Periodic homogenization in the context of structured deformations

An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral type featuring contributions of bulk and interfacial terms, is proved to possess an integral representation in terms of relaxed bulk and interfacial energy densities. These energy densities, in turn, are obtained via asymptotic cell formulae defined by suitably averaging, over larger and larger cubes, the bulk and surface contributions of the initial energy. The integral representation theorem, the main result of this paper, is obtained by mixing blow-up techniques, typical in the context of structured deformations, with the averaging process proper of the theory of homogenization.


INTRODUCTION
The mathematical modeling of materials has interested scientists for many centuries. In the last decades, the accuracy of these models has considerably increased as an effect of more sophisticated measure instruments, on the experimental side, and of the availability of sound mathematical abstract frameworks, on the theoretical side. Classical theories of continuum mechanics provide a good description of many phenomena such as elasticity, plasticity, and fracture, and are susceptible of incorporating fine structures at the microscopic level. The mathematical process through which the effects of the microstructure emerge at the macroscopic level is called homogenization: this procedure provides an effective macroscopic description as the result of averaging out the heterogeneities.
Structured deformations [14] provide a mathematical framework to capture the effects at the macroscopic level of geometrical changes at submacroscopic levels. The availability of this framework, especially in its variational formulation [9], leads naturally to the enrichment of the energies and force systems that underlie variational and field-theoretic descriptions of important physical phenomena without having to commit at the outset to any of the existing prototypical mechanical theories, such as elasticity or plasticity. A (first-order) structured deformation is a pair (g, G) ∈ SBV (Ω; R d ) × L 1 (Ω; R d×N ) =: SD(Ω), where g : Ω → R d is the macroscopic deformation and G : Ω → R d×N is the microscopic deformation tensor. As opposed to classical theories of mechanics, in which g and its gradient ∇g alone characterize the deformations of the body Ω, the additional geometrical field G captures the contributions at the macroscopic level of the smooth submacroscopic changes. The difference ∇g − G captures the contributions at the macroscopic level of slips and separations occurring at the submacroscopic level (which are commonly referred to as disarrangements [14]). Heuristically, the disarrangement tensor M := ∇g − G is an indication of how non classical a structured deformation is: should M = 0 and if g is a Sobolev field, then the field G is simply the classical deformation gradient; on the contrary, if M = 0 , there is a macroscopic bulk effect of submacroscopic slips and separations, which are phenomena involving interfaces. This fact will be made precise in the Approximation Theorem 2.5.
In order to assign an energy to a structured deformation (g, G) ∈ SD(Ω), the proposal has been made in [9] to take the energetically most economical way to reach (g, G) by means of SBV fields u n : according to the Approximation Theorem [9, Theorem 2.12], we say that a sequence {u n } ⊂ SBV (Ω; R d ) converges to (g, G) if and v u (x) denote the jump of u and the normal to the jump set for each x ∈ S u , the jump set. In mathematical terms, the process just described to assign the energy to a structured deformation (g, G) ∈ SD(Ω) reads In the language of calculus of variations, the operation described in (1.3) is called relaxation and the main results in [9] was to prove that the functional I admits an integral representation, that is, there exist functions H : R d×N ×R d×N → [0, +∞) and h : R d ×S N −1 → [0, +∞) such that I(g, G) = Ω H(∇g(x), G(x)) dx + Ω∩Sg h([g](x), ν g (x)) dH N −1 (x). (1.4) In this work, we focus on heterogeneous, hyperelastic, defective materials featuring a fine periodic microstructure. Our scope is to provide an asymptotic analysis of the energies associated with these materials, as the fineness of their microstructure vanishes, in the variational context of structured deformations [9].
The initial energy functionals that we consider involve a bulk contribution and a surface contribution, each of which is described by an energy density which depends explicitly on the spatial variable in a periodic fashion, namely the energy associated with a deformation u ∈ SBV (Ω; R d ) has the expression are Q -periodic in the first variable ( Q being the unit cube in R N ), and ε > 0 is the length scale of the microscopic heterogeneities (see Assumptions 3.1 for the precise assumptions on W and ψ ). We will perform a relaxation analogous to that in (1.3) for the ε -dependent initial energy (1.5).
