ρ(τ_-j_ρ(x)))=
    ∇_u_ρ(τ_-j_ρ(x)).

Thus, by using the change of variables y=τ_-j_ρ(x) and recalling that the determinant of the Jacobian is 1  (see (<ref>)), we get the following chain of identities:

    ∑_j∈ S^ρ_k∫_ Q_j^ρ
    f(x,∇_U_ρ(x)) d x
    =
    ∑_j∈ S^ρ_k∫_ Q_j^ρ
    f(x,∇_u_ρ(τ_-j_ρ(x)
    )) d x
    
    =   ∑_j∈ S^ρ_k∫_ Q^k_ρ
    f(τ_j_ρ(y),∇_u_ρ(y)
    ) d y
    =∑_j∈ S^ρ_k∫_ Q^k_ρ
    f(y,∇_u_ρ(y)
    ) d y,

where in the last identity above we have used the periodicity assumption on f (see assumption (<ref>)). The integrals in the last term of (<ref>) do not depend anymore on j, then 

    ∑_j∈ S^ρ_k∫_ Q^k_ρ
    f(y,∇_u_ρ(y)
    ) d y   =
    card(S^ρ_k)
    ∫_ Q^k_ρ
    f(y,∇_u_ρ(y)
    ) d y
       ≤card(S^ρ_k) (C_q+ρ) |Q^k_ρ|,

where the last inequality follows from (<ref>).

Put together (<ref>),(<ref>) and (<ref>), we get the following estimate:

    ∫_S^ρ_k
    f(x,∇_v_ρ(x)) d x
    ≤card(S^ρ_k) (C_q+ρ) |Q^k_ρ|.


It remains to estimate the integral on the complementary of S^ρ_k by using that v_ρ(x)=q·π_m(x) for all x∈ Q^k\S^ρ_k by definition, hence

    ∫_Q^k\S^ρ_k
    f(x,∇_v_ρ(x)) d x
    ≤∫_Q^k\S^ρ_k
    f(x,q) d x≤
    C_2(|q|^α+1) |Q^k\S^ρ_k|.

Estimates (<ref>) and  (<ref>), together with the fact that v_ρ is admissible for μ_q(Q^k), give

    μ_q(Q^k)/|Q^k|≤F(v_ρ,Q^k)/|Q^k|≤card(S^ρ_k) (C_q+ρ) |Q^k_ρ|/|Q^k|+
    C_2(|q|^α+1) |Q^k\S^ρ_k|/|Q^k|,
 
where in the last inequality we have used that S^ρ_k⊂ Q^k and
| S^ρ_k|=card(S^ρ_k) |Q^k_ρ|, which together  imply
card(S^ρ_k) |Q^k_ρ|/|Q^k|≤ 1.

To conclude we  claim that the following limit holds true:

    lim_k→+∞|Q^k\S^ρ_k|/|Q^k|=0.

Then, by simply taking the limsup as k→ +∞ in the inequality (<ref>) and using claim 
 (<ref>), we get

    lim sup_k→ +∞μ_q(Q^k)/|Q^k|≤(C_q+ρ),

which conclude the proof as ρ→ 0^+.


It remains only now to prove claim
 (<ref>).
 By a simple rescaling we can actually show that this limit is the same as the one shown in the proof of Lemma 2.21 in <cit.>. In fact, set ε=1/k, then by using the properties of dilations (Lemma <ref>)
 
    Q^ε=δ_ε(Q)=
     δ_1/k_ρ(
      δ_k_ρ(
      δ_ε(Q)
      )
     )=
     δ_ε/k_ρ(Q^k_ρ).

Set Q:=δ_ε(Q^k_ρ)=
 δ_ε/k_ρ(Q^1/ε),  by using the properties of dilations and left-translations one can easily check that

    τ_δ_ε(j)
    (Q^ε)⊂Q  ⟺  τ_δ_k_ρ(j)
    (Q^k_ρ).

Thus by using the limit proved in  <cit.> we conclude the proof.





We define f_0:^m→
as

    f_0(q):=C_q,

where C_q is the limit proved in Theorem <ref>.

From Lemma <ref>, one can show that f_0 keeps the properties of f simply by passing to the limit as k→ +∞. More precisely


Given f:^N×^m→  measurable,
the following properties hold:


  (i) if assumption (<ref>) is satisfied, then 

    C_1|q|^α≤ f_0(q)≤ C_2 (|q|^α+1),

where C_1,C_2 and α are the same constants given in  (<ref>).

  (ii)  if  assumption (<ref>) is satisfied,
then for all q_1,q_2∈^m 

    f_0(λ q_1+(1-λ)q_2)
    ≤λ f_0(q_1)+(1-λ)f_0(q_2),
      λ∈ (0,1).





