Analysis and homogenisation of a linear Cahn–Larché system with phase separation on the microscale

We consider the process of phase separation of a binary system under the influence of mechanical stress modelled by the Cahn–Larché system, where the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level. After linearisation, we prove existence and uniqueness of a weak solution by a Galerkin approach. As discretisation in space leads to a linear differential–algebraic system of equations, we adjust known solution theory for such equations to a weak setting. This approach may be of interest more generally for coupled elliptic–parabolic systems. A‐priori estimates enable us to pass to the homogenisation limit of the linear system rigorously using the concept of two‐scale convergence. A comparison with the formally homogenised full nonlinear problem shows that both systems lead to models of distributed‐microstructure type in the limit and that homogenisation and linearisation commutes.


INTRODUCTION
Phase-separation processes are classically modelled by the Cahn-Hilliard equation.In the case of binary systems (two phases), the nonlinear fourth-order parabolic equation describes the evolution of an order parameter denoting the relative concentration  = (, ) of two phases E and C, say, so that  = 0 and  = 1 in space-time points (, ) of pure phases E and C, respectively.The Cahn-Hilliard equation describes the  −1 -gradient flow of an energy potential incorporating a term associated with the mixture, typically modelled by a double-well potential, and a term penalising phase interfaces.When elastic effects need to be taken into account as well, the energy functional is extended by an elastic-energy term and the gradient flow is considered in  −1 × ( 2 )  for tuples (, ), where  = (, ) is the displacement vector with values in ℝ  .The result is the Cahn-Larché system, consisting of an extended Cahn-Hilliard equation coupled with the (linearised) equations of elasticity [1].
In a number of situations, the processes-phase separation and mechanics-happen on different spatial and temporal scales.For example, in phase-separation experiments on Langmuir-Blodgett film balances, monolayers of different lipid phases decompose under mechanical deformations induced by a teflon barrier.The phase separation occurs on length scales of the order of several microns, which differs from the scale of the mechanical deformation induced by the teflon barrier by about five orders of magnitude [2].We also refer to [3], [4] and [5] for phase separation in lipid membranes and to [6] in particular for modelling of lipid decomposition by the Cahn-Hilliard equation.Similarly, phase separation in alloys, for which the Cahn-Hilliard model was originally derived, also happens on much smaller scales than mechanical deformations of the workpieces made of these alloys [7].
A multiscale Cahn-Larché system taking into account these different length scales was recently derived in the context of phase-separation experiments making use of formal homogenisation techniques [8].The result was a model of distributedmicrostructure type, in which a macroscopic elastic equation is coupled to an extended Cahn-Hilliard equation to be solved in (local) representative unit cells associated with each macroscopic point.The homogenisation process was formal in the sense that asymptotic expansions were assumed valid and the limit system was obtained by matching terms of same order of the (small) homogenisation parameter .Numerical simulations showcasing typical results demonstrated the applicability of the homogenised model.
It is the aim of this article to make the homogenisation process mentioned above rigorous.As the Cahn-Larché system is highly nonlinear and, more importantly, the -scale model derived in [8] is degenerate in the homogenisation limit  → 0, we restrict to a linearised model, where the linearisation is about a given solution of the Cahn-Larché system.For this linearised model, which is a system of a parabolic fourth-order equation and an elliptic second-order equation, we show existence of a weak solution by a Galerkin discretisation in space adapting results from the theory of differentialalgebraic equations.The approach is kept general so that it can be easily adapted to other parabolic-elliptic problems.The homogenisation process is carried out in the context of (rigorous) two-scale convergence leading to a linear limit system of distributed-microstructure type.Comparing this limit system with the linearised version of the formally derived limit system of [8] shows that linearisation and homogenisation commute for this problem.
The article is organised as follows: Based on the modelling of phase-separation experiments on Langmuir-Blodgett film balances of [8], we introduce the nonlinear Cahn-Larché system with phase separation on the microscale and linearise it about a given solution in Section 2. The well-posedness of the linear Cahn-Larché system is shown in Section 3 including the -independent a-priori estimates required for the homogenisation process, where the generalisations of the solution theory for differential-algebraic systems are discussed in Section 3.2.1.The passage to the homogenisation limit is presented in Section 4. The general two-scale compactness results are discussed in Section 4.1 and the final limit system is found in Section 4.2.A short summary and discussion of the results are given in Section 5.

