On the integral representation of variational functionals on $BD$
Marco Caroccia Matteo Focardi Nicolas Van Goethem
math.AP 49J45, 46E30


 Following the global method for relaxation we prove an integral representation result for a large class of variational functionals naturally defined 
 on the space of functions with Bounded Deformation. Mild additional continuity assumptions are required on
 the functionals. 
 

Annotation-Free Cardiac Vessel Segmentation via Knowledge Transfer from Retinal Images
    Fei Yu1 Jie Zhao2 Yanjun Gong3 Zhi Wang3 Yuxi Li3 Fan Yang3 Bin Dong4,1,2 Quanzheng Li5,2 Li Zhang1,2()
    December 30, 2023
===========================================================================================================







§ INTRODUCTION

In linearized elasticity one route is to consider the displacement (or the velocity) field u as basic model variable. In this case, the deformation (or strain) tensor of an elastic body is given by the symmetric gradient of u, i.e., e(u):=1/2(∇ u+∇^tu). Therefore, the study of well-posedness of the PDE system of linear elasticity was at the origin of the study of the differential operator e(u)  and in particular its coerciveness properties, first analysed by Korn in 1906 <cit.> and followed by plenty of refinements  to this date (see for instance <cit.> for a survey). In linearized elasticity, the variational approach consists in minimizing the stored elastic energy (which is quadratic in the strain) minus the work of the external forces.
However, as soon as elasto-plasticity is considered, two main problems are faced: first, the observed stress-strain relation in plasticity is not linear anymore, resulting in a less-than quadratic, sometimes linear relation between the stored elastic energy and the strain. Here we refer to the pioneer work by Suquet on well posedness in perfect plasticity <cit.> itself based on preliminary work on the distributional operator of bounded deformation published in <cit.>. Specifically, in the above quoted works the authors study the properties of the differential operator Eu:=1/2(Du+D^tu) where D stands for the distributional derivative that generalizes the gradient ∇ to account for discontinuous fields u . In this way the space BD(Ω) of function with Bounded Deformation on the open subset Ω of ^n has been introduced as the space of L^1(Ω;^n) vector fields u whose symmetrized distributional derivative Eu is a Radon measure (see <cit.>, see also section <ref> to which we refer for the notation used in this introduction on BD maps). 
Moreover, e(u) is proven to be the density of the absolutely continuous part of Eu. 

The second issue arising in plastic problems is that concentration phenomena observed in plasticity require some weak notion of deformation that allow for slip or boundary concentration of strain for instance. Indeed, these effects are well handled in BD(Ω) by the so-called singular part of the deformation measure field.  It should also be said, that these aforementioned two issues are related, since a linear growth of the stored elastic energy prevents coerciveness in Sobolev spaces. Thus, bounds in the non-reflexive space L^1 require to consider limit of sequences in the space of Radon measures, and hence, again, justifies the choice of the space BD(Ω) when dealing with elasto-plastic models. For these models, the associated general bulk stored elastic energy reads as the integral

    F_0(u):=∫_Ω f_0(x,u(x),e(u)(x))x̣,

where f_0 has linear growth in the third variable and satisfies suitable assumptions (see Section <ref>),
and u∈ BD(Ω) is such that Eu is absolutely continuous with respect to Ł^nΩ, namely u∈ LD(Ω). 

To account also for singular effects a more general energy expression reads as

    F_1(u):=∫_Ω f_1(x,u(x),e(u)(x))x̣
    +∫_J_u g_1(x,u^-(x),u^+(x),ν_u(x))dℋ^n-1

where f_1 has linear growth in the third variable, f_1 and g_1 satisfy suitable assumptions (see Section <ref>),
and u∈ SBD(Ω), the subspace of BD(Ω) where the singular part E^su of the measure Eu is 
concentrated on the (n-1)-rectifiable set of approximate discontinuity points J_u
(see Section <ref> for the precise notation).



It is a classical problem in the Calculus of Variations to determine the lower semicontinuous envelope of the energies in (<ref>) and (<ref>) in order to find the limits of minimizing sequences lying in the larger space BD(Ω). 
More precisely, let F be the functional either in (<ref>) or in (<ref>) if u belongs to LD(Ω) or SBD(Ω)  respectively, and +∞ otherwise on L^1(Ω;^n). 
Then, the L^1(Ω;^n) lower semicontinuous envelope of the functional F, that is the greatest functional less or equal than F which is L^1 lower semicontinuous, is given by 

    (u):=inf{lim inf_j→+∞F(u_j): u_j→ u  in L^1(Ω;^n)},

provided some coercivity assumptions on the integrands are imposed (cf. <cit.>).

A suitable localized version of the functional  to the family of open subsets of Ω turns out to be a variational functional according to Dal Maso and Modica <cit.> naturally defined on the space BD(Ω) (see Section <ref> below and the discussion in what follows). 
Therefore, more generally, we consider variational functionals in this sense and prove for them in Theorem <ref> an integral representation result following closely the celebrated 
global method for relaxation developed in Bouchitté, Fonseca and Mascarenhas <cit.> to deal with the 
analogous problem for functionals defined either on Sobolev spaces or on the space BV of functions with Bounded Variation (see <cit.> for an extensive survey of the subject and an exhaustive bibliography).

The integral representation result in Theorem <ref> will be applied to the above mentioned relaxation problems for the 
functionals of the type (<ref>) and (<ref>). 
More precisely, in Theorems <ref> and <ref> we show that the resulting L^1 lower semicontinuous envelope 
 has indeed an integral representation of the type

    (u) =∫_Ω     f(x,u(x),e(u)(x))x̣
       +∫_J_ug(x,u^-(x),u^+(x),ν_u(x))̣̋^n-1(x)
    +∫_Ω  f^∞(x,u(x),Ẹ^cu/|̣E^cu|(x))|̣E^cu|

for all u∈ BD(Ω). Here, f^∞ denotes the (weak) recession function of the integrand f.
Moreover, a characterization of the energy densities f and g is given in terms of asymptotic Dirichlet problems involving F itself with boundary values related to the infinitesimal behaviours of the function u around the base point x. 


Apart from the usual lower semicontinuity (and therefore locality), growth conditions and measure theoretical properties to be satisfied by the functional  (see assumptions (H1)-(H3) 
in Section <ref>), we impose two conditions expressing continuity of the energy functional with respect to specific family of rigid motions. 
More precisely, continuity with respect to translations both in the dependent and independent variable is stated in (H4). Such a condition is used for instance in <cit.> 
in the BV setting to express the energy density of the Cantor part in terms of the recession function f^∞ of the bulk energy density.
Additionally, in the current BD setting we need to require further assumption (H5), that expresses continuity of the energy with respect to infinitesimal rigid motions. In turn, this condition implies that the bulk energy density depends only on the symmetric part of the relevant matrix. Condition (H5) is crucial for our arguments
both from a technical side and conceptually as we discuss in details in Section <ref>. 

In this respect, we emphasize that all integral representation, relaxation, lower semicontinuity results 
available in literature for energies defined on BD(Ω) (see e.g. <cit.>) 
are based on a stronger version of (H5) that imposes invariance of the energy with respect to infinitesimal rigid motions (cf. Remark <ref>). From a mechanical perspective such a condition reflects a restriction on the material behaviour. Therefore, it is preferable to avoid it, also because of its controversy in the continuum mechanics community (see <cit.>). 

Note that the quoted invariance property with respect to superposed infinitesimal rigid motions would imply 
the integrands in (<ref>) to be independent of u. In our result, though, this explicit dependence is kept, as was the case in the BV(Ω) setting <cit.>. Finally, let us point out that assumption (H5) is actually not needed to give a partial integral representation result on the subspace SBD(Ω), or more generally on BD(Ω) but only for the volume and surface terms of the energies, as already noticed in <cit.>.

Further possible applications of our main theorem are in the field of homogenization problems 
(cf. Section <ref>), or more generally to problems in which the determination 
of variational limits in terms of Γ-convergence of energies defined on 
BD are involved (see e.g. <cit.>, <cit.>). 

We mention that integral representation results for energies defined on distinguished subspaces of BD 
(in particular satisfying a different set of growth conditions different from (H2)) have been recently 
obtained either in the superlinear case in the 2 dimensional framework in <cit.>, 
or in the space of Caccioppoli affine functions in <cit.>. 



Let us now summarize the contents of the paper. In Section <ref> we state Theorem <ref>
the main result of the paper, all the preliminaries needed to prove it are provided in Section <ref>. 
Section <ref> focuses on the analysis of the Cantor part of the energy and more precisely on its integral representation. 
In turn, those results are used in Section <ref> to establish Theorem <ref>. 
Applications of Theorem <ref> to several issues related to energies with linear growth defined on BD are 
studied in Section <ref>.
More precisely, the relaxation of variational integrals is the topic of Section <ref>,
the lower semicontinuity either of bulk or of bulk and surface energies is investigated in Section <ref>, 
eventually Section <ref> deals with the periodic homogenization of bulk type energies.
In the final Section <ref> an example of a quasiconvex function f:𝕄^n× n→[0,+∞) being bounded
by the norm of the symmetric part of the relevant matrix, but on the other hand depending non-trivially also on the skew-symmetric part, 
is provided following a celebrated example by Müller <cit.>. The issue of relaxation on BD for the associated 
bulk energy functional is discussed, highlighting the role of assumption (H5) to deduce Theorem <ref> and related open problems.



§ MAIN RESULT




 §.§ Basic notation

The unitary vectors of the standard coordinate basis of ^n will be denoted by ,…,.
𝕄^n× n stands for the set of all n× n matrices and 
𝕄_sym^n× n, 𝕄_skew^n× n for the subsets of all symmetric 
and skew-symmetric matrices, respectively. 

For a given a set E we adopt the notation E(x_0,r):=x_0+rE for the rescaled copy of size r>0 translated in x_0. 
In particular, Q^ν(x_0,r) stands for any cube centered at x_0, with edge length r and with one face orthogonal 
to ν. 
We also adopt the convention that, whenever 


ν is omitted then ν=e_i for some i∈{1,…,n}, and the corresponding cube Q(x_0,r) is oriented according to the coordinate directions.

In what follows, we shall often consider a function Ψ:(0,+∞)→ [0,+∞) such that 
lim_t→ 0^+Ψ(t)=0, referring to it as a modulus of continuity. 

Finally, throughout the paper Ω will denote a non empty, bounded, open subset of ^n with Lipschitz boundary,
and 𝒪(Ω), 𝒪_∞(Ω) denote the families of all the open 
subsets of Ω and of all the open subsets of Ω with Lipschitz boundary, respectively.



 §.§ Framework and main result


We consider a class of local energies typically arising in variational problems: 
ℱ:L^1(Ω;^n)×𝒪(Ω)→[0,+∞], 
and BD(Ω) is the set of maps with Bounded Deformation
(for the precise definition, the notation used in what follows, and several properties of BD 
functions see Section <ref>). 
We assume that the following properties are in force on ℱ:


  (H1) (·, A) is strongly L^1(A;^n) lower semicontinuous for all A∈𝒪(Ω)[In the rest of the paper, we shall write L^1 lower semicontinuous 
in place of strongly L^1 lower semicontinuous for the sake of simplicity.];

  (H2) There exists a constant C>0 such that for every 
(u,A)∈ BD(Ω)×𝒪(Ω),
        
    1/C|Eu|(A)≤(u,A) ≤ C (Ł^n(A)+|E u|(A));


  (H3) ℱ(u,·) is the restriction to 𝒪(Ω) of a Radon measure for every u∈ BD(Ω);

  (H4) There exists a modulus of continuity Ψ such that 
        
    |ℱ(v+u(· - x_0),x_0+A) - ℱ(u,A)|≤Ψ(|x_0|+|v|)(Ł^n(A)+|E u|(A) )

       for all (u,A,v,x_0)∈ BD(Ω)×𝒪(Ω)×^n×Ω, with x_0+A⊂Ω;

  (H5) There exists a modulus of continuity Ψ  such that 
 
    |ℱ(u+(·-x_0),A)-ℱ(u,A)|≤Ψ(||diam(A))(Ł^n(A)+|Eu|(A))

 for every (u,A,) ∈ BD(Ω)×𝒪(Ω)×𝕄_skew^n× n, and for all x_0∈ A.



It is well-known that assumption (H1) implies locality of (·,A) for all A∈𝒪(Ω).
Namely, if u=v Ł^n a.e. on A then (u,A)=(v,A). 



Hypothesis (H5) implies that the energy depends only on the symmetric gradient (see formula (<ref>) and Section <ref> 
for more details).


Following the global method for relaxation introduced by Bouchitté, Fonseca and Mascarenhas in <cit.>
we consider the local Dirichlet problem
    
    (u,A):=inf{ℱ(v,A):  v∈ BD(Ω), v|_∂ A=u|_∂ A},

    where (u,A)∈ BD(Ω)×𝒪_∞(Ω) is given, 
    and prove the ensuing result.


Let ℱ:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] be satisfying (H1)-(H5). 
Then, for all (u,A)∈ BD(Ω)×𝒪(Ω)

    (u,A)=   ∫_A f(x,u(x),e(u)(x)) x̣+ ∫_J_u∩ A g(x,u^-(x),u^+(x),ν_u(x)) ̣̋^n-1(x)
       + ∫_A f^∞(x,u(x),Ẹ^cu/|̣E^cu|(x))|̣E^cu|(x),

where for all (x_0,,v,v^-,v^+,ν)∈Ω×𝕄_sym^n× n×(^n)^4
    
    f(x_0,v,):=lim sup_→ 0(v+(·-x_0), Q(x_0,))/^n,


    g(x_0,v^-,v^+,ν):=lim sup_→ 0( u_v^-,v^+,ν(·-x_0), Q^ν(x_0,))/^n-1,


    f^∞(x_0,v,):= lim sup_t→+∞ f(x_0,v,t )-f(x_0,v,0)/t,

and 
    
    u_v^-,v^+,ν(y):={[        v^+  if y·ν≥ 0;        v^- otherwise. ].


Note that f^∞ is classically termed the weak recession function, in contrast to the strong recession function for which the limit is assumed to
exists (see the comments in Section <ref>). 
Its finiteness is guaranteed by the linear growth of f (see (<ref>)).

We point out that the analogous result to Theorem <ref> for functionals defined on the space BV of functions with bounded variation 
has been proven under the sole assumptions (H1)-(H4) (cf. <cit.>). 
A detailed discussion on the need of assumption (H5) in the BD setting is the topic of Section <ref>.
Several comparisons with the BV case are discussed in Remarks <ref> and <ref>.




§ PRELIMINARIES




 §.§ Some results of geometric measure theory


In the forthcoming blow-up procedure, it will be mandatory to obtain  limits satisfying additional structural properties. 
To this aim we introduce some useful concepts of geometric measure theory. Let U⊆^n be either open or closed. Here and in what follows ℳ_loc(U;^k) stands for the sets of all ^k-valued Radon measures on U, and  ℳ(U;^k) for the subset of all ^k-valued finite Radon measures on U. 

Let μ_i, μ∈ℳ_loc(U;^k), following <cit.> we say that μ_i locally weakly* converge to μ, 
and we write μ_iμ in ℳ_loc(U;^k), if
    
    lim_i→+∞∫_Uφμ̣_i =∫_Uφμ̣    for all φ∈𝒞_c^0(U).

If μ_i, μ∈ℳ(U;^k), we say that μ_i weakly* converges to μ if the condition above
holds for all φ∈𝒞_0^0(U). The following property holds true.

Let {μ_i}_i∈ be locally weakly* converging to μ in ℳ_loc(U;^k), 
and {|μ_i|}_i∈ be locally weakly* converging to λ in ℳ_loc(U). 
Then, λ≥|μ|, and μ(E)=lim_i→∞μ_i(E) for every relatively 
compact Borel subset E of U with |λ|(∂ E)=0.

    
We use standard notations for the push-forward of measures, and in particular, given 
μ∈ℳ_loc(^n;^k), x∈^n and >0, we will often consider the push forward with the map 
F^x,(y):=y-x/ defined as
	
    F_#^x,μ(A):=μ(x+ A).

Preiss' tangent space (μ,x) at a given point x∈^n, is defined as the subset of non zero measures 
 ν∈ℳ_loc(^n;^k) 
such that ν is the local weak* limit 
of 1c_iF_#^x,_iμ, for some sequence {_i}_i∈↓ 0 
as i↑+∞ and for some positive sequence {_i}_i∈ 
(see <cit.>, <cit.>, <cit.>). To ensure that the total variation is preserved 
along the blow-up limit procedure we recall the ensuing result. 
 
Let μ∈ℳ_loc(^n;^k). 

Then, for |μ|-a.e. x∈^n and for every 
bounded, open, convex set K the following assertions hold

	
  (a) There exists a tangent measure γ∈(μ,x) such that |γ|(K)=1, 
	|γ|(∂ K)=0;
	
  (b) There exists 
	{_i}_i∈↓ 0 as i↑+∞ such that 
	F^x,ε_i_#μ/F^x,_i_#|μ|(K)γ in ℳ(K;^k).
	 

Finally, with the help of the next result, we will be able to select a blow-up with a partial affine structure. 

Let μ∈ℳ_loc(^n;^k) be a Radon measure. Then for |μ|-a.e. x∈^n any ν∈(μ,x) satisfies the following properties


  (a) For any convex set K, F^y,ρ_#ν/F^y,ρ_#|ν|(K)∈(μ,x)  for all y∈ν and ρ>0;

  (b) (ν,y)⊆(μ,x) for all y∈ν;


Note that the original result in <cit.> is proven for k=1. However, the good properties of tangent space of measures (i.e. <cit.> or
<cit.>) allow to immediately extend its validity for generic k.


