Homogenization of Perforated Elastic Structures

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Introduction
This paper deals with the linearized elasticity problem posed in different periodic domains. These domains are obtained by reproducing a representative cell of size ε in such a way that one can get beam-like, plate-like or N -dimensional structures. It is assumed that a part of their exterior boundary denoted by Γ ε is fixed.
The ε-cells are made of elastic materials. The reference cell is denoted by C (Fig. 1). We assume that C has a Lipschitz boundary and that the interior of the closure of the union of two contiguous cells is connected. Under these assumptions, the whole periodic structure might have a non-Lipschitz boundary. Throughout this article, the cell C is included in the unit parallelotope of R N (resp. R 3 ), and one can replace this parallelotope by any bounded domain having the paving property with respect to a discrete group of rank N (resp. 3).
Our aim is to investigate the asymptotic behavior of these elastic periodic structures as ε tends to 0. Since these structures might be non-Lipschitz, one of the main difficulties is to obtain a priori estimates. The classical extension approach (see [25]) and Korn's inequalities for Lipschitz domains (see [9,10]) cannot be used. Thus, in order to derive a priori estimates we use interpolations as suggested in [14,Sect. 5.5]. This makes it possible to prove Korn's inequalities with constants independent of ε. Note that in case of beam-like and plate-like domains the derivation of Korn's inequalities is also based on the decomposition of beam or plate displacements. These decompositions have been introduced in [2,18].
To derive the limit problems, we use the periodic unfolding method introduced in [11]. This method has been applied to a vast number of problems such as problems in perforated domains [5,6,13,16], transmission problems [17], contact problems [20,22], problems including a thin layer [21], problems in domains with "rough boundary" [1,3,4], to name but a few. In our work, in contrast to earlier works for plate-like or beam-like structures [14,19,21,22,24], we simultaneously proceed to the homogenization and reduction of dimension. The periodic unfolding method used in this paper includes the following steps: -introducing and applying appropriate unfolding operators, depending on the problem, -obtaining a priori estimates for the displacements, then uniform estimates for the unfolded displacements, which, in turn, are used to pass to a weak limit in appropriate spaces over a fixed domain, -establishing an unfolded limit problem from which a homogenized problem is derived.
As a general reference for the homogenization of elasticity problems in 3D periodically perforated domains with Lipschitz boundary we refer to [25]. In case of a plate-like domain we mention [14,Chapter V] where the interaction of homogenization and domain reduction, involving two small parameters such as plate thickness δ and periodicity ε, in its large dimensions was investigated. For similar results in case of a beam-like domain we refer to [19]. The novelty of this paper is the extension of the results to non-Lipschitz perforated domains.
The paper is organized as follows. Sections 2, 3 and 4 deal with periodically perforated 3D, plate-like and beam-like domains, respectively. We begin every of these sections by introducing the notation and describing the specific type of a periodic domain. Then, for every type of a periodic domain, we introduce the unfolding operator, we derive weak limits of the fields, we specify the limit problem for characterize the limit fields. Moreover, at the end of every section, there is a conclusion in which we provide an approximation of the solution to the elasticity problem.
The proofs of Korn's inequalities for different types of domains, namely N -dimensional, plate-like and beam-like, are given in Appendix A. The proofs of all lemmas and propositions are given in Appendix B.

N -Dimensional Periodic Domain
This section deals with the asymptotic behavior of the solution to the linearized elasticity problem for ε-periodically perforated N -dimensional structures as ε → 0. At first, we explain the notation, introduce the structure and state the elasticity problem. Then, we introduce the unfolding operator and its properties. And finally, we derive the unfolded limit problem and the homogenized problem.

