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 underlying 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 measuring 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 The main result of this work is the following theorem, which provides a representation result analogous to that in (1.4), for structured deformations (g, G) ∈ SD p (Ω) := SBV (Ω; R d ) × L p (Ω; R d×N ). Indeed, the convergence * SD and the uniform control on the L p norm of the gradients ∇u n required in (1.6) imply that the gradients in (1.1) converge weakly in L p (Ω; R d×N ) to G, instead of converging weakly-* in the sense of measures (see our Approximation Theorem 2.5 below). Therefore, it makes sense to actually define 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. Consequently, I {εn} hom , itself, is independent of {ε n }, and we write I hom in place of I {εn} hom . The independence of h hom (λ, ν) from the specific choice of the cube Q ν can be deduced from Proposition 3.5. In (1.11) and (1.12), 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 (1.13), 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 (1.11) and (1.12) 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., [6]). 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 (1.11) 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, convergence (1.1) generally entails 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 (1.11), where we point out that both the initial bulk and surface energy densities contribute to H hom , with 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 (1.12).
The proof of (1.10), to which whole Sect. 4 is devoted, is obtained by computing the Γ-limit in (1.8) by combining blow-up techniquesà la Fonseca-Müller [19,20] 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 (1.10) are indeed given by (1.11) and (1.12). 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 [7,11]).
We would like to close this introduction by mentioning two alternative possibilities for identifying relaxed energies for this problem that have analogues in the context of dimension reduction (see [8], where ε denotes the thickness of a body in a preassigned direction), namely to carry out successively the "partial relaxations" that fix either ε or n, i.e., (i) first relax with respect to structured deformations and second homogenize, or (ii) first homogenize and then relax with respect to structured deformations. These alternative possibilities have interest not only from the point of view of variational analysis, but they also may enlarge the class of multiscale problems in mechanics to which homogenization and relaxation to structured deformations can be applied. Consider, for example, a system for which the variables ∇u ZAMP Periodic homogenization in the context Page 5 of 30 173 and [u] for simple deformations u are expected to vary only over length scales much larger than the period ε of the microstructure. In this case, it would seem reasonable to first homogenize and then relax to structured deformations. This iterated relaxation procedure presumably would, in general, assign a larger energy to structured deformations, and such examples provide a motivation for future research on the alternatives (i) and (ii).
We will collect some preliminary results in Sect. 2, where we also prove Approximation Theorem 2.5 which guarantees the nonemptiness 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 definitions (1.11) and (1.12). For the reader's convenience, we present in Appendix A some technical measure-theoretical results which are by now standard.

Notation
We will use the following notations 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; • 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 [16,Theorem 1.33] for a more general statement).
Observe that (i) above entails

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.

3)
where C N > 0 is a constant depending only on N . 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 (1.8).
Moreover, there exists C > 0 such that, for all n ∈ N, In particular, this implies that, up to a subsequence, is easily seen to approximate (g, G) in the sense of (2.5). In fact, 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).
In light of convergence (2.5), class (1.9) of admissible sequences for relaxation problem (1.8) can be written as which is not empty because of the Approximation Theorem just proved.

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 (1.11) and (1.12), respectively.
(iv) There exist C W > 0, and c W > 0 such that W (x, ξ) C W |ξ| p − c W for every ξ ∈ R d×N and a.e. x ∈ Ω. (v) There exist c ψ , C ψ > 0 such that, for every ( (vi) There exists a function ω ψ : [0, +∞) → [0, +∞) such that ω ψ (s) → 0 as s → 0 + such that for every is positively homogeneous of degree one, i.e., for every λ ∈ R d and t > 0, We make the following observations. (A) The p-Lipschitz continuity in (ii) jointly with (i) and (iii) implies that W has p-growth from above in the second variable, namely that there exists C W > 0 such that for every ( 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.  can be weakened to where ψ 0 is the positively homogeneous function of degree one defined by As a consequence of this weakening, to recover the boundedness of the BV norm of the approximating sequences, the class R p of admissible sequences for the relaxation introduced in (1.6) must be adapted to include also the uniform control sup n∈N u n BV (Ω;R d ) < +∞ ; moreover, the relaxed energy density H hom in (1.11) must be redefined with ψ 0 in place of ψ, see [9,Remark 3.3].
We now present a translation invariance property of H hom and h hom .
Proof. The proof of both (3.3)  The next proposition shows that if the initial bulk and surface energy densities W and ψ are convex in the gradient and jump variable, respectively, then asymptotic cell formula (1.11) for the homogenized bulk energy density reduces to a cell formula in the unit cube.
. and extended by periodicity is Q-periodic and satisfies

(3.5)
Then the following properties hold true: (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 [7, 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) from (i) in Theorem 2.8].
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 SBV (Ω; 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 end, let u ∈ C surf (λ, ν; kQ ν ) be a competitor for g k (λ, ν) and consider the functionū : 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 defineũ : 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 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, the first inequality being obvious. Recalling definition (1.12) of h hom (λ, ν), this gives the equality h hom = h hom of (iii) and concludes the proof.

