Multiscale Modelling and Analysis of Signalling Processes in Tissues with Non-Periodic Distribution of Cells

In this paper, a microscopic model for a signalling process in the left ventricular wall of the heart, comprising a non-periodic fibrous microstructure, is considered. To derive the macroscopic equations, the non-periodic microstructure is approximated by the corresponding locally periodic microstructure. Then, applying the methods of locally periodic homogenization (the locally periodic (l-p) unfolding operator, locally periodic two-scale (l-t-s) convergence on oscillating surfaces and l-p boundary unfolding operator), we obtain the macroscopic model for a signalling process in the heart tissue.

to a positive angle at the endocardium. In the microscopic model of a signalling process, we consider the diffusion of signalling molecules in the extracellular space and their interaction with receptors located on the surfaces of muscle cells. There are two main challenges in the multiscale analysis of microscopic problems posed in domains with non-periodic perforations: (i) the approximation of the non-periodic microstructure by a locally periodic one and (ii) derivation of limit equations for the non-linear equations defined on oscillating surfaces of the microstructure. First, assuming the C 2 -regularity for the rotation angle γ , we define the locally periodic microstructure which approximates the original non-periodic plywood-like structure. Similar approximation of non-periodic plywood-like microstructure by locally periodic one was considered in [7,29]. Then, applying techniques of locally periodic homogenization (locally periodic two-scale convergence (l-t-s) and l-p unfolding operator), we derive macroscopic equations for the original microscopic model. The l-p two-scale convergence on oscillating surfaces and l-p boundary unfolding operator allow us to pass to the limit in the non-linear equations defined on surfaces of the locally periodic microstructure.
In this paper, we consider a simple model describing the interactions between processes defined in the perforated domain and on the surfaces of the microstructure. However, the techniques presented here can be also applied to more general microscopic models as well as to other non-periodic microstructures, provided the variations in the microscopic structure are sufficiently regular.
Previous results on homogenization in locally periodic media constitute the multiscale analysis of a heat-conductivity problem defined in domains with non-periodically distributed spherical balls [3,8,31], and elliptic and Stokes equations in non-periodic fibrous materials [4,6,7,29]. Formal asymptotic expansion and two-scale convergence defined for periodic test functions, [27], were used to derive macroscopic equations for models posed in domains with locally periodic perforations, i.e., domains consisting of periodic cells with smoothly changing perforations [5,10,11,22,23,33].
The paper is organized as follows. In Section 2, the microscopic model for a signalling process in a tissue with non-periodic plywood-like microstructure is formulated. In Section 3, we prove the existence and uniqueness results for the microscopic model and derive a priori estimates for a solution of the microscopic model. The approximation of the microscopic equations posed in the domain with non-periodic microstructure by the corresponding problem defined in a domain with locally periodic microstructure is given in Section 4. Then, applying the l-p unfolding operator, l-t-s convergence on oscillating surfaces, and l-p boundary unfolding operator we derive the macroscopic model for a signalling process in the heart muscle tissue. In Appendix, we summarize the definitions and main compactness results for the l-t-s convergence and l-p unfolding operator.

Microscopic Model for a Signalling Process in Heart Tissue
In this work, we consider a receptor-based microscopic model for a cellular signalling process in cardiac tissue. A signalling system is important for proper function of cells and appropriate respond to changes in the extracellular environment. Many regulatory events in cardiac tissue are mediated via surface receptors, located in cardiac cells membrane, that transmit signals through the activation of GTP binding proteins (G proteins) [34]. Cardiac cells (myocytes) in the heart wall tissue are joined in a linear arrangement to form muscle fibres. In the left ventricular wall of the heart the orientation of layers of muscle fibre changes with position through the wall. The layers of parallel aligned muscle fibres are rotated from −60 • at epicardium to +70 • at endocardium [25] and create a non-periodic plywood-like microstructure. For simplicity, we assume that the individual muscle fibres are not connected to each other. However, it is possible to consider a periodic distribution of connections between the fibres. In the mathematical model for a signalling process in cardiac tissue, we consider the binding of signalling molecules to receptors located on the cell membrane, which through the activation of G proteins (not considered in our simple model) results in the activation of a cell signalling pathway. We consider the diffusion, production and decay of ligands (signalling molecules) c and binding of ligands to the membrane receptors. We shall distinguish between free receptors r f and bound receptors r b , which correspond to receptor-ligand complexes. We assume that the receptor-ligand complex can dissociate and result in a free receptor and ligand. We also consider the production of new free receptors and natural decay of free and bound receptors.