 §.§ Preliminaries on BD
 
We recall next some basic properties of the space BD needed for our purposes. We refer to <cit.> for classical theorems, 
while for the fine properties we refer to <cit.> (see also <cit.>). 



The space of functions with Bounded Deformation on Ω, BD(Ω), is the set of all maps u∈ L^1(Ω;^n) whose 
symmetrized distributional derivative Eu is a matrix-valued Radon measure. It is a Banach space equipped with the norm 
u_BD(Ω):=u_L^1(Ω,^n)+|Eu|(Ω), where |μ| stands for the total variation of the Radon measure μ (see <cit.>). 
A sequence {u_j}_j∈ is said to strictly converge to u in BD(Ω) if

    u_j→ u  L^1(Ω;^n)  |Eu_j|(Ω)→ |Eu|(Ω),

as j→∞. 

As shown by Ambrosio, Coscia and Dal Maso in <cit.>, BD(Ω) maps are approximately differentiable 
Ł^n-a.e. in Ω, the jump set is ^̋n-1-rectifiable, and Eu can be decomposed as
    
    E u = e(u)Ł^nΩ + (u^+(x)-u^-(x))⊙ν_u ^̋n-1 J_u + E^cu,

where e(u)=∇ u+∇ u^t/2, 
∇ u is the approximate gradient of u, 
u^+-u^- denotes the jump of u over the jump set J_u, 
with u^± the traces left by u on J_u, 
ν_u is a unitary Borel vector field normal to J_u (here, a⊙ b:=1/2(a⊗ b+b⊗ a), a, b∈^n, 
denotes the symmetrized tensor product), E^cu is the Cantor part of E u defined as E^cu:=E^su(Ω∖ J_u) 
and E^su:=E u - e(u)Ł^nΩ (cf. <cit.>). 
Let S_u be the complement of the set of points of approximate continuity of u, <cit.> implies that 
|Eu|(S_u∖ J_u)=0, so that E^cu=Eu C_u, where 

    C_u:={x∈Ω∖ S_u: lim_r↓ 0|Eu|(B_r(x))/r^n=+∞, lim_r↓ 0|Eu|(B_r(x))/r^n-1=0}.

The limits in the definition of C_u can be taken with respect to any family K(x,r), with K a bounded, open, convex set containing 
the origin. We shall often use the previous characterization of E^cu throughout the paper.

The space of special functions of bounded deformation is then defined as

    SBD(Ω)={u∈ BD(Ω): E^cu=0}.

The space  𝒞^∞(Ω;^n)∩ W^1,1(Ω;^n) is dense in BD(Ω) for the strict topology on BD(Ω).
Moreover, for Ω an open bounded set with Lipschitz boundary, there exists a surjective, bounded, linear trace operator
γ: BD(Ω)⟶ L^1(∂Ω,^n) satisfying the following integration by parts formula: for every 
u∈ BD(Ω) and φ∈𝒞^1(Ω̅),

    ∫_Ω u ⊙∇φx̣+∫_ΩφẸu=∫_∂Ωφ γ(u)⊙νH^n-1,

with ν the unit external normal to ∂Ω. The trace operator is continuous if BD(Ω) is endowed with the strict topology. For notational simplicity, in what follows γ(u)
will be denoted simply by u itself.

With the same assumptions on Ω, one also has the following embedding result: BD(Ω)↪ L^q(Ω,^n) is compact
for every 1≤ q<n/n-1.
In view of compactness, the following holds for Ω a bounded extension domain (cf <cit.>, <cit.>): if  {u_j}_j∈ is bounded in BD(Ω) 
there exists a subsequence that converges to some u∈ BD(Ω) with respect to the L^1(Ω;^n) topology. 


 We recall next a Poincaré inequality for BD maps which has been proven in <cit.>, 
 (see also <cit.> and also <cit.>).
 To this aim consider the space of infinitesimal rigid motions 

    ℛ:={ x+v : v∈^n, ∈𝕄_skew^n× n} .


 Let A be a bounded, open, connected set with Lipschitz boundary, then 
 
    ℛ={ϑ∈𝒟'(A;^n):  Eϑ=0}.

 Moreover, let ℜ:BD(A) →ℛ 
 be a linear continuous map which leaves the elements of ℛ fixed. 
 Then there exists a constant c=c(A,ℜ) such that for all u∈ BD(A)
    
    u-ℜ[u]_L^n/n-1(A;^n)≤ c|E u|(A).


In particular, for a bounded, open convex set K, denoting by ν_∂ K the unit normal vector to ∂ K,
let ℜ_K: BD(K) →ℛ be the map defined as
	
    ℜ_K[u] (y):= 𝕄_K[u]y+b_K[u],

with
	
    b_K[u]:=_K u x̣ ,

	and

    𝕄_K[u]:=1/2Ł^n(K)∫_∂ K(
    u⊗ν_∂ K - ν_∂ K⊗ u)̣̋^n-1 .

For BV maps we express the quantity 𝕄_K in (<ref>) in terms of the skew-symmetric part of 
the total variation measure.

Let A be a bounded open set with Lipschitz boundary. Then, for all u∈ BV(A;^n)

    ∫_∂ A(u ⊗ν_∂ A - ν_∂ A⊗ u)̣̋^n-1
    =Du(A)-(Du)^t(A) .

In particular, for all bounded, open, convex sets K, and 
for all u∈ BV(K;^n) we have

    𝕄_K[u]=1/2ℒ^n(K)(Du(K)-(Du)^t(K)) .



To prove the first equality we use the divergence theorem for scalar BV functions <cit.>

    ∫_A v div Φ  dx+∫_AΦ· dDv=
    ∫_∂ Av Φ·ν_∂ Ạ̋^n-1 ,

for all v∈ BV(A), Φ∈𝒞^1(A̅;^n), to get for any fixed y∈^n

    ∫_∂ A(u ⊗ν_∂ A) y ̣̋^n-1=
    ∫_∂ A (ν_∂ A· y)u ̣̋^n-1=Du(A)y ,

and moreover

    ∫_∂ A(ν_∂ A⊗ u) y ̣̋^n-1=
    ∫_∂ A (u· y) ν_∂ Ạ̋^n-1=(Du)^t(A)y .

The very definition of 𝕄_K[u] in (<ref>) and the previous computation
provide the conclusion.

 
ℛ is an invariant set for ℜ_K.


If u is affine, i.e. u= x+v, ∈𝕄_skew^n× n and v∈^n, 
we have b_K[u] =v by a simple computation. Furthermore, by taking into account 
that Du=, Lemma <ref> implies
	
    𝕄_K[u]
    
    
    =1/2(-^t)=.



As the trace theorem implies the continuity of ℜ_K:BD(K)→ℛ, thanks to the previous result, 
Theorem <ref> yields the existence of a constant c>0 
depending only on K such that for every u∈ BD(K) 
	
    u-ℜ_K[u]_L^n/n-1(K;^n)≤ c |E u|(K).



 Let u∈ BD( K), where >0 and K a bounded, open, convex set K containing the origin. 
 Then, for a constant c depending only on K we have
 
    u-ℜ_ K[u]_L^1( K;^n)≤ c  |E u|( K) .

This follows from Hölder inequality, (<ref>) and a scaling argument by considering 
u_(y):=u( y), y∈ K, and noting that ℜ_ K[u]( y)=ℜ_K[u_](y), 
u-ℜ_ K[u]_L^1( K;^n)=^nu_-ℜ_K[u_]_L^1(K;^n), and 
|E u|( K)=^n-1|Eu_|(K).





 §.§ On the Cantor part of the symmetrized distributional derivative

Recently, the fine properties of BD functions have been complemented with the analog of Alberti's rank-one theorem in the BV setting.
More precisely, we recall the fundamental contribution by De Philippis and Rindler (cf. <cit.>).

 Let u∈ BD(Ω). Then, for |E^cu|-a.e. x∈Ω 
    
    Ẹ u/|̣E u|(x)=η(x)⊙ξ(x)/|η(x)⊙ξ(x)|

for some ξ, η:Ω→𝕊^n-1 Borel vector fields.
 
Next, we state a rigidity result for BD maps with constant polar vector that follows from that established 
in <cit.> by taking into account that the measure on the right hand side below is 
in addition positive (see also <cit.>). 


 
 If w∈ BD_loc(^n) is such that for some η, ξ∈^n
	
    E w= η⊙ξ/|η⊙ξ||E w|,

then 

 
  (i) if η≠±ξ 
 	
    w(y)=α_1(y·ξ)η+α_2(y·η)ξ+ y+v,

for some α_1,α_2∈ BV_loc(), 
∈𝕄_skew^n× n, v∈^n;


  (ii) if η=±ξ
 	
    w(y)=α(y·ξ)ξ
    	
    	+ y+v,

for some α∈ BV_loc(), 

∈𝕄_skew^n× n, v∈^n.



























The next Lemma will be particularly useful when dealing with the anti-symmetric part of the gradient 
in the Cantor part of the measure E u (see (<ref>) and (<ref>) for the definitions of 
b_K and 𝕄_K, respectively).

Let K be a bounded, open, convex set containing the origin. 
For any u∈ BD(Ω) and for |E^c u|-a.e. x_0∈Ω 
    
    lim_→ 0 b_K(x_0,)[u]=u(x_0),     lim_→ 0 |𝕄_K(x_0,)[u]|=0.



Let u∈ BD(Ω) be fixed. As noticed in the preliminaries |E^c u|-a.e. x_0∈Ω is a point of approximate continuity for u, 
thus b_K(x_0,)[u]→ u(x_0) as ↓ 0.

For the second part of the statement, we use the computation in <cit.> implying that for 
|E^c u|-a.e. x_0∈Ω 
    
    lim_→ 0_B(x_0,)|u-d_B(x_0,)[u]|ỵ=0,

where
    
    d_B(x_0,)[u]:=_∂ B(x_0,)u(x)̣̋^n-1(x).

Let x_0∈ C_u be a point for which (<ref>) holds, and recall then that ^1-n|Eu|(B(x_0,))→ 0 as ↓ 0.
Define for any v∈ BD(Ω) and for  sufficiently small 
    
    ℜ_^*[v](y):= d_B(x_0,)[v]+𝕄_K(x_0,)[v](y-x_0).

Arguing as in the proof of Proposition <ref> it is immediate to see that ℛ is an invariant set for 
ℜ_^*. In particular, thanks to Theorem <ref> we infer for all v∈ BD(Ω) that 
    
    v-ℜ_^*[v]_L^1(B(x_0,);^n)
     ≤ c  |Ev|(B(x_0,))

where the constant c is independent from  (this is obtained with a scaling argument similar to that in
Remark <ref>). In particular, it follows that
    
    ∫_B(x_0,) |u(y)-d_B(x_0,)[u]-𝕄_K(x_0,)[u] (y-x_0)|ỵ≤ c  |E u|(B(x_0,)).

Therefore, by the triangular inequality we have
    
    _B(x_0,)|𝕄_K(x_0,)[u] (y-x_0)|ỵ≤_B(x_0,)|u(y)-d_B(x_0,)[u]|ỵ+ C|Eu|(B(x_0,))/^n-1,

and thus by the choice of x_0 it follows
    
    _B(0,1)|𝕄_K(x_0,)[u]z|ẓ=_B(x_0,)|𝕄_K(x_0,)[u](y-x_0)|ỵ→ 0.

Notice that, the quantity 𝕄↦_B(0,1)|𝕄z|ẓ defines a norm on 𝕄^n× n, 
and thus for some constant C depending only on the dimension, we have
    
    |𝕄_K(x_0,)[u]|≤ C_B(0,1)|𝕄_K(x_0,)[u]z|ẓ→ 0.





 §.§ Change-of-base formulas

It is well-known that the chain rule formula does not hold in general for BD maps. 
We provide a simple variation of it that will be useful throughout the paper.

Let ∈𝕄^n× n be invertible, let w∈ BD(Ω) and set
	
    w̃(y):= w(^ty).

Then, w̃∈ BD(^-tΩ) and

    Ew̃   = ||^-1 (^-t_# Ew) ^t.

Moreover, if K is an open convex set and v∈ BD(K) we have

    ℜ_^-tK[ṽ](y)=ℜ_K[v] (^t y)   for all y∈^n.



Let φ∈𝒞^∞(Ω;^n)∩ BD(Ω) and define 
φ̃(y):=φ(^t y). 
Clearly, φ̃∈𝒞^∞(^-tΩ;^n), with ∇φ̃ (y)= 
∇φ (^t y) ^t and e(φ̃)(y)=  e(φ) (^t y) ^t.
Hence, the symmetrized distributional derivative of φ̃ is given by
	
    E φ̃ =||^-1(^-t_#(e(φ)Ł^nΩ))^t .

Finally, if w∈ BD(Ω) we conclude by approximation of w by smooth maps in the BD strict topology.

The last assertion follows from a direct computation.




We shall often use Lemma <ref> to reduce ourselves to the case in which the two vectors ξ, η
in the polar decomposition of (<ref>) are actually given by  and . To this aim the following remarks
are useful. Let w∈ BD(K) be given by 
	
    w(y):=ψ_1(y·η) ξ+ψ_2(y·ξ)η

for some ψ_1, ψ_2∈ BV() and for ξ,η∈^n non-parallel unit vectors, i.e. η≠±ξ. 
Consider any invertible matrix ∈𝕄^n× n such that η=, ξ=, 
and the associated function 
	
    w̃(y):= (ψ_1(^ty·η)ξ+ψ_2(^ty·ξ)η)= ψ_1(y·)+ψ_2(y·).
	
Then, w̃∈ BD(^-tK) with Ew̃=||^-1 (^-t_# Ew) ^t. Furthermore, since

    ^-t_# Ew=η⊙ξ/|η⊙ξ|^-t_#|Ew|

we then have

    Ew̃= ||^-1
    ⊙/|η⊙ξ|^-t_# |E w|,

in turn implying both

    Ew̃= ⊙/|⊙||Ew̃|

and 

    |Ew̃|=||^-1|⊙|/|η⊙ξ|_#^-t|E w|.

In particular, we conclude that 
	
    |E w̃|(^-tK)=||^-1|⊙|/|η⊙ξ|  |E w|(K).








 §.§ On the cell problem defining .
 The next two results clarify the link between 
 and ℱ. They have been originally proved in <cit.> in the BV setting 
and then straightforwardly adapted to the BD setting in <cit.>.

Let u∈ BD(Ω), and set μ:=Ł^n+|E^s u|. Then, for any bounded, open, convex set K containing 
the origin we have
    
    lim_r→ 0(u, K(x_0,r))/μ(K(x_0,r))=lim_r→ 0(u,K(x_0,r))/μ(K(x_0,r))    for μ-a.e. x_0∈Ω.



There exists a constant C>0 such that for any u_1,u_2∈ BD(Ω), A∈𝒪_∞(Ω) 
    
    |(u_1;A)-(u_2;A)|≤ C ∫_∂ A |u_1-u_2|̣̋^n-1.


Finally, we refine Lemma <ref> as a consequence of assumptions (H4) and (H5).
Following <cit.>, for all K bounded, open, convex set containing the origin, 
for every 
(,v_0,v,x_0,x)∈𝕄^n× n×

(^n)^2× (Ω)^2 
and for every >0 small enough, hypothesis (H4) implies 

    |(v+v_0+(·-x-x_0),K(x+x_0,))-   (v_0+(·-x_0),K(x_0,))|   
    ≤ C_KΨ(|x|+|v|)(1+|+^t|/2)^n,

and, in turn, hypothesis (H5) implies
 
    |(v_0+(· -x_0),K(x_0,))- (v_0++^t/2(· -x_0),K(x_0,))|
               ≤ C_KΨ( diam(K)|-^t|/2)(1+|+^t|/2)^n

for some constant C_K>0 depending on K only.





§ ANALYSIS OF THE BLOW-UPS OF THE CANTOR PART



In this section we show how to select a suitable blow-up limit at Cantor type points. To this aim we fix some notation: 
let u∈ BD(Ω), we may assume u to be extended to a map in BD(^n) being Ω Lipschitz 
(cf. <cit.>). By a slight abuse of notation we denote by u the extended function.

With fixed a bounded, open, convex set K containing the origin, 
for every >0 and x∈Ω set K(x,):=x+ K. For  sufficiently small, consider the associated rescaled 
functions u_K,x,:K→^n given by

    u_K,x,(y):=
    u(x+ y)- ℜ_K[u(x+·)](y)/|E u|(K(x,))/Ł^n(K(x,))),

where ℜ_K is defined in (<ref>). Clearly, we may assume that |E u|(K(x,))>0 
for all ∈(0,dist(x,∂Ω)), otherwise u would be an infinitesimal rigid 
motion around x (cf. Eq. (<ref>) in Theorem <ref>).
We analyze first basic compactness properties of the rescaled family {u_K,x,}_>0. 


For |Eu|-a.e. x∈Ω there exist a sequence _i↓ 0, a map w∈ BD(K), and a measure
γ∈(Eu,x) such that 


  (i) {u_K,x,_i}_i∈ converges to w strictly in BD(K), ℜ_K[w]=0


  (ii) {1/Ł^n(K)Eu_K,x,_i}_i∈ converges to γ weakly* in ℳ(K̅;^n× n), 

|γ|(K)=|γ(K)|=1, |γ|(∂ K)=0, |γ|∈(|Eu|,x), 
γ=dEu/d|Eu|(x)|γ| |γ|-a.e. in ^n, and Ew=Ł^n(K)γ K. 