Notation and Geometric Setting
Let Ω ⊂ R N , N ∈ N \ {0, 1}, be a bounded domain with a Lipschitz boundary and Γ be a subset of ∂Ω with non-zero measure. We assume that there exists an open set Ω with a Lipschitz boundary such that Ω ⊂ Ω and Ω ∩ ∂Ω = Γ .
We will use the following notations through this section: -Y . = (−1/2, 1/2) N is the unit cube, -C ⊂ Y is a domain with Lipschitz boundary such that the interior C ∪ (C + e i ) , i = 1, N, is connected, s is the space of N × N symmetric matrices, -for a.e. x ∈ R N one has Note that N i=1 Ξ ε,i ⊂ Ξ ε and that the domains Ω * ε , Ω * ε are connected. We are interested in the elastic behavior of a structure occupying the domain Ω * ε which is fixed on a part of its boundary Γ ε = Γ ∩ Ω * ε . The space of all admissible displacements is V ε = u ∈ H 1 (Ω * ε ) N | ∃u ∈ H 1 (Ω * ε ) N such that u |Ω * ε = u and u = 0 in Ω * ε \ Ω * ε .
This means that the displacements belonging to V ε "vanish" on the part Γ ε of ∂Ω * ε . Fig. 2 Domains Ω, Ω , Ω * ε , Ω * ε and sets Ξ ε , Ξ ε Remark 1 Note that the domain Ω * ε might be non-Lipschitz (see Fig. 2). In this case, one cannot extend the displacements to the holes of this domain as it is proposed in [25].

Statement of the Elasticity Problem
For a displacement u ∈ H 1 (Ω * ε ) N , we denote by e the linearized strain tensor (or symmetric gradient) e(u) .
Let a ij kl ∈ L ∞ (C), i, j, k, l = 1, N be the components of the elasticity tensor. These functions satisfy the usual symmetry and positivity conditions: a ij kl (X) = a jikl (X) = a klij (X) for a.e. X ∈ C; -for any τ ∈ M N s , there exists c 0 > 0 such that a ij kl (X)τ ij τ kl ≥ c 0 τ ij τ ij for a.e. X ∈ C. (2. 2) The constitutive law for the material occupying the domain Ω * ε is given by the relation between the linearized strain tensor and the stress tensor where the coefficients a ε ij kl are given by Let f be in L 2 (Ω 1 ) N , one defines the applied forces f ε by The unknown displacement u ε : Ω * ε → R N is the solution to the linearized elasticity system in the strong formulation where ν ε is the outward normal vector to ∂Ω * ε .
The variational formulation of (2.5) is given by

The Unfolding Operator
As mentioned above, for the derivation of the limit problem we use the periodic unfolding method. This method requires the introduction of an unfolding operator depending on the geometry of the problem. One of the main properties of this operator is that it replaces the integrals over the periodically oscillating domain Ω * ε by integrals over the "almost fixed" domain Ω ext ε × C which includes the whole domain Ω and the periodicity cell C. Moreover, it allows us to decompose any function into a main part without micro-oscillations and a remainder which takes the micro-oscillations into account. Below, in a similar way as for domains with holes (see [14]), we introduce a specific unfolding operator and give its properties.

Definition 1 For every measurable function
Below are some properties of T * ε , which are similar to those of the unfolding operators introduced in [14]. That is due to the fact that For more properties see [14].

Weak Limits of the Fields and the Limit Problem
Set Denote by H 1 N,per (C) the subspace of the periodic functions belonging to H 1 by H 1 N,per,0 (C) the subspace of the functions in H 1 N,per (C) with zero mean The proof of the following lemma is given in Appendix B.1.

Lemma 1
The solution u ε of problem (2.5) satisfies The proof of the proposition below is also postponed to Appendix B.1. Proposition 2 (The unfolded limit problem) Let u ε be the solution of problem (2.5). There exist u ∈ H 1 Γ (Ω) N and u ∈ L 2 (Ω; H 1 N,per,0 (C)) N such that (2.11) and the pair (u, u) is the unique solution to the following unfolded problem:

Homogenization
In this section, we give the expressions of the microscopic field u in terms of the macroscopic displacement u. First, taking v = 0 as a test function in (2.12), we obtain This shows that the displacement u can be written in terms of the elements of the tensor e(u). Denote by M np the N × N symmetric matrix with following coefficients where δ ij is the Kronecker symbol.
Since the tensor e(u) has N 2 components, we introduce the N 2 correctors which are solutions to the following cell problems Observe that χ np = χ pn n, p = 1, N. As a consequence, the function u can be written in the form e np (u)(x) χ np (X) for a.e. (x, X) ∈ Ω × C.
(2.14) Theorem 1 (The homogenized limit problem) The limit displacement u ∈ H 1 Γ (Ω) N is the unique solution of the following homogenized problem: Thus, the operator in problem (2.15) is elliptic and by virtue of the Lax-Milgram theorem this problem admits a unique solution.

