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Homogenization for a Variational Problem with a Slip Interface Condition

This paper is concerned with the homogenization of a variational boundary value problem governing the small vibrations of a periodic mixture of an elastic solid and a slightly viscous fluid, with a slip interface condition. The method of two-scale convergence is used to obtain the macroscopic behaviour of the solution and to identify the role played by said interface condition.


Introduction
The homogenization for the variational boundary value problem of the stiff type that governs the small vibrations of a periodic mixture of an elastic solid and a slightly viscous fluid, with no discontinuity of the displacement in the interface between the two phases, was developed by Nguetseng in [11]. In this paper, we carry out the analysis for the case in which a slip boundary condition at the solidfluid interface is allowed.
We consider the mixture of an elastic solid and a slightly viscous fluid, in the framework of small motions linearized with respect to a rest state, where the geometric distribution of the solid and fluid parts is periodic, with characteristic length of the period given by ǫ, with 0 < ǫ << 1. Mixtures in mechanics are of great interest in physical applications, see for example, [8,9,13,11,2,7].
A variety of different problems arise according to the orders of the viscosity coefficients and the topological properties of the mixture. In [12], the authors used the energy method (see [5]) to show that, whether or not the fluid phase is connected, if the elasticity coefficients together with the viscosity coefficients are O(ǫ 0 ), the limit of the displacement, as ǫ → 0, does not depend on the local variables. In this paper, we will take the elasticity coefficients to be O(ǫ 0 ), and the viscosity coefficients to be O(ǫ 2 ) i.e. µǫ 2 and ηǫ 2 with constant µ and η. In the formal analysis, seen in [8] or Chapter 8 of [13], it is concluded that, if the fluid part is strictly contained in the period of reference, and therefore it is not connected, the formal limit of the displacement in the mixture does not depend on the local variables. On the other hand, if the fluid part intersects each face of the period of reference, and it is connected, the formal limit of the displacement depends on the local variables. The formal analysis results above were rigorously proved in [11] using the method of 2-scale convergence (see [10,3]). Differing from [11], connectedness doesn't play a role in the analysis developed and the results obtained in this paper.
The novelty of this paper is that the results obtained in [11] are generalized to the case in which there is a slip interface condition. Though the results of the present paper are similar to those of [11], dealing with the interface term (2.13) is not trivial. New technical lemmas are required in order to carry out the limiting process.
This paper is organized as follows. In Section 2, we present the set up of the mathematical problem. In Section 3, we derive (3.6), the variational formulation of the boundary value problem that governs the small vibrations of a solid-fluid mixture with a slip boundary condition on their interface . In Section 4, we prove the existence and uniqueness of the solution to our problem for a fixed ǫ. The main general convergence and extension results can be found in Section 5. In Section 6, we prove the necessary uniform estimates to find the macroscopic equation. The derivation for the local problems (for u 1 (x, y) and u r (x, y) in Lemma 6.5) is done in Section 7. In Section 8, we derive the homogenized problem. Finally, in Section 9, we present our conclusions.

Background
Following the notation in [11], we consider the space R 3 of the variables y = (y 1 , y 2 , y 3 ) to be a periodic set, with unit cell Y = − 1 2 , 1 2 3 , decomposed as: where Y s and Y f are open sets in R 3 , where Y s represents the part of Y occupied by the solid and Y f represents the part of Y occupied by the fluid, and Γ is the smooth surface separating them. We denote byỸ s (respectively,Ỹ f ) the Y -periodic extension of Y s (respectively, Y f ), that is, the union of all the (Y s ∪(Y s ∩ ∂Y ))+ k, k ranging over Z 3 (similarly forỸ f ), where Y s is the closure of Y s in R 3 (respectively, Y f ), and ∂Y the boundary of the cube Y . Similarly, we denote byΓ the Y -periodic extension of Γ. Following [1], we assume the following hypotheses: (i) Y s and Y f have strictly positive measures on Y .
(ii)Ỹ f andỸ s are open sets with boundary of class C 1 , and are locally located on one side of their boundary. Moreover,Ỹ s is connected, so Y s ∩ ∂Y = ∅. (iii) Y s is an open connected set with a locally Lipschitz boundary.
Remark 2.1. Observe that, as a consequence of (ii), Y s has an intersection with each face of the cube Y which has strictly positive surface measure.
