The Periodic Unfolding Method in Homogenization

. The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs in [ Pro-ceedings of the Narvik Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakk¯otosho, Tokyo, 2006, pp. 119–136]). In the present paper we go into all the details of the method and include complete proofs, as well as several new extensions and developments. This approach is based on two distinct ideas, each leading to a new ingredient. The ﬁrst idea is the change of scale, which is embodied in the unfolding operator. At the expense of doubling the dimension, this allows one to use standard weak or strong convergence theorems in L p spaces instead of more complicated tools (such as two-scale convergence, which is shown to be merely the weak convergence of the unfolding; cf. Remark 2.15). The second idea is the separation of scales, which is implemented as a macro-micro decomposition of functions and is especially suited for the weakly convergent sequences of Sobolev spaces. In the framework of this method, the proofs of most periodic homogenization results are elementary. The unfolding is particularly well-suited for multiscale problems (a simple backward iteration argument suﬃces) and for precise corrector results without extra regularity on the data. A list of the papers where these ideas appeared, at least in some preliminary form, is given with a discussion of their content. We also give a list of papers published since the publication [Cio-ranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104], and where the unfolding method has been successfully applied.


Introduction.
The notion of two-scale convergence was introduced in 1989 by Nguetseng in [58], further developed by Allaire in [1] and by Lukkassen, Nguetseng, and Wall in [55] with applications to periodic homogenization. It was generalized to some multiscale problems by Ene and Saint Jean Paulin in [38], Allaire and Briane in [2], Lions et al. in [52] and Lukkassen, Nguetseng, and Wall in [55].
In [24], we expanded on this idea and presented a general and quite simple approach for classical or multiscale periodic homogenization, under the name of "unfolding method." Originally restricted to the case of domains consisting of a union of ε-cells, it was extended to general domains (see the survey of Damlamian [34]). In the present work, we give a complete presentation of this method, including all of the proofs, as well as several new extensions and developments. The relationship of the papers listed above with our work is discussed at the end of this introduction.
The periodic unfolding method is essentially based on two ingredients. The first one is the unfolding operator T ε (similar to the dilation operator), defined in section 2, where its properties are investigated. Let Ω be a bounded open set, and Y a reference cell in R n . By definition, the operator T ε associates to any function v in L p (Ω), a function T ε (v) in L p (Ω × Y ). An immediate (and interesting) property of T ε is that it enables one to transform any integral over Ω in an integral over Ω × Y . Indeed, by Proposition 2.6 below Ω×Y T ε (w)(x, y) dx dy ∀w ∈ L 1 (Ω).
Proposition 2.14 shows that the two-scale convergence in the L p (Ω)-sense of a sequence of functions {v ε } is equivalent to the weak convergence of the sequence of unfolded functions {T ε (v ε )} in L p (Ω × Y ). Thus, the two-scale convergence in Ω is reduced to a mere weak convergence in L p (Ω × Y ), which conceptually simplifies proofs.
In section 2 are also introduced a local average operator M ε and an averaging operator U ε , the latter being, in some sense, the inverse of the unfolding operator T ε .
The second ingredient of the periodic unfolding method consists of separating the characteristic scales by decomposing every function ϕ belonging to W 1,p (Ω) in two parts. In section 3 it is achieved by using the local average. In section 4, the original proof of this scale-splitting, inspired by the finite element method (FEM), is given. The confrontation of the two methods of sections 3 and 4 is interesting in itself (Theorem 3.5 and Proposition 4.8). In both approaches, ϕ is written as ϕ = ϕ ε 1 + εϕ ε 2 , where ϕ ε 1 is a macroscopic part designed not to capture the oscillations of order ε (if there are any), while the microscopic part ϕ ε 2 is designed to do so. The main result states that, from any bounded sequence {w ε } in W 1,p (Ω), weakly convergent to some w, one can always extract a subsequence (still denoted {w ε }) such that w ε = w ε 1 +εw ε 2 , with (1.2) (i) w ε 1 w weakly in W 1,p (Ω), w weakly in L p (Ω; W 1,p (Y )), (iii) T ε (w ε 2 ) w weakly in L p (Ω; W 1,p (Y )), where w belongs to L p (Ω; W 1,p per (Y )). In section 5 we apply the periodic unfolding method to a classical periodic homogenization problem. We point out that, in the framework of this method, the proof of the homogenization result is elementary. It relies essentially on formula (1.1), on the properties of T ε , and on convergences (1.2). It applies directly for both homogeneous Dirichlet or Neumann boundary conditions without hypothesis on the regularity of ∂Ω. For nonhomogenous boundary conditions (or for Robin-type condition), some regularity of ∂Ω is required for the problem to make sense, in which case the method applies also directly (see Remark 5.12).