THE MATHEMATICAL MODEL
We give a brief description of the -scale (microscopic) model derived in [8], which is the starting point for our considerations.Let Ω ⊂ ℝ  be an open and bounded domain with Lipschitz continuous boundary Γ ∶= Ω having pairwise disjoint parts Γ 0 , Γ g and Γ s such that Γ = Γ 0 ∪ Γ g ∪ Γ s and a finite time interval  = (0, ).This choice is motivated by the film-balance experiments: In two dimensions, Ω is a rectangle describing the domain occupied by the lipid monolayer, which is bounded by the lateral boundaries of the film balance Γ s , the movable teflon barrier Γ g and the boundary opposite the teflon barrier Γ 0 .We assume the evolving microstructure of the pattern to have an intrinsic length scale associated with it.For this purpose, we introduce a characteristic macroscopic length scale , representing the order of magnitude of the size of the film balance and corresponding to the macroscopic process, and a characteristic microscopic length scale , which corresponds to the order of magnitude of the scale on which the phase separation is observable, and we write It is clear that it holds  ≪ 1.Then, the order parameter, which describes the microstructure, depends on .We denote this with an  in the index and write   and, analogously,   for the displacement.

The Cahn-Larché system
If only small deformations are considered a linearised theory is applicable so that we only consider infinitesimal strains defined by In general, we have different elastic properties of the two phases.Thus, the elasticity tensor (  ), characterising the stiffness of the phases, naturally depends on the order parameter   .The stress tensor is thus given by More precisely, we assume the two phases to have different elastic properties and hence we denote the elasticity tensor describing the elastic properties of the softer phase by  E and the tensor of the harder phase by  C .(The labelling is motivated by the labelling in [8].)Each of the two pure lipid phases is isotropic, and so are the two component tensors.
Then, for the lipid mixture, we consider an elasticity tensor depending on the relative concentration of the mixture, which is simply an interpolation of the two component tensors.The interpolation function  ∶ [0, 1] → [0, 1] should be defined such that With this we have also determined that  = 0 corresponds to the elastically softer phase and  = 1 corresponds to the elastically stiffer phase.We assume positive definiteness for the individual component tensors, that is, for each  ∈ {E, C} we assume the existence of positive numbers   > 0 such that for any symmetric matrix  ∈ ℝ × .Furthermore, we assume the usual symmetry conditions in linear elasticity theory, that is, for   = (  ℎ ) 1≤,ℎ,,≤ ,  ∈ {E, C}, we require Obviously, the interpolated tensor defined by ( 4) is also positive definite and fulfils the symmetry condition (7).By (  ), we denote the eigenstrain.In general, this refers to a strain which is present in the absence of any applied stress.This phenomenon occurs in the presence of inhomogeneities, such as thermal expansions, or as in our case, with phase transitions and leads to self-generated internal stress [9].The eigenstrain is often referred to as stress-free strain and, just like the elastic material parameters, it may be different for each phase.A natural choice is a multiple of the identity where the scalar-valued function ( ⋅ ) specifies the eigenstrain behaviour at a particular phase state and  ∈ ℝ × is the second-order identity tensor.According to (8), the eigenstrain is uniform in all directions, which is a common choice, see for example, [9][10][11].
Assuming that the mechanical equilibrium is reached much faster than the diffusion takes place and using representation (8) for the eigenstrain, then, since we can write the Cahn-Larché system as follows: )) in Ω × , (9) where (  ) is the free-energy density of the mixture, which we model by a double-well potential, The  2 -scaling arises from a nondimensionalisation and we refer to [8] for details.
The boundary conditions are chosen as follows, where  denotes the outer unit normal and  the unit tangential vector on Γ.At any time, the lipid monolayer remains on the film balance and cannot pass over the edges.Thus, we choose no-flux conditions for the relative concentration   and the chemical potential   on the whole boundary Γ, where The force applied by the controllable barrier and compressing the lipid monolayer is modelled by applying a boundary force  on Γ g , hence, On the opposite boundary part Γ 0 , we do not allow for any deformation and hence we require Furthermore, on the lateral boundary part Γ s we set and a free-slip condition as well, that is, These conditions describe that the monolayer cannot expand past the lateral edges and does not adhere there when compressed.We complete the system with an appropriate initial condition for   , describing the initial homogeneous relative concentration of the mixture, that is, the initial homogeneous state of the monolayer.Well-posedness is discussed in [8] with reference to [12].This is the system, which was homogenised formally in [8].For future reference, we note that the limit system was found to be given by where  is the representative unit cell of the microscale and the details of the notation can be found in Section 4.2.