We now prove the main result of the paper.


Given a  bounded  domain A⊂^N with Lipschitz boundary, u:A→ and  the functional F(u,A) defined in (<ref>). Let us assume that (<ref>),  (<ref>) and  (<ref>)  hold true, and  u_0(x)=q·π_m(x)+a for some q∈^m  and  a∈.
Define the rescaled functionals F_ε introduced in (<ref>) and let us consider the corresponding minimisation problems for u-u_0∈ W^1,α_,0(A) (see (<ref>))
then 

    lim_ε→ 0^+ m(F_ε,u_0,A)
    =m(F_∞,u_0,A),
 where the limit functional F_∞ can be characterised as 

    F_∞(u,A):=
    {∫_A f_0(∇_ u)  d  x,     
    u∈  W_^1,α (A),
    
    +∞,     else,
    .

and  f_0:^m→ defined  as f_0(q)=C_q with  C_q constant given in Theorem <ref>.

Moreover the limit function f_0 is still measurable, convex and satisfies the same growth condition (<ref>) satisfied by f.



Applying Theorem <ref> we deduce that F_ε Γ-converge to some functional F_∞.
Let us now prove that the limit functional F_∞ can be identified as the integral functional associated to f_0 given in (<ref>). 
Choose as substantial family A_ρ:=[-ρ,ρ]^N, and
fix t=1/ε,  at the moment let us  assume the following claim:

    μ_q(δ_t(A_ρ))=t^𝒬
    m(F_ε, l_q,A_ρ),

with 𝒬 homogeneous dimension. 

By using that all the  previous results in Section 5 can be obtained replacing Q with the cube A_ρ,  we have

    m(F_∞,l_q,A_ρ)/|A_ρ|   =
    lim_ε→ 0^+m(F_ε,l_q,A_ρ)/|A_ρ| 
    =lim_t→ +∞1/|A_ρ|μ_q(δ_t(A_ρ))/t^𝒬
       =
    lim_t→ +∞μ_q(δ_t(A_ρ))/|δ_t(A_ρ)|
    =C_q,
      ∀ ρ>0.

By Theorem <ref>, passing to the limit as ρ→ 0^+,  we conclude f_0(q)=C_q.


It only remains to check claim (<ref>).
At this purpose, we use the change of variables y=δ_1/t(x); hence recalling definition (<ref>)
and using that δ_t(u) is the scaled function defined as δ_t(u)(x)=u(δ_t(x)), we have

    μ_q(δ_t(A_ρ))
       =min{∫_δ_t(A_ρ)f(x,∇_ u(x)) d x |  u-l_q∈ W^1,α_,0(δ_t(A_ρ))
    }
       =t^𝒬min{∫_A_ρf(δ_t(y),∇_ u(δ_t(y))) d y | δ_t(u)-l_tq∈ W^1,α_,0(A_ρ)
    },

by simply using that l_q(δ_t(y))=q·π_m(δ_t(y))=q· t π_m(y)=(tq)·π_m(y) and (<ref>). Defining the function w:=1/tδ_t(u) and  using   Lemma <ref>, we have that

    ∇_w(y)=1/t ∇_(δ_t(u))(y)=1/t  t ∇_u(δ_t(y))=∇_u(δ_t(y)).

Moreover informally we have  that, for y∈∂ A_ρ,  1/tu(δ_t(y))=1/tl_q(δ_t(y))=l_q(y). Hence 
 (<ref>)  gives
 
    μ_q(δ_t(A_ρ))
       =t^𝒬min{∫_A_ρf(δ_t(y),∇_ w(y)) d y |  w-l_q∈ W^1,α_,0(A_ρ)
    }
       =t^𝒬
    m(F_1/t, l_q,A_ρ),

 which proves claim (<ref>).

The properties for the limit function f_0 are proved in Lemma <ref>.




§ APPLICATIONS AND GENERALIZATIONS

We conclude listing further directions in which we are presently 
working, for some of  which we obtained already some partial results.









 §.§ Homogenisation for  functionals associated to Carnot groups
and the subelliptic p-Laplacian.


As mentioned in the introduction all the proofs never use the specific structure of the Heisenberg group but they instead  use properties true for all Carnot groups. So all the results apply without any modification to the general case of Carnot groups.