where d f and d b are the decay rates, β denotes the dissociation rate for the receptor-ligand complex, α is the binding rate, the function p models the production of free receptors, and the function F describes the production and decay of ligands.
To define the plywood-like microstructure of the cardiac muscle tissue of the left ventricular wall, we consider a function γ ∈ C 2 (R), with −π/2 ≤ γ (x) ≤ π/2 for x ∈ R and define the rotation matrix around the x 3 -axis as where γ (x) denotes the rotation angle with the x 1 -axis. Denote R x := R(γ (x 3 )).
We consider an open, bounded subdomain ⊂ R 3 , with Lipschitz boundary, representing a part of the cardiac muscle tissue and the x 3 -axis to be orthogonal to the layers of parallel-aligned muscle fibres. We assume that the radius of the muscle fibres depends on the position in the tissue and define the characteristic function of a fibre by , with 0 < ρ 0 ≤ ρ(x R ) ≤ ρ 1 < ∞ and ρ(x R )a ≤ 2/5 for all x ∈ , i.e., a = 2/(5ρ 1 ).
By small parameter ε we denote the characteristic size of the microstructure of cardiac tissue, given as a ratio between the characteristic diameter of muscle fibres and characteristic size of the cardiac tissue. Notice that for plywood-like microstructure the axis of each fibre can be defined by a rotated around the x 3 -axis line, parallel to the x 1 -axis and passing through a point of an ε-grid in the plane x 1 = const. Thus, for j ∈ Z 3 , we define Notice that x ε j,3 = εj 3 and the third variable is invariant under the rotation R x ε j . This ensures that for each fixed εj 3 we obtain a layer of parallel aligned fibres. Then the perforated domain * ε , corresponding to the extracellular space of cardiac tissue, is defined as * We denote 0,1 = {y ∈ R 3 : y 1 = ±1/2}. Then, assumptions on ρ and a ensure that = ∅ for any m, n ∈ ε with n 2 = m 2 or n 3 = m 3 . Hence, * ε is connected. This corresponds to our assumption that muscle fibres do not touch each other and are not directly connected, and the interactions between the muscle fibres are facilitated through the extracellular matrix. Now, using the definition of ϑ, the characteristic function of muscle fibres in cardiac tissue reads and the extracellular space is characterised by The surfaces of muscle cells, i.e., the boundaries of the microstructure, are denoted by Notice that the changes in the microstructure of * ε are defined by changes in the periodicity given by the linear transformation (rotation) R(x) and by changes in the shape of the microstructure (changes in the radius of muscle fibres) given by the linear transformation To determine the non-constant reaction rates for binding and dissociation processes on cell membranes, we consider α, β ∈ C 1 ( ; C 1 0 (Y 1 )), extended in y-variable by zero to R 3 , and define Then, the microscopic model for a signalling process in cardiac tissue reads where the dynamics in the concentrations of free and bound receptors on cell surfaces is determined by two ordinary differential equations with initial conditions defined as For simplicity of the presentation we shall assume that the diffusion coefficient A and the decay rates d f , d b are constant. We also assume that the functions F and p are independent of x ∈ . The dependence of A, d f , d b , F and p on the microscopic and macroscopic variables can be analyzed in the similar way as for α ε and β ε .
Notice that the C 1 -regularity of α and β is required for the approximation of the integrals defined on the boundaries of the non-periodic microstructure by the integrals defined on the boundaries of the corresponding locally periodic microstructure.