Linearisation
For the rigorous homogenisation procedure, we derive a linear (scaled) Cahn-Larché system.Let  , and  , denote general solutions of system ( 9)-( 17), such that Then, we consider  , + ℎ c and  , + ℎ ũ , for a small ℎ > 0 and functions c , ũ having the same multiscale character as described in Section 2.1, to obtain a linear system for c and ũ .Neglecting second-order terms, the linear equations for c and ũ are as follows.
c =  denoting the linearised stress tensor.The boundary conditions are also linearised, which leads to denoting the linearised chemical potential.In what follows, whenever we talk about the linear Cahn-Larché system, we mean the equations ( 21) and (22) completed by the boundary conditions (23) and a suitable initial condition for c .Furthermore, we drop all tildes for ease of notation and more clarity.From now on, we denote the solutions of the linearised scaled Cahn-Larché system by   and   , the solutions of the nonlinear scaled Cahn-Larché system (9), (10) completed with the corresponding initial and boundary conditions, are still referred to as  , and  , .Before we analyse the well-posedness of the linearised system, we need to specify further technical details.We consider the interpolated tensor  defined by (4) with constant tensors   ,  ∈ {E, C} corresponding to two phases C and E, such that there exist   ,   ∈ ℝ, with 0 <   <   , such that   ∈ (  ,   , ), where ( α, β, Ω) denotes the space of fourthorder tensors, bounded by positive constants α and β (see (6)), fulfil the symmetry condition (7) and are periodic on a rectangular domain Ω ⊂ ℝ  .Then, there exist ,  ∈ ℝ, with 0 <  < , such that ( , ( ⋅ , )) ∈ (, , Ω), (24) for a.e. ∈  and there exist two numbers  ′ ,  ′′ > 0 such that for any  ∈ ℝ × .For the eigenstrain () = (), we first choose the same type of interpolation as for the elasticity tensor, that is, with constants  E ,  C ∈ ℝ describing the eigenstrain behaviour of the corresponding lipid phase, and with the interpolation function (⋅) defined by ( 5).