We prove the conclusion for all Lebesgue points of the polar vector Ω∋ x↦dEu/d|Eu|(x).
A simple computation shows that 
	
    E u_K,x,=Ł^n(K) F^x,_#E u/F^x,_#|E u|(K) ,

(recall the definition of F^x,_# in (<ref>)) so that |Eu_K,x,|(K)=Ł^n(K).
Lemma <ref> yields for a sequence _i↓ 0 that 
{1/Ł^n(K)Eu_K,x,_i}_i∈ converges weakly* in ℳ(K̅;M^n× n) 
to some γ∈(Eu,x), with |γ|(K)=1,
|γ|(∂ K)=0, |γ|∈(|Eu|,x) and such that γ=dEu/d|Eu|(x)|γ|
|γ|-a.e. in ^n (for the last two claims see <cit.>, <cit.>), 
then |γ(K)|=1. 


Moreover, by taking into account Remark <ref> we get

    u_K,x,_L^1(K;^n)=Ł^n(K)/ |E u|(K(x,))u(x+·)-ℜ_ K[u(x+·)](·)_L^1( K;^n) ,

so that for some constant depending only on K

    u_K,x,_L^1(K;^n)≤ c Ł^n(K) .

The compact embedding BD(K)↪ L^1(K;^n) yields that we can extract a subsequence (not relabeled) with u_K,x,_i
converging in L^1(K;^n) to some limit map w belonging to BD(K) satisfying ℜ_K[w]=0. 
Therefore, Eu_K,x,_i converge to Ew weakly* in ℳ(K;^n× n), and thus 
Ew=Ł^n(K)γ K. The strict convergence in BD(K) of {u_K,x,_i}_i∈ to w 

then follows at once.

In what follows, any map w given by Proposition <ref> will be termed blow-up limit. 
If x is a point of approximate differentiability or a jump point, usually introduced with a different definition 
of the rescaled maps, the blow-up limit is well-known to be unique. 
In turn, this implies that the Radon-Nikodým derivative of the functional  with respect to |Eu| in such points can be 
straightforwardly characterized in terms of asymptotic Dirichlet problems with boundary values given by the blow-up limit itself 
(cf. Lemma <ref>). 
In contrast, if x∈ C_u is a point satisfying (<ref>), usually referred to as a Cantor type point, 
the blow-up limit is in general not unique. In order to overcome this difficulty, a double blow-up procedure is performed. 
By means of this argument, we can reduce ourselves to the case of a two dimensional BV map which is affine in one direction. 


The strategy of the proof is a slight variation of <cit.>, which is originally worked out 
in the context of generalized Young measures. We basically follow the lines of such proof by incorporating also the need
of selecting a sequence preserving the mass along the blow-up process. 
We shall improve upon the structure of blow-ups in Proposition <ref> in section <ref>.






 §.§ A double blow-up procedure


We introduce some notation necessary for the blow-up procedure. Given a couple of vectors ξ, η∈𝕊^n-1 
(possibly ξ=±η), consider an orthonormal basis ζ_i of span{ξ,η}^⊥ 
 (thus for n=2 either span{ξ,η}^⊥={0} if ξ≠±η or i=1 if ξ=±η, 
 and for n≥ 3 either 1≤ i≤ n-2 if ξ≠±η or 1≤ i≤ n-1 if ξ=±η), 
 then for all ρ>0 define the bounded, open, convex set containing the origin
 
    P^ξ,η_ρ:= {y∈^n: |y·η|≤ρ2,  |y·ξ|≤12, 
     |y·ζ_i|≤12  for 1≤ i≤ n-2},

 if ξ≠±η, and otherwise if ξ=±η
 
    P^ξ,η_ρ:= {y∈^n: |y·η|≤ρ2, 
     |y·ζ_i|≤12  for 1≤ i≤ n-1}.

 We underline that the role of η and ξ is not symmetric in the definition of  P_ρ^ξ,η
(in this respect see the comments right before Case 1 in the ensuing proof). Moreover, we do not highlight the dependence of 
P_ρ^ξ,η on ζ_i not to further overburden the notation. In any case the specific choice of ζ_i is not relevant 
for the arguments that follow.

With this notation at hand, we can state the key result to prove the integral representation of the Cantor part (recall the definitions
of F^x,_#Eu given in (<ref>) and that of u_K,x, given in (<ref>)).


Let u∈ BD(Ω). Then for |E^c u|-a.e. x∈Ω and for every ρ>0 there exist an infinitesimal sequence {_i}_i∈,
vectors ξ, η∈𝕊^n-1, bounded, open, convex set containing the origin P^x_ρ:=P^ξ,η_ρ (cf. (<ref>) and (<ref>)), and a map v_ρ∈ BV(P^x_ρ;^n) such that 
u_P^x_ρ,x,_i converge to v_ρ strict in BD(P^x_ρ), where

    v_ρ(y)= ψ̅_ρ( y·η) ξ + (y·ξ) β̅_ρ η+_ρy +v_ρ,

for some ψ̅_ρ∈ BV_loc(), β̅_ρ∈, 
v_ρ∈^n, and _ρ∈𝕄_skew^n× n.
Moreover, v_ρ satisfies

    ℜ_P^x_ρ[v_ρ]=0,    
    	|E v_ρ|(P^x_ρ)=Ł^n(P^x_ρ).

and
	
    Ẹ v_ρ/|̣E v_ρ |(y)=η⊙ξ/|η⊙ξ|    |E v_ρ|-a.e. on P^x_ρ.



We divide the proof in two steps, each corresponding to a blow-up procedure.


First blow-up. 
We perform a first blow-up in a point x satisfying all the conditions listed in (I)-(III) that follows. 
More precisely, consider the subset of points x∈ C_u (so that x is a point of approximate continuity 
of u (see (<ref>))) having the following additional properties:


  (I) Ẹ u/|̣E u|(x)= Ẹ^c u/|̣E^c u|(x)= η(x)⊙ξ(x)/|η(x)⊙ξ(x)|, 
for some vectors ξ(x),η(x)∈𝕊^n-1, and (|Eu|,x)=(|E^cu|,x);

  (II) Proposition <ref> holds true with K:=^t Q(0,ρ) if η(x)≠±ξ(x),
where 𝔹 is any invertible matrix  such that 𝔹η(x)= and 𝔹ξ(x)=,  
and K:=P_ρ^ξ(x),η(x) if η(x)=±ξ(x): we extract a subsequence (not relabeled) such that 
u_K,x,_i converge to some map w strict in BD(K), with ℜ_K[w]=0, 
{1/Ł^n(K)Eu_K,x,_i}_i∈ converges weakly* in ℳ(K̅;^n× n) to some 
γ∈(Eu,x) such that |γ|(K)=|γ(K)|=1, |γ|(∂ K)=0, 
|γ|∈(|Eu|,x), γ=dEu/d|Eu|(x)|γ| 
|γ|-a.e. in ^n, and Ew=Ł^n(K)γ K;




  (III) (|Ew|, z)⊆(|E^c u|,x) for all z∈|Ew|.

Notice that the set of points where either (I) or (II) or (III) fails is |E^c u|-negligible thanks to 
Theorem <ref>, to the locality of Preiss' tangent space to a measure, to 
Proposition <ref> itself and to Theorem <ref>. 

Proposition <ref> describes the structure of the blow-up limit w in (II) in details: 
 
 
  * if η(x)≠±ξ(x): we can find two maps α_1,α_2∈ BV_loc(), 
 _ρ∈𝕄_skew^n× n and v_ρ∈^n, such that 
 
    w(y)=α_1(y·η(x)) ξ(x)+ α_2(y·ξ(x)) η(x)+_ρy+v_ρ ,



  * if η(x)=±ξ(x): we can find a map α∈ BV_loc(), 
_ρ∈𝕄_skew^n× n and v_ρ∈^n, such that
 
    w(y)=α(y·ξ(x)) ξ(x)+_ρy+v_ρ .


Thus, if η(x)=±ξ(x) we conclude by setting v_ρ:=w and with  P_ρ^x:=P_ρ^ξ(x),η(x). 
Otherwise, if η(x)≠±ξ(x),  we are forced to take a second blow-up to prove that (at least) one between the 
α_k's can be taken affine.  


Second blow-up if η(x)≠±ξ(x).
First, we change variables by means of the invertible matrix 𝔹 introduced in item (II) above.
Following Remark <ref>, with 𝕃̃_ρ:=𝔹𝕃_ρ𝔹^t and 
ṽ_ρ:=𝔹v_ρ,  we consider the associated map 

    w̃(y)=α_1(y·) + α_2(y·)
    +𝕃̃_ρ y+ṽ_ρ .

We blow-up w̃ around a suitable point y∈^-tK=
Q(0,ρ), distinguishing two cases depending on the distributional 
derivatives of the α_k's. We note that the vectors η and ξ in the statement correspond exactly 
to the two vectors provided by the polar decomposition of E^cu at x, 
η(x) and ξ(x), respectively, if D^s α_1≠ 0. In this case we 
also set P_ρ^x:=P_ρ^ξ(x),η(x), the latter set being introduced in (<ref>).
Instead, if D^s α_1=0 and D^s α_2≠ 0,
then η corresponds to ξ(x) and ξ to η(x), and 
P_ρ^x:=P_ρ^η(x),ξ(x).
Finally, if both D^s α_1=D^s α_2=0, we choose
η=η(x) and ξ=ξ(x) and set P_ρ^x:=P_ρ^ξ(x),η(x)
(actually, the opposite choice would be fine as well).



Case 1: either D^s α_1≠ 0 or D^s α_2≠ 0. Without loss of generality 
we may assume that D^s α_1 ≠ 0. Set P̃_ρ^x:= ^-t P_ρ^x 
(the matrix  has been introduced above), and select a point y∈ Q(0,ρ) such that 


  (a) Ẹw̃/|̣E w̃|(y)
=⊙/|⊙|, and δ^-n|Ew̃|(P̃^x_ρ(y,δ))→+∞ 
as δ↓ 0;

  (b) Proposition <ref> holds true: w̃_P̃_ρ^x,y,δ_i, for 
some sequence δ_i↓ 0, converge to a map g̃_ρ strict in BD(P̃_ρ^x) with 
ℜ_P̃^x_ρ[g̃_ρ]=0, 
and {1/Ł^n(P̃_ρ^x)Ew̃_P̃_ρ^x,y,δ_i}_i∈ converges weakly* in 
ℳ(P̅̃̅_ρ^x;^n× n) to some γ̃∈(Ew̃,y) such that 
|γ̃|(P̃_ρ^x)=|γ̃(P̃_ρ^x)|=1, 
|γ̃|(∂P̃_ρ^x)=0, |γ̃|∈(|Ew̃|,y), and 
γ̃=Ẹw̃/|̣E w̃|(y)|γ̃| |γ̃|-a.e. on ^n,
Eg̃_ρ=Ł^n(P̃_ρ^x)γ̃P̃_ρ^x; 


  (c) lim_r→ 0_y·-r^y·+r
| α_2'(s)- α_2'(y·)| ṣ =
lim_r→ 01/2r|D^s  α_2|(y·-r,y·+r)=0.
	 
Since for Ł^1-a.e. t∈ it holds

    lim_r→ 0_ t-r^t+r | α_2'(s)- α_2'(t)| ṣ =0,  lim_r→ 01/2r|D^s α_2|(t-r,t+r)=0,

conditions (c) is true for |D^s  α_1| ⊗Ł^n-1-a.e. y∈ Q(0,ρ). 
First, note that |D^sα_1|⊗Ł^n-1 is non trivial by assumption; then if I⊆ is the subset of 
points of full Ł^1 measure for which the previous two conditions hold true, we conclude that 

    (|D^s  α_1| ⊗Ł^n-1)((-ρ2,ρ 2)×((-ρ2,ρ 2)∖ I)
    ×(-ρ2,ρ2)^n-2)=0.

On the other hand, conditions (a)-(b) hold |E^sw̃|-a.e. on Q(0,ρ) in view of 
Theorem <ref>, the decomposition of Ew in (<ref>), and 
Proposition <ref> itself. 
Since the measures |D^s α_1|⊗Ł^n-1 and |Dα_1| ⊗ |D^s α_2|⊗Ł^n-2 are mutually singular,  
the measure |D^s α_1|⊗Ł^n-1 is absolutely continuous with respect to |E^s w̃|, 
thus we conclude that (a)-(c) hold for |D^sα_1|⊗Ł^n-1-a.e. y∈ Q(0,ρ).


Then, fix a point y∈ Q(0,ρ) for which (a)-(c) are satisfied,
we show that the second blow-up limit g̃_ρ satisfies

    g̃_ρ(z)=ψ_ρ(z·) +𝕃̃_ρ z+ṽ_ρ ,

for some ψ_ρ∈ BV_loc(), ṽ_ρ∈^n, and _ρ∈𝕄^n× n_skew.
We make a slight abuse of notation since the latter two quantities maybe different from those in the definition of w̃,
but as their role is inessential we keep the same symbols.
To this aim, we split E w̃_P̃^x_ρ,y,δ_i as follows

    E w̃_P̃^x_ρ,y,δ_i=E w̃^(1)_i+E w̃^(2)_i ,

where we have set

    w̃^(1)_i(z):=α_1((y+δ_i z)·e_1)e_2-
    ℜ_P̃^x_ρ[α_1((y+δ_i ·)·e_1)e_2](z)/δ_i|E w̃|(P̃^x_ρ(y,δ_i))/Ł^n(P̃^x_ρ(y,δ_i)) ,

and

    w̃^(2)_i(z):=α_2((y+δ_i z)·e_2)e_1-
    ℜ_P̃^x_ρ[α_2((y+δ_i ·)·e_2)e_1](z)/δ_i|E w̃|(P̃^x_ρ(y,δ_i))/Ł^n(P̃^x_ρ(y,δ_i)) .

First, we claim that as i↑+∞

    |Ew̃^(2)_i|(P̃^x_ρ)
    → 0 .

Indeed, noting that

    E w̃^(2)_i=Ł^n(P̃^x_ρ) ⊙/|E w̃|(P̃^x_ρ(y,δ_i))(δ_i^n α_2'((y+δ_i · )·)Ł^nP̃^x_ρ
    +F^y,δ_i_#((Ł^1⊗ D^s α_2 ⊗Ł^n-2)P̃^x_ρ)) .

On setting t_i:= 1|Ew̃|(P̃^x_ρ(y,δ_i)), and noting that t_i δ_i^n is infinitesimal 
in view of the choice of y above, if φ∈𝒞^0_c(P̃^x_ρ;^n) and 
α:= α_2'(y·), we conclude that
	
    t_i δ_i^n   |∫_P̃^x_ρφ(z):(⊙) α'_2((y+δ_i z)·)ẓ|
    	≤ t_i  ∫_P̃^x_ρ(y,δ_i)|φ(x-y/δ_i)| 
    	| α'_2(x·)| x̣
    
    	   ≤ t_iφ_∞∫_P̃^x_ρ(y,δ_i) | α'_2(x·)- α| x̣
    	+ t_iδ_i^n|α|φ_∞ℒ^n(P̃^x_ρ) 
       
    	= ρφ_∞t_i δ_i^n
    	_(y·)-δ_i2^(y·)+δ_i2  | α'_2(s)- α| ṣ
    	+ o(1) ,

which vanishes as i↑+∞ (cf. item (b) above). Arguing similarly, we get that

    t_i   |∫_P̃^x_ρφ(z) F̣_#^y,δ_i((Ł^1⊗ D^s α_2 ⊗Ł^n-2)P̃^x_ρ)(z)|
       ≤ρφ_∞Ł^n(P̃^x_ρ)t_iδ_i^n-1
    |D^s α_2|(y·-δ_i2,y·+δ_i2)

is infinitesimal for i↑+∞, as well (cf. item (c) above).

In conclusion, (<ref>) implies that w̃_i^(2) converges to 0 strongly in BD(P̃^x_ρ) in view of Remark <ref>, 
and then the limit of w̃_P̃^x_ρ,y,δ_i is equal to that of 
w̃^(1)_i. 
Moreover, the latter coincides with the L^1(P̃^x_ρ;^n) limit of 

    α_1((y+δ_i z)·e_1)e_2-
    ℜ_P̃^x_ρ[α_1((y+δ_i ·)·e_1)e_2](z)/δ_i|E(α_1(··e_1))|(P̃^x_ρ(y,δ_i))/Ł^n(P̃^x_ρ(y,δ_i)) .

as |Ew̃_P̃^x_ρ,y,δ_i|(P̃^x_ρ)
=ℒ^n(P̃^x_ρ) and |Ew̃^(2)_i|(P̃^x_ρ) 
is infinitesimal. Thus, by the blow-up theory for BV functions (cf. <cit.>), 
Lemma <ref> and Remark <ref> we conclude (<ref>) with

    g̃_ρ(z)=ψ_ρ(z·) -1/√(2)(⊗-⊗)z .




Case 2: D^sα_1=D^sα_2=0. 
In this case w̃∈ W^1,1(P̃^x_ρ,ℝ^n), where P̃_ρ^x:= ^-t P_ρ^x
(the matrix  has been introduced in item (II) above).
Arguing as in Case 1, we select a point y∈ Q(0,ρ) such that 
	

  (a') Ẹw̃/|̣E w̃|(y)=⊙/|⊙|;

  (b') Proposition <ref> holds true: w̃_P̃_ρ^x,y,δ_i, for 
some sequence δ_i↓ 0, converge to a map g̃_ρ strict in BD(P̃_ρ^x) with 
ℜ_P̃^x_ρ[g̃_ρ]=0, and
{1/Ł^n(P̃_ρ^x)Ew̃_P̃_ρ^x,y,δ_i}_i∈ converges weakly* in 
ℳ(P̅̃̅_ρ^x;^n× n) to some γ̃∈(Ew̃,y) such that 
|γ̃|(P̃_ρ^x)=|γ̃(P̃_ρ^x)|=1, 
|γ̃|(∂P̃_ρ^x)=0, |γ̃|∈(|Ew̃|,y), 
γ̃=Ẹw̃/|̣E w̃|(y)|γ̃| |γ̃|-a.e. on ^n,
and Eg̃_ρ=Ł^n(P̃_ρ^x)γ̃P̃_ρ^x;

  (c') lim_r→ 0_(y·e_k)-r^(y·e_k)+r 
	| α_k'(s)- α_k'(y·e_k)| ṣ =0 for k∈{1,2};

  (d') | α_1'(y·)+ α_2'(y·)|≠ 0.
	 