We shall consider a weak solution of the problem (1) and (2), defined in the following way. (1) and (2) are functions c ε , r ε f ,

Existence, Uniqueness, and a Priori Estimates for a Weak Solution of the Microscopic Problem (1) and (2)
In a similar way as in [9,21,30], we can prove the existence, uniqueness, and a priori estimates for a weak solution of problem (1)- (2). Notice that for the derivation of a priori estimates a trace estimate, uniform in ε, where the constant C depends on Y 1 , Y 0 , K and is independent of ε and j ∈ ε . Then, considering the change of variables x = εR x ε j y + x ε j = εR x ε j (y + j) and summing up over where the constantμ depends on Y 1 , Y 0 , R and K and is independent of ε. (1) and (2) satisfying the following a priori estimates

Lemma 1 Under Assumption 1 there exists a unique non-negative weak solution of the microscopic problem
where the constant μ is independent of ε, , withμ being the constant in the trace inequality (4).
Proof (Sketch) As in [30] the existence of a solution of the microscopic problem (1) and (2) for each fixed ε > 0 is obtained by applying fixed point arguments and Galerkin method.
Also, using the same arguments as in [30] we obtain that c ε (t, x) ≥ 0 for (t, x) ∈ * ε,T and r ε To derive a priori estimates, we consider the structure of the microscopic equations. For non-negative solutions, by adding the equations for r ε f and r ε b , we obtain Then the Lipschitz continuity of p and the non-negativity of r ε f and r ε b imply the boundedness of r ε f and r ε b on ε T . Considering c ε as a test function in (3) and using the trace inequality (4), we obtain the estimates for c ε . Testing (2) by ∂ t r ε f and ∂ t r ε b , respectively, yields the estimates for the time derivatives of r ε f and r ε b . In the derivation of the a priori estimate for ∂ t c ε we use the equation for ∂ t r ε f to estimate the non-linear term on the boundary ε , i.e., Considering (c ε − M 1 e M 2 t ) + as a test function in (3), where M 1 and M 2 are as in the formulation of the lemma, we obtain * ε Using the non-negativity and boundedness of β ε and r ε f , along with the trace inequality (4), the last integral can be estimated as for any δ > 0, where the constants μ 1 , μ 2 and μ δ depend on and on the transformation matrices R and K, but are independent of ε. More specifically, . Using the non-negativity of c ε and r ε f , the Lipschitz continuity of F , and the assumptions on M 1 and M 2 , and applying the Gronwall inequality yield estimate (6).
To show the uniqueness of a solution of the microscopic problem (1) and (2), we consider the equations for the difference of two solutions (c ε 1 , r ε f,1 , r ε b,1 ) and (c ε 2 , r ε f,2 , r ε b,2 ). The nonnegativity of α ε , r ε f,j , and c ε j , along with the boundedness of r ε f,j , ensures Testing the sum of the equations for r ε and using the estimate from above yield Combining the last two inequalities and applying the Gronwall inequality imply the estimates for r ε l, Considering (c ε − S) + , with some S > 0, as a test function in (3) and using the boundedness of r ε f and r ε b we obtain , |F (0)|}, μ 1 is some positive constant, and * ,S ε (t) = {x ∈ * ε : c ε (t, x) > S}. Then, Theorem II.6.1 in [20] yields the boundedness of c ε in (0, T ) × * ε for every fixed ε. Considering now (3) for two solutions (c ε 1 , r ε f,1 , r ε b,1 ) and (c ε 2 , r ε f,2 , r ε b,2 ), we obtain the estimates for c ε , shown above, and applying the Gronwall inequality, we conclude that The assumptions on the non-periodic microstructure of * ε and the regularity of the transformation matrices R and K ensure the following extension result.
where μ depends on Y 1 , Y 0 , R and K and is independent of ε and j ∈ ε .
where μ depends on Y 1 , Y 0 , R and K and is independent of ε.