WELL-POSEDNESS OF THE LINEAR CAHN-LARCHÉ SYSTEM
In this section, we examine the linearised Cahn-Larché system ( 21)- (23).A key point in the analysis, which seems to be of general interest for showing existence of solutions of systems involving equations of different type, is the semidiscretisation in space leading to a system of equations which can be interpreted as a differential-algebraic equation (DAE) in a weak functional-analytical setting.We discuss existence of solutions of DAE systems in this framework before applying it to the linear Cahn-Larché system, which represents a coupled system of partial differential equations of elliptic and parabolic type.We first fix some assumptions and state the weak formulation.After, we give an a-priori estimate for every  > 0. Furthermore, for every  > 0, we proof the existence and uniqueness of a weak solution using theory about linear differentialalgebraic equations.As we want to work in a weak setting, we introduce now some function spaces, specify some assumptions and thus also state the notation we use.Then, we will have all tools available to state the weak formulation of the linear scaled Cahn-Larché system.We denote by (, ) Ω = ∫ Ω () () d and (, ) Ω, = ∫  0 (() , ()) Ω ds the scalar products on  2 (Ω) and  2 ((0, ),  2 (Ω)) for  ∈ [0, ], respectively, and the abbreviation ‖ ⋅ ‖ Ω ∶= ‖ ⋅ ‖  2 (Ω) for the standard norm on  2 (Ω) as well as ‖‖ 2 Ω, ∶= ∫  0 (() , ()) Ω d.Furthermore, we define the function space equipped with the norm which is equivalent to the standard  2 -norm on (Ω), and provided with the standard norm on [ 1 (Ω)]  .For the unknown functions, we need the function spaces (Ω) ∶=  2 (, (Ω)) and (Ω) ∶=  2 (, (Ω)).
Standard norms of matrix-or vector-valued function are to be understood in an averaged componentwise sense, for example, Assuming for the initial value  in ∈  2 (Ω) and  ∈  2 (, [ −1∕2 (Γ  )]  ) for the boundary force, we can state the equations ( 21) and (22) in their weak form: and for any (, ) ∈ (Ω) × (Ω) and almost every  ∈ .
Before beginning with the existence analysis of this system, we show that the time derivative of a function   satisfying equation ( 27) is really an element of  2 (,  ′ (Ω)).
Proof.For almost every  ∈  we have First, we take a closer look at the trace terms for which we use the identity tr   =   ∶ .For the norm of the linearised stress tensor, we get where we used the inequalities of Minkowski and Hölder as well as the boundedness of  and its first derivative.With (30), we obtain Analogous to this, we get For the next term, applying the same inequalities as above, we get and the last term of the right-hand side of (29) we estimate in an analogous way.The remaining first two terms in (29) are estimated using Hölder's inequality.Altogether we obtain and ‖‖ (Ω) = 1, the right-hand side is bounded for almost every  ∈ .□

A-priori estimate
We begin the existence analysis with showing an a-priori estimate necessary for the existence proof.In order to enable the limit passage in the sense of two-scale convergence in Section 4, the constants are carefully tracked in an -independent way.

Proposition 2 (Boundedness).
There exists a constant  > 0, independent of , such that for almost every  ∈ .
Proof.Starting with equation (28), we show first that . Therefore, we use   as test function in (28) and after rearranging terms, we get We estimate the left-hand side of (35) by using the positive definiteness of  as well as Korn's and Poincaré's inequalities, which gives Next we consider the terms on the right-hand side of (35), which can be estimated by Hölder's and Young's inequalities and using the boundedness of .We get and for a  > 0. For the boundary term, we obtain where   > 0 is the constant from the trace inequality and further, we used Young's inequality.Combining now (36) -(39), we absorb the terms with   and (  ) and get Integration with respect to time from 0 to , with  ∈ (0, ] and  small enough, gives for some constants , We estimate the terms on the right-hand side of (42) successively, using Hölder's and Young's inequalities.The first one gives Similar to (31), we treat the terms including the traces of the stress tensors.With (30) and Young's inequality, we get For the last two terms from the right-hand side of (42) we obtain and  2 1 2 Absorbing the  4 ‖Δ  ‖ Ω, -terms, we get for some constants ,  1 > 0. For  > 0 small enough, the left-hand side of ( 43) is positive and with (41) we get ) for some constants , C > 0, which do not depend on .Now, applying Gronwall's inequality we obtain for a constant  > 0 independent of .Due to the regularity assumptions on  and the initial data  in , the right-hand side of (44) is bounded for a.e. ∈ .Since   () and  2 Δ  () are bounded in  2 (Ω) for a.e. ∈ , the scaled gradient ∇  () is bounded in [ 2 (Ω)]  , for a.e. ∈ , since whereby the boundary integral vanishes because of the no-flux condition ∇  ⋅  = 0 on Ω.Then, with Integration with respect to time then gives the desired result.Estimates (41) and (44) now finally yield the boundedness of   in (Ω), Altogether we finally obtain for a constant  > 0, which does not depend on .□