In the space R 3 of the variables x = (x 1 , x 2 , x 3 ), we consider a smooth bounded open set Ω, with boundary ∂Ω, and we set: where 0 < ǫ ≪ 1. Observe that Ω s ǫ is connected. The following notation for linear strain of v = v i (x) and w = w i (y) , is used throughout the paper: We denote by div y the divergence operator with respect to y, and by div x , or simply div, the same operator with respect to x. If V is a vector space, the vector space of the same name written in boldface V represents the corresponding product In this paper, all the vector spaces considered are over the complex field C. The Einstein summation convention is used throughout the rest of the paper, δ ij is the Kronecker symbol, and C represents a universal constant, which is independent of variable quantities such as ǫ, t, ..., and that may change value from line to line. For a measurable set A, we denote by |A| its measure. Consider the elastic moduli of the solid a ijkl , 1 ≤ i, j, k, l ≤ 3, and the viscosity coefficients of the fluid ηǫ 2 , µǫ 2 , where η and µ are real numbers, which satisfy the following conditions: We define the following Y -periodic functions: where ρ s , ρ f are positive constants and γ = c 2 0 ρ f , c 0 > 0. We will give the physical definition of these constants later in this paper. We use the standard notation: and define the forms: Lastly, assume the force f = f i ∈ L 2 loc (0, +∞; L 2 (Ω)) is independent of ǫ and satisfies the following bound: , for almost all 0 < t < ∞. Following Chapter 8 of [13], we consider the small linearized perturbation of a mixture of an elastic body and a viscous isotropic fluid, from a reference rest state. As in [11], the governing equations for the displacement u ǫ in the solid phase are: For the fluid phase, the following equations are satisfied: The positive constants ρ s , ρ f are the densities of the solid and the fluid, respectively, in the reference state at rest. Also, the constitutive equation for the acoustic pressure p ǫ = −c 2 0 ρ f ∇ · u ǫ implies that (2.10) can be rewritten as: The initial conditions are given by u ǫ (0) = 0 and v ǫ (0) = ∂u ǫ ∂t (0) = 0. Let n be the unit normal vector of Γ ǫ pointing toward the solid part. Let Γ s ǫ and Γ f ǫ denote the solid side and the fluid side of Γ ǫ , respectively. In terms of this notation, the jump · f s is defined as: We assume the following conditions at the interface Γ ǫ : ∂u ǫ ∂t f s (slip condition), (2.13) where α > 0 is the slip constant. For more information about the ǫ-scaling for the slip constant in (2.13), see [2,6].

Variational Formulation Derivation
We denote by dσ(y), for y ∈ Y , and by dσ ǫ (x), for x ∈ Ω, the surface measures on Γ and Γ ǫ , respectively. Observe that: , with the norm, induced from the appropriate inner product, suggested by the interface term (3.5), given by: Notice that V is a closed subspace of H 1 (Ω f ǫ ∪ Ω s ǫ ) by the trace theorem. The counterpart of V for functions defined in the unit cell Y is given by: . The role of the interface term in the norm will be made clear later.
For the solid phase, (2.7) and (2.8) lead to: for all w ∈ V , where n s is the unit normal vector of Γ ǫ pointing out of Ω s ǫ (toward the fluid part). By the symmetry of a ijkl in (2.1), the above can be rewritten as: where the minus sign in the last integral is due to the fact that n = −n s .
For the fluid phase, (2.9) and (2.11) imply for all w ∈ V , we have: Observe that a ijkl E kl (u ǫ )n j = (σ ∼ s n) i is the total solid stress acting on the interface and, similarly, the terms in the boundary integral in (3.4) can be regarded as (σ ∼ f n) i . Summing the boundary integrals in (3.3) and (3.4), we can use (2.12) to obtain: (3.4) and (3.5), the variational formulation for our problem is as follows: Find u ǫ , function of t with values in V , such that: for all w ∈ V ; or equivalently: The sesquilinear forms b ǫ and c ǫ (given by (2.4) and (2.5), respectively) can be written explicitly as: where, by (2.1), we have: Let us denote byv(λ) the Laplace transform of a function v(t). Omitting the argument λ inû ǫ (λ) andf (λ), and considering any fixed λ such that Reλ > λ 0 > 0, with λ 0 large enough, the variational formulation of problem (3.6) in the Laplace transform domain reads as follows: Findû ǫ ∈ V such that, for all w ∈ V , we have: where the form a ǫ (û ǫ , w) is defined as follows: For convenience, the analysis in the rest of the paper will be based on (3.8), the "transformed version" of (3.6).