Section 6 is devoted to a corrector result, which holds without any additional regularity on the data (contrary to all previous proofs; see [11], [30], and [59]). This result follows from the use of the averaging operator U ε . The idea of using averages to improve corrector results first appeared in Dal Maso and Defranceschi [33]. We also give some error estimates and a new corrector result for the case of domains with a smooth boundary (obtained by Griso in [42], [43], [44], and [45]). These results are explicitely connected to the unfolding method and improve on known classical ones (see [11] and [59]).
The periodic unfolding method is particularly well-suited for the case of multiscale problems. This is shown in section 7 by a simple backward iteration argument. This problem has a long history; one of the first papers on the subject is due to Bruggeman [19]. Its mathematical treatment by homogenization goes back to the book of Bensoussan, Lions, and Papanicolaou [11], where for this problem, the method of asymptotic expansions is used. For more recent references of multiscale homogenization and its applications, we refer to the books of Braides and Defranceschi [17], Milton [57], and the articles by Damlamian and Donato [35], Lukkassen and Milton [54], Lukkassen [53], Braides and Lukkassen [18], Babadjian and Baía [6], and Barchiesi [8].
The final section gives a list of papers where the method has been successfully applied since the publication of [24].
To conclude, let us turn back to the papers quoted at the beginning of this introduction and point out their relationships with our results. The dilation operation from Arbogast, Douglas, and Hornung [5] was defined in a domain which is an exact union of εY -cells. It consists in a change of variables, similar to that used in Definition 2.1 below. By this operation, any integral on Ω can be written as an integral over Ω × Y . Some elementary properties of the dilation operator in the space L 2 were also contained in Lemma 2 of [5].
The same dilation operator was used by Bourgeat, Luckhaus, and Mikelic in [16] under the name of "periodic modulation." Proposition 4.6 of [16] showed that if a sequence two-scale converges and its periodic modulation converges weakly, they have the same limit.
In the context of two-scale convergence, Allaire and Conca [3] defined a pair of extension and projection operators (suited to Bloch decompositions) which are adjoint of each other. They are similar to our operators T ε and U ε and the equivalent of property (2.12) and Proposition 2.18(ii) below, are proved in Lemma 4.2 of [3]. These properties were exploited by Allaire, Conca, and Vanninathan in [4] for a general bounded domain by extending all functions by zero on its complement.
In [48], Lenczner used the dilation operator (here called "two-scale transformation") in order to treat the homogenization of discrete electrical networks (by nature, the domain is a union of ε-cells). The convergence of the two-scale transform is called two-scale convergence (this would be confusing except that it was shown to be equivalent to the original two-scale convergence). As an aside, a convergence similar to (1.2)(iv) was also treated. In Lenczner and Mercier [49], Lenczner and Senouci-Bereksi [50], and Lenczner, Kader, and Perrier [51], this theory was applied to periodic electrical networks.
Finally, Casado Díaz and Luna-Laynez [21], Casado Díaz, Luna-Laynez, and Martin [22] and [23] used the dilation operator in the case of reticulated structures. In this framework, they obtained the equivalent of (3.7)(i) of Theorem 3.5 below. to Y , and set Then for each x ∈ R n , one has Figure 1).
We use the following notations: The set Ω ε is the largest union of ε(ξ + Y ) cells (ξ ∈ Z n ) included in Ω, while Λ ε is the subset of Ω containing the parts from ε ξ + Y cells intersecting the boundary ∂Ω (see Figure 2). Definition 2.1. For φ Lebesgue-measurable on Ω, the unfolding operator T ε is defined as follows: Observe that the function T ε (φ) is Lebesgue-measurable on Ω × Y and vanishes for x outside of the set Ω ε .
As in classical periodic homogenization, two different scales appear in the definition of T ε : the "macroscopic" scale x gives the position of a point in the domain Ω, while the "microscopic" scale y (= x/ε) gives the position of a point in the cell Y . The unfolding operator doubles the dimension of the space and puts all of the oscillations in the second variable, in this way separating the two scales (see Figures 3,4 and Figures 5,6).