Note that conditions (a')-(c') hold for Ł^n-a.e. y∈ Q(0,ρ), and (d') for a set of positive Lebesgue
measure in Q(0,ρ) as |Ew̃|(Q(0,ρ))=|⊙|/|η⊙ξ|Ł^n(Q(0,ρ))≠ 0
thanks to (<ref>) and (II). 
As a consequence of all these conditions, 
δ^-n|Ew̃|(P̃^x_ρ(y,δ))→|e(w̃)(y)|ℒ^n(P̃_ρ^x)
≠ 0 as δ↓ 0.
Thus, by blowing-up the function w̃ at one such point along the sequence of radii 
δ_i given by (b'), we may infer that, up to extracting a subsequence not relabeled,
w̃_P̃^x_ρ,y,δ_i converge strictly in BD(P̃^x_ρ) to 


    g̃_ρ(z)=α_1'(y·)+α_2'(y·)/2|e(w̃)(y)|((z·) +(z·) ) .






To get the latter formula, we use that in this setting α_1 and α_2 are W^1,1_loc(), and that condition (c') 
and Lemma <ref> hold true.


Conclusion in case η(x)≠ξ(x). 
Both in case 1 and 2 we have selected a point 
y∈ Q(0,ρ), a function g̃_ρ∈ BD(P̃_ρ^x) 
and a measure γ̃∈Tan(Ew̃,y)
satisfying: g̃_ρ is affine in (at least) one direction between  and  (cf. (<ref>)) 

and 
















|γ̃|(P̃_ρ^x)=|γ̃(P̃_ρ^x)|=1, 
|γ̃|(∂P̃_ρ^x)=0, |γ̃|∈(|Ew̃|,y), 
γ̃=⊙/|⊙||γ̃| |γ̃|-a.e. on ^n,
and Eg̃_ρ=Ł^n(P̃_ρ^x)γ̃P̃_ρ^x.

Lemma <ref> implies that
Tan(Ew,^ty)=^-1(_#^tTan(Ew̃,y))^-t. Therefore, by setting 
g_ρ:=^-1g̃_ρ(^-t·) and
γ:=^-1(^t_#γ̃)^-t we deduce that 
γ∈Tan(Ew,^t y)⊆(E^c u,x) and that
|γ|∈Tan(|Ew|,^t y)⊆(|E^c u|,x) in view of condition (III). Moreover, 
Eg_ρ=Ł^n(P_ρ^x)γ P_ρ^x.

Let {r_i}_i∈ be a sequence of radii and {c_i}_i∈ 
of positive constants  such that c_i F^x,r_i_# E u converge locally weakly* to γ in 
ℳ_loc(^n;^n× n), and c_i F^x,r_i_#|Eu| converge locally weakly* 
to |γ| on ℳ(^n). Then, |γ|(∂P_ρ^x)=0,
by taking into account that |γ̃|(∂P̃_ρ^x)=0. 
Thus, Lemma <ref> 
yields
	
    c_i F^x,r_i_# |Eu|(P_ρ^x) → 
    	|E g_ρ|(P_ρ^x)(<ref>)=|||η⊙ξ|/|⊙|Ł^n(P̃_ρ^x)=|η⊙ξ|/|⊙|Ł^n(P_ρ^x) .

Hence, Ł^n(P_ρ^x) F^x,r_i_#E u/F^x,r_i_#|E u|(P_ρ^x)|⊙|/|η⊙ξ| Eg_ρ
in ℳ(P_ρ^x;^n× n), and
Ł^n(P_ρ^x) F^x,r_i_# |E u|/F^x,r_i_# |E u|(P_ρ^x)|⊙|/|η⊙ξ| |Eg_ρ|
in ℳ(P_ρ^x).
Furthermore, note that the rescaled maps u_P_ρ^x,x,r_i converge strongly in 
L^1(P_ρ^x;^n) to a map v_ρ∈ BD(P_ρ^x) and Eu_P_ρ^x,x,r_i converge weakly* to 
Ev_ρ in ℳ(P_ρ^x,^n× n), up to a subsequence not relabeled. 
Therefore, we find Ev_ρ= |⊙|/|η⊙ξ| E g_ρ as measures on P_ρ^x 
(cf. Remark <ref>). In view of 
Eq. (<ref>) in Theorem <ref>, 
there is an infinitesimal rigid motion such that 

    v_ρ(z)=|⊙|/|η⊙ξ| g_ρ(z)+_ρ z+v_ρ .

In addition, |E v_ρ|(P_ρ^x)=Ł^n(P_ρ^x), so that in particular u_P_ρ^x,x,r_i converge 
to v_ρ strictly in BD(P_ρ^x).
We complete the proof of (<ref>) by deducing that 
ℜ_P_ρ^x[v_ρ]=0 from ℜ_P_ρ^x[u_P_ρ^x,x,r_i]=0
for all i and the above mentioned strict convergence in BD(P_ρ^x).


Notice that the parallelogram P^x_ρ produced by Proposition <ref>, along which the sequences u_P_ρ^x,x, converges strictly in BD(P_ρ^x) to a function v_ρ affine in one direction, has been chosen in a way that the short edge (the edge of size ρ) is oriented exactly in the direction of non-affinity of v_ρ. The non-affine direction is exactly the one we need to control in order to conclude the representation Theorem <ref> and thus, the fact that the parallelograms P^x_ρ are well oriented plays a crucial role in the argument that will follow. 





 §.§ Finer analysis of the blow-up limits
 
We proceed next with the investigation 
of some properties of the blow-up limits provided by Proposition <ref> 
that follow by exploiting their structure evidenced in Eqs. (<ref>)-(<ref>).
Similar results are available in the BD setting in case the base point is either a point of approximate differentiability 
or a jump point. The analogue of the ensuing result is also well-known for BV functions (see for instance <cit.>).

We will state some technical lemmas that will allow us to identify in a more precise way the blow-up limits.
To this aim, for a function ψ∈ BV((a,b)) we denote by ψ(a), ψ(b) the right and left traces in a, b∈, respectively.
We start with the case ξ≠±η.

Let {v_ρ}_ρ>0⊆ BV(P ^ξ,η_ρ;^n), ξ, η∈^n∖{0} 
with ξ≠±η, be a sequence of functions such that
	
    v_ρ(x)=ψ̅_ρ(x·η) ξ+ 
    	
    	(x·ξ)β̅_ρ η
    	+_ρ x+v_ρ

for some ψ̅_ρ∈ BV((-ρ2,ρ2)), β̅_ρ∈,
v_ρ∈^n, _ρ∈𝕄_skew^n× n. 
Assume also that 
	
    ℜ_P^ξ,η_ρ[v_ρ]=0,     |E v_ρ|(P^ξ,η_ρ)=Ł^n(P^ξ,η_ρ) ,


Then, v_ρ can be re-written as
	
    v_ρ(x)= ψ_ρ(x·η) ξ+
    	x·ξ/2|η⊙ξ|η,

for some ψ_ρ∈ BV((-ρ2,ρ2)) with zero average such that 

{ψ_ρ(·)ρ}_ρ>0 is uniformly bounded in 
L^∞((-ρ2,ρ2)) and

    D ψ_ρ((-ρ2,ρ2))
      =ρ/2|η⊙ξ| .



	
Set R_ρ:=(-ρ2,ρ2)×(-12,12)^n-1,
and let 𝔹∈𝕄^n× n be any invertible matrix such that 𝔹η=, 𝔹ξ= and mapping R_ρ onto P_ρ^ξ,η. 
Namely, P_ρ^ξ,η=𝔹^t R_ρ. By invoking Lemma <ref> and Remark <ref> we can infer that
    
    ṽ_ρ(y):=𝔹 v_ρ(𝔹^t y)

satisfies ṽ_ρ∈ BD(R_ρ) and
    
    ṽ_ρ(y)=ψ̅_ρ(y·) +
        (y·)β̅_ρ  +
    	𝕃̃_ρ y+ṽ_ρ,

with 𝕃̃_ρ=𝔹𝕃_ρ𝔹^t, ṽ_ρ=𝔹v_ρ. 
Moreover ℜ_R_ρ[ṽ_ρ]=0, and

	
    |E ṽ_ρ|(R_ρ)=|⊙|/|η⊙ξ|ρ,      E ṽ_ρ = ⊙/|⊙| |E ṽ_ρ|.
	


Step 1: Identification and
properties of ψ_ρ and β_ρ.

Condition ℜ_R_ρ[ṽ_ρ]=0 is equivalent to
𝕄_R_ρ[ṽ_ρ]=b_R_ρ[ṽ_ρ]=0
(see (<ref>) for the definition of b_R_ρ and (<ref>) for that of 𝕄_R_ρ).
In turn, from these equalities we get that
	
    ṽ_ρ+(_R_ρψ̅_ρ (y·)ỵ)
    	=ṽ_ρ+(_-ρ2^ρ2ψ̅_ρ(t)  ṭ)=0 .

Thanks to Lemma <ref> we get

    𝕄_R_ρ[( ψ̅_ρ(y·) -_-ρ2^ρ2ψ̅_ρ(t)  ṭ)]
    =1/2ρ(ψ̅_ρ( ρ2) -ψ̅_ρ(-ρ2))(⊗-⊗) ,

and 

    𝕄_R_ρ[ (y·)β̅_ρ]=   β̅_ρ/2  (⊗-⊗) .

Therefore, recalling that 𝕄_R_ρ[_ρy]=_ρy, we conclude that for every y∈ R_ρ

    _ρy+𝕄_R_ρ[ṽ_ρ]y=
    _ρy+κ_ρ(⊗-⊗)y=0 ,

where we have set 	

    κ_ρ:= 1/2ρ(ψ̅_ρ(ρ2)-ψ̅_ρ(-ρ2)-β̅_ρρ).

In particular, by defining
	
    ψ_ρ(t):=ψ̅_ρ(t)-κ_ρ t - _-ρ2^ρ2ψ̅_ρ(t) ṭ, 
    	    β_ρ:=β̅_ρ+κ_ρ,

	ψ_ρ∈ BV((-ρ2,ρ2)) has zero average, and 
	
    ṽ_ρ(y)= ψ_ρ(x·)  + (x·) β_ρ.

Moreover, from the very definitions of ṽ_ρ, ψ_ρ and β_ρ we see that
	
    E ṽ_ρ=(⊙)(D ψ̅_ρ⊗Ł^n-1 +β̅_ρŁ^n)
    	=(⊙)(D ψ_ρ⊗Ł^n-1 +β_ρŁ^n).

Let I⊆ (-ρ2,ρ2) be Ł^1-measurable, then

    ⊙/|⊙|   |E ṽ_ρ| (I×  (-12,12)^n-1)
    	=E ṽ_ρ (I×  (-12,12)^n-1)
       
    	=(⊙)( D ψ̅_ρ (I)+   β̅_ρŁ^1(I) )  
    	=(⊙)( D ψ_ρ (I)+   β_ρŁ^1(I) ).

In particular, if I=(-ρ2,ρ 2) by exploiting (<ref>) we conclude 
	
    Dψ̅_ρ((-ρ2,ρ2))+β̅_ρρ
    	=Dψ_ρ((-ρ2,ρ2))+β_ρρ
    	=ρ/|η⊙ξ|.



Step 2: The value of β_ρ.
Let :={I⊆(-ρ2,ρ2)  Ł^1 -measurable:Ł^1(I)>0}, then from (<ref>) we deduce that 
	
    inf_Dψ̅_ρ(I)/Ł^1(I)+β̅_ρ≥0,     inf_D ψ_ρ(I)/Ł^1(I)+β_ρ≥0  .

Hence, t↦ψ̅_ρ(t)+β̅_ρt and t↦ψ_ρ(t)+β_ρt 
are monotone non-decreasing functions. 














Thus, thanks to the definition of κ_ρ we infer that D ψ_ρ (-ρ2,ρ2)=β_ρρ as

    D ψ_ρ   ((-ρ2,ρ2))=D ψ̅_ρ((-ρ2,ρ2))-κ_ρρ
       (<ref>)=1/2(D ψ̅_ρ((-ρ2,ρ2))+β̅_ρρ)
    (<ref>)=1/2(D ψ_ρ((-ρ2,ρ2))+β_ρρ).

In conclusion, we get 
	
    D ψ_ρ((-ρ2,ρ2))
    	=
    	β_ρρ=ρ/2|η⊙ξ|.



Step 3: Inverse change of variables and conclusion. 
By combining (<ref>) and (<ref>) we are thus led to
    
    ṽ_ρ(y)=ψ_ρ(x·) + (x·) /2|η⊙ξ| .





Due to the very definition, v_ρ(y):=𝔹^-1ṽ_ρ (𝔹^-t y), thus we get that
    
    v_ρ(y)=ψ_ρ(x·η)ξ + x·ξ/2|η⊙ξ|η .

Finally, by taking into account that  t↦ h_ρ(t):=ψ_ρ(t)+β_ρt 
is monotone non-decreasing with zero average we get (by <cit.> and a simple scaling argument)

    h_ρ_L^∞((-ρ2,ρ2))≤ |Dh_ρ|((-ρ2,ρ2))=ρ/|η⊙ξ|.


The statement for ψ_ρ then follows at once.
	
Similarly, we can characterize the case ξ=±η.

Let {v_ρ}_ρ>0⊆ BV(P^ξ,η_ρ;^n), ξ, η∈^n∖{0} 
with ξ=±η, be a family of functions such that
	
    v_ρ(x)=ψ̅_ρ(x·η) η+_ρ x+v_ρ

for some v_ρ∈^n, _ρ∈𝕄_skew^n× n, 
ψ̅_ρ∈ BV((-ρ2,ρ2)). 
Assume also that, 
	
    ℜ_P^ξ,η_ρ[v_ρ]=0,    |E v_ρ|(P^ξ,η_ρ)=
    	Ł^n(P^ξ,η_ρ) .

Then, v_ρ can be re-written as
	
    v_ρ(x)=    ψ_ρ(x·η) η

for a non-decreasing function ψ_ρ∈ BV((-ρ2,ρ2)) with zero average. 
Moreover, {ψ_ρ(·)ρ}_ρ>0 is uniformly bounded in 
L^∞((-ρ2,ρ2)), and

    |D ψ_ρ|((-ρ2,ρ2))
     =ρ/|η⊙η| .



Let us summarize the results contained in Proposition <ref>, Lemma <ref> and
Lemma <ref> in the following statement (see (<ref>) for the definition of u_K,x,, and
Eqs. (<ref>), (<ref>) for those of P_ρ^ξ,η).

Let u∈ BD(Ω). Then for |E^cu|-a.e. x∈Ω 
there exist vectors ξ, η∈𝕊^n-1, {ρ_j}_j∈, ρ_j↓ 0 as j↑+∞, 
and for all j∈ a sequence {_i,j}_i∈, with _i,j↓ 0 as i↑+∞, 
a bounded, open, convex set containing the origin P^x_j:=P^ξ,η_ρ_j and a function v_j such that 
u_P^x_j,x,_i,j converge to v_j strictly in BD(P^x_j) as i↑+∞, with
	
	
  (a) if ξ≠±η: 

    v_j(y):=ψ_j(y·η)ξ+y·ξ/2|η⊙ξ|η

for some map ψ_j∈ BV((-ρ_j2,ρ_j2)) with zero average such that
	
    Dψ_j((-ρ_j2,ρ_j2))
    =ρ_j/2|η⊙ξ|,    
    sup_j∈ψ_j (·)ρ_j_L^∞((-ρ_j2,ρ_j2))<+∞;


  (b) if ξ=±η: 

    v_j(y)=ψ_j(y·η)η

for some non-decreasing map ψ_j∈ BV((-ρ_j2,ρ_j2)) with zero average such that

    |Dψ_j|((-ρ_j2,ρ_j2))
    =ρ_j/|η⊙η|,    
    sup_j∈ψ_j (·)ρ_j_L^∞((-ρ_j2,ρ_j 2))<+∞.


 

We prove the result for the subset of points F for which Proposition <ref> holds true.
In particular, |E^c u|(Ω∖ F)=0. One such point x being fixed, note that given any infinitesimal 
sequence {ρ_j}_j∈ we can extract a subsequence (not relabeled) along which the maps v_ρ_j 
provided by Proposition <ref> are affine in one single direction (either η 
or ξ) provided in the statement.
Without loss of generality we denote such a direction by ξ to be coherent with the notation of 
Proposition <ref> itself.

Assume first ξ≠±η. Thanks to Proposition <ref> we can find a sequence of scales 
{_i,j}_i∈ such that the rescaled maps converge strictly in BD(P_j^x) to a map v_ρ_j  as
in the statement there. By using Lemma <ref> we conclude.

Finally, if ξ=±η we argue similarly by using Lemma <ref> rather than Lemma <ref>.






§ PROOF OF THE MAIN RESULT

We first recall the results in <cit.>. 
The original statement concerns integral representation of the volume and jump energy 
densities of functionals satisfying (H1)-(H4) and a more stringent version of (H5) (cf. Remark <ref>) 
and for functions in the subspace SBD(Ω). In what follows we state the result for the full space 
BD(Ω). Indeed, the same proof works with no difference since it is obtained via 
the global method for relaxation, hinging on a blow-up argument and the 
characterization of the energy densities in terms of the Dirichlet cell formulas defining 
. We notice that (H4) and (H5) are actually not needed for the integral representation of the bulk and surface 
terms of the energy.