Proof of Theorem 1.1
This section is entirely devoted to the proof of Theorem 1.1. The proof is achieved by obtaining upper and lower bounds for the Radon-Nikodým derivatives of the functional I {εn} hom defined in (1.8) with respect to the Lebesgue measure L N and to the Hausdorff measure H N −1 in terms of the homogenized bulk and surface energy densities H hom and h hom defined in (1.11) and (1.12), respectively. To keep the notation lighter, and in view of the fact that the dependence on the vanishing sequence {ε n } is only illusory, in the following we will just write I hom .

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 relabeled subsequence) μ n * μ.
We will prove that dμ dL N (x 0 ) H hom (∇g(x 0 ), G(x 0 )). Let {r k } be a vanishing sequence of radii such that μ(∂Q(x 0 ; r k )) = 0; then we have 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)), (4.8) becomes (4.9) 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, (4.9) becomes where γ k := x 0 /ε n(k) . By (4.6) we have that ∇v k (y) = ∇ x u n(k) (x 0 + r k y) − ∇g(x 0 ); from (4.5) and 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 (4.11), such that 12) and such that so that (4.10) becomes

(4.13)
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 (4.12) 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. 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 (1.11)), we need to integrate over integer multiples of Q. To this aim, we extend U k to the cube ( s k + 1)Q by settinĝ Notice that, by the first condition in (4.14) 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 (4.16), the definition of U k and the second condition in (4.12) thatÛ (1.13)). Then, using (3.1) and the linear growth of ψ (see (v)), we can continue with (4.15) and obtain where we have used Proposition 3.3 for the last equality.
The bulk energy density: upper bound. Here we prove that (1.13)). Let us consider a sequence of radii r j → 0 as j → ∞, and let h j ∈ SBV (Q rj (x 0 ); R d ) be a function provided by Theorem 2.3 such that finally, let {h j,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 } is a diverging sequence of integers 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 lim n→∞ u j,n = g in L 1 (Q rj (x 0 ); R d ) for every j ∈ N; (4.22) moreover, recalling (4.19), we have so that, by applying the Riemann-Lebesgue lemma to the sequence kQ y → ∇u (n) (y) := ∇u(m n y), we obtain that ∇u (n) converges weakly in L p (kQ; R d×N ) to kQ ∇u(y) dy, yielding The convergences in (4.22) and (4.24) show that the sequence {u j,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 → I hom (g, G; A) is the trace of a Radon measure on the open subsets of Ω (see Proposition A.2), 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 (4.20), 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 (4.20), and by changing variables according to where we have defined γ n := x 0 /ε n and m n := r j /kε n , and used the decomposition to get the first equality; the second equality follows from the kQ-periodicity of u and from the Qperiodicity 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 (4.1) and (4.18) together, we obtain that for all the points x 0 ∈ Ω satisfying the conditions stated at the beginning of Sect. 4.1, thus proving the first part of integral representation (1.10).

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 (1.8) is a measure. In particular, (see (A.2)) there exists C > 0 such that Finally, let us define w k := u k + v − v k , so that w k − SDp (g, 0) and therefore Thus, by invoking (ii) and Hölder's inequality for the volume integrals, and first the sub-additivity 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 In view of this, without loss of generality, we will consider G = 0 for the rest of the proof. The lower bound (see (4.28)) below will be obtained considering g of the type s λ,ν in (1.15), with (λ, ν) ∈ (R d \ {0}) × S n−1 ; the upper bound (see (4.37) 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 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 ) (4.31) 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 ; R d ) ∩ L ∞ (Q σ ; R d ) and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (i) {w n } is equi-bounded; (ii) w n → s λ in L 1 (Q σ ; R d ), as n → ∞; (iii) Qσ |∇w n (x)| p dx → 0, as n → ∞;  Indeed, (4.32)(i) follows by construction and (4.32)(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 (4.29). In order to prove (4.32)(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) We now prove (4.32)(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 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 sequencê w n,k , so thatv n,k := w n +ŵ n,k ∈ SBV (Q σ ; R d ) is such that lim k→∞v n,k = w n in L 1 (Q σ ; R d ) and ∇v n,k = 0 a.e. in Q σ ; moreover, invoking (4.32)(ii), we have that lim n→∞ lim k→∞v n,k = s λ in L 1 (Q σ ; R d ). (4.33)