In particular, we aim at assigning an energy to structured deformation (g, G) where the geometric field G is p-integrable for some p > 1 . As proved in [9], this has the effect that the submacroscopic slips and separations diffuse in the bulk and contribute to determining the relaxed bulk energy density, whereas the relaxed surface energy density is only determined by optimizing the initial surface energy density. The mechanical interpretation of this fact, which is also the motivation for our choice, is that the relaxed surface energy is not influenced by the elastic energy in the limit. We define the class of admissible sequences for the relaxation by and for every sequence ε n → 0 , we define The main result of this work is the following theorem, which provides a representation result analogous to that in (1.4).
Theorem 1.1. Let p > 1 and let us assume that Assumptions 3.1 hold; let u ∈ SBV (Ω; R d ) and let E ε (u) be the energy defined by (1.5). Then, for every (g, G) ∈ SD(Ω), the homogenized functional I hom (g, G) defined in (1.7) admits the integral representation (1.8) The relaxed energy densities H hom : are given by the formulae for every A, B ∈ R d×N , and for every (λ, ν) ∈ R d × S N −1 , where Q ν is any rotated unit cube so that two faces are perpendicular to ν .
The independence of h hom (λ, ν) from the specific choice of the cube Q ν can be deduced from Proposition 3.5. In (1.9) and (1.10), we have defined, for A, B ∈ R d×N , (λ, ν) ∈ R d × S N −1 , and R, R ν ⊂ R N cubes, is the elementary jump of amplitude λ across the hyperplane perpendicular to ν . In formula (1.11), we denote by SBV # (R; R d ) the set of R d -valued SBV functions with equal traces on opposite faces of the cube R . We notice that (1.9) and (1.10) are asymptotic cell formulae, as it is expected in the context of homogenization when no convexity assumptions are made on the initial energy densities (see, e.g., [7]). In the special case of functions W and ψ which are convex in the gradient and jump variable, respectively, we are able to show that (1.9) reduces to a cell problem in the unit cell (see Proposition 3.4 below); whether the same result holds for h hom is still unknown.
Since, in a structured deformation (g, G) ∈ SD(Ω), the field G is generally different from ∇g , it is clear that the convergence (1.1) is obtained at the expenses of the discontinuity sets of u n diffusing in the bulk, namely H N −1 (S un ) → +∞ as n → ∞ (so that the hypotheses of Ambrosio's compactness theorem in SBV [2] are in general not satisfied). This is reflected in the form of the relaxed bulk energy density in (1.9), where we point out that both the initial bulk and surface energy densities contribute to the definition of H hom , both undergoing the bulk rescaling. On the contrary, the coercivity assumption (see Assumption 3.1-(iv) below) yields an L p constraint on the gradients of the approximating sequences which avoids the appearance of any bulk contributions in the relaxed surface energy density h hom in (1.10).
The proof of (1.8), to which the whole Section 4 is devoted, is obtained by computing the Γ-limit in (1.7) by combining blow-up techniques à la Fonseca-Müller [18,19] with rescaling techniques typically used in homogenization problems. This will be especially visible in the construction of the recovery sequences for proving that the densities in (1.8) are indeed given by (1.9) and (1.10). To deduce the upper bound for the homogenized surface energy density h hom we also make use of comparison results in a Γ-convergence setting (see [8,11]).
We will collect some preliminary results in Section 2, where we also prove the Approximation Theorem 2.5 which guarantees the non-emptiness of the class R p introduced in (1.6). Section 3 contains the precise formulation of the standing assumptions on the initial energy densities W and ψ and a collection of results on the homogenized energy densities H hom and h hom which can be deduced from the definitions (1.9) and (1.10). For the reader's convenience, we present in Appendix A some technical measure-theoretical results which are by now standard.
denotes the jump of u on S u , and ν u is the unit normal vector to S u ; finally, ⊗ denotes the dyadic product; for Q ⊂ R N a cube, we denote by SBV # (Q; R d ) the set of R d -valued SBV functions with equal traces on opposite faces of Q ; • L p (Ω; R d×N ) is the set of matrix-valued p-integrable functions; for p > 1 we denote by p ′ its Hölder conjugate. • for p 1 , SD p (Ω) := SBV (Ω; R d ) × L p (Ω; R d×N ) is the space of structured deformations (g, G) (notice that SD 1 (Ω) is the space SD(Ω) introduced in [9]); • C represents a generic positive constant that may change from line to line.