   Let  be satisfying (H1)-(H3). Then, for every u∈ BD(Ω) 
    
        
  (a) for Ł^n-a.e. x_0∈Ω 
            
    lim_→ 0(u,Q(x_0,))/^n= f(x_0,u(x_0),∇ u(x_0)),

            where f denotes the function defined in (<ref>); 
        
  (b) for ^̋n-1-a.e. x_0∈  J_u 
             
    lim_→ 0(u,Q^ν_u(x_0)(x_0,))/^n-1= 
                g(x_0,u^-(x_0),u^+(x_0),ν_u(x_0))

            where g denotes the function defined in (<ref>), 
    


It should be noted that Lemma <ref> and the lower semicontinuity of the integral on W^1,1 
implies that the Borel functions f and g are respectively quasiconvex
(see <cit.>, see also formulas (<ref>), (<ref>) below) and BV elliptic (see <cit.>).

By taking into account (H2), we conclude that there exists a constant C>0 such that 
for every (x,v,)∈Ω×^n×𝕄^n× n

    1/C|+^t/2|≤ f(x,v,)
     ≤ C(1+|+^t/2|),

and that for every (x,v^-,v^+)∈Ω×(^n)^2

    1/C|(v^+-v^-)⊙ν|≤ g(x,v^-,v^+,ν)≤ 
     C|(v^+-v^-)⊙ν|.

Several other properties of f and g can be inferred according to the invariance 
properties satisfied by the functional  (cf. <cit.>). 
For instance, assumption (H5) implies that f depends only on the symmetric part of 
the relevant matrix.
Indeed, from (<ref>) we immediately deduce, for all (x_0,v_0,)∈Ω×^n×𝕄^n× n, 
(thanks also to item (a) in Lemma <ref>) that
    
    f(x_0,v_0,)   =lim_→ 0(v_0+(· -x_0),Q(x_0,))/^n
       =lim_→ 0(v_0++^t/2(· -x_0),Q(x_0,))/^n
        =f(x_0,v_0,+^t/2).

Therefore, in this case we deduce that f is symmetric quasiconvex. Namely, for every bounded open set D⊂^n, for Ł^n a.e. x∈Ω,
for all (v,)∈^n×𝕄^n× n_sym and for all φ∈𝒞^1_c(D;^n)
	
    f(x,v,)≤_D f(x,v,+e(φ)(y))ỵ,

	or, equivalently, for all φ∈𝒞^1(Q^ν(0,1);^n) that are Q^ν(0,1)-periodic, 

it holds
	
    f(x,v,)≤_Q^ν(0,1) f(x,v,+e(φ)(y))ỵ.



If, in addition, we strengthen (H5) to
 
    ℱ(u+(·-x_0)+v,A)=ℱ(u,A)

for every (u,A,v,,x_0) ∈ BD(Ω)×𝒪(Ω)×^n×𝕄_skew^n× n×Ω, 
 then the cell formulas imply f(x,v,+^t/2)=f(x,+^t/2) and 
 g(x,v^-,v^+,ν)=g(x,v^+-v^-,ν). 




 §.§ Preliminary lemmas


We now exploit the result in Section <ref>, in particular 
Proposition <ref>, 
to deduce the asymptotic behavior of the energy around |E^cu|-a.e. point along
the same line developed in <cit.>. 
We keep the notation introduced in Proposition <ref> and to simplify it 
we set ψ_j:=ψ_ρ_j and P_j^x_0:=P_ρ_j^x_0 (for the definition of
ℜ_K see (<ref>)).

Let  satisfy (H1)-(H4). Then, for every u∈ BD(Ω) and 
for |E^c u|-a.e x_0∈Ω there exist a sequence {ρ_j}_j∈ infinitesimal as j↑+∞, 
and for all j∈ an infinitesimal sequence {_i,j}_i∈ as i↑+∞, such that 
    
    (u,·)/|̣E u|(x_0)
            =lim_j → +∞lim sup_i→ +∞(w_i,j, P_j^x_0(x_0,_i,j)) /|E u|(P_j^x_0(x_0,_i,j))

    where
    
    w_i,j(y):=ℜ_P_j^x_0(x_0,_i,j)[u] (y-x_0)
        +|E u|(P_j^x_0(x_0,_i,j))/Ł^n(P_j^x_0(x_0,_i,j)))η⊙ξ/|η⊙ξ|(y-x_0).


 We consider the subset of points of C_u for which Proposition <ref> (and hence Proposition <ref>) is valid. 
For one such point x_0∈ C_u consider the corresponding 
vectors ξ and η∈𝕊^n-1. 
Note that by Lemma <ref> for any j∈
    
    (u,·)/|̣E u|(x_0)=lim_i→ +∞(u, P_j^x_0(x_0,_i,j))/|E u|(P_j^x_0(x_0,_i,j)).



Case 1: η≠±ξ. 
By Proposition <ref> we have that for every j∈
	
    u_i,j:=u_P_j^x_0,x_0,_i,j→ v_j:=ψ_j(y·η) ξ +y·ξ/2|η⊙ξ|η     strict in BD(P_j^x_0).

Define, for some constant c_j to be specified in what follows, the functions
    
    v_i,j(y):=w_i,j(y)+_i,j|E u|(P_j^x_0(x_0,_i,j))/Ł^n(P_j^x_0(x_0,_i,j)) c_j ξ.

As by Remark <ref>

    ℜ_P^x_0_j[u(x_0+_i,j· )](y-x_0/_i,j)
     =ℜ_P^x_0_j(x_0,_i,j)[u](y-x_0),

we have by Lemma <ref> 
    
    |(u,P_j^x_0(x_0,_i,j))-(v_i,j,P_j^x_0(x_0,_i,j))|/|E u |(P_j^x_0(x_0,_i,j))
       ≤ C∫_∂ P_j^x_0(x_0,_i,j)|u(y)-v_i,j(y)|/|E u |(P_j^x_0(x_0,_i,j))̣̋^n-1(y)
       = C/ρ_j∫_∂ P_j^x_0|u_i,j(y) - η⊙ξ /|η⊙ξ| y -c_jξ|̣̋^n-1(y).

By taking the superior limit in i we thus get

    lim sup_i→+∞   |(u,P_j^x_0(x_0,_i,j))-(v_i,j,P_j^x_0(x_0,_i,j))|/|E u |(P_j^x_0(x_0,_i,j))
       ≤C /ρ_j∫_∂ P_j^x_0| ψ_j(y·η)- (y·η)/2|η⊙ξ|-c_j|̣̋^n-1(y)
       =C /ρ_j|ψ_j(ρ_j2)- ρ_j/4|η⊙ξ|-c_j  |
            +C /ρ_j|ψ_j(-ρ_j2)+ρ_j/4|η⊙ξ|-c_j  |
            
       +C /ρ_j∫_-ρ_j2^ρ_j2| ψ_j(t)- t/2|η⊙ξ| -c_j|ṭ,

recalling that
P_j^x_0={y∈^n: |y·η|≤ρ_j2,  |y·ξ|≤12, |y·ζ_i|≤12   i=1,…,n-2},
and that ξ and η depend on x_0.
    
    
By choosing c_j:=ψ_j(-ρ_j2)+ρ_j/4|η⊙ξ|, since ψ_j(ρ_j2)-ψ_j(-ρ_j2)
=ρ_j/2|η⊙ξ|, the first two summands in the last inequality are then null. Therefore, we have

    lim sup_i→+∞   |(u,P_j^x_0(x_0,_i,j))-(v_i,j,P_j^x_0(x_0,_i,j))|/|E u |(P_j^x_0(x_0,_i,j))
       ≤C /ρ_j∫_-ρ_j2^ρ_j2| ψ_j(t)- t/2|η⊙ξ| -c_j|ṭ≤ C(ψ_j_L^∞(-ρ_j/2,ρ_j/2)+ρ_j+c_j)≤ Cρ_j ,

where we have used that ψ_j(·)ρ_j is equi-bounded in L^∞((-ρ_j2,ρ_j2)), 
and that ψ_j(±ρ_j2) are boundary trace values to infer c_j≤ Cρ_j, for some universal constant C>0. 
In conclusion, we have proved that 
    
    lim_j→+∞lim sup_i→+∞|(u,P_j^x_0(x_0,_i,j))-(v_i,j,P_j^x_0(x_0,_i,j))|/|E u |(P_j^x_0(x_0,_i,j))=0,

so that (<ref>) yields    
    
    (u,·)/|̣E u|(x_0)=lim_j→+∞lim sup_i→+∞(v_i,j,P_j^x_0(x_0,_i,j))/|E u |(P_j^x_0(x_0,_i,j)).

Finally, recalling the definition of w_i,j in (<ref>) and of v_i,j in (<ref>), by estimate (<ref>) we have
    
    | (v_i,j   
      ,P_j^x_0(x_0,_i,j))- (w_i,j,P_j^x_0(x_0,_i,j))|
       ≤ C_P_j^x_0Ψ(_i,j|E u|(P_j^x_0(x_0,_i,j))/Ł^n(P_j^x_0(x_0,_i,j))c_j)
       (Ł^n(P_j^x_0(x_0,_i,j))+|Eu|(P_j^x_0(x_0,_i,j))).

and (<ref>) then follows at once from (<ref>) by letting i↑+∞, 
in view of the choice x_0∈ C_u. 


Case 2: ξ=±η. Suppose, without loss of generality that ξ=η. We argue as in Case 1.
For the sequences {ρ_j}_j∈, {_i,j}_i∈ provided by Proposition <ref> we have that 

    u_P_j^x_0,x_0,_i,j→ v_j:=ψ_j(y·η) η     strict in BD(P_j^x_0).

By setting
    
    v_i,j(y):= w_i,j(y)+_i,j|E u|(P_j^x_0(x_0,_i,j))/Ł^n(P_j^x_0(x_0,_i,j)) c_j η ,

for 

    c_j:=ψ_j(-ρ_j/2)+ρ_j/2|η⊙η| ,

we conclude that 

    lim_j→ +∞lim sup_i→ +∞(v_i,j,P_j^x_0(x_0,_i,j))/|Eu|(P_j^x_0(x_0,_i,j))=
    lim_i→ +∞(u,P_j^x_0(x_0,_i,j))/|Eu|(P_j^x_0(x_0,_i,j) ).
 
We again combine this equality with (<ref>) to conclude.

We now use assumption (H5) to prove a lower bound for the cell formula  computed
on affine functions as done in <cit.>.

Let  satisfy (H1)-(H5). For all v∈^n, ξ'∈^n∖{0}, η∈𝕊^n-1, x_0∈Ω 
and for every sequence (t_i,_i)  such that t_i→ +∞ and _i t_i → 0, 
and for every ρ>0, it holds 
        
    f(x_0,v,η⊙ξ')-f(x_0,v,0)≤lim inf_i→ +∞(v+t_i η⊙ξ' (·-x_0), P_ρ^ξ,η(x_0,_i))/t_iŁ^n(P_ρ^ξ,η(x_0,_i))

where ξ:=ξ'|ξ'|, P_ρ^ξ,η is defined either in (<ref>) or (<ref>) according to whether ξ≠±η or not, and f is the 
volume energy density defined in item (a) of Lemma <ref>.


We start off noting that η⊗ξ'=
η⊙ξ'+, where := 1/2(η⊗ξ'-ξ⊗η')∈𝕄^n× n_skew.
Then, in view of (H5) formula (<ref>) implies
    
    |(v+t_i η⊗ξ'(· -x_0), P_ρ^ξ,η(x_0,_i)) 
        -(v+t_i η⊙ξ' (·-x_0), P_ρ^ξ,η(x_0,_i))|
               ≤ C_P_ρ^ξ,ηΨ(_i t_i ||)(1+t_i|ξ'|)_i^n.

Hence, 
    
    lim inf_i→ +∞(v+ t_i η⊙ξ' (·-x_0), P_ρ^ξ,η(x_0,_i))/t_iŁ^n(P_ρ^ξ,η(x_0,_i))   =lim inf_i→ +∞(v+t_i η⊗ξ' (·-x_0), P_ρ^ξ,η(x_0,_i))/t_iŁ^n(P_ρ^ξ,η(x_0,_i))
       ≥ f(x_0,v,η⊗ξ')-f(x_0,v,0).

For the last inequality we have used <cit.>. Moreover, since by (<ref>) 
    
    f(x_0,v,η⊗ξ')  = f(x_0,v,η⊙ξ'),

the conclusion follows at once.

Before proving Theorem <ref>, we note that the continuity estimate on  contained in (<ref>), deduced as a consequence of (H4), implies both

    |f(x_0+x,v+v_0,)-f(x_0,v_0,)|≤Ψ(|x|+|v|)(1+||)

and

    |g(x_0+x,v_0+v,v_1+v,ν)-
    g(x_0,v_0,v_1,ν)|≤Ψ(|x|+|v|)|v_0-v_1|

for all (x_0,x,v_0,v_1,v,ν,)∈(Ω)^2×(^n)^3×𝕊^n-1×𝕄^n× n_sym.

These properties are instrumental already in the BV setting to express the Radon-Nikodým derivative of  
at u with respect to |E^cu| in terms of an energy density computed on relevant quantities related to the base 
function u itself.
In particular, by taking such properties into account, one can prove that the recession function f^∞ of 
the bulk energy density f is actually the energy density of the Cantor part.
 
 

 §.§ Proof of the integral representation result
 
 

The representation of the volume and surface energy densities is dealt with in Lemma <ref>.

We then turn to the representation of the energy density of the Cantor part. 
For |E^cu|-a.e. point x_0∈ C_u we may apply Lemma <ref> 
    (in what follows we keep the notation introduced there) and find infinitesimal sequences 
    {ρ_j}_j∈, {_i,j}_i∈ such that
    
    (u,·)/|̣E^cu|(x_0)=(u,·)/|̣E u|(x_0)
        =lim_j→ +∞lim sup_i→+∞(w_i,j,P_j^x_0(x_0,_i,j))/|E u|(P_j^x_0(x_0,_i,j)).

On setting

    v_i,j:=_P_j^x_0(x_0,_i,j)u(x)x̣,     _i,j :=𝕄_P_j^x_0(x_0,_i,j)[u]

and t_i,j:=|E u|(P_j^x_0(x_0,_i,j))/Ł^n(P_j^x_0(x_0,_i,j)), we have
w_i,j=v_i,j+_i,j(· -x_0)+t_i,jη⊙ξ/|η⊙ξ| (·-x_0).
Note that t_i,j→ +∞ as i→ +∞ for all j∈ as x_0∈ C_u. Moreover, recall that 
x_0 is a point of approximate continuity of u. 

Next we note that 

    |   (w_i,j,P_j^x_0(x_0,_i,j))-
    (u(x_0)+t_i,jη⊙ξ/|η⊙ξ| (·-x_0),P_j^x_0(x_0,_i,j))|
       ≤|(w_i,j,P_j^x_0(x_0,_i,j))-
    (v_i,j+t_i,jη⊙ξ/|η⊙ξ| (·-x_0),P_j^x_0(x_0,_i,j))|
       +|(v_i,j+t_i,jη⊙ξ/|η⊙ξ| 
    (·-x_0),P_j^x_0(x_0,_i,j))-
    (u(x_0)+t_i,jη⊙ξ/|η⊙ξ| (·-x_0), P_j^x_0(x_0,_i,j))|
       (<ref>), (<ref>)≤ 
    C_P_j^x_0(Ψ(_i,jdiam(P_j^x_0) |_i,j|)+
    Ψ(|v_i,j-u(x_0)|))(1+t_i,j)_i,j^n.

By taking into account v_i,j→ u(x_0) and _i,j|_i,j|→ 0 as i↑ +∞
thanks to Lemma <ref>, the latter estimate combined with (<ref>) leads to
    
    (u,·)/|̣E^cu|(x_0)   =(u,·)/|̣E u|(x_0)
       
        =lim_j→ +∞lim sup_i→+∞(
         u(x_0)+t_i,jη⊙ξ/|η⊙ξ| (·-x_0),P_j^x_0(x_0,_i,j))/Ł^n(P_j^x_0(x_0,_i,j)) t_i,j.

With fixed j∈ and λ>0, by applying Lemma <ref> with 
ξ'=λξ/|η⊙ξ| and t_i:=t_i,jλ, Eq. (<ref>) implies 
    
    (u,·)/|̣E^cu|(x_0)≥f(x_0,u(x_0),λη⊙ξ/|η⊙ξ|)-f(x_0,u(x_0),0)/λ.

Hence, by taking the superior limit as λ↑+∞ we infer
    
    (u,·)/|̣E^cu|(x_0)≥ 
            f^∞(x_0,u(x_0),η⊙ξ/|η⊙ξ|).


On the other hand, using as a competitor in the cell problem defining 

    (u(x_0)+ t_i,jη⊙ξ/|η⊙ξ| (· -x_0), P_j^x_0(x_0,_i,j))
 
the affine map u(x_0)+t_i,jη⊙ξ/|η⊙ξ| (·-x_0) itself, we can apply item (a) in Lemma <ref> to deduce
    
    (u(x_0)+t_i,jη⊙ξ/|η⊙ξ| (·-x_0), P_j^x_0(x_0,_i,j))/t_i,jŁ^n(P_j^x_0(x_0,_i,j))≤ℱ(u(x_0)+t_i,jη⊙ξ/|η⊙ξ| (·-x_0), P_j^x_0(x_0,_i,j))/t_i,jŁ^n(P_j^x_0(x_0,_i,j))
       = _P_j^x_0(x_0,_i,j)1/t_i,jf(x,u(x_0)+t_i,jη⊙ξ/|η⊙ξ|(x-x_0),t_i,jη⊙ξ/|η⊙ξ|)x̣
       (<ref>)≤1/t_i,j
    f(x_0,u(x_0),t_i,jη⊙ξ/|η⊙ξ|)
    +2Ψ(_i,j+_i,jt_i,j)
       ≤1/t_i,j(f(x_0,u(x_0),t_i,jη⊙ξ/|η⊙ξ|)
    -f(x_0, u(x_0),0))+C/t_i,j.