Proof (Sketch) The proof follows the same lines as in the periodic case, see e.g. [15,19]. The only difference here is that the extension depends on the Lipschitz continuity of K and R and the uniform boundedness from above and below of | det ) and obtain the estimates in (7). Notice that due to the definition of K(x), the fibre radius varies between different fibres in the plywood-like structure of the heart tissue, but is constant along each individual fibre. Thus, apart from the end parts of the fibres near ∂ , we have to extend u only in the directions orthogonal to the fibres. In the definition of * ε we consider those j that εR x ε j (Y 1 + j) ⊂ . Hence the extension for the end parts of the fibres near ∂ , i.e., for such , we obtain that the constant μ in (7) is independent of x ε j , ε, and j ∈ ε . Then scaling and R x ε j (Y 1 + j) by ε and summing up over j ∈ ε in (7) imply (8). In the case when the boundary ∂ crosses the fibres in a non-orthogonal way, we would obtain only a local extension to a subdomain δ = {x ∈ : dist(x, ∂ ) > δ} for any fixed δ > 0.

Derivation of Macroscopic Equations
To derive macroscopic equations for the microscopic problem posed in a domain with the non-periodic plywood-like microstructure, we approximate it by a problem defined in the domain with the corresponding locally periodic microstructure and apply the methods of locally periodic two-scale convergence (l-t-s) and l-p unfolding operator (see Appendix for the definitions and convergence results for l-t-s convergence and l-p unfolding operator). Notice that the regularity assumptions on the orientation angle γ are essential for the construction of an appropriate locally periodic microstructure for the non-periodic plywood-like structure.
To define the locally periodic microstructure related to the original non-periodic one, we consider, similarly to [8,29], the partition covering of by a family of open nonintersecting cubes { ε n } 1≤n≤N ε of side ε r , with 0 < r < 1, such that For each x ∈ R 3 , we consider a transformation matrix D(x) ∈ R 3×3 and assume that D, D −1 ∈ Lip(R 3 ; R 3×3 ) and 0 < d 0 ≤ | det D(x)| ≤ d 1 < ∞ for all x ∈ . The matrix D will be defined by the rotation matrix R and its derivatives and the specific form of D will be given later.
Then, the locally periodic microstructure is defined by considering a covering of ε n by parallelepipeds εD x ε n Y such that and points x ε n ,x ε n ∈ ε n , for n = 1, . . . , N ε , are arbitrary chosen, but fixed. Here Y = (0, 1) 3 , D x := D(x), and D x ε n = D(x ε n ) for 1 ≤ n ≤ N ε . Then, the perforated domain with locally periodic microstructure is given by * . . , N ε , and the transformation matrix K will be specified later. We shall also denotê ε n =x ε n + Int The boundaries of the locally periodic microstructure are defined as where x ε n ,K = K x ε n and = ∂Y 0 \ 0,1 . For the problem analyzed here, we shall consider x ε n = x ε n . The following calculations illustrate the motivation for the locally periodic approximation and determine formulas for the transformation matrices D and K. For n = 1, . . . , N ε , we choose such κ n ∈ Z 3 that for x ε n = R x ε n εκ n we have x ε n ∈ ε n . In the definition of covering of ε n by shifted parallelepipeds, we consider a numbering of ξ ∈ ε n and write ε n ⊂ x ε n + Then for 1 ≤ j ≤ I ε n we consider k n j = κ n + ξ j and x ε k n j = R k n j εk n j . Here R k n j := R x ε k n j and R κ n := R x ε n . Using the regularity assumptions on the function γ and considering the Taylor expansion of R −1 around x ε n , i.e. around εκ n,3 , we obtain where W x ε n = W (x ε n ) with W (x) = (I − ∇R −1 (γ (x 3 ))x). The notation of the gradient is understood as ∇R −1 (γ (x))x = ∇ z (R −1 (γ (z))x)| z=x . Thus, for x, x ε n ∈ ε n , since |x − x ε n | ≤ Cε r , the distance between R −1 is of the order sup 1≤j ≤I ε n |ξ j ε| 2 ∼ ε 2r .