• For every x ∈ R N , the symbol ⌊x⌋ ∈ Z N denotes the integer part of the vector x, namely that vector whose components are the integer parts of each component of x. We denote by x the fractional part of x, i.e., x := x − ⌊x⌋ ∈ [0, 1) N .

Function spaces.
The following proposition serves as a definition of Lebesgue points for L p functions (see [15,Theorem 1.33] for a more general statement).
Observe that (i) above entails 2.3. The approximation theorem in SD p (Ω). In this section we prove the approximation theorem for structured deformations in SD p (Ω). This result will be useful for the proof of our homogenization Theorem 1.1 and rests on the following two statements.
One of the main results in the theory developed by Del Piero and Owen was the Approximation Theorem, stating that any structured deformation can be approximated, in the L ∞ sense, by a sequence of simple deformations (see [14] for the details, in particular Theorem 5.8). For structured deformations (g, G) ∈ SD(Ω), the corresponding result is obtained in [9, Theorem 2.12]. Here we prove a version in SD p (Ω), which is the natural framework for the integral representation of the functional I hom defined in (1.7).

Theorem 2.5 (Approximation Theorem). For every
and Moreover, there exists C > 0 such that, for all n ∈ N, In particular, this implies that, up to a subsequence, Proof. Let (g, G) ∈ SD p (Ω) and, by Theorem 2.3 with f := ∇g − G, let v ∈ SBV (Ω; R d ) be such that ∇v = ∇g − G. Furthermore, letv n ∈ SBV (Ω; R d ) be a sequence of piecewise constant functions approximating v , as per Lemma 2.4. Then, the sequence of functions is easily seen to approximate (g, G) in the sense of (2.5). In fact, u n → g in L 1 (Ω; R d ) and ∇u n (x) = G(x) for L N -a.e. x ∈ Ω. Estimate (2.6) follows from the inequality in (2.3) and from (2.4); finally, (2.5) and (2.6) imply (2.7).

STANDING ASSUMPTIONS AND PROPERTIES OF THE HOMOGENIZED DENSITIES
In this section we present the hypotheses on the initial energy densities W and ψ and we prove some properties of the homogenized densities H hom and h hom defined in (1.9) and (1.10), respectively.
is positively homogeneous of degree one, i.e., for every λ ∈ R d and t > 0 ,

Remark 3.2.
We make the following observations. (i) The p-Lipschitz continuity in (ii) jointly with (i) and (iii) imply that W has pgrowth from above in the second variable, namely that there exists On the contrary, p-growth from above jointly with the quasiconvexity of the bulk energy density in the gradient variable (which is the natural assumption in equilibrium problems in elasticity) returns the p-Lipschitz continuity. (ii) Condition (v) does not allow for a control on the H N −1 -measure of the jump set, which, in the spirit of the Approximation Theorem 2.5, is crucial in the context of structured deformations.
(iii) Conditions (v) and (viii) imply Lipschitz continuity of the function λ → ψ(x, λ, ν), i.e., for every (x, ν) ∈ R N × S N −1 and for every λ 1 , λ 2 ∈ R d , (iv) Conditions (vii), (viii), and (ix) are natural ones for fractured materials; in particular, condition (ix) allows one to identify ψ(x, λ, ν) with ψ(x, λ ⊗ ν), for a suitable function ψ : We now present a translation invariance property of H hom and h hom . The next proposition shows that under if the initial bulk and surface energy densities W and ψ are convex in the gradient and jump variable, respectively, then the homogenized bulk energy density is obtained via a cell formula in in the unit cube and is not asymptotic anymore.
Proposition 3.4. Let W and ψ satisfy Assumptions 3.1, let us assume that the functions ξ → W (x, ξ) and λ → ψ(x, λ, ν) are convex for every x ∈ R d and every ν ∈ S N −1 , and let .