Since, t_i,j→ +∞ as i→ +∞ for all j∈, we deduce that
    
    lim_j→+∞lim sup_i→+∞(u(x_0)+t_i,jη⊙ξ/|η⊙ξ| (·-x_0), P_j^x_0(x_0,_i,j))/Ł^n(P_j^x_0(x_0,_i,j))) t_i,j≤ f^∞(x_0,u(x_0),η⊙ξ/|η⊙ξ|),

that combined with (<ref>) and (<ref>) finally leads to
    
    (u,·)/|̣E^cu|(x_0)=
        f^∞(x_0,u(x_0),Ẹ^cu/|̣E^cu|(x_0)).


A standard monotone approximation technique provides the following extension of Theorem <ref> 
(see Section <ref> for a similar argument).

Let :L^1(Ω;^n)×𝒪(Ω)→[0,+∞] be satisfying (H1), (H3), (H4), (H5) 
and in place of (H2)


  (HH2) there exists a constant C>0 such that for every (u,A)∈ BD(Ω)×𝒪(Ω)
        
    0≤(u,A) ≤ C (Ł^n(A)+|E u|(A)).


Then, the conclusions of Theorem <ref> hold for all u∈ BD(Ω).


Let δ>0, and consider the functionals _δ:L^1(Ω;^n)×𝒪(Ω)→[0,+∞]  
be defined by _δ(u,A):=(u,A)+δ|Eu|(A). Since _δ satisfies all the conditions (H1)-(H5) of 
Theorem <ref> there are two functions f_δ and g_δ such that _δ can be represented as in 
the statement there. The family of functionals _δ is pointwise increasing in δ, therefore there exist the pointwise 
limits f of f_δ and g of g_δ as δ↓ 0 (note that from the very definition of recession function 
f^∞_δ(x,v,𝔸)=f^∞(x,v,𝔸)+δ|𝔸|). 
As _δ(·,A) is pointwise converging to (·,A) for δ↓ 0 on BD(Ω) for all 
A∈𝒪(Ω), we conclude that the integral representation with energy densities f, g and f^∞ 
for the bulk, surface and Cantor parts, respectively, holds for . 






§ SOME APPLICATIONS


Following <cit.> we provide some applications of the integral representation 
Theorem <ref> to the topics of relaxation of bulk energies, of bulk and interfacial energies, to that of L^1 lower semicontinuity of functionals defined on BD, and to that of homogenization of bulk energies. 

Throughout the section, for all ν∈𝕊^n-1 and r>0  we denote the cubes Q^ν(0,r) by Q^ν_r, 
and moreover Q(0,r) by Q_r.




 §.§ Relaxation and L^1 lower semicontinuity of bulk energies



In this section we address the issue of giving an explicit expression to the L^1 lower semicontinuous envelope of a linearly growing functional defined on 
smooth maps, for instance LD(Ω).

Theorem <ref> below generalizes to BD(Ω) the results proven in <cit.> and <cit.> on SBD(Ω). In particular, in <cit.> a continuous autonomous integrand f_0 (i.e. depending only on the symmetric gradient) is considered, the integral representation is then given in terms of the symmetric quasiconvex envelope of f_0  (see the definition below) and its associated recession function.  In addition, Theorem <ref> also generalizes 
partially the results on BD(Ω) established in <cit.> and <cit.>, <cit.>. Note that in the former, also a Dirichlet boundary condition is considered, while
in the last two integral representations of the weakly* lower semicontinuous envelope of functionals with linear growth at infinity are provided for more general PDEs constraints on the approximating sequences. 

We stress that in the ensuing result, the strong recession function is not required to exist, and that the integrand is allowed 
to depend also on x and v. Moreover, global continuity is replaced by the weaker condition (H2'). 




Let f_0:Ω×^n×𝕄^n× n_sym→[0,+∞) be a Borel integrand satisfying:


  (H1') there exists a constant C>0 such that 
for every (x,v,)∈Ω×^n×𝕄^n× n_sym 

    1/C|| ≤ f_0(x,v,)≤ C(1+||);

 
  (H2') there exists a constant C>0 such that for every >0 there exists 
 δ>0 such that
 
    |x-x_0|+|v-v_0|≤δ⟹
      |f_0(x,v,)-f_0(x_0,v_0,)|≤ C(1+||),

 for every 
 (x,v,v_0,)∈Ω×(^n)^2×𝕄^n× n_sym.

Let then _0:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] be the functional defined by 

    _0(u,A):=inf{lim inf_j→+∞F_0(u_j,A): u_j→ u  in L^1(Ω;^n)},

namely the L^1(Ω;^n) lower semicontinuos envelope of the functional

    F_0(u,A):=∫_Af_0(x,u(x),e(u)(x))x̣   if u∈ LD(Ω)
       
    
                           +∞   otherwise on L^1(Ω;^n).

We denote by  the cell formula defined in (<ref>) and related to ℱ_0, and
recall the notation u_v^-,v^+,ν introduced in (<ref>). 
We also recall that f^∞ stands for the (weak) recession function as defined by (<ref>).  
 
 Let F_0:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] be the functional defined in (<ref>).
 Then, assuming (H1')-(H2'), the functional _0:L^1(Ω;^n)×𝒪(Ω)→[0,+∞]
 defined in (<ref>) is represented by
 
    _0(u,A)   =∫_Af(x,u(x),e(u)(x))x̣
       
      +∫_J_u∩ Ag(x,u^-(x),u^+(x),ν_u(x))̣̋^n-1(x)
        +∫_Af^∞(x,u(x),Ẹ^cu/|̣E^cu|(x))|̣E^cu|(x),

for all (u,A)∈ BD(Ω)×𝒪(Ω), where for every 
(x_0,v,)∈Ω×^n×𝕄^n× n_sym

    f(x_0,v,)
     =lim sup_→0inf_w∈ LD(Q_1)
     w|_∂ Q_1= y|_∂ Q
    ∫_Q_1f_0(x_0,v+ w(y),e(w)(y))ỵ,

and for every (x_0,v^-,v^+,ν)∈Ω×(^n)^2×𝕊^n-1

    g(x_0,v^-,v^+,ν)=lim sup_→0inf_w∈ LD(Q^ν_1)
    w|_∂ Q^ν_1=u_v^-,v^+,ν|_∂ Q^ν_1∫_Q^ν_1  f_0(x_0,w(y),1/e(w)(y))ỵ.



Assumption  (H2') implies that _0 satisfies (H4).
Instead, in the BV-setting, the ensuing weaker assumption (H3') replaces  (H2') in <cit.>:

 
  (H3') there exists a constant C>0 such that for every >0 there exists 
 δ>0 such that
 
    |v-v_0|≤δ⟹
      |f_0(x,v,)-f_0(x,v_0,)|≤ C(1+||),

 for every (x,v,v_0,)∈Ω×(^n)^2×𝕄^n× n_sym.

The latter condition and a truncation argument 
(cf. <cit.>) are employed both to simplify the minimum problems defining f and g with respect to the v-variable 
(cf. equations <cit.>, and also to dispense with the analogue of (H4). Therefore, since the integral representation result in the BV setting cannot be directly applied, 
the quoted truncation argument and the analogue of Lemma <ref> are central to give an 
explicit formula for the energy density of the Cantor part of the relaxed functionals analogous to _0 
(cf. Eq. <cit.> and 
<cit.>). 
Instead, since truncations are not permitted in the current BD setting, we need the stronger assumption (H2') to enforce (H4).



In order to prove the integral representation of _0 over the subspace SBD(Ω) 
only, assumption (H2') is actually not needed and (H1') can be weakened. Indeed, to that aim one can allow for f_0 to depend also on 
the skew-symmetric part of the given matrix in view of Lemma <ref> (cf. Section <ref>). 
Clearly, formulas (<ref>) defining f and (<ref>) defining g have to be changed accordingly.


In view of the density of W^1,1(Ω;^n) in BD(Ω) with respect to the strict topology 
with assigned boundary trace (cf. <cit.>), the space LD can 
be substituted with W^1,1 in the minimum problems defining f and g. 

Therefore, the same conclusions of Theorem <ref> can be drawn if we consider the functional 
F'_0(u,A):=F_0(u,A) for u∈ W^1,1(Ω;^n), and +∞ otherwise on L^1(Ω;^n). 
Then the space of test maps for the minimum problems defining f and g in (<ref>) and (<ref>) respectively, 
is W^1,1(Ω;^n).

The main steps to prove Theorem <ref> are similar to those exploited for the analogous result in the BV setting in <cit.> 
to which we refer. Therefore, we provide only a sketch of those proofs for which some changes are needed. 

Assume (H1')-(H2'). Then, _0 satisfies (H1)-(H5), and for all (u,A)∈ BD(Ω)×𝒪_∞(Ω) 

    (u,A)=_0(u,A):=inf{F_0(w,A): w∈ BD(A), w=u  on ∂ A}.



Assumptions (H1), (H2) and (H5) are easily checked to be satisfied by _0 in view of (H1')
and the very definition of _0. Instead, (H4) follows from (H2').
For what concerns (H3) one can argue similarly to <cit.>.

The inequality (u,A)≤_0(u,A) follows from _0≤ F_0. 
For the opposite, one uses the very definition of _0 as the relaxation of F_0 together 
with the version of the De Giorgi's averaging/slicing lemma stated in 
<cit.>.

By means of the alternative characterization of  provided by _0, of 
Theorem <ref>, of the results in Section 4 and of a change of variable, Theorem <ref> follows at once.

If an additional quantified closeness condition between f computed on large gradients and f^∞ is added, 
formula (<ref>) defining g can be simplified. The ensuing claim (<ref>) is 
well-known in the in the BV case (cf. <cit.>). 

 Under the assumptions and notation of Theorem <ref> and in addition 
 
  
  (H4') on setting for any L>0 and x_0∈Ω

    ω_f_0(x_0,L):=sup_v∈^n, t≥ L 
    ∈𝕄^n× n_sym: ||=1|f_0^∞(x_0,v,)
    -1/tf_0(x_0,v,t)| ,
  
then ω_f_0(x_0,L) is infinitesimal as L→+∞,





 
the function g in Eq. (<ref>) in the conclusions of 
Theorem <ref> is characterized alternatively by

    g(x_0,v^-,v^+,ν)=
    inf_w∈ LD(Q^ν_1)
    w|_∂ Q^ν_1=u_v^-,v^+,ν|_∂ Q^ν_1∫_Q^ν_1f_0^∞   (x_0,w(y),e(w)(y))ỵ.


Furthermore, we deal with the v-independent case for which (H4') is actually not needed (cf. <cit.> 
for the analogous result in the BV setting). We start off with a preliminary result.

Under the assumptions and notation of Theorem <ref>, if the integrand f_0 satisfies
f_0(x,v,)=f_0(x,) for every (x,v,)∈Ω×^n×𝕄^n× n_sym, then

    g(x_0,v^-,v^+,ν)=f^∞(x_0,(v^+-v^-)⊙ν)

for every (x_0,v^-,v^+,ν)∈Ω×(^n)^2×𝕊^n-1. 


We start off defining

    F_0(u,A):=∫_Af(x,e(u)(x))x̣   if u∈ LD(Ω)
     
                           +∞   otherwise on L^1(Ω;^n),

where f is given by (<ref>). Note that f is v-independent as
_0(u+z,A)=_0(u,A) for all (u,A,z)∈ BD(Ω)×𝒪(Ω)×^n.
Moreover, f(x_0,)≤ f_0(x_0,), for all (x_0,)∈Ω×𝕄^n× n_sym, by using 
the linear function y itself as a test in (<ref>) and (H2').

Denote by ℱ_0 the L^1(Ω;^n) lower semicontinuous envelope of F_0.
Then, F_0≤ F_0 implies that ℱ_0≤_0 on L^1(Ω;^n), and since _0=F_0 on LD(Ω)
and _0≤F_0 otherwise, we conclude that 
_0≡ℱ_0.
Therefore, equality (<ref>) defining g holds with f in place of f_0 in the minimum problem there. 
In passing, we point out that the invariance of f_0 implies that g=g(x,v^+-v^-,ν), as well
(cf. Remark <ref>).

Let us first prove that g(x_0,v^+-v^-,ν)≥ 
f^∞(x_0,(v^+-v^+)⊙ν).
Let {ν_1,…,ν_n-1,ν} form an orthonormal basis of ^n and set 

    𝒜^ν:={w∈ W^1,1(Q^ν_1;^n):     w|_{x∈∂ Q^ν_1: x·ν=±12}=0,
       w is 1-periodic in ν_i, 1≤ i ≤ n-1},

and

    ℬ^ν:={w∈ W^1,1(Q^ν_1;^n):     (w-u_v^-,v^+,ν)|_{x∈∂ Q^ν_1: x·ν=±12}=0,
       w is 1-periodic in ν_i, 1≤ i ≤ n-1},

In particular, note that ℬ^ν=ζ(·)+𝒜^ν, where

    ζ(x):=v^++v^-/2+(v^+-v^-)x·ν.

We then argue as follows

    g   (x_0,v^+-v^-,ν)
    =lim sup_→0inf_w∈ LD(Q^ν_1)
    w|_∂ Q^ν_1=u_v^-,v^+,ν|_∂ Q^ν_1∫_Q^ν_1 f(x_0,1/e(w)(y))ỵ
       =lim sup_→0inf_w∈ W^1,1(Q^ν_1;^n)
    w|_∂ Q^ν_1=u_v^-,v^+,ν|_∂ Q^ν_1∫_Q^ν_1 f(x_0,1/e(w)(y))ỵ
       ≥lim sup_→0inf_w∈ℬ^ν∫_Q^ν_1 f(x_0,1/e(w)(y))ỵ
       =lim sup_→0inf_w∈𝒜^ν∫_Q^ν_1 f(x_0,1/
    (v^+-v^-)⊙ν+1/e(w)(y))ỵ ,

where for the second equality we have used Remark <ref>, and for the last, Eq. (<ref>). Using the characterization of symmetric quasiconvexity expressed in (<ref>) we conclude from (<ref>) that 

    g(x_0,v^+-v^-,ν)≥lim sup_→0 f(x_0,1/(v^+-v^-)⊙ν)=
    f^∞(x_0,(v^+-v^-)⊙ν).

To prove the opposite inequality, consider the affine function ζ in (<ref>), 
extend it by 1-periodicity in the directions ν_i, 1≤ i≤ n-1, and extend 
it further by v^± if ± x·ν≥12 (with a slight 
abuse we keep the same notation for the extended function). 
Next let w∈ W^1,1(Q^ν_1;^n) have the same trace of u_v^-,v^+,ν
on ∂ Q^ν_1 (cf. <cit.>), extend it by 1-periodicity 
in the directions ν_i, 1≤ i≤ n-1, and then extend it by u_v^-,v^+,ν
on the complement set (again we keep the same notation for the extended function).
Let r and δ∈(0,1) and fix a cut-off function φ∈ W^1,∞_0(Q^ν_1;[0,1]) such that φ|_Q^ν_r≡ 1.
Define ζ_r,δ:=φ ζ(·δ)+(1-φ)w(·δ)∈ W^1,1(Q^ν_1;^n) 
and use it as a test function in the minimum problem in (<ref>). Note that ζ_r,δ=v^± if ± x·ν≥12.
A simple computation yields,

    e(ζ_r,δ)=1/δ(φ e(ζ)(·δ)+(1-φ)e(w)(·δ))
    +∇φ⊙(ζ(·δ)-w(·δ)),
 
thus using (H1'), a simple scaling argument and periodicity give for some ω_δ↓ 0 as δ↓ 0

    g(   x_0,v^+-v^-,ν)≤lim sup_→0∫_Q^ν_1 f(x_0,1/e(ζ_r,δ)(y))ỵ
    
    =   lim sup_→0(∫_Q^ν_1∩{|x·ν|≤δ2} f(x_0,1/e(ζ_r,δ)(y))ỵ+ f(x_0,0))
    ≤   (1+ω_δ)lim sup_→0∫_Q^ν_1δ f(x_0,1/δ(v^+-v^+)⊙ν)ỵ
       +C(1+ω_δ)∫_(Q^ν_1∖ Q^ν_r)∩{|x·ν|≤δ2}(1/δ|e(ζ)(yδ)|+1/δ|e(w)(yδ)|
    +|∇φ||ζ(yδ)-w(yδ)|)ỵ
    ≤    (1+ω_δ)f^∞(x_0,(v^+-v^-)⊙ν)
    
    +C∫_Q^ν_1∖ Q^ν_r(|e(ζ)|+|e(w)|)ỵ
       +Cδ/1-r∫_Q^ν_1∖ Q^ν_r|ζ(y)-w(y)|ỵ.

We conclude 

    g(x_0,v^+-v^-,ν)≤ f^∞(x_0,(v^+-v^+)⊙ν),

by letting first δ↓ 0 and then r↑ 1 in the latter inequality.

In view of the previous corollary we are able to characterize explicitly the relaxed functional _0 in the v-independent case.
Additionally, as a consequence, we are also able to deal with the issue of L^1 lower semicontinuity on BD. In particular, we improve upon <cit.> 
and <cit.> (in the curl-curl case according to terminology used there) dispensing with 
the existence of the strong recession function. 
Note that f_0 is allowed to depend on x and that  the full continuity of f_0 is not required, being replaced by the weaker assumption (H2').  

To state the result recall that given f:𝕄^n× n_sym→ a Borel function, 
its symmetric quasiconvex envelope SQf:𝕄^n× n_sym→ is defined to be
	
    SQf():=sup{h()   |  h≤ f,  h is symmetric quasiconvex}.