Conclusion
We summarize the result of this section: for ε-periodic porous materials with a known structure, for e.g. structures made of beams whose thicknesses are of order ε, or dense packages of small compressed balls, the solution to the linearized elasticity problem (2.5)-(2.6) in a heterogeneous 3D domain is approximated by where u is the solution of the homogenized problem (2.16) and where the correctors χ np are given by (2.13). In (2.17), the sum represents the warpings of the cells.

Periodic Plate
This section is devoted to the study of the asymptotic behavior of the solution to the linearized elasticity problem for a ε-periodic plate-like structure as ε → 0. Note that this structure is 3-dimensional and only periodic in two directions. In the third direction it is "thin", that is, its thickness is of the same order ε as the period of the other two dimensions. The section is organized in a similar way as the previous one. It can be considered as an extension of the results obtained for the homogenization of a periodic plate (see [8], [14,Chap. 11], [22], [23], [26,Sect. 3.2] and also [24] for a shell).

Notation and Geometric Setting
We consider a bounded domain ω in R 2 with Lipschitz boundary. As in Sect. 2, we introduce γ , a subset of ∂ω with a non-zero measure. We assume that there exists a bounded domain ω with Lipschitz boundary such that ω ⊂ ω and ω ∩ ∂ω = γ.
In this section we use the following notations: Note that the domain Ω * ε is a connected open set, and if ε is small enough, we have The space of all admissible displacements is denoted by V ε :

Statement of the Elasticity Problem
We are interested in the elastic behavior of a structure occupying the domain Ω * ε and fixed on the part Γ ε of its boundary, Γ ε . 3 . We define the applied forces f ε as follows Again, the unknown displacement u ε : Ω * ε → R 3 is the solution to the linearized elasticity system in the strong formulation, where ν ε is the outward normal vector to ∂Ω * ε . The variational formulation of problem (3.2) is given by

The Unfolding-Rescaling Operator
Below, we introduce the unfolding operator for a plate-like structure and state its properties. Note that, since this structure is periodic only in two directions and it is "thin" in the third one, the unfolding operator is a "rescaling" operator in the third direction. As a consequence, the asymptotic reduction from 3D plate-like structure to 2D takes place. The reduction of dimension is done by the standard scaling to a fixed thickness (see the pioneer papers [7,15]).

Definition 2
For every measurable function u : Ω * ε → R 3 the unfolding operator T * ε is defined as follows: Below we recall some properties of T * ε (for further results see [14]).

Weak Limits of the Fields and the Limit Problem
Denote by H 1 γ (ω) the space of functions in H 1 (ω) that vanish on γ , and by H 2 γ (ω) the space of functions in H 2 (ω) that vanish on γ and their first derivatives vanish on γ as well Since we are dealing with a plate-domain, we use the decomposition of the displacements of a plate (see [18] and Sect. A.2 of Appendix A). Any displacement u ∈ V ε can be decomposed as where U stands for the displacement of the mid-surface of the plate restricted to Ω * ε ∩ {x 3 = 0}, R(x ) ∧ x 3 e 3 represents the small rotation of a "fiber" from x ∈ Ω * ε ∩ {x 3 = 0} and u is the warping of the "fibers". Here In the next step, we compute the strain tensor of the displacement u, using the decomposition (3.6) (3.7) Further, we extend U , R by 0 to ω \ ω int 3ε and the field u by 0 to Ω ε \ Ω int ε . The following lemma is proved in Appendix B.2.
The proof of the following lemma is postponed to Appendix B.2.

10)
and Since the fields U α , R, U 3 and the gradient U 3 vanish in ω \ ω, we obtain Lemma below is proven in Appendix B.2.

Lemma 4 For a subsequence, still denoted {ε}, we have
The following proposition provides the first main result of this section. Its proof is given in Appendix B.2. Proposition 4 (The unfolded limit problem) Let u ε be the solution to (3.2). Then the following convergences hold: 3 are the solution to the following unfolded problem:

Homogenization
In this section, we give the expressions of the microscopic displacement u in terms of the membrane displacements U m and the bending U 3 .
This shows that the microscopic displacement u can be written in terms of the tensors E M , E B . Define we introduce 6 correctors which are the unique solutions to the following cell problems 3 . As a consequence, the function u from (3.16) is given in terms of U as follows (3.19) This substitution allows us to separate the scales and formulate the second main result: Theorem 2 (The homogenized limit problem) The limit field is the unique solution to the homogenized problem Taking into account the variational problems (3.18) satisfied by the correctors, the problem (3.20) with the homogenized coefficients given by (3.21) is obtained by a simple computation. Now, we prove the ellipticity of the operator in Problem (3.20). Using the formulas (3.21) for the homogenized coefficients, we obtain Then, in view of (2.2) and following the proof of [14,Lemma 11.19], we obtain Thus, the operator of problem (3.20) is elliptic and this problem has a unique solution.