and extended by periodicity is Q -periodic and satisfies Then the following properties hold true: where c ψ and C ψ are the constants in Proof. The proof of items (i) and (ii) is essentially the same as that of [ The continuity of h hom can be obtained by arguing in the following way: (a) one shows that the function h hom (λ, ·) is continuous on S N −1 , uniformly with respect to λ, when λ varies on bounded sets; (b) one shows that for every ν ∈ S N −1 , the function h hom (·, ν) is continuous on R d ; (c) one shows that h hom is continuous in the pair (λ, ν). The proof of point (a) above relies on the fact that for every fixed λ ∈ R d , formula (3.5) does not depend on the cube Q ν once the direction ν is prescribed, by (i). For the proof of point (b), we can argue as in [8, proof of Proposition 2.2, Step 6] (here we exploit Assumptions 3.1-(v) and the Lipschitz continuity of ψ , see (3.2)). Point (c) can be obtained by arguing as in [12, proving (ii) To conclude the proof of (ii), we need to prove (3.6). The estimate from above can be easily obtained from the very definition of h hom in (3.5), by using Assumptions 3.1-(v). Concerning the estimate from below, it is sufficient to observe that the functional is lower semicontinuous with respect to the convergence u n −⇀ SDp (g, 0), with g a pure jump function, as it follows from the lower semicontinuity of the total variation with respect to the weak-* convergence and, again, from Assumptions 3.1-(v).
To prove (iii), we take inspiration from the proof of [10, Lemma 2.1]: we show that h hom is an infimum over the integers. Together with (i), we will conclude that h hom = h hom , as desired. Let g T : so that we can write (3.5) as h hom (λ, ν) = lim sup T →+∞ g T (λ, ν).
We start by proving a monotonicity property of g T over multiples of integer values of T , namely we prove that, for every (λ, ν) ∈ R d × S N −1 , for every h, k ∈ N. (3.8) To this aim, let u ∈ C surf (λ, ν; kQ ν ) be a competitor for g k (λ, ν) and consider the function u : hkQ ν → R d defined byū obtained by replicating u by periodicity in the (N − 1)-dimensional strip perpendicular to ν and extending it to 0 and λ appropriately. It is immediate to see thatū ∈ C surf (λ, ν; hkQ ν ), so that (3.8) follows. We now consider two integers 0 < m < n and a function u ∈ C surf (λ, ν, mQ ν ); we definẽ u : nQ ν → R d byũ and notice thatũ ∈ C surf (λ, ν; nQ ν ). Then, invoking Assumptions 3.1-(v), so that, by infimizing first overũ ∈ C surf (λ, ν; nQ ν ) and then over u ∈ C surf (λ, ν; mQ ν ), we obtain Using [n/m]m in place of m and (3.8), we get By (i), h hom (λ, ν) = lim T →+∞ g T (λ, ν), so that, by taking the limit as n → ∞ in the inequality above, we can write this yields, by taking the infimum over the integers, We start by observing that any (g, G) ∈ SD(Ω) for which there exists a sequence {u n } ∈ R p (g, G) is indeed an element of SD p (Ω). In particular, the functional (g, G) → I hom (g, G) is finite if and only if (g, G) ∈ SD p (Ω). Therefore, we have I hom : SD(Ω) → [0, +∞] defined by We notice that by the Approximation Theorem 2.5, the class R p defined above is not empty.
Since if (g, G) ∈ SD(Ω) \ SD p (Ω) there is nothing to prove, we will drop the tildes in the proof.

The bulk energy density.
We tackle here the bulk energy density H hom . In the next two subsections, we assume that x 0 ∈ Ω is a point of approximate differentiability for g and a Lebesgue point for G, namely, Theorem 2.2(i) and (iii) hold for g and (2.1) holds for G (notice that L N -a.e. x 0 ∈ Ω satisfies these properties).