The following property of T ε is a simple consequence of Definition 2.1 for v and w Lebesgue-measurable; it will be used extensively: Another simple consequence of Definition 2.1 is the following result concerning highly oscillating functions. Proposition 2.
2. For f measurable on Y , extended by Y -periodicity to the whole of R n , define the sequence {f ε } by Then For example, with f (y) = 1 for y ∈ 0, 1 / 2 , f ε is the highly oscillating periodic function, with period ε from Figure 5.
+∞[, and f ε be defined by (2.3). It is well-known that {f ε | Ω } converges weakly in L p (Ω) to the mean value of f on Y , and not strongly unless f is a constant (see Remark 2.11 below). The next two results, essential in the study of the properties of the unfolding operator, are also straightforward from Definition 2.1.
Proposition 2.5. For p ∈ [1, +∞[, the operator T ε is linear and continuous from L p (Ω) to L p (Ω × Y ). For every φ in L 1 (Ω) and w in L p (Ω), Proof. Recalling Definition 2.2 of Ω ε , one has Hence, each integral in the sum on the right-hand side successively equals By summing over Ξ ε , the right-hand side becomes Ω ε φ(x) dx, which gives the result. Property (iii) in Proposition 2.5 shows that any integral of a function on Ω is "almost equivalent" to the integral of its unfolded on Ω × Y ; the "integration defect" arises only from the cells intersecting the boundary ∂Ω and is controlled by its integral over Λ ε .
The next proposition, which we call unfolding criterion for integrals (u.c.i.), is a very useful tool when treating homogenization problems.
Based on this result, we introduce the following notation.
We now investigate the convergence properties related to the unfolding operator when ε → 0. For φ uniformly continuous on Ω, with modulus of continuity m φ , it is easy to see that So, as ε goes to zero, even though T ε (φ) is not continuous, it converges to φ uniformly on any open set strongly included in Ω. By density, and making use of Proposition 2.5, further convergence properties can be expressed using the mean value of a function defined on Ω × Y .
Definition 2.8. The mean value operator M Y : L p (Ω × Y ) → L p (Ω) for p ∈ [1, +∞], is defined as follows: Observe that an immediate consequence of this definition is the estimate Then (iii) For every relatively weakly compact sequence {w ε } in L p (Ω), the correspond- (v) Suppose p > 1, and let {w ε } be a bounded sequence in L p (Ω). Then, the following assertions are equivalent: Proof. (i) The result is obvious for any w ∈ D(Ω). If w ∈ L p (Ω), let φ ∈ D(Ω). Then, by using (iv) from Proposition 2.5, from which statement (i) follows by density.
In view of (i), one can pass to the limit in the right-hand side to obtain For p = 1, one uses the extra property satisfied by weakly convergent sequences in L 1 (Ω), in the form of the De La Vallée-Poussin criterion (which is equivalent to relative weak compactness): there exists a continuous convex function Φ : R + → R + such that lim t→+∞ Φ(t) t = +∞, and the set Unfolding the last integral shows that For the case where the measure of Ω is not finite, a similar argument shows that the equiintegrability at infinity of the sequence {w ε } carries over In view of (i), one can pass to the limit in the right-hand side to obtain This identity implies the required equivalence.
. Convergence (2.9) follows from Proposition 2.9(iii). 1 Remark 2.11. In general, in the case where Λ ε is not null set (for every ε), the strong (resp. weak) convergence of the sequence {T ε (w ε )} does not imply the corresponding convergence for the sequence {w ε }, since it gives no control of the sequence Then, for any ϕ in C c (Ω), one has Proof. The result follows from the fact that, in both cases, the sequence {u ε v ε φ} satisfies the u.c.i. by the Hölder inequality.
Remark 2.13. A consequence of (iii) of Proposition 2.9, together with (iv) of Proposition 2.5, is the following. Suppose the sequence {w ε } converges weakly to w in L p (Ω). Then the sequence {T ε (w ε )} is relatively weakly compact in L p (Ω × Y ), and each of its weak-limit points w satisfies M Y ( w) = w.
The next result reduces two-scale convergence of a sequence to a mere weak L p (Ω × Y )-convergence of the unfolded sequence.