Conclusion
We summarize the results of this section. The solution to the linearized elasticity problem (3.2) (in the strong form), or (3.3) (in the weak/variational form) is approximated by where U is the solution of the homogenized 2D-problem with constant effective coefficients (3.21) and χ M αβ , χ B αβ ∈ H 1 2,per (C) 3 with α, β = 1, 2 are 6 3D displacement correctors, the solutions of auxiliary problems (3.18) on the periodicity cell (see Fig. 3).
As usual for a plate, we first recognize a Kirchhoff-Love displacement plus here a second term which represents the warpings of the cells.

Periodic Beam
In this section, we study the asymptotic behavior of the solution to the linearized elasticity problem for ε-periodic beam-like structure as ε → 0. This structure is 3-dimensional and periodic in one dimension. In two other directions the structure is "thin", that is, its size in each of these directions, is of order ε. The section is organized in a similar way as the previous ones. It can be considered as an extension of the results of [19] to beam-like structures with a boundary that does not have to be a Lipschitz boundary.

Notation and Geometric Setting
Let C ∈ R 3 be a bounded domain with Lipschitz boundary and let L be a fixed positive constant. In this section, we also assume that the interior of C ∪ (C + e 3 ) is connected and C ∩ (C + e 3 ) = ∅. The beam-like structure is introduced in the following way: We choose as centerline of the structure the segment whose direction is e 3 and place the origin at the center of mass of the first cell (thus the centers of mass of the other cells are also on this segment). The orthonormal basis (e 1 , e 2 , e 3 ) is chosen in such a way that C x 1 x 2 dx = 0, and we set Concerning the directions e 1 and e 2 , it is important to note that they do not necessary correspond to the principal axes of inertia.
The space of all admissible displacements is denoted by V ε

Statement of the Elasticity Problem
As before, we are interested in the elastic behavior of a structure occupying the domain Ω * ε and fixed on the part Γ ε of its boundary. Let f and g be in L 2 (0, L) 3 , we define the applied forces f ε ∈ L 2 (Ω * ε ) 3 by for a.e. x ∈ Ω * ε . (4.1) The unknown displacement u ε : Ω * ε → R 3 is the solution to the linearized elasticity system where ν ε is the outward normal vector to ∂Ω * ε . The variational formulation of problem (4.2) is given by

The Unfolding-Rescaling Operator
Below, we introduce the unfolding operator for a beam-like structure and provide its properties. Note that since this structure is only periodic in one direction and is "thin" in the other two directions, the unfolding operator is a "rescaling" operator in two direction. As a consequence, the asymptotic reduction from 3D beam-like structure to 1D takes place.

Weak Limits of the Fields and the Limit Problem
Denote As in [18], we decompose the displacement field u ∈ V ε in the following way: where U , R ∈ H 1 Γ (0, L) 3 . The displacement u belongs to V ε .
The field U stands for the displacement of the centerline of the structure. The term R(x 3 ) ∧ (x 1 e 1 + x 2 e 2 ) represents the small rotation of the cross-section at the point x 3 of the centerline, whereas the last term u(·, x 3 ) is the warping of the cross-section at the point x 3 of the centerline.
The strain tensor of the displacement u is In order to simplify the expression of the strain tensor e(U e ), we define a new triplet (u, U, Θ) (see also [19]) by for a.e. x 3 ∈ (0, L).
From now on, we have a new decomposition of the field U e (x) (4.8) and the strain tensor of the displacement U e is Note that the boundary conditions for the terms of this new decomposition are The estimates for the functions from the decomposition of u ε can be obtained using Lemma 6 and Lemmas 15, 16 from Appendix A. They are given in the following lemma: and The proofs of the following two lemmas are given in Appendix B.3.