The bulk energy density: lower bound. Let {u n } ∈ R p (g, G; Ω), and let µ n ∈ M + (Ω) be the Radon measure defined by Without loss of generality, we can assume that sup n∈N µ n (Ω) < +∞, so that there exists µ ∈ M + (Ω) such that (up to a not relabelled subsequence) µ n * ⇀ µ. We will prove that dµ dL N (x 0 ) H hom (∇g(x 0 ), G(x 0 )). (4.4) Let {r k } be a vanishing sequence or radii such that µ(∂Q(x 0 ; r k )) = 0 ; then we have where we have changed variables in the last equality. Upon defining, for every y ∈ Q , u 0 (y) := ∇g(x 0 )y (4.6) and u n,k (y) := u n (x 0 + r k y) − g(x 0 ) r k − u 0 (y), x 0 + r k y ε n , ∇u n,k (y) + ∇g(x 0 ) dy where we have used the positive 1 -homogeneity of ψ (see (vii)). By writing and using the 1-periodicity of W and ψ (see (i)), (4.11) becomes (4.12) We now choose n(k) so that, setting s k := r k /ε n(k) , we have that lim k→∞ s k = +∞. By defining v k (y) := u n(k),k (y) for every y ∈ Q , (4.12) becomes dµ dL N (x 0 ) = lim k→∞ Q W (s k y + γ k , ∇v k (y) + ∇g(x 0 )) dy where γ k := x 0 /ε n(k) . By (4.9) we have that ∇v k (y) = ∇ x u n(k) (x 0 + r k y) − ∇g(x 0 ); from (4.8) and (4.10) the sequence {v k } satisfies v k → 0 in L 1 (Q; R d ) and ∇v k ⇀ G(x 0 ) − ∇g(x 0 ) in L p (Q; R d×N ) as k → ∞. (4.14) It is now possible 1 to replace the sequence {v k } with a sequence {w k } ⊂ SBV (Q; R d ) still satisfying the convergences in (4.14), such that (4.15) and such that 1 This is achieved, following the strategy in [9, Proposition 3.1 Step 2], by constructing a double-indexed sequence that gradually makes a transition from v k to its limit. The transition takes place across a suitably located frame of vanishing thickness 1/m (independent of k ) and is obtained via convex combination (see also the construction in [9, Lemma 2.21] where suitable truncations of the approximating sequences are considered; [9, Lemma 2.20] (see Lemma A.3) states that it is possible to work on bounded sequences). A sequencew k is then obtained by a diagonalization argument. Finally, the condition on the average in (4.15) is enforced by a further modification of the sequencew k into w k by modification with a linear function on cubes that invade Q . The difference between our problem and that in [9] is the explicit dependence on the spatial variable that we have; nonetheless, our assumptions (ii) and (v) allow us to estimate the vanishing terms independently of the spatial variable.
so that (4.13) becomes (4.16) By changing variables, setting z := s k y and U k (z) := s k w k (x/s k ), so that where we have used the positive 1 -homogeneity of ψ once again (see (vii)).
In order to comply with the definition of H hom (∇g(x 0 ), G(x 0 )) (see (1.9)), we need to integrate over integer multiples of Q . To this aim, we extend U k to the cube (⌊s k ⌋ + 1)Q by settingÛ Notice that, by the first condition in (4.17) no further jumps are created, so that [Û k ](z) = [U k ](z) for every z ∈ SÛ k = S U k . Moreover, if follows from (4.19), the definition of U k and the second condition in (4.15) that so that {Û k } ∈ C bulk p ∇g(x 0 ), G(x 0 ); (⌊s k ⌋+1)Q (see (1.11)). Then, using (3.1) and the linear growth of ψ (see (v)), we can continue with (4.18) and obtain where we have used Proposition 3.3 for the last equality.
The bulk energy density: upper bound. Here we prove that Let k ∈ N \ {0} and u ∈ C bulk p (∇g(x 0 ), G(x 0 ); kQ) (see (1.11)). Let us consider a sequence of radii r j → 0 as j → ∞, and let h j ∈ SBV (Q rj (x 0 ); R d ) be the function provided by Theorem 2.3 such that finally, let {h j,n } n be a piecewise constant approximation of h j in L 1 (Q rj (x 0 ); R d ) provided by Theorem 2.4. We notice that, thanks to Proposition 2.1 and Theorem 2.2, For every j, n ∈ N, we define the function u j,n ∈ SBV (Q rj (x 0 ); R d ) by where {m n } n is a diverging sequence of integer numbers to be defined later. By defining kQ ∋ y := k(x − x 0 )/r j , and by applying the Riemann-Lebesgue Lemma to the sequence of functions kQ ∋ y → u (n) (y) := u(m n y), we obtain that u (n) converges weakly in L p (kQ; R d ) to kQ u(y) dy , so that moreover, recalling (4.22), we have The convergences in (4.25) and (4.27) show that the sequence {u j,n } n is admissible for the definition of I hom (g, G; Q rj (x 0 )), for every j ∈ N.
Recalling that the localization O(Ω) ∋ A →Ĩ hom (g, G; A) is the trace of a Radon measure on the open subsets of Ω (see Proposition A.2) and that for every (g, G) ∈ SD p (Ω) we have that I hom (g, G) =Ĩ hom (g, G), we can estimate where we have used the subadditivity and the positive 1 -homogeneity of ψ (see (vii) and (viii)) to obtain the second inequality. Now, using, in order, (v), the estimate in (2.3) and (2.4), and finally (4.23), the last three integrals above vanish as first n → ∞ and then j → ∞.