This calculation together with the estimates below will ensure that the non-periodic plywood-like structure can be approximated by the corresponding locally periodic microstructure, comprising Y x ε n -periodic structure in each ε n for n = 1, . . . , N ε , and | ε n | ∼ ε 3r for an appropriate r ∈ (0, 1). Here where (x 3 )). The transformation matrix K is defined as K(x) = W −1 (x)K(x) and the boundary of the muscle fibres in the locally periodic approximation is given by The definitions of R, W and γ ensure that the transformation matrices D and K are Lipschitz continuous, as well as 0 Since ϑ is independent of the first variable, we consider in the definition of W (x) the shift only in the second variable. Notice that if the original microstructure would be locally periodic, i.e. R(γ (x 3 )) = R(γ (x ε n,3 )) for x ∈ ε n and some x ε n ∈ ε n , then the matrix W would be constant in each ε n and we would obtain In the approximation of the problem posed in the domain with the non-periodic plywoodlike structure, we shall use the following lemma, proven in [6], that facilitate the estimate for the difference between the values of the characteristic function at two different points. Deriving estimates for the difference of solutions of the original microscopic problem and the corresponding locally periodic approximation and applying techniques of locally periodic homogenization, we obtain the following macroscopic equations for the microscopic problem (1) and (2). (1) and (2) converges to a solution c ∈ L 2 (0, T ; H 1 ( )) ∩ H 1 (0, T ; L 2 ( )) and r l ∈ H 1 (0, T ; L 2 ( ; L 2 ( x ))),

Theorem 1 A sequence of solutions of the microscopic problem
for (t, x) ∈ (0, T ) × and y ∈ x , where the macroscopic diffusion coefficient A is defined as with w j , for j = 1, 2, 3, which are solutions of the unit cell problems div(A(∇ y w j + e j )) = 0 in Y * x,K , A(∇ y w j + e j ) · n = 0 on x , w j Y x − periodic, Y * x w j d y = 0 .

(12)
Here (10), and Proof Using calculations from above, we consider a domain with a locally periodic microstructure characterised by the periodicity cell Y x ε n = D x ε n Y in each ε n , with n = 1, . . . , N ε and the shift x ε n ∈ ε n in the covering of ε n by D x ε n (Y + ξ), with ξ ∈ ε n . Then, the characteristic function of the extracellular space * ε in a tissue with locally periodic microstructure is defined by χ * The boundaries of the locally periodic microstructure are denoted by Notice that non-periodic changes in the shape of the perforations (radius of muscle fibres) are approximated by the same transformation matrix K(x). This is consistent with the results obtained in [10,11,23,33]. However spatial changes in the periodicity are approximated by D x = R x W x . The reaction rates (binding and dissociation rates) are defined in terms of locally periodic microstructure in the following way To show that we can approximate the problem (1) and (2) by a microscopic problem defined in the domain with the locally periodic microstructure, we have to prove that the difference between the characteristic function of the original domain χ * ε and of the locally periodic perforated domain χ * ε converges to zero strongly in L 2 ( ) as ε → 0. Also, we have to show that the difference between boundary integrals and their locally periodic approximations converges to zero as ε → 0. This will ensure that as ε → 0 the sequence of solutions of the original microscopic problem (1)-(2) will converge to a solution of the macroscopic equations obtained by homogenization of the corresponding problem defined in the domain with locally periodic microstructure.