Clearly, f is symmetric quasiconvex if and only if f=SQf. 
Finally, if f:Ω×𝕄^n× n_sym→ [0,+∞) we write 
SQf(x,·):=SQ(f(x,·)) for all x∈Ω.


Let f_0:Ω×𝕄^n× n_sym→[0,+∞) be a Borel function satisfying (H1')-(H2'). 
Let F_0:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] be the functional defined in (<ref>)
corresponding to f_0. 

Then, for all (u,A)∈ BD(Ω)×𝒪(Ω)

    _0(u,A)=∫_A SQf_0(x, e(u)(x))x̣
    +∫_A(SQf_0)^∞(x,Ẹ^su/|̣E^su|(x))|̣E^su|.


In particular, if f_0:Ω×𝕄^n× n_sym→[0,+∞) is a Borel symmetric quasiconvex satisfying (H1')-(H2'), 
the functional _0:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] 
defined by

    _0(u,A):=∫_A f_0(x, e(u)(x))x̣
    +∫_A f_0^∞(x,Ẹ^su/|̣E^su|(x))|̣E^su|

if u∈ BD(Ω) and +∞ otherwise, is L^1(Ω;^n) lower semicontinuous.


We start off noting that by (<ref>), by inequality SQf_0≤ f_0, and by the symmetric quasiconvexity of SQf_0 we get

    f(x_0,)   = inf_w∈ LD(Q_1)
     w|_∂ Q_1= y|_∂ Q_1∫_Qf_0(x_0,e(w)(y))ỵ
       ≥inf_w∈ LD(Q_1)
     w|_∂ Q_1= y|_∂ Q_1∫_Q_1 SQf_0(x_0,e(w)(y))ỵ≥ SQf_0(x_0,).

As noticed in Corollary <ref>, the bulk energy density f is symmetric quasiconvex and f≤ f_0. 
Thus, by the very definition of SQf_0, we get f≤ SQf_0. Therefore, the representation for _0 in (<ref>) is attained
thanks to Corollary <ref>. 

Finally, the L^1(Ω;^n) lower semicontinuity of _0 in (<ref>) follows at once. 





 §.§ Relaxation of bulk and interfacial energies


In this section we consider linear functionals defined on the subspace SBD(Ω) and provide a relaxation result for them. 
To our knowledge this is the first result of this kind.

We introduce the notation required for our purposes following <cit.>. 
Let f_1:Ω×^n×𝕄^n× n_sym→[0,∞) and
g_1:Ω×(^n)^2×𝕊^n-1→[0,∞) be continuous integrands such that 



  (H1”) there exists a constant C>0 such that for every 
(x,v,)∈Ω×^n×𝕄^n× n_sym 

    1/C||
     ≤ f_1(x,v,)≤ C(1+||);


  (H2”) there exists a constant C>0 such that for every >0 there exists 
δ>0 such that

    |x-x_0|+|v-v_0|≤δ⟹
     |f_1(x,v,)-f_1(x_0,v_0,)|≤ C(1+||),

for every (x,x_0,v,v_0,)∈(Ω)^2×(^n)^2×𝕄^n× n_sym;


  (H3”) there exist C>0, such that for every (x,v^-,v^+,ν)∈Ω×(^n)^2×𝕊^n-1 

    1/C|v^+-v^-|≤ g_1(x,v^-,v^+,ν)
    ≤ C|v^+-v^-|.


  (H4”) there exist C>0, such that for every (x,x_0,v^-,v^+,v_0,ν)∈(Ω)^2×(^n)^3×𝕊^n-1;

    |x-x_0|+|v_0|≤δ⟹
     |g_1(x,v^-+v_0,v^++v_0,ν)-g_1(x_0,v^-,v^+,ν)|≤ C|v^+-v^-|.


Let then _1:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] be the functional defined by 

    _1(u,A):=inf{lim inf_jF_1(u_j,Ω): u_j→ u  in L^1(Ω;^n)},

namely the L^1(Ω;^n) lower semicontinuos envelope of the functional

    F_1(u,A):=∫_Af_1(x,u(x),e(u)(x))x̣
        +∫_J_u∩ Ag_1(x,u^-(x),u^+(x),ν_u(x))̣̋^n-1(x)   if u∈ SBD(Ω)
       
    
                           +∞   otherwise on L^1(Ω;^n).

Denote by  the cell formula defined in (<ref>) related to ℱ_1. We provide next an integral representation
result for _1

Let F_1:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] be the functional defined in (<ref>).
 Then, assuming (H1")-(H4"), the functional _1:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] 
defined in (<ref>) is represented by 
 
    _1(u,A)   =∫_Af(x,u(x),e(u)(x))x̣
       
      +∫_J_u∩ Ag(x,u^-(x),u^+(x),ν_u(x))̣̋^n-1(x)+∫_A f^∞(x,u(x),Ẹ^cu/|̣E^cu|(x))|̣E^cu|(x),

 for all (u,A)∈ BD(Ω)×𝒪(Ω), 
where for every (x_0,v,)∈Ω×^n×𝕄^n× n_sym

    f(x_0,v,)=lim sup_→0   inf_w∈ SBD(Q_1)
     w|_∂ Q_1= y|_∂ Q_1{∫_Q_1f_1(x_0,v+ w(y),e(w)(y))ỵ
       +∫_J_w∩ Q_11/ g_1(x_0,v+ w^-(y),v+ w^+(y),ν_w(y))̣̋^n-1(y)},

and for every (x_0,v^-,v^+,ν)∈Ω×(^n)^2×𝕊^n-1

    g(x_0,v^-,v^+,ν)=lim sup_→0   inf_w∈ SBD(Q^ν_1)
     w|_∂ Q^ν_1=u_v^-,v^+,ν|_∂ Q^ν_1{∫_Q^ν_1 f_1(x_0,v+ w(y),1/e(w)(y))ỵ
       +∫_J_w∩ Q^ν_1g_1(x_0,w^-(y),w^+(y),ν_w(y))̣̋^n-1(y)}.



















In the BV setting under study in <cit.> conditions (H3") and (H4") are additionally used to simplify 
the analogue of formulas (<ref>) for f and (<ref>) for g thanks to the truncation argument quoted in 
Remark <ref> (cf. equations <cit.>). 


Formulas (<ref>) for f and (<ref>) for g can be expressed in terms of the recession functions of f_1 at ∞, f_1^∞, 
and the recession function of g_1 at 0, g_1^0, provided some further technical conditions in the spirit of (H3') in Corollary <ref>
are imposed (cf. Eq. <cit.>).


The proof of Theorem <ref> is similar to the corresponding one of <cit.>.
First, we note that arguing as in Lemma <ref> one can prove the following result.

Assume (H1")-(H4"). Then, _1 satisfies (H1)-(H5), and for all (u,A)∈ BD(Ω)×𝒪_∞(Ω) 

    (u,A)=_1(u,A):=inf{F_1(w,A): w=u  on ∂ A}.


The alternative characterization of  via _1, 
Theorem <ref>, the results in Section 4 and a change of variable 
provide the proof of Theorem <ref>.

We give next an explicit application of Theorem <ref>. 

Let f_1:Ω×𝕄^n× n_sym→[0,+∞) be a Borel, symmetric quasiconvex function satisfying (H1")-(H2").
 
If F_1:L^1(Ω;^n)×𝒪(Ω)→[0,+∞] is the functional in (<ref>) corresponding to 
f_1 and g_1=f_1^∞, then 

    _1(u,A)=∫_A f_1(x, e(u)(x))x̣
    +∫_Af^∞_1(x,Ẹ^su/|̣E^su|(x))|̣E^su|

if u∈ BD(Ω) and +∞ otherwise.


We start off defining

    F_1(u,A):=∫_Af_1(x,e(u)(x))x̣   if u∈ LD(Ω)
     
                           +∞   otherwise on L^1(Ω;^n).

Denote by ℱ_1 the L^1(Ω;^n) lower semicontinuous envelope of F_1. 
Note that _1≤ F_1≤F_1, therefore 
_1≤ℱ_1.
On the other hand, Corollary <ref> shows that  
ℱ_1 coincides with the right hand side of (<ref>), therefore  ℱ_1≤ F_1, and then ℱ_1≤_1. 





 §.§ Homogenization

In this section we briefly show how to apply Theorem <ref> to a homogenization problem in BD.
More precisely, we identify the Γ(L^1(Ω;^n))-limit of the family of functionals
F_δ:L^1(Ω;^n)×𝒪(Ω)→[0,+∞], δ>0, given by 
	
    F_δ(u,A):=∫_A f_0(x/δ,e(u)(x))x̣   if u∈ LD(Ω)
    
    	+∞   otherwise on L^1(Ω;^n),

where f_0:^n×𝕄^n× n_sym→ [0,+∞) is any Borel function satisfying
	
	
  (Hom)  f_0(· ,) is Q_1-periodic for all ∈𝕄^n× n_sym and
		
    1/C||≤ f_0(x,)≤ C(1+||)

	for all (x,)∈^n×𝕄^n× n_sym and for some universal constant C>0.
	
Homogenization type problems are well-known, the literature on the topic is very rich, we refer to the book <cit.> for an exhaustive introduction in particular in the case of variational energies defined on Sobolev spaces (see also <cit.> for more classical results), to <cit.> for homogenization issues related to energies defined on suitable subspaces of 
SBV, to <cit.>, to <cit.> and
to <cit.> for energies defined on BV, and to <cit.> for linear growth functionals in the setting of 𝒜-quasiconvexity under additional conditions both 
on the differential operator 𝒜, ruling out the 
curl-curl setting considered in this section, and on the density f_0, i.e. f_0(x,·) is Lipschitz continuous uniformly in x∈^n.

For energy densities f_0(x,·) convex for all x∈^n, the homogenization of the energies 
in (<ref>) on BD(Ω) has been investigated in <cit.> by duality methods. In this subsection, we extend such a result to the general case under the sole assumption (Hom) thanks to the global method 
for relaxation. In this perspective, Corollary <ref> and <cit.> will be instrumental. 

In addition, let us remark that the argument used in Theorem <ref> below works also in the setting of homogenization 
of bulk energies defined on BV, thus recovering the results contained in <cit.>, for 
f_0(x,·) convex for all x∈^n, and in 
<cit.>, for f_0(x,·) continuous for 
Ł^n-a.e. x∈^n, via the global method for relaxation without any additional assumption on f_0 than (Hom). 
Finally, we remark that <cit.> deals with the analogous problem in BV for energies consisting more
generally of the sum of a volume and of a surface term. We shall not deal with that more general setting here, that will be the object of 
further studies. Despite this, our result for the homogenization of bulk integrals seems to be new for what the regularity of f_0 is concerned in the BV setting, as well.


We recall for the reader's convenience the definition of Γ(d)-convergence for a family of functionals 
F_δ:(X,d)→∪{±∞} defined on a metric space (X,d): {F_δ}_δ>0 
Γ(d)-converges to :(X,d)→∪{±∞} if for every infinitesimal sequence δ_i
the ensuing two conditions are satisfied


  (i) For all x∈ X and for all x_i d⟶ x it holds
lim inf_i → +∞  F_δ_i (x_i)≥(x);


  (ii) For all x∈ X there exists x_i d⟶ x such that 
lim sup_i→ +∞ F_δ_i(x_i)≤(x).

We refer to the books <cit.>, <cit.>, and to the survey <cit.> 
for several properties of Γ-convergence in metric spaces. 











In what follows, 


we use the short hand notation u_v,ν for the piecewise constant function u_v,0,ν 
defined in (<ref>). 


We are now ready to prove the following homogenization result. 

Let f_0:^n×𝕄^n× n_sym→ [0,+∞) be a Borel function satisfying (Hom). 

The family F_δ:L^1(Ω;^n)×𝒪(Ω)→ [0,+∞] defined 
in (<ref>) Γ(L^1(Ω;^n))-converges to the functional
_:L^1(Ω;^n)×𝒪(Ω)→ [0,+∞] given for 
(u,A)∈ BD(Ω)×𝒪(Ω) by 

    _(u,A):=   ∫_A f_(e(u)(x))x̣
    +∫_A f_^∞(Ẹ^s u/|̣E^s u|(x))|̣E^s u|

and +∞ otherwise on L^1(Ω;^n), where f_:𝕄^n× n_sym→[0,+∞) is defined by

    f_():= lim_T→ +∞1/T^ninf_w∈ LD(Q_T)
     w|_∂ Q_T= y|_∂ Q_T∫_Q_T f_0(x,e(w)(x))x̣ ,

and where f_^∞ denotes the recession function of f_.

The first part of the proof of Theorem <ref> is a simplification of that of <cit.> 
since we deal only with bulk energies (cf. Steps 1 and 2). For this reason, we give a concise proof providing precise references 
whenever details are omitted. Instead, the second part is based upon a homogenization type argument contained in 
<cit.> (cf. Step 3).

We divide the proof in intermediate steps for the sake of convenience. 

We start off recalling some abstract compactness properties that follows from the general theory of 
Γ-convergence (cf. <cit.>). 
In view of the growth condition on f_0 in (Hom), with given any sequence δ_i↓ 0, 
we can extract a subsequence δ_i_j such that the Γ(L^1(Ω;^n))-limit of 
F_δ_i_j(·,A) exists for all A∈𝒪(Ω) (cf. <cit.>, and <cit.>). 
For the sake of notational convenience we denote the subsequence δ_i_j simply by δ_j and by 
 the corresponding Γ(L^1(Ω;^n))-limit. 

In what follows, we shall show that  does not depend on the chosen subsequence, thus proving that the Γ(L^1(Ω;^n))-limit  
of the family {F_δ}_δ>0 exists by Urysohn's property (cf. <cit.>). 
To prove that, we shall first show that each functional  introduced above satisfies the assumptions of the integral 
representation Theorem <ref>, and then we shall identify its energy densities with f_ for the absolutely 
continuous part, and with f^∞_ for the singular part.


Step 1: Integral representation of .
We start off noting that <cit.>
(see also <cit.>) implies that  satisfies (H4) and (H5), namely one can prove that 

    (u(·-x_0)+· + v,x_0+A)=(u,A)

for all (u,A,v,x_0,)∈ BD(Ω)×𝒪(Ω)×(^n)^2×𝕄^n× n_skew,
with x_0+A⊆Ω.
The very definition of  gives immediately (H1), (Hom) implies (H2), and finally arguing as in 
<cit.> together with <cit.> 
yields (H3). 

In particular, we can apply Theorem <ref> to deduce that  can be represented as

    (u,A)=∫_A    f(e(u)(x))x̣
        +∫_J_u∩ A g(u^+(x)-u^-(x),ν_u(x))̣̋^n-1(x)
    +∫_A f^∞(Ẹ^c u/|̣E^c u|(x))|̣E^c u|(x)

where f and g are identified by the cell formulas (<ref>) and (<ref>), respectively.
Indeed, on account of Eq. (<ref>) with x_0=0, Remark <ref> implies that 
f(x,v,)=f(x,) and 
g(x,v^-,v^+,ν)=g(x,v^+-v^-,ν). Instead, if in Eq. (<ref>)
one chooses =0 and v=0, we infer that f and g do not depend on x.



Finally, since  is translation invariant we may assume that 0∈Ω, this will simplify the 
notation in the sequel.




Step 2: f=f_. 
First we claim that the limit defining f_() exists finite for all 
∈𝕄^n× n_sym. Indeed, with fixed 
∈𝕄^n× n_sym, the proof of such a claim 
follows by applying the global ergodic theorem in <cit.> (or <cit.>)
to the ℤ^n-invariant subadditive process 𝒮:𝒪_∞(^n)→[0,+∞] defined by 
𝒮(A):=( x;A)
(for more details cf. Eq. (4.3.15) in <cit.>
and <cit.>). In particular, we note that the conclusions of <cit.> 
are true also for subadditive processes defined only on bounded open sets with Lipschitz boundary, 
as outlined in the remark at the end of <cit.>.

For (u,A)∈ BD(Ω)×𝒪_∞(Ω) and 
j∈, consider the intermediate cell problems
	
    _δ_j(u,A):=inf{F_δ_j(u,A): w∈ LD(A),   w=u 
    	 on ∂ A} .

With fixed x_0∈Ω 
using the version of the De Giorgi's averaging/slicing lemma developed in 
<cit.> one can argue as in <cit.> to establish the convergence 

    lim inf_j→+∞_δ_j(u,Q(x_0,r))=(u,Q(x_0,r))

for ℒ^1-a.e. r∈(0,2/√(n)d(x_0,∂Ω)), where  is the cell formula for 
defined in Eq. (<ref>). 



Thus, by taking into account (<ref>) and (<ref>) we conclude that
	
    f()=lim_→ 0lim inf_j→+∞_δ_j( x,Q_)/^n.

We prove f≥ f_. With fixed ∈𝕄^n× n_sym, 
select δ_j_, with δ_j_→ 0 as ↓ 0, and v_∈ LD(Q_), 
with v_|_∂ Q_=  x|_∂ Q_, such that
	
    f()=lim_→ 01/^n F_δ_j_ (v_,Q_).

Set T_:=δ_j_↑+∞ and
w_(y):=  1/δ_j_ v_(δ_j_ y) ∈ LD(Q_T_), then
w_|_∂ Q_T_= x|_∂  Q_T_ and by a change of 
variables the very definition of f_() yields
	
    f()= lim_→ 01/^n F_δ_j_ (v_,Q_)= 
    	lim_→ 01/T_^n∫_Q_T_f_0(z,e(w_)(z)) ẓ≥ f_().

For the converse inequality f≤ f_, for every 
T>0 choose u_T ∈ LD(Q_T) such that 
u_T|_∂ Q_T= x|_∂ Q_T and
	
    f_()=lim_T→ +∞1/T^n∫_Q_T f_0(x,e(u_T)(x))x̣ .