We are left with one volume integral and one surface integral; by adding and subtracting W (x/ε n , ∇g(x 0 ) + ∇u(m n k(x − x 0 )/r j )) in the volume integral and using (ii), Hölder's inequality, and (4.23), and by changing variables according to we have where we have defined γ n := x 0 /ε n and m n := ⌊r j /kε n ⌋, and used the decomposition 1 m n r j kε n z = 1 m n r j kε n + r j kε n z = z + 1 m n r j kε n z to get the first equality; the second equality follows from the kQ -periodicity of u and from the Q -periodicity in the first variable of W and ψ (see (i)). Upon noticing that m −1 n r j /kε n → 0 as n → ∞, we can extract a subsequence j → n(j) such that m −1 n r j /kε n z < 1/j , so that, upon diagonalization and invoking (iii) and (vi), we can write moreover, by using the definition of infimum in H τ hom in (3.3) (for τ = γ n(j) ), both k ∈ N and u ∈ SBV # (kQ; R d ) can be chosen in such a way that where we have used the translation invariance property of H hom (see Proposition 3.3) to obtain the last equality. Putting (4.4) and (4.21) together, we obtain that for all the points x 0 ∈ Ω satisfying the conditions stated at the beginning of Section 4.1, thus proving the first part of the integral representation (1.8).

The surface energy density.
We tackle here the surface energy density h hom . From now on, we consider a point x 0 ∈ S g . Recalling Proposition A.2, for every U ∈ O(Ω) and for every (g, G) ∈ SD p (U ), the functional U → I hom (g, G; U ) in (1.7) is a measure. In particular, (see (A.2)) there exists C > 0 such that Observe that (4.29) guarantees that, for every g ∈ SBV (Ω; R d ), the computation of the Radon-Nikodým derivative dI hom (g, G) d|D s g| (x 0 ) does not depend on G. Indeed, let us consider {u k } ∈ R p (g, G; U ) a recovery sequence for I hom (g, G; U ) and, by Theorems 2.3 and 2.4, let us consider v ∈ SBV (U ; R d ) such that ∇v = −G and piece-wise constant functions v k ∈ SBV (U ; (g, 0) and therefore Thus, by invoking (ii) and Hölder's inequality for the volume integrals, and first the subadditivity of ψ (see (viii)) then the linear growth of ψ (see (v)) for the surface integrals, we can estimate where C > 0 is a suitable constant. By virtue of the estimate in (2.3) and by (2.4), the two surface integrals in the last line above are bounded by the volume integral, so that, by exchanging the roles of I hom (g, G; U ) and I hom (g, 0; U ), we arrive at the conclusion that for every U ∈ O(Ω). In turn, this guarantees that, for H N −1 -a.e. x 0 ∈ S g , In view of this, without loss of generality, we will consider G = 0 for the rest of the proof. The lower bound (see (4.31)) below will be obtained considering g of the type s λ,ν in (1.13), with (λ, ν) ∈ (R d \ {0}) × S n−1 ; the upper bound (see (4.40) below) will be obtained considering g taking finitely many values, that is g ∈ BV (Ω; L) where L ⊂ R d is a set with finite cardinality. In particular, the upper bound will also hold for functions of the type g = s λ,ν . To conclude, the general case will be obtained via standard approximation results as in [9,Theorem 4.4, Step 2] (stemming from the ideas [5,Proposition 4.8]), so that this part of the proof (which relies on the continuity properties of h hom , see Proposition 3.5) will be omitted.
The surface energy density: lower bound. In this section we prove that and .