Existence of weak solutions
In this subsection, we show the existence and uniqueness of a weak solution of the considered linear system.The proof is provided by a Galerkin approximation.Since the finite-dimensional system that is created in the course of this represents a linear differential-algebraic equation (DAE), we first introduce some aspects of general theory about solvability of linear DAEs in a weak setting.
In [13], the author studies coupled systems of partial differential and differential-algebraic equations in Hilbert spaces, so-called abstract differential-algebraic systems of the type (48) but with matrices which are continuous in time.Among other things, the unique solvability is proven by use of a Galerkin method.In what follows, we summarise some results of the theory of linear differential-algebraic equations concerning existence and uniqueness of solutions of linear DAEs according to [13] and extend them to matrices with  ∞ time regularity as in (48).The concept is based on decoupling the DAE into a dynamic part, which represents an ordinary differential equation, and an algebraic part.The first definition tells us when the matrices () and () are well matched in a certain way.This is important when decoupling a system as stated above into a dynamic and an algebraic part.
for a.e. ∈ ( 0 , ) and its derivative with respect to time,  ′ , exists and is bounded almost everywhere.Next, we present an index concept for the considered linear DAE.Laxly spoken, the index of a DAE indicates how much it differs from an ordinary differential equation.Following [13,14], we introduce a projector-based index, which is compatible with working in a weak setting.As the Cahn-Larché system fits this index concept with index  = 1, we only consider this case and, moreover, we adjust it from [13] to matrices with  ∞ time regularity.As we will see later, the decoupling of the linear DAE, which is based on this index concept, is based on the decomposition of ℝ  realised by projectors.For the sake of notational simplicity, from now on we drop the time argument  from the matrices.
Let  − denote the reflexive generalised inverse of , that is, To determine  − uniquely we set where  is the projector from Definition 1 and  0 =  −  0 .Now we can state the existence result we want to work with, which was proved by [13] for the case of continuous matrices ,  and .We adapt the existence result to our case closely following the ideas of [13].For the decoupling of the DAE into its dynamic and its algebraic part, we refer to [15].

Existence of weak solutions of the linear Cahn-Larché system
To prove the existence of a weak solution of the scaled linear Cahn-Larché system, we consider the system in a form, where the influence of the unknowns   and   is separated in both equations.Therefore, we write the equations ( 27), (28) together with the initial condition as which holds for all  ∈ (Ω),  ∈ (Ω) and with bilinear forms  ch (⋅ , ⋅),  ch (⋅ , ⋅) describing the influence of   and   on the extended Cahn-Hilliard equation, respectively, and  m (⋅ , ⋅),  m (⋅ , ⋅) describing the influence of   and   on the mechanical equilibrium equation, respectively.
In what follows, we proof this result in four steps using a Galerkin approach and the theory of linear differentialalgebraic equations in a weak setting introduced in the previous subsection.
Step 1: Galerkin equations We consider a Galerkin scheme, that is, finite dimensional subspaces   = span { According to the previously presented theory about linear DAEs, there exists a unique solution if the differential-algebraic equation has a properly stated leading term and if it has index 1.To show this, we first identify the setting and write the Galerkin equations in the form of an initial-value differential-algebraic system: with  ∈ ℝ + ,  ∈ ℝ + and  ∈  ∞ (, ℝ (+)×) ),  ∈  ∞ (, ℝ ×(+) ),  ∈  ∞ (, ℝ (+)×(+) ).We identify Next, we check if the conditions of Theorem 1 are satisfied.Equation ( 79) has a properly stated leading term.It is ker  = {0}, since the stiffness matrix ((  ,   ) Ω ) 1≤,≤ is regular and im  = ℝ  .Hence, ker  ⊕ im  = ℝ  .Furthermore, we can simply choose  =   as constant projector onto im  along ker .Notice that  0 =  is singular.Let  0 be the projection onto the kernel of  0 = .If the matrix  1 =  +  0 is regular the equation ( 79) has index  = 1 and hence, there exists a unique solution.We have ) .
Due to the property of the basis functions   , 1 ≤  ≤ , the matrix ((  ,   ) Ω ) 1≤,≤ is regular.Hence, it is sufficient to show that the matrix ( m  ) 1≤,≤ , which corresponds to the mechanical equation, is regular.This is equivalent to the wellknown fact that there exists a unique solution of the Galerkin scheme for the equation of linear elasticity with the applied boundary conditions.Therefore, the differential-algebraic system (79), (80) has index  = 1 and consequently, according to Theorem 1, there exist a unique solution of the Galerkin equations ( 76), (77).Thus, there exists a unique solution (  ,   ) of the equivalent equations ( 73), ( 74), (75), which fulfil (78).□ Step 2: Estimate for approximate solutions Proposition 4.There exists a constant  > 0, independent of  and , such that Proof.For fixed ,  ∈ ℕ we set  =   and  =   in (73) and (74).Then the result follows directly from the estimate in § 3.1.□
Proof.With the a priori estimates established in Proposition 4, the convergence of the sequence of approximate solutions to the solution of the original system is standard for this linear problem and, thus, further details are omitted.□ Step 4: Uniqueness of the solution Proposition 6 (Uniqueness).There exists at most one solution (  ,   ) of system ( 69), ( 70), (71).