For the difference between χ * ε and χ * ε , we have We notice that ε 3 |J ε n | ≤ Cε 3r and |N ε | ≤ Cε −3r . For the first integral, we have To estimate the second integral, we use Lemma 3. Since in each ε n the length of fibres is of order ε r , applying estimate in Lemma 3, equality (9), and the estimates N ε ≤ Cε −3r and |J ε n | ≤ Cε 3(r−1) , we conclude that Thus, for r ∈ (2/3, 1), we have I 1 → 0 and I 2 → 0 as ε → 0. To estimate the difference between boundary integral we have to extend c ε , r ε f , and r ε b from * ε to . For c ε , we can consider the extension as in Lemma 2. Then, using the extendedc ε and the fact that the reaction rates and the initial data are defined on whole we can extend r ε f and r ε b to as solutions of the ordinary differential equations withc ε instead of c ε The non-negativity of c ε and the construction of the extension ensure thatc ε is non-negative. Then, in the same way as for r ε f and r ε b , using the properties of p and the non-negativity of the coefficients and initial data, we obtain the non-negativity ofr ε f andr ε b . Thus, adding the equations forr ε f andr ε b , we obtain the boundedness ofr ε f andr ε b in T , i.e.
Notice that ε −1 in the estimates for ∇r ε f and ∇r ε b will be compensated by ε in the estimate for the difference between neighbouring points in non-periodic and locally periodic domains, respectively, i.e.
Then, for the boundary integrals, we have Considering the regularity of K and R and the uniform boundedness from below and above of | det K|, and using the trace estimate for the L 2 ( )-norm of a H ς (Y )-function, with ς ∈ (1/2, 1), the first integral we can estimates as , with j ∈ J ε n and n = 1, . . . , N ε . Here, we used the short notations Using the regularity of γ , K, and α, and applying a priori estimates for c ε and r ε f , together with (15), we obtain for 0 < ς 1 < 1/2, with ς + ς 1 = 1, Conducting similar calculations as for I 3 and using the fact Similarly, we obtain that The definition of * ε , ε , α ε , and β ε implies that the original non-periodic problem is approximated by equations posed in a domain with locally periodic microstructure. Hence we can apply the methods of locally periodic two-scale convergence (l-t-s) and l-p unfolding operator to derive the limit equations.
Using the extension of c ε , we have that the sequences {c ε }, {∇c ε } and {∂ t c ε } are defined on T and we can determine Then, the convergence results for the l-p unfolding operator and l-t-s convergence, see [29,30] or Appendix, imply that there exist subsequences (denoted again by c ε , r ε f and r ε b ) and the functions c ∈ L 2 (0, T ; H 1 ( )), ∂ t c ∈ L 2 ( T ), c 1 ∈ L 2 ( T ; H 1 per ( Y x )), r f , r b ∈ H 1 (0, T ; L 2 ( ; L 2 ( x ))) such that The coefficients α ε and β ε can be defined as locally periodic approximations of α and β, given by (13), i.e., See Appendix or [29] for the definition of the locally periodic approximation L ε . The regularity assumptions on α, β, K, and γ ensure that α, β ∈ C( ; C per ( Y x )).
Considering ψ ε (t, x) = ψ 1 (t, x) + εL ε ρ (ψ 2 )(t, x), with ψ 1 ∈ C 1 ( T ) and ψ 2 ∈ C 1 0 ( T ; C 1 per ( Y x )), as a test function in (3) (see Appendix or [29] for the definition of L ε ρ ) and applying l-p unfolding operator and l-p boundary unfolding operator imply The regularity assumptions on γ and K ensure that and +∞). Applying the results from [30] Using the a priori estimates for c ε , r ε f and r ε b , the strong convergence of T ε L (c ε ) in L 2 ( T ; H 1 (Y )), the strong convergence and boundedness of T b,ε L ( α ε ), the weak convergence and boundedness of T b,ε L (r ε f ), together with the regularity of D, γ , and K, and the strong convergence of Similar arguments along with the Lipschitz continuity of F and the strong convergence of as ε → 0. Using the convergence results (16), the strong convergence of T ε L (ψ ε ) and T ε L (∇ψ ε ) and the fact that | * ε | → 0 as ε → 0, taking the limit as ε → 0, and considering the change of variables y = D xỹ forỹ ∈ Y and y = D x K xȳ forȳ ∈ , we obtain where w j are solutions of (12). Choosing ψ 2 (t, x, y) = 0 for (t, x) ∈ T and y ∈ Y x yields the macroscopic equation for c.