Existence and Uniqueness of the solution to the Variational Formulation (3.8)
First, we state well-known Lax-Milgram lemma.
represents the dual pairing between V and V ′ .
To recast (3.8) into a problem to which the Lax-Milgram lemma can be easily applied, we divide both sides of (3.8) by λ = 0 to obtain: Note that Re(λ) and Re( 1 λ ) have the same sign. We begin by showing the coercivity of 1 λ a(w, w), for w ∈ V , with Re(λ) > 0 and ǫ > 0. To do this, we will make use of the following lemmas.  [13]). Korn's inequality. Given a bounded set Υ with ∂Υ smooth, there exists γ ′ > 0, such that the following estimate holds: The following lemma, which is a modification of Lemma 1.2, page 87 of [13], will also be useful.
Then, for all w ∈ V , there exists K = K(ǫ) > 0 such that the following estimate holds: Proof. We prove this by contradiction. Suppose (4.2) is not true. Then, there exists a sequence w k ∞ k=1 in V, with w k Hence, by Lemma 4.2, there exists w * ∈ V such that w k → w * weakly in V and w k → w * strongly in L 2 (Ω). Therefore, w * L 2 (Ω) = 1 and w k V ≤ C. By Proposition 1.1 on page 8 of [13], we have w * V ≤ lim inf k w k V . From the latter, it follows that: which, in turn, implies that: Therefore w * is a rigid body motion in each phase. Since w * | ∂Ω s ǫ ∩∂Ω = 0, we have that w * = 0 in Ω s ǫ . By the interface integral in (4.3), we must also have w * = 0 in Ω f ǫ . This contradicts w * L 2 (Ω) = 1. The proof above demonstrates the importance of including the interface jump in the norm of the space V in (3.1). Without the interface term, the lemma would not be true.
By the properties of the coefficients a ijkl , η and µ in (2.2) and (2.3), we have: Also, as long as ∂Ω s ǫ ∩ ∂Ω = ∅ , by Lemma 4.3, there exists K > 0 such that: for all w ∈ V . Using (4.5), (4.6), and (4.7) in (4.4), we have: Therefore, for all w ∈ V , we have: This proves the coercivity of the sesquilinear form in (4.1). The boundedness of this form can be checked easily by a repeated application of the Cauchy-Schwarz inequality. Therefore, by Lemma 4.1 (Lax-Milgram lemma), there exists a unique solution of (4.1) and, hence, of (3.8), for any fixed ǫ > 0 and λ, with Re(λ) > 0.

5.1.
Notation. For clarity of exposition, we follow the same notation introduced in [11]. Consider R 3 , with the variables y = (y 1 , y 2 , y 3 ), and let Y o be an open set contained in the unit cell Y = (−1/2, 1/2) 3 , subject to the following conditions: a) The sets ∂Y o ∩ {y : have nonzero measure with respect to the 2-dimensional measure. Moreover, these sets are symmetrical with respect to the plane {y : y j = 0}. b) Y o contains a cylinder Q j of length 1, whose axis is parallel to the y j -axis, for j = 1, 2, 3. c) If Y o = Y , we denote by Γ the separation surface between Y o and the interior of Y \ Y o , and we assume that Γ is smooth.
Note thatỸ o is a connected open set in R 3 with smooth boundaryΓ. On the other hand,Q j is a union of infinite cylinders contained inỸ o (with same radius, and with axes parallel to the y j -axis) and periodically distributed in the space.
5.1.1. Function spaces. The following function spaces are used in the paper.
• K (Ω): the set of all restrictions to Ω of continuous functions on R 3 with compact supports. • D(Ω): space of C ∞ functions with compact support in Ω.
In the case Y o = Y , we will write L 2 p (respectively, H 1 p ) instead of L 2 p (Y ) (respectively, H 1 p (Y )).

5.2.
Statement of convergence theorems. Consider R 3 , with the variables x = (x 1 , x 2 , x 3 ), and let Ω ⊂ R 3 be a smooth bounded open set. We introduce the parameter ǫ, 0 < ǫ ≪ 1, which is regarded as the ratio of the characteristic size in the microscale to that in the macroscale; and we define Ω o ǫ = Ω ∩ ǫỸ o , where Y o satisfies the conditions a), b), and c) in Section 5.1.