Homogenization
In this section, we derive the representation of the microscopic displacement u in terms of the macroscopic fields u, U and Θ.
Taking (v, V, T) = 0 as a test function in (4.27), we obtain This shows that the microscopic displacement u can be written in terms of the tensor E. Define The tensors E(u, U, Θ) have 6 components and we introduce 6 correctors χ u m , χ U α , χ Θ ∈ H 1 1,per,0 (C) 3 , α = 1, 2, m= 1, 2, 3, which are the unique solutions to the following cell problems Taking into account the variational problems (4.28) satisfied by the correctors, the problem (4.30) with the homogenized coefficients given by (4.31) is obtained by a simple computation.
Then, in view of (2.2) and following the proof of [14, Lemma 11.19], we obtain
The first term in the previous formula is a Bernoulli-Navier displacement completed by the displacements εu, and the term εu 3 stands for the stretching-compression of the structure. The remaining terms represents the warpings of the cells.
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A.1 Korn's Inequality on N -Dimensional Domains
See Sect. 2.1 for the principal notations. We also denote (see Fig. 5) where M Ω int δ (ϕ) is the mean value of the function ϕ in the domain Ω int δ .
Below, in every cell we compare a displacement to a rigid displacement. Then, in a second step, we compare the rigid displacements obtained in two neighboring cells. After that, we build a global displacement in order to obtain a Korn's type inequality.
Let C i = interior(C ∪ C + e i ) and Φ be a displacement in W 1,p (C i ) N , p ∈ (1, +∞) and i ∈ {1, . . . , N}. Applying Korn's inequality in C and C + e i yields two rigid displacements R i,0 , R i,1 , given by where the constant depends only on C.

Lemma 9
The following estimates hold: where the constant C depends only on C.
Proof Since the domain C i is connected and has a Lipschitz boundary, it satisfies Korn's inequality. Hence, there exists a rigid displacement R i , where the constant C depends on C i . Hence, by (A.3) and (A.5) Taking into account the inequality (A.6), we obtain Subtracting yields (A.4) 1 . Now we prove (A.4) 2 . First observe that (A.8) Besides, we have The previous estimate together with (A.8) and (A.7) gives Similarly, we obtain Hence (A.4) 2 holds. Thus Lemma 9 is proved.
such that (using (A.3) and after ε-scaling) (A.9) As above we obtain the following estimates for every ξ ∈ Ξ ε,i : where C ξ i = interior (C + ξ) ∪ (e i + ξ + C) . An immediate consequence of Lemma 9, we have Lemma 10 The following estimates hold: where the constant C depends only on C.
Let ξ be in Ξ ε . If all the vertices of the parallelotope ε(ξ + Y ) belong to Ξ ε , we extend the field a (or B, resp.) to this parallelotope as the Q 1 interpolate of its values at the vertices of the parallelotope.
We obtain a field, still denoted a (or B, resp.), defined at least in Ω int 2ε √ N . This field belongs to

Lemma 11
For every displacement u ∈ W 1,p (Ω * ε ) N the following estimates hold where the constants do not depend on ε.
Proof Since the boundary of Ω int 2ε √ N is uniformly Lipschitz, Korn's inequality and (A.11) 3 give a rigid displacement R such that Then (A.11) 2 and the previous estimate lead to .
The estimate (A.10) 1 in Lemma 10 together with the previous estimate and Lemma 8 yield Now, from Proposition 9 and (A.11) 3 , there exits a ∈ R such that The estimates (A.14), (A.10) 2 together with the previous estimate and Lemma 8 yield which yields an estimate, independent of ε, for the components of R . Thus the estimate (A.13) follows.

A.2 Korn's Inequality on a Plate-Like Domain
In this subsection, the proofs of the lemmas are similar to the proofs of those in the previous subsection.
In the same way as in Lemma 10 we get the following lemma:

Lemma 12
The following estimates hold: (A.17) The constant C depends only on C.
As in the previous subsection, using Q 1 interpolation we extend the fields U and R to the whole domain ω int 3ε and obtain two fields U ∈ W 1,∞ (ω int 3ε ) 3 and R ∈ W 1,∞ (ω int 3ε ) 2 satisfying Below, we use the plate decomposition from [18]. We define the displacement U e as

Lemma 13 For every displacement
The constant C depends only on C.
Proof The estimates (A.18) are the consequences of (A.17) and the fact that the fields U and R are piecewise linear in every cell.