HOMOGENISATION
For the homogenisation process, we use the method of two-scale convergence.We first recall some well-known results and show two extensions necessary for the homogenisation of the linearised Cahn-Larché system.These are then used to upscale the system rigorously.

Two-scale convergence
In what follows, we briefly summarise the required essentials of two-scale convergence, going back to Nguetseng [16] and Allaire [17].Except for the two Propositions proven at the end of this subsection, the results can be found in [17], [18] and [19], which we refer to for more details.Unless stated otherwise, Ω ⊂ ℝ  is a bounded and open set and  ∈ (1, ∞) and we set  = [0, 1)  as the reference cell for simplicity.Furthermore, whenever we extract a subsequence, for brevity, we always denote it by the same symbol as the sequence itself.We start with the definition of two-scale convergence in   (Ω).
Definition 3 (Two-scale convergence).A function  = (, ) in   (Ω × ), which is -periodic in  and which satisfies is called an admissible test function.
⇀  0 .A sequence of functions   in   (Ω) is said to two-scale converge strongly to a limit  0 ∈   (Ω × ) if   two-scale converges to  0 in   (Ω) and Then, we write   2s.
⟶  0 .Moreover, if the -periodic extension of  0 belongs to   (Ω,  # ()), where the subscript # denotes -periodicity, we have We remark that all admissible test functions two-scale converge strongly by definition.We are interested in criteria which enable us to conclude that a given sequence in   (Ω) is two-scale convergent.The next compactness theorem ensures the existence of a two-scale limit of a sequence bounded in   (Ω) or  1, (Ω).
For dealing with the homogenisation of the linearised Cahn-Larché system, two additional results are required, which we prove in what follows.
Proposition 7 (Compactness of 2nd order derivatives).Let   ,       and  2  2       all be bounded sequences in   (Ω).Then, there exists a function  0 ∈   (Ω;  ⇀  2      0 , for ,  = 1, … , . Proof.From [17] we already know that for sequences   and       bounded in   (Ω), there exists a function  0 ∈   (Ω;  1, # ()), such that, up to a subsequence,   and       two-scale-converge to  0 and     0 , respectively.Since  2  2       is also bounded in   (Ω), we can extract a subsequence, still denoted by  2  2       , and there exists a function  ∈   (Ω × ) such that this subsequence two-scale converges to , that is, The next theorem enables to pass to the limit of products of several two-scale convergent sequences.This result is an extension of the well-known result that the product of one strongly two-scale convergent with one weakly two-scale convergent sequence converges towards the product of their two-scale limits in the sense of distributions, see for example, [17,19].for every  ∈  ∞ (Ω).
Proof.We show the proof of this result for the product of two strongly and one weakly two-scale converging sequences, that is,  = 2.The proof of the general case, for  > 2, can be continued successively.
For  = 1, 2, let  ) (2) It is easy to see that this decomposition can be continued successively if further strongly converging consequences are added to the product on left-hand side of (87).We multiply by , integrate over Ω and subtract the right-hand side of (86) from both sides and obtain with the triangle inequality () () − (,