The following theorem will be very useful. The proof for the first part can be found in [10]) and for the second part, it can be found in [11].
for any test function ψ = ψ(x, y), with ψ ∈ D(Ω, C ∞ p (Y )), see [10,3]. 5.3. Two-scale convergence on the surface. To handle the interface term in the weak formulation, we will also need the following theorems, which generalize results about two-scale convergence to sequences in L 2 (Γ ǫ ). Their proofs can be found in [4].
Theorem 5.5. [4] Let u ǫ be a sequence in L 2 (Γ ǫ ) such that the surface integral satisfies the bound: Then, there exist a subsequence (still denoted by ǫ) and a two-scale limit g(x, y) ∈ L 2 (Ω; L 2 (Γ)), such that u ǫ (x) two-scale converges to g(x, y), in the sense that: The following generalizes the corrector result of the usual two-scale convergence: Lemma 5.6. Let u ǫ be a sequence in L 2 (Γ ǫ ) which two-scale converges to a limit g(x, y) ∈ L 2 (Ω; L 2 (Γ)). Then, the measure u ǫ dσ ǫ converges, in the sense of distributions in Ω, to the function u(x) = Γ g(x, y) dσ(y) belonging to L 2 (Ω), and we have: Assume further that g(x, y) is smooth and that: then: In the case where u ǫ is the trace on Γ ǫ of some function in H 1 (Ω), a link can be established between its usual and its surface two-scale limits.
Theorem 5.7. Let u ǫ be a sequence of functions in H 1 (Ω) such that: Then, the trace of u ǫ on Γ ǫ satisfies the estimate: and up to a subsequence, it two-scale converges, in the sense of Theorem 5.5, to a limit g(x, y), which is the trace on Γ of the usual two-scale limit, a function in L 2 (Ω; H 1 p (Y )). More precisely: for any continuous function φ(x, y) ∈ C[Ω; C p (Y )].
The following extension theorems play a crucial role in establishing the uniform bounds of solutions, which are required for the two-scale convergence.

Extension theorems. Define
: where d designates the Euclidean metric and Ω is the closure of Ω in R N .
Theorem 5.8 (Theorem A of [11]). For each ǫ < ǫ o (ǫ o is a suitable constant), there exists an extension operator T ǫ ∈ L(V s , H 1 0 (Ω 1 )) (i.e., T ǫ is continuous linear and T ǫ u = u on Ω s ǫ , for all u ∈ V s ) such that: where the constant C does not depend on ǫ.
A similar extension theorem can be established for V f . Theorem 5.9 (Theorem B of [11]). There exists an extension operator T p ∈ L(H 1 p (Y s ), H 1 p ) such that T p w = w almost everywhere in Y s , for all w ∈ H 1 p (Y s ) and: where the constant C does not depend on ǫ.
A similar extension theorem can be established for Y f .

Necessary Estimates.
In this section, we present the estimates and convergence results necessary to obtain the macroscopic equations of our problem. Several of the results found here are the same as in [11], and are included for completeness of the presentation.
The following lemma plays an important role in the various estimates established later in this section. Lemma 6.1. Let ρ * = min {ρ f , ρ s } and z ǫ (t) = e −rt u ǫ (t), where r > 0 is a fixed real number and u ǫ is the solution (3.6). We have, for r > max 1, m 2 , where m is the growth rate of f defined in (2.6), that: for almost all 0 < t < ∞ and for all ǫ > 0.
Proof. From the definition of z ǫ (t), we have: Plugging these into (3.6), and taking w = ∂z ǫ ∂t , we obtain: Rearranging terms and applying the estimate of f in (2.6) lead to: since r and α are non-negative. Due to the fact that: we can simplify (6.2) to obtain Because z ǫ | t=0 = ∂zǫ ∂t t=0 = 0, we have: Note that the bound does not depend on ǫ. Therefore, for z ǫ with r > 1, (6.1) must be true for t > 0 a.e. and for all ǫ > 0.