For every >0 and j∈, let T_,j=δ_j>0, and set w_,j(z):= 
δ_ju_T_,j(
z/δ_j).
For every >0 note that T_,j↑+∞ as 
j↑+∞, w_,j|_∂ Q_= z|_∂ Q_, and thus by changing variables that 
	
    f_()   =lim_j→ +∞1/T_,j^n∫_Q_T_,j 
    	f_0(z,e(u_T_,j)(z))ẓ
       =lim_j→ +∞1/^n∫_Q_ f_0(z/δ_j,e(w_,j)(z))ẓ≥lim inf_j→+∞_δ_j( z,Q_)/^n .

Therefore, Eq. (<ref>) provides the inequality f()≤ f_() by letting ↓ 0 in the inequality above.

In particular, since the argument above is independent from the chosen subsequence δ_j, we conclude that 
for all (u,A)∈ LD(Ω)×𝒪(Ω) 

    (u,A)=∫_A f_(e(u)(x))x̣

for all functionals  arising as Γ(L^1(Ω;^n))-limits of subsequences of the family {F_δ}_δ>0.
Therefore, by (<ref>) the Γ-liminf ^-
of the family {F_δ}_δ>0, defined as

    ^-(u,A):=inf{lim inf_j→ +∞ F_δ_j(u_j,A): 
    u_j→ u   L^1(Ω;^n),  δ_j→ 0} ,

satisfies

    ^-(u,A)=∫_A f_(e(u)(x))x̣

for all (u,A)∈ LD(Ω)×𝒪(Ω).
In particular, by definition ^-(·,A) is L^1(Ω;^n)
lower semicontinuous and 

    (u,A)≥^-(u,A)

for all (u,A)∈ LD(Ω)×𝒪(Ω), where 
 is any  Γ(L^1(Ω;^n))-limit
 of a subsequence F_δ_j.






Step 3: g=f_^∞.
We start off proving the inequality g≤ f_^∞. For all (u,A)∈ L^1(Ω;^n)×𝒪(Ω) set

    F(u,A):=∫_A f_(e(u)(x))x̣   if u∈ LD(Ω)
                     +∞   otherwise on L^1(Ω;^n),

then by Steps 1 and 2 it follows that (u,A)≤ F(u,A). 
Denoting by  the L^1(Ω;^n) lower semicontinuous envelope of F, 
then (u,A)≤(u,A). Hence, we conclude g≤ f^∞_ in view of Corollary <ref> 
that provides the explicit expression of . 

To prove g≥ f_^∞ we argue as follows. 
For every >0, we introduce the family of auxiliary functionals 
F_δ_j^(), ^(): L^1(1/Ω;^n)×𝒪(1/Ω)→[0,+∞] 
defined by

    F_δ_j^()(u,A):=∫_A f_0(/δ_jx, e(u)(x))x̣   if u∈ LD(1/Ω)
    
    	+∞   otherwise on L^1(1/Ω;^n),

and

    ^()(w,A)   :=∫_A f_(e(w)(x))x̣
       +∫_J_w∩ A1/g( (w^+(x)-w^-(x)),ν_w(x))H^n-1(x)+∫_Af_^∞(Ẹ^cw/|̣E^cw|(x))|̣E^cw| .

For w∈ BD(Ω) set w_(y):=1/w( y)∈ BD(1/Ω), then a simple scaling argument yields that 
(cf. Remark <ref>)

    F_δ_j(w,A)=^nF_δ_j^()(w_,1/A)   and   (w,A)=^n^()(w_,1/ A) ,

so that ^()(u,O)=Γ(L^1(1/Ω;^n))-lim_j→+∞ F_δ_j^()(u,O)
for all (u,O)∈ BD(1/Ω)×𝒪(1/Ω). 
In addition, for all (v,ν)∈^n×𝕊^n-1, by (<ref>) we deduce that 

    g(v,ν)=lim_→ 0(u_v,ν,Q^ν_)/^n-1=
    lim_→ 0inf{^()(w,Q^ν_1): w=1/ u_v,ν on ∂ Q^ν_1} .

We next claim that for all >0

    inf{^()(w,Q^ν_1): w=1/ u_v,ν on ∂ Q^ν_1}
    
    ≥ f_(v⊙ν/) .

Given this for granted, we infer that

    g(v,ν)≥lim sup_→ 0  f_(v⊙ν/)=
    f_^∞(v⊙ν) .


Thus, to conclude we are left with the proof of (<ref>).
To this aim we use a construction introduced in <cit.>, and adapt their argument to our setting. By scaling we assume that Q_2⊂Ω, and
for the sake of simplicity we assume ν=. In particular, 
^()(w,Q_1)=^()(w(·+e),(0,1)^n)  
for every >0 and w∈ BD(Q_1), where e:=1/2(+…+).
Therefore, in place of (<ref>) we shall equivalently prove that for all >0

    inf{^()(w,(0,1)^n): w=1/ 
    u_v,ν(·-e)  on ∂ (0,1)^n}
    
    ≥ f_(v⊙ν/) .

We introduce next some notation: ⌊ t⌋ stands for the integer part of t∈, 
and with a slight abuse set ⌊ x⌋:=(⌊ x_1⌋,…,⌊ x_n⌋) 
for all x∈^n. Then, for every w∈ BD((0,1)^n) with w=1/u_v,ν(·-e) on ∂ (0,1)^n, set 

    w_j(x):=1/j(w(jx-⌊ jx⌋)
    +1/⌊ jx_1⌋v)

for all j∈ and for all x∈ (0,1)^n.
The sequence {w_j}_j∈ is bounded in BD((0,1)^n) and converges to the affine function 
w_∞(x):=x_1/v strongly in 
L^1((0,1)^n;^n) (we do not highlight the dependence of w_j and w_∞ on  for notational convenience). Indeed, 
it is easy to check that 1/j⌊ jx_1⌋ converges to 
x_1 strongly in L^∞((0,1)^n;^n), and that by Q_1-periodicity of w

    1/j∫_(0,1)^n|w(jx-⌊ jx⌋)|x̣=
    1/j^n+1∫_(0,j)^n|w(y-⌊ y⌋)|ỵ=1/j∫_(0,1)^n|w(y)|ỵ .

Let Q^(1),…,Q^(j^n) be the standard decomposition of (0,1)^n into j^n congruent subcubes of 
side 1j with sides parallel to the coordinate hyperplanes. 
Since by construction |Ew_1|(π)=0 for every coordinate hyperplane of the form 
π={x∈^n: x_i=c∈ℤ^n}, for some i∈{1,…,n}, and w_j(x)=1/jw_1(jx), then 
|Ew_j|((0,1)^n∩∂ Q^(k))=0, for all k∈{1,…,j^n}. Then, |Ew_j|((0,1)^n)=|Ew|((0,1)^n).

By translation invariance of ^( j) and by (0,1j)^n-periodicity of w_j we have that 

    ^( j)(w_j,(0,1j)^n)=^( j)(w_j,Q^(i))

for all i∈{1,…,j^n}. 
Recalling that f_()≤ C(1+||) for all 
∈𝕄_sym^n× n, then ^( j)(u,A)≤ C(Ł^n(A)+|Eu|(A)) for all 
(u,A)∈ BD((0,1)^n)×𝒪((0,1)^n). Therefore, ^( j)(w_j,(0,1)^n∩∂ Q^(k))=0, and 
thus from (<ref>) we get that 

    ^( j)(w_j,(0,1)^n)=j^n^( j)(w_j,(0,1j)^n) .

On the other hand, by changing variables (x= yj) we get that

    ^( j)(w_j,(0,1j)^n)=j^-n^()(w,(0,1)^n) .

In conclusion, Eqs. (<ref>) and (<ref>) yield that for all 
>0 and for all j∈

    ^()(w,(0,1)^n)= ^( j)(w_j,(0,1)^n) .

Since F^()_δ_j(·,(0,1)^n)=F_δ_j(·,(0,1)^n) for all j∈ and >0,

then ^() is a Γ(L^1(Ω;^n))-limit of a (suitable) subsequence of the family {F_δ}_δ>0.
Thus, by the L^1(Ω;^n) lower semicontinuity 
of the Γ-liminf functional 
^-(·,(0,1)^n), we conclude that 

    ^()(w,(0,1)^n)   (<ref>)=lim inf_j→+∞^( j)(w_j,(0,1)^n)
       (<ref>)≥lim inf_j→+∞^-(w_j,(0,1)^n)≥^-(w_∞,(0,1)^n)
    (<ref>)=f_(v⊙/) .

In turn, from this (<ref>) follows at once.


In the convex case it is well-known that the homogenized bulk energy density f_ can be alternatively expressed for all ∈𝕄^n× n_sym as

    f_(𝔸)=inf_w∈ LD_loc(^n)
    wQ_1-periodic∫_Q_1f_0(x,𝔸+e(w)(x))x̣

(see <cit.> and <cit.>). 








§ COMMENTS ON THE ASSUMPTION OF INVARIANCE UNDER SUPERPOSED RIGID BODY MOTION

 

In this section we comment on the need of assumption (H5) in the BD setting. As noticed in formula (<ref>), assumption (H5) implies that the bulk 
energy density f of , and then in turn its recession function f^∞, does 
not depend on the skew-symmetric part of the relevant matrix. 
This piece of information has been substantially used in the proof of Theorem <ref> to give a lower bound of the Radon-Nikodým derivative of 
 at u∈ BD(Ω) with respect to |E^cu| (cf. (<ref>)), the upper bound instead being always true. 
As far as we have understood, this seems not to be a mere technical issue as we try to explain in what follows. 
First we notice that there exist one-homogeneous, nonconvex, quasiconvex functions f satisfying for all  ∈𝕄^n× n

    1/C|+^t|/2≤ f()≤ C|+^t|/2

and depending non trivially on the skew-symmetric part of the relevant matrix . 
The example is obtained by a slight modification of the one-homogeneous, nonconvex, quasiconvex function exhibited by 
Müller <cit.> that we briefly recall. Given a matrix ∈𝕄^2× 2 consider

    h():=|_11-_22|+|_12+_21|
    +min{|_11+_22|,|_12-_21|}

and set Qh to be the quasiconvexification of h, i.e. 
	
    Qh():=sup{h'()   |  h'≤ h,  h' is quasiconvex}.

Clearly, Qh is quasiconvex by definition and it is easy to see that it is one-homogeneous and satisfying
0≤ Qh()≤ C|+^t|/2 for all  ∈𝕄^2× 2. 
To prove the nonconvexity of Qh, Müller shows that for the rank-2 matrix

    _0:=
    [  1 -1;  1  1 ]

Qh(_0)>0 is satisfied. On the other hand, it is easy to show that the convex envelope of 
h is null on _0. In addition, to prove that Qh depends on the skew-symmetric part 
of the matrix  it suffices to notice that  

    _0+_0^t/2=Id,     _0-_0^t/2=
    [  0 -1;  1  0 ]

and that h(Id)=0, in turn implying Qh(Id)=0.
Finally, in order to satisfy the growth condition in (<ref>) we define the function 
f to be equal to 

    Qh()+|+^t|/2

for >0 sufficiently small. Indeed, the set 

    I:={≥0:𝕄^n× n∋↦ 
    Qh()+|+^t|/2 is convex}

is closed (and potentially empty), since convexity is stable under pointwise convergence, and 0∈[0,+∞)∖ I.
In passing, we recall that an example of similar nature has been exhibited in the superlinear case in 
<cit.> with a polyconvex, non-convex energy density.  

Hence, the full integral representation result for the corresponding functional

    (u,A):=inf{lim inf_j∫_Af(∇ u_j(x))x̣: u_j→ u
     in L^1(Ω;^2), u_j∈ LD(Ω)}

can be proven by means neither of Theorem <ref>, since assumption (H5) 
is violated (while (H1)-(H4) are easily checked to be valid), nor of any of the results available 
in the related literature (cf. <cit.>,  <cit.>, <cit.>, <cit.>, <cit.>).
On the other hand, thanks to Lemma <ref> (cf. <cit.>), 
the quasiconvexity and 1-homogeneity of f itself imply that for all (u,A)∈ SBD(Ω)×𝒪(Ω) it holds

    (u,A)=∫_Af(∇ u(x))x̣+∫_J_u∩ Af((u^+(x)-u^-(x))⊗ν_u(x))̣̋^1

(see also Theorem <ref>, Remark <ref> and Corollary <ref>).

More generally, such representations of the volume and surface parts of the energy hold for all u∈ BD(Ω) 
in view of Lemma <ref> itself.

In addition, for all (u,A)∈ BV(Ω;^2)×𝒪(Ω) we claim that

    (u,A)=∫_Af(∇ u(x))x̣+∫_J_u∩ Af((u^+(x)-u^-(x))⊗ν_u(x))̣̋^1
    +∫_Af(Ḍ^cu/|̣D^cu|(x))|̣D^cu| .

To prove this, for δ∈(0,) consider the functional 

    _δ(u,A):=inf{lim inf_j→+∞∫_A f_δ(∇ u_j)
    
    x̣: u_j→ u
     in L^1(Ω;^2), u_j∈ W^1,1(Ω;^2)},

where 𝕄^2× 2∋↦ f_δ():=f()+δ|-^t|/2, 
and note that it satisfies the assumptions of the integral representation result 
<cit.>. To this aim, set 

    F_δ(u,A):=∫_A f_δ(∇ u)x̣,   and    F(u,A):=∫_A f(∇ u)x̣,

if u∈  W^1,1(Ω;^2) and +∞ otherwise on L^1(Ω;^2). In particular,  turns out to be the 
L^1 lower semicontinuous envelope of F (cf. Remark <ref>).
Furthermore, by its very definition f_δ is quasiconvex, one-homogeneous and δ||≤ f_δ()≤ C|| 
for all ∈𝕄^2× 2, for some C independent from δ. 
Therefore, <cit.> gives that for all (u,A)∈ BV(Ω;^2)×𝒪(Ω)

    _δ(u,A)=∫_Af_δ(∇ u(x))x̣+∫_J_u∩ Af_δ((u^+(x)-u^-(x))⊗ν_u(x))̣̋^1
    +∫_Af_δ(Ḍ^cu/|̣D^cu|(x))|̣D^cu|,

and moreover _δ(u,A)=+∞ if u∈ L^1∖ BV(Ω;^2).

Being δ↦ F_δ monotone increasing and converging to F as δ↓ 0  
in view of the pointwise convergence of f_δ to f, the Γ-limit in the strong L^1(Ω;^2) 
topology of both (F_δ)_δ and (_δ)_δ is exactly  
(cf. <cit.>, <cit.>, see also 
<cit.>).
Moreover, δ↦_δ is monotone increasing with pointwise limit as δ↓ 0
the functional  given by the 
right hand side of (<ref>) for all (u,A)∈ BV(Ω;^2)×𝒪(Ω) and +∞ 
otherwise on L^1(Ω;^2).
The equality in (<ref>) then follows at once in view of the L^1(A;^2) lower semicontinuity of (·,A) 
itself on BV(A;^2) for all A∈𝒪(Ω) (cf. <cit.>, <cit.>). 
In addition, the functional  turns out to provide the L^1_loc(Ω;^2) lower semicontinuous envelope of 
F on (u,A)∈ BV(Ω;^2)×𝒪(Ω) in view of <cit.>.

Actually, as a byproduct, it follows that for all (u,A)∈ BD(Ω)×𝒪(Ω)

    (u,A)=inf{lim inf_j→+∞(u_j,A):u_j→ u  in L^1(Ω;^2), u_j∈ BV(Ω;^2)} .
















Summarizing, for the functional  related to the integrand f introduced above, 
we can prove an integral representation result on maps u∈ BV(Ω;^2) (see (<ref>)) with volume density 
expressed in terms of the full approximate gradient of u and with surface density depending on the full tensor 
product of the jump and the approximate normal to the jump set of u (cf. (<ref>) and 
(<ref>)). 
For BD(Ω) maps which are not BV(Ω;^2) we are not able to provide the integral representation of the Cantor
term. 
The expression of  for BV maps in (<ref>) 


suggests that despite the recent progresses obtained by De Philippis and Rindler in <cit.>, some other structure properties of the Cantor part of the symmetrized distributional derivative are still missing for BD maps
(see <cit.> for further discussions on the topic).
In Theorem <ref> we use assumption (H5) to rule out such kind of difficulties. 

As already noticed, (H5) is not needed for the integral representation on the subspace of functions SBD(Ω).
The latter comment is coherent with the needed assumptions in case of functionals defined on SBD^p(Ω), p>1 
(see <cit.>).


Finally, we note that all the (few) known examples of functions u in BD(Ω)∖ BV(Ω;^n) are such that 
∇ u∉ L^1(Ω;𝕄^n× n), though, with a slight abuse of notation, D^su∈ℳ(Ω;𝕄^n× n) 
(cf. <cit.>, <cit.>, <cit.>, 
<cit.>, <cit.>, <cit.>). 


 More generally, considering a generic functional  satisfying (H1)-(H4), similar conclusions as those discussed above can be drawn
 for what concerns the integral representation on the subspace BV(Ω;^n) and the analogous of the relaxation formula (<ref>).






 §.§ Acknowledgements

The work of M.C. has been supported by the grant “Calcolo delle variazioni, Equazione alle derivate parziali, Teoria geometrica della misura, Trasporto ottimo” 
co-founded by Scuola Normale Superiore di Pisa and the University of Firenze. M.F. acknowledges the support of GNAMPA of INdAM. N.V.G. was supported by FCT - Fundação para a Ciência e a Tecnologia, starting grant “Mathematical theory of dislocations: geometry, analysis, and modelling” (IF/00734/2013) and by the FCT project UIDB/04561/2020. 

The first and second authors would like to thank G. De Philippis for interesting conversations on the subject.


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