Using the strong convergence of T b,ε L (c ε ) in L 2 ( T ; L 2 ( )), estimates (5a)-(5b) and (6), and the Lipschitz continuity of p we obtain that {T b,ε L (r ε j )} is a Cauchy sequence in L 2 ( T ; L 2 ( )) for j = f, b, and hence upto a subsequence, T b,ε L (r ε j ) → r j (·, ·, D x K x ·) strongly in L 2 ( T ; L 2 ( )). Then applying the l-p boundary unfolding operator to the equations on ε and taking the limit as ε → 0 we obtain the equations for r f and r b . The proof of the uniqueness of a solution of the macroscopic problem is similar to the corre-sponding proof for the microscopic problem, and hence the convergence of the whole sequences of solutions of the microscopic problem follows.
Remark 1 Notice that for the proof of the homogenization results it is sufficient to have a local extension of c ε from * ε to δ , with δ = {x ∈ : dist(x, ∂ ) > δ} for any fixed δ > 0, and hence, the local strong convergence of T ε L (c ε ), i.e., the strong convergence in L 2 (0, T ; L 2 loc ( ; H 1 (Y ))).
Remark 2 For numerical computations of the cell problems (12) and the ordinary differential equations (11), defining the dynamics of receptor densities, approaches from the two-scale finite element method [24] or the heterogeneous multiscale method [1,2,17,18] can be applied.

Appendix: Definition and Convergence Results for the l-t-s Convergence and l-p Unfolding Operator
We shall consider the space C( ; C per ( Y x )) given in a standard way, i.e. for any ψ ∈ C( ; C per (Y )) the relation ψ(x, y) = ψ(x, D −1 x y) with x ∈ and y ∈ Y x yields ψ ∈ C( ; C per ( Y x )). In the same way the spaces L p ( ; C per ( Y x )), L p ( ; L q per ( Y x )) and C( ; L q per ( Y x )), for 1 ≤ p ≤ ∞, 1 ≤ q < ∞, are given. Due to the assumptions on D, i.e. D ∈ Lip( ) and 0 < d 0 ≤ | det D(x)| ≤ d 1 < ∞ for all x ∈ , we obtain that the function u : → C( Y x ) is well-defined. The separability of C per ( Y x ) for each x ∈ and the Weierstrass approximation for continuous functions u : → C per ( Y x ) ensure the separability of C( ; C per ( Y x )). Also, we have the following relation for the norm ψ C( ;C per ( Y x )) := sup x∈ sup y∈ Y x |ψ(x, y)|: The assumptions on D and K ensure that u(x) ∈ L 2 ( x ) for a.e. x ∈ and u 2 L 2 ( x ) dx < ∞ are well-defined, separable Hilbert spaces, see e.g. [16,26,32].
We recall here the definition of locally periodic two-scale (l-t-s) convergence and l-p unfolding operator, see [29,30] for details. Definition 2 ([29]) Let u ε ∈ L p ( ) for all ε > 0 and 1 < p < ∞. We say the sequence {u ε } converges l-t-s to u ∈ L p ( ; L p ( Y x )) as ε → 0 if u ε L p ( ) ≤ C and for any ψ ∈ L q ( ; C per ( Y x )) where L ε is the l-p approximation of ψ and 1/p + 1/q = 1.

Definition 4 ([30])
For any Lebesgue-measurable on function ψ the locally periodic unfolding operator (l-p unfolding operator) T ε L is defined as for x ∈ and y ∈ Y .

Definition 5 ([30])
For any Lebesgue-measurable on ε function ψ the l-p boundary unfolding operator T b,ε L is defined as ψ εD x ε n D −1 x ε n x/ε Y + εD x ε n K x ε n y χˆ ε n (x) for x ∈ and y ∈ .