To show the uniform boundedness with respect to the V norm, defined in (3.2), we use the following lemma. Lemma 6.2. There exists a positive constant C, independent of ǫ, such that: Proof. For all w ∈ V , we have: where the constant c is the V -elliptic constant for the solid elasticity tensor a ijkl defined in (2.2). By the extension result in Theorem 5.8, there exist operators T f ǫ and T s ǫ that extend w to Ω 1 from Ω f ǫ and Ω s ǫ , respectively, such that the following estimates are valid with positive constants C f and C s , independent of ǫ: where V s,f and Ω 1 are defined in (5.2) and (5.3), respectively. Since the extended functions belong to H 1 0 (Ω 1 ), Korn's inequality implies that: where the positive constant D ′ depends only on C s , C f and the Korn's constant of Ω 1 . Finally, (6.11) becomes: It then follows from Lemma 6.2, (6.8), (6.9), and (6.10) that: Also, from (6.5) we obtain: We now state a really important lemma. Lemma 6.3. There is some ǫ o such that: Proof. By (3.7), (6.8), and (2.2), we have: From Korn's inequality for H 1 0 (Ω 1 ), Theorem 5.8, and the inequality above, we have: Lemma 6.4. We can extract a subsequence of {û ǫ } such that: Moreover, we have: (6.6). Proof. Since û k ǫ ǫ>0 is bounded in L 2 (Ω), (6.15) follows immediately by Lemma 5.2. Property (6.16) follows as a consequence of (6.12), (6.15), Remark 5.3, with an integration by parts argument similar to the one used in Proposition 1.14 in [3]. From (6.13) and (6.16), we have as ǫ → 0, taking k = l: from where we obtain (6.17). As for (6.18), it follows from (6.5) and (6.15).
Because of the uniform boundedness of the first derivatives in the solid phase (6.14), more can be said about the two-scale limit ofû ǫ as follows.
Lemma 6.5. A subsequence can be extracted from the one in Lemma 6.4, such that: for 1 ≤ k, l ≤ 3; for all ψ ∈ L 2 p and all φ ∈ K (Ω), where u = u k ∈ H 1 0 (Ω), . Moreover, the limit w o in Lemma 6.4 decomposes as follows: with u r ∈ L 2 (Ω; H 1 p (Y f ∪Y s )), u r (x, y) = 0 for y ∈ Y s and x fixed, and div y u r = 0. Proof. This follows from using Lemma 6.3 and applying Lemma 6.4 and Theorem 5.2 by letting Y o = Y s . Lemma 6.6. Let ǫ be the subsequence involved in Lemma 6.4. Then, as ǫ ↓ 0, the following holds: with w ∈ H 1 p (Y f ), we obtain the result, due to (6.17) and ψ ǫ = ǫ b ǫ ijkl ∂w i ǫ ∂x j . Now, in order to link u 1 and u r , we take advantage of the uniform boundedness of divû ǫ (6.13) and study the behavior of the acoustic pressurep ǫ defined as: Then we have: for all v ∈ K(Ω). Since D kl ǫ L 2 (Ω) ≤ C, for all ǫ > 0 and 1 ≤ k, l ≤ 3, the sequence D kl ǫ has a weak limit in the sense of Lemma 5.2, which we denote by D kl . Taking w ǫ = 1 and φ = v in Lemma 5.2, we obtain: and, by Lemma 6.5, we can conclude: By letting p 0 (x, y) = −γD kk (x, y), for (x, y) ∈ Ω × Y f , we obtain the following lemma.
Lemma 6.7. As ǫ ↓ 0 (ǫ a subsequence from the one in Lemma 6.5), we have, for all ψ ∈ L 2 p , all φ ∈ K (Ω) and x ∈ Ω: Proof. The lemma follows from (6.21) and (6.20)). If k = l, we have: To obtain (6.22), by (6.5) we have, for all v ∈ D(Ω), that: By taking w s = 0, (7.1) becomes: for all w ∈ H 1 p (Y f ) with w·n = 0 on ∂Y f and, by a density argument and integration by parts, we obtain Y f ∇ y p 0 (x, y) · w dy = 0, hence p 0 does not depend on y. In other words, p 0 ∈ L 2 (Ω). To further simplify (7.1), we observe that, for all w ∈ V Y such that w · n is continuous across Γ, we have: Therefore, for any fixed x ∈ Ω, (7.1) can be written as: where q(·, ·) represents the sesquilinear form given by: We have that u 1 (x, ·) is the unique solution of the variational problem (7.2). To prove this, we need to check that q(·, ·) is coercive on H 1 p (Y s )/C 3 , which means that there exists c > 0 such that: But (7.4) follows from (2.1), (2.2), and an application of Korn's inequality for H 1 p (Y s )/C 3 . Next, we calculate u 1 in terms of u and p 0 . To do this, we seek a solution of the form: with χ, χ j i ∈ H 1 p (Y s )/C 3 , 1 ≤ i, j ≤ 3, real-valued vector functions, independent of x. It can be verified with a straightforward calculation that the vectors χ and χ j i satisfy the following equations: respectively, which are uniquely defined by (7.4) and are independent of λ.