Theorem 5
For every displacement u ∈ H 1 (Ω * ε ) 3 , there exists a rigid displacement R such that The constant C does not depend on ε.
Proof From Proposition 9, there exits Then, the previous estimate, (A.17) 1 and Proposition 8 yield Furthermore, (A.18) 2 and (A.19) lead to Proceeding as above, there exists a 3 ∈ R such that ξ ∈Ξε From (A.18) 2 we also obtain Since the boundary of ω int 3ε is uniformly Lipschitz, Korn's inequality for 2D gives a rigid displacement r(x 1 , These estimates and (A.18) 2 imply that Then, as above, we obtain ξ ∈Ξε By choosing R(x) = a + b ∧ x and using (A.16) we complete the proof of the theorem.
Let γ be a subset of ∂ω with a non-zero measure. Assume that there exists a domain ω with Lipschitz boundary such that ω ⊂ ω and ω ∩ ∂ω = γ. where Theorem 6 For every displacement u in V ε the following estimates hold The constant C does not depend on ε.
Proof Since u belongs to V ε , there exists u ∈ H 1 (Ω * ε ) 3 such that u = u |Ω * ε , u = 0 in Ω * ε \ Ω * ε . Then, applying Theorem 5 with u in place of u, and Ω in place of Ω, gives a rigid displacement R such that As a consequence of the two previous theorems, we obtain the following result Corollary 1 For every displacement u in V ε the following estimates hold: (A.23) The constants C do not depend on ε.

A.3 Korn's Inequality on a Beam-Like Domain
In this subsection, the notations are those of Sect. 4.1.

(A.24)
Remark 3 By construction, the fields U and R are piecewise constant.
In the same way as in Lemmas 10-12 we get

Lemma 14
The following estimates hold: (A.25) The constant C depends only on C. Define Now, using Q 1 interpolation, we extend the fields U and R to fields U , R belonging to W 1,∞ (0, L) 3 and such that Let us introduce the displacement U e as follows:

Lemma 16
For every displacement u ∈ V ε the following estimates hold: and The constant C in (A.28), (A.29) does not depend on ε.
Proof We extend u by 0 to the cell ε − e 3 + C). Then, proceeding as in Lemma 9 we obtain As a consequence of the previous lemma and (A.24), we have the following decomposition of a displacement u ∈ V ε : where the displacement U e is given by and where the displacement u ∈ V ε satisfies the estimates (see [18]) The constant C in (A.30) does not depend on ε.
In order to obtain the limit problem (2.12), we use the same approach as in [13,Theorem 4.3]. Let us introduce the following fields Then, applying T * ε to v ε , gives The solution to the variational problem (2.12) is unique. Indeed, if there are two solutions (u 1 , u 1 ) and (u 2 , u 2 ) to this problem, denote U = u 1 − u 2 and U = u 1 − u 2 . Taking into account the respective equalities from (2.12) and choosing the test functions U, U , we obtain Ω×C a ij kl e kl (U ) + e X,kl ( U) e ij (U ) + e X,ij ( U dx dX = 0. So e( U) = −e(U ) and thus the field U is an affine function with respect to X. Since U is periodic with respect to X and belongs to L 2 (Ω; H 1 N,per,0 (C)) N , it is equal to 0 (because its mean value on the cell is equal to 0). Hence, e(U ) = 0 and due to the boundary conditions we obtain U = 0. Finally, the whole sequences in (B.1) converge to respective limits. Now, we prove the strong convergences (2.11) 2,3 . By Proposition 1, (2.6) and (2.12) we have And finally, from the previous two estimates Using this estimate, we obtain (3.8).

Proof of Lemma 3
In order to prove (i)-(ii), we note that from the estimates (3.8) and (A.22) 1 in Corollary 1 and Lemma 8, it follows that there exist functions U ∈ H 1 (ω ) 3 and R ∈ H 1 (ω ) 2 such that the following convergences hold Now we prove that the fields U α , R, U 3 and ∇U 3 vanish in ω \ ω.
Let O be an open subset such that O is strictly included in ω \ ω. Since u ε vanishes in Ω * ε \ Ω * ε , then the fields U ε , R ε vanish in ω ε \ ω int 3ε . If ε is small enough then O ⊂ ω ε \ ω int 3ε . Thus, by construction, the fields U ε,α , R ε , U ε, 3 and ∇U ε,3 vanish in O. As a consequence, their weak limits also vanish in O. Since this holds for every open set O strictly included in ω \ ω, this is also satisfied in the full set ω \ ω. Estimate (A.18) 2 in Lemma 13 leads to → 0 strongly in L 2 (ω ).
Unfolding the left-hand side of (3.3) and taking into account that by virtue of (3.8), (B.6) and Cauchy-Schwarz inequality Unfold the right hand side of (3.3) 1