We now substitute the sequence {u n } by a new sequence {ū n } ∈ SBV (Ω; R d ) ∩ L ∞ (Ω; R d ) (this is possible thanks to Lemma A.3) with the following properties:ū n −⇀ SDp (s λ , 0); given {m n } a diverging sequence of integers such that β n := m n ε n → 0 as n → ∞, it holds (the latter convergence is due to the metrizability of the weak convergence on bounded sets); and for every η > 0 , (4.33) For τ ∈ {0} × Z N −1 , let x n,τ := β n τ and Q n,τ := x n,τ + β n Q σ = β n (Q σ + τ ). Let τ (n) be the index corresponding to a 'minimal cube' such that E εn (ū n ; Q n,τ (n) ) E εn (ū n ; Q n,τ ) (4.34) for every τ ∈ {0} × Z N −1 and Q n,τ ⊂ Q σ . We now define Q n := Q n,τ (n) , x n := x n,τ (n) , and, for every x ∈ Q σ , we let w n (x) :=ū n (x n + β n x). We claim that w n ∈ SBV (Q σ ; where {α n } is a suitable vanishing sequence. (4.35) Indeed, (4.35)(i) follows by construction and (4.35)(ii) is obtained by changing variables according to y = x n + β n x ∈ Q n , by the definition of w n , by the choice of τ , by the inclusion Q n ⊂ Q σ , and finally by the first limit in (4.32). In order to prove (4.35)(iii), we observe that, by the boundedness of the energy, there exists a constant C > 0 such that where the second inequality is due to the fact the cubes Q n,τ are disjoint; the third inequality follows from counting them; in the fourth inequality we have used the non-negativity of ψ ; in the last inequality we have exploited (iv). Next, observe that (using the change of variables y = x n + β n x ∈ Q n and the inclusion Q n ⊂ Q σ again) Qσ |∇w n (x)| p dx = Qσ |β n ∇ū n (x n + β n x)| p dx β p−N n Qσ |∇ū n (y)| p dy, (where in the second integrand we computed the gradient of the composed function), whence We now prove (4.35)(iv) with α n = ε n /β n . To this end, we observe that x n /ε n = m n τ (n) ∈ {0} × Z N −1 and so, by using the change of variables y = x n + β n x ∈ Q n , the non-negativity of W , and the periodicity of ψ (see (i)), we obtain Thus (4.35)(iv) follows from (4.33) since The sequence {w n } can now be modified into a new sequence {ṽ n } such that ∇ṽ n = 0 a.e. in Q σ as follows: for every n ∈ N, we approximate via Theorem 2.5 the pair (0, −∇w n ) ∈ SD p (Q σ ) by a sequenceŵ n,k , so thatv n,k := w n +ŵ n,k ∈ SBV (Q σ ; R d ) is such that and ∇v n,k = 0 a.e. in Q σ ; moreover, invoking (4.35)(ii), we have that Hence, by a standard diagonalization argument, by definingṽ n :=v n,k(n) , we have that v n −⇀ SDp (s λ , 0), ∇ṽ n = 0 , and (4.37) The next step is to modify the sequence {ṽ n } into a new sequence {v n } such that v n | ∂Qσ = s λ | ∂Qσ . This can be achieved by defining the function where {r n } ⊂ (0, 1) is a sequence such that lim n→∞ r n = 1 − and, by (4.36), (4.39) Clearly ∇v n = 0 a.e. in Q σ and, again by (4.36), lim n→∞ v n = s λ in L 1 (Q σ ; R d ). Moreover, by (3.2), (4.37), and (4.35)(iv), we have and the latter limit is 0 by the choice of r n , as consequence of (4.38) and (4.39). We conclude the proof by extending, without relabeling it, v n to the whole unit cube Q by defining it as s λ in Q \ Q σ , so that the previous inequality becomes By Proposition 3.5 and the change of variables y = α −1 n x ∈ α −1 n Q , the functionv n (y) := v n (α n y) belongs to C surf (λ, e 1 ; α −1 n Q) (see (1.12)), so that, recalling that we had set λ = [g](x 0 ) and letting η → 0 , we obtain (4.31).