We set: Ys div y χ j i dy, β = Ys div y χ dy.
Note that β = q(χ, χ) ≥ 0 (see (7.6)). Equation (7.5) allows us to write p 0 (x) in terms of u(x) and u r (x) as follows. By substitution of (7.5) into (6.22), we get: where δ and Π are given by: 7.2. The local problem for u r . Because of the properties of u r described in Lemma 6.5, we consider the following space: W is a closed vector subspace of V Y , and we have u r ∈ L 2 (Ω; W ) (see Lemma 6.5). Now, we use the test function w ǫ φ, w ∈ W and φ ∈ D(Ω) in (3.8), to obtain: Then, by letting ǫ ↓ 0 in (7.10), and using Lemmas 6.4, 6.6, and 6.7, we obtain: With a straightforward calculation, applying Lemma 6.4 and Lemma 6.7, and integrating by parts, we have: For the interface integral term, we have the following lemma.
The uniform boundedness of the interface integral in (6.10) is still one order shy of the assumption stated in Theorem 5.7. To get a stronger uniform bound, we apply the following scaling argument. Fix an ǫ-periodic cell in Ω, say the cell indexed by k, the trace theorem implies: Hence, by summing over all k, we arrive at the bound needed for Theorem 5.7: By Theorem 5.5, we have that there exists v ∈ L 2 (Ω, L 2 (Γ)) such that: for all φ(x, y) ∈ C[Ω, C p (Y )]. Following the proof of Proposition 2.6 in [4] (see Theorem 5.7), for any vector-valued smooth test function ϕ(x, y), we have: Passing to the two-scale limit in each term, we obtain: Therefore,
An application of integration by parts to the second integral above leads to: (h(x, y) − ζ(x, y))ϕ (x, y) · n dσ(y)dx = 0, which implies that ζ(x, y) Γ f = h(x, y). Recall that ζ(x, y) = u(x) + u r (x, y) for y ∈ Y f . Then, we obtain the following two-scale convergence result: (u(x) + u r (x, y))φ(x)(w f (y) − w s (y))dσ(y)dx. (7.16) The lemma is then proved by subtracting (7.15) from (7.16). From (7.11)-(7.14), the local problem for u r is obtained as follows We note that the above problem is coercive in the V Y norm and, hence, has a unique solution. It is the weak formulation of the following cell problem (7.18)    λ 2 ρ f u i r (x, y) + 2λµ 2µe ij (u r )n j = αu i r on Γ, ∀w ∈ W.
In this section, we derive the boundary value problem for w 0 , the weak limit, in the sense of Lemma 5.2, ofû ǫ . We also prove its uniqueness and establish the convergence of u ǫ to u 0 (see (6.6)).
8.1. Macroscopic equations. We take w ∈ D(Ω) in (3.8), and concentrate on passing to the limit as ǫ ↓ 0, using Lemmas 5.2, 6.4, 6.19 and 6.5. Observe that, if w ∈ D(Ω), the interface term drops automatically because w f = w s on Γ ǫ , which is a Hilbert space with the norm: We have u r ∈ H(Ω; W ), and we would like to extend (8.4) to all w r ∈ H(Ω; W ). To achieve this, we need to following lemma.
The macroscopic equation in the case of inclusions looks exactly like the one in [11]; but in the case of connected geometry (8.4) the macroscopic equation contains the boundary term, and therefore this term also appear in the homogenized problem for w 0 (8.5).
The homogenized equations (8.5) are posed in six dimension space. Since u r in (7.18) is linearly proportional to the force term: we could have defined the auxiliary matrix-valued variable θ, such that: u i r (x, y) = θ ip (y)F p (x). By substituting this expression into (7.17), the following equations for θ(y) can be easily obtained: w · e p dy, p = 1, 2, 3, ∀w ∈ W.
This will lead to homogenized equations in three dimensions, instead of six. However, in contrast to the auxiliary variables introduced for u 1 , that satisfy the equations (7.6) and (7.7), θ is dependent on λ. This means that the homogenized problem in time-domain will contain memory terms with the inverse Laplace transform of θ being the kernel function.