As it is well-known by Euler-Lagrange equations, we can connect minima of functionals  to solutions of PDEs. Whenever uniqueness holds this correspondence is one-to-one. Then our results can be used to study homogenisation for several subelliptic PDEs and in particular for the subelliptic p-Laplacian, which is defined, for  1<p<+∞, as 

    div_(<𝒜∇_u,∇_u>^p-2/2𝒜∇_u
    )
    =0,

where 𝒜(x) is a m× m symmetric matrix satisfying the usual  ellipticity condition.
Equations of this form have been studies by many authors, see e.g. <cit.> and references therein.
The functional associated to the p-Laplacian is

    F_p(u,A)=
    {∫_A|𝒜(δ_1/ε(x))∇_u(x)|^p  d  x,      
    u∈ W^1,p_(A)
    
    +∞,     else.
    .

Note that F_p satisfies all our conditions for all 1<p<+∞. Then we can apply Theorem <ref>. It remains now to show that the limit functional has still the structure of a functional associated to a subelliptic p-Laplacian equation (work in preparation).




 §.§ Stochastic functionals.

Another generalisation  is the case of 
 random functionals, i.e. integral functionals of the form

    u↦
    F_ε(u, A)=∫_Af(δ_1/ε(x),ω,∇  u(x)) dx,

where ω belongs to a probability space and 
the integrand f(x,ω,p) is stationary and ergodic  with respect to left translations. 
For a precise definition of stationary ergodic in the setting of Carnot groups we refer to
 <cit.>, where the authors  prove an homogenisation result for stochastic Hamilton-Jacobi equations.
 The general stationary ergodic case
 will be treated in a forthcoming  paper, but we sketch here a proof for the simpler situation of short correlated random variables. 

More precisely, we assume that the random integrand f(x,ω, p) satisfies  (<ref>) and (<ref>) uniformly in ω  and (<ref>) in law, i.e. the random integrand and its translations are not equal, but have the same law as random variables. In addition, we require that there exits a constant C>0 such that f(x,ω,p) and f(y,ω,p) are  independent, if d_h(x,y)>C,  where by d_h(x,y) we indicate the homogeneous distance in Carnot groups, i.e. for example in  1-dimensional Heisenberg d_h(x,y)=|y^-1*x|_h where |x|_h:=((x_1^2+x_2^2)^2+x_3^2)^1/4.
Note that this is different from being short correlated in the Euclidean distance.

Under these assumptions one can show along the lines of <cit.> that 

    lim_k→ +∞μ_q(ω, Q^k)/|Q^k|=C_q,

in probability to a constant C_q>0, and conclude convergence of the functionals in probability to an integral functional with constant integrand f_0(q)=C_q.

As a first step,  defining

    μ_q(Q):=𝔼(μ_q(ω,Q)),

one can show along the lines of Section 5 that 

    lim_k→ +∞μ_q(ω, Q^k)/|Q^k|=C_q,

for some constant C_q>0.
Note that because of the invariance in law, Lemma   <ref> holds for μ but not for μ(ω,·) with ω fixed.

Now fix k_0≫1 so large  that 

    |μ_q(Q^k)/|Q^k|-C_q|<δ/4,  for all  k≥ k_0

 and  now fix k>>k_0, we use the construction in step 2 of the proof of 
Theorem <ref> to show that

    μ(ω, Q^k)/|Q^k|≤|Q^k|/|Q^k_0|∑_j∈ S_k^k_0μ(ω, τ_jk_0(Q^k_0))/|Q^k_0|+o( 1).

The r.h.s. is a normalised sum over (k/k_0)^𝒬 independent, identically distributed random variables with mean close to C_q. By the weak law of large numbers, we 
have that for δ>0 and k sufficiently large the quantity

    β(δ):=ℙ({ω | μ(ω, Q^k)/|Q^k|>C_q+δ/4 })
 is small. Now define

    α(δ):=ℙ({ω | μ(ω, Q^k)/|Q^k|<C_q-√(δ)}).

We have

    C_q   ≤   𝔼(μ(ω, Q^k))/|Q^k|+δ/2
       ≤   α(δ)(C_q-√(δ)) +C_2(|q|^α+1)β(δ)
       +   ℙ({ω |  C_q-√(δ)≤μ(ω, Q^k)/|Q^k|<C_q+δ/4})+δ/2
       ≤   α(δ)(C_q-√(δ)) +C_2(|q|^α+1)β(δ) +(1-α-β)(C_q+δ/4)+δ/2
       ≤    C_q-α(δ)√(δ) +(3/4) δ+β(δ)C_2(|q|^α+1).

As we can make β(δ) arbitrarily small  by choosing k big, this implies that for such k also α(δ)→ 0 in order to avoid the contradiction C_q<C_q.


Acknowledgments. The first author was partially supported by EPSRC via grant EP/M028607/1.
The third and the fourth author are members of GNAMPA-INdAM and have been partially supported also by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games”. They warmly thank University of Cardiff for the kind hospitality.






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