The surface energy density: upper bound. In this section we prove that dI hom (g, 0) d|D s g| by following the lines of [8,Proposition 6.2]. Recall that by the preliminary discussion we made at the beginning of the section we will restrict ourselves to the case of piecewise constant functions g , that is g ∈ BV (Ω; L), where L ⊂ R d has finite cardinality; naturally, such a function g is also an element of SBV (Ω; R d ). We will obtain estimate (4.40) by using the abstract representation result contained in Theorem A.4, for which we need to prove that our (localized) functional I hom : As a consequence of (4.29), for every g ∈ BV (Ω, L) and for every U ∈ O(Ω), the inequality I hom (g, 0; U ∩ S g ) C|D s g|(U ∩ S g ) holds true, giving (i). By Proposition A.2, for every g ∈ BV (Ω; L), the set function O(Ω) ∋ U → I hom (g, 0; U ∩ S g ) is a measure, giving (ii). From the definition (1.7) of I hom and from the locality property of the (sequence of) energies {E εn } , we obtain that I hom (g, 0; U ∩ S g ) = I hom (g 1 , 0; U ∩ S g1 ) whenever g = g 1 a.e. in U ∈ O(Ω). Indeed, it suffices to notice that the competitors for I hom (g, 0) and I hom (g 1 , 0) are the same. Therefore, condition (iii) is satisfied. To show that condition (iv) holds, let us consider a sequence {g n } ⊂ SBV (Ω; L) such that g n → g pointwise a.e.; then, the fact that L has finite cardinality entails that g n → g in L 1 (Ω; R d ), and therefore that g n * ⇀ SD (g, 0) ∈ SD(Ω). Since g → I hom (g, 0; U ) is lower semicontinuous for every U ∈ O(Ω) by definition of Γ-liminf, the desired inequality I hom (g, 0; U ) lim inf n→∞ I hom (g n , 0; U ) follows immediately. It remains to prove (v): as a matter of fact, we will prove a stronger condition, as it is obtained in the proof of [8,Proposition 4.2]. This translation invariance result, which is obtained following the argument in [8,Lemma 3.7], provides then a sufficient condition for (v). We claim that, for every z ∈ R N and every U ∈ O(Ω), we have I hom (g, 0; U ) = I hom (g(· − z), 0; U + z). (4.41) Indeed, let z ∈ R N be given and observe that it can be approximated by means of a sequence of integers in the sense that there exists {z n } ⊂ Z N such that ε n z n → z as n → ∞. Let now U ∈ O(Ω) be fixed, let {u n } ∈ R p (g, 0; U ) be a recovery sequence for I hom (g, 0; U ), and define v n := u n (· − ε n z n ) : U + z n → R d . Then we have E εn (u n ; U ) = U W x + ε n z n ε n , ∇u n (x) dx + Let now V ⊂⊂ U , so that, for n sufficiently large we may assume U + ε n z n ⊇ V + z ; hence, invoking the non-negativity of W and ψ , which yields I hom (g, 0; U ) I hom (g(· − z), 0; V + z), since v n * ⇀ SD (g(· − z), 0). By the arbitrariness of V ⊂⊂ U we obtain that I hom (g, 0; U ) I hom (g(·−z), 0; U +z). The reverse inequality can be obtained with the same reasoning, by defining v n := u n (· + εz n ). The translation invariance (4.41) is proven, and this implies condition (v).
On the other hand, we can get a precise estimate from above arguing as in [8]. whence dI hom (g, 0) d|D s g| (x 0 ) h hom ([g](x 0 ), ν g (x 0 )) for H N −1 -a.e. x 0 ∈ S g , which is (4.40) when g ∈ BV (Ω, L). Putting (4.31) and (4.40) together and keeping (4.30) into account, we obtain that, for g = s λ,ν ∈ BV (Ω; L) and for all G ∈ L p (Ω; R d×N ), the equality dI hom (g, G) d|D s g| (x 0 ) = h hom ([g](x 0 ), ν g (x 0 )) holds for all the points x 0 ∈ Ω satisfying the conditions stated at the beginning of Section 4.2. To conclude, the equality in the general case, that is, for every g ∈ SBV (Ω; R d ), is obtained via standard approximation results as in [9,Theorem 4.4, Step 2] (stemming from the ideas [5, Proposition 4.8]), so that this part of the proof, which relies on the continuity properties of h hom stated in Proposition 3.5, will be omitted. Theorem 1.1 is now completely proved.

APPENDIX A. SOME TECHNICAL RESULTS
This appendix contains some technical results that are either reported here with no proof or proved for the reader's convenience since their proof is quite standard but in some measure different from the analogous results in the literature.
We start by showing that, for every U ∈ O(Ω) and for every (g, G) ∈ SD p (U ), the localization U →Ĩ hom (g, G; U ) of the functionalĨ hom of (4.2), defined as I hom (g, G; U ) := inf lim inf n→∞ Then there exists a unique continuous function f : Ω × L × L × S N −1 → [0, Λ] such that f (x, i, j, ν) = f (x, j, i, −ν), and the function p → f x, i, j, p |p| |p| is convex in R N for every x ∈ Ω, i, j ∈ L , and F (u; U ) is representable as for every u ∈ BV (Ω; L) and for every U ∈ O(Ω).