𝑥 𝜖 )
] ()   () () d (,   ) () − We consider the last term in (88) and pass to the limit, first for  → 0 and after for  → ∞.We get  ] (2) − and Proof.The proof consists of several steps.First, we pass to the limit in the weak form of the linear Cahn-Larché system.Afterwards, we proof the uniqueness of the solutions of the resulting weak homogenised system and, in a third step, we derive the strong formulation of the homogenised system.□

Homogenisation process
We start by identifying the precise form of the two-scale limits of the sequences of the unknowns.We have already proven that   and ]  for the mechanical equation (28).Then, Proposition 8 enables us to pass to the limit.The proof shows that (86) also applies when choosing  = ( ⋅ , ⋅∕) from  ∞ (Ω,  ∞ # ()).Considering the cubic interpolation of the elasticity tensor (4), several terms of products of sequences appear.The most critical terms to deal with include products of one weakly two-scale convergent sequence with three strongly two-scale convergent sequences and the required convergences of the sequences  , and  , are sufficient to pass to the limit.Hence, for  → 0, we get where we denote the limit of the stress tensor by The limit passage of the remaining terms of the equation is analogous.Passing to the limit in the term with the time derivative first requires integration by parts with respect to time since     is only bounded in  2 (, ((Ω)) ′ ).Re-integration then results in the limit of the time derivative corresponding to the time derivative of the limit function  0 .In summary, we can now read off a variational formulation for the two-scale limit functions ( 0 ,  0 ,  1 ) ∈  2 (Ω × , which holds for all (,  0 ,  1 ) ∈  ∞ (Ω;  ∞ # ()) × [ ∞ (Ω)]  × [ ∞ (Ω;  ∞ # ())]  .By density, the above equations still hold for all (,  0 ,  1 ) ∈  2 (Ω,  2 # ()) × (Ω) × [ 2 (Ω,  1 # ()∕ℝ)]  and since the limits  ,0 ,  ,0 and  ,1 are essentially bounded with respect to space and time, the integrals are well-defined.

Strong form of the homogenised system
To finish the proof of Theorem 4, we derive the strong form of the homogenised system above.This is accomplished by choosing special test functions and integration by parts.First, choosing  0 ≡ 0 in (108) and integrating by parts with respect to  yields equation (92).Then, choosing  1 ≡ 0 in (108) and integrating by parts, we obtain the macroscopic equation (93).At this step, we applied the boundary conditions (95) -(98).Finally, twofold integration by parts of equation ( 107) leads to equation (91) of the homogenised system and our proof is done.
As it is usual, the unknown  1 can be eliminated from equations (91), ( 92), (93) and therefore the homogenised twoscale system can be decoupled into a macroscopic and a microscopic equation by expressing  1 in terms of  0 .For this purpose we introduce the cell problems: For each ,  = 1, … , , a vector-valued function   is required, which solves where   = (   ) 1≤≤ ∈ ℝ  is defined by where   = (  ℎ ) 1≤,,,ℎ≤ is a fourth-order tensor with components defined by   ℎ =   ( ℎ ). (114) Note that the inverse tensor  −1 of  exists due to the positive definiteness of  and is uniquely defined through  −1 =  −1  = .Here  is the symmetric fourth-order identity tensor with components

0
in    (Ω × ) and  ∈  ∞ (Ω).We split the product of the sequences as follows