Proposition 2.14. Let {w ε } be a bounded sequence in L p (Ω), with p ∈]1, +∞[. The following assertions are equivalent: Proof. To prove this equivalence, it is enough to check that, for every ϕ in a set of admissible test functions for two-scale convergence (for instance, Remark 2.15. Proposition 2.14 shows that the two-scale convergence of a sequence in L p (Ω), p ∈]1, +∞[, is equivalent to the weak−L p (Ω × Y ) convergence of the unfolded sequence. Notice that, by definition, to check the two-scale convergence, one has to use special test functions. To check a weak convergence in the space L p (Ω × Y ), one simply makes the use of functions in the dual space L p (Ω × Y ). Moreover, due to density properties, it is sufficient to check this convergence only on smooth functions from D(Ω × Y ).

2.2.
The averaging operator U ε . In this section, we consider the adjoint U ε of T ε , which we call averaging operator. In order to do so, let v be in L p (Ω × Y ), and let u be in L p (Ω). We have successively, This gives the formula for the averaging operator U ε .
As a consequence of the duality (Hölder's inequality) and of Proposition 2.5(iv), we get the following.
Proposition 2.17. Let p belong to [1, +∞]. The averaging operator is linear and continuous from L p (Ω × Y ) to L p (Ω) and , which is meaningless, in general, by a function which always makes sense. Notice that this implies, in particular, that the largest set of test functions for two-scale convergence is actually the set It is immediate from its definition that U ε is almost a left-inverse of T ε , since

In particular, for every
. Then, the following assertions are equivalent: . Then, the following assertions are equivalent: Proof. (i) This follows from Proposition 2.9(ii) by duality for p > 1. It still holds for p = 1 in the same way as the proof of Proposition 2.9(ii). Indeed, if the De La Vallée-Poussin criterion is satisfied by the sequence {Φ ε }, it is also satisfied by the sequence {U ε (Φ ε )}, since for F convex and continuous, Jensen's inequality implies that (ii) The proof follows the same lines as that of (i)-(ii) of Proposition 2.9.
As for the converse (b)⇒(a), Proposition 2.9(ii) implies that The implication (c)⇒(d) follows from (iii) and the second condition of (c).
Remark 2.19. The statement of Proposition 2.18(iii) does not hold with weak convergences instead of strong ones, contrary to an erroneous statement made in [24]. In view of (2.11) and Proposition 2.
But the converse of this last implication cannot hold.
If the converse were true, it would imply that T ε (w ε ) converges weakly to both w and Remark 2.20. Assertions (iii)(b) and (iv)(d) are corrector-type results. Remark 2.21. The condition (iii)(a) is used by some authors to define the notion of "strong two-scale convergence." From the above considerations, condition (c) of Proposition 2.18(iv) is a better candidate for this definition.

The local average operator M ε .
In this section, we consider the classical average operator associated to the partition of Ω by ε-cells Y (setting it to be zero on the cells intersecting the boundary ∂Ω).
Remark 2.23. It turns out that the local average M ε is connected to the unfolding operator T ε . Indeed, by the usual change of variable cell by cell, (ii) Suppose that p is in [1, +∞]. For φ ∈ L p (Ω) and ψ ∈ L p (Ω), In particular, for every φ ∈ L p (Ω), The same holds true for the weak- * topology in L ∞ (Ω).
Proof. The proofs of (i) and (ii) are straightforward. The proof of (iii) is a simple consequence of (ii) of Proposition 2.9. For the proof of (iv), let φ be in L p (Ω), with p ∈ [1, +∞[ (p = 1), and use (2.14) and (2.15) For p = 1, in the same way as the proof of Proposition 2.9(ii) and Proposition 2.18(i), if the De La Vallée-Poussin criterion is satisfied by the sequence {v ε }, it is also satisfied by the sequence {M ε (v ε )}, since for F convex and continuous, Jensen's inequality implies that which ends the proof.
Proof. Since w does not depend on y, one has U ε (w) = M ε (w) which, by Proposition 2.25(iii), converges strongly to w. The conclusion follows from Proposition 2.18(iii), respectively, (iv).

Unfolding and gradients.
This section is devoted to the properties of the restriction of the unfolding operator to the space W 1,p (Ω). Some results require no extra hypotheses, but many others are sensitive to the boundary conditions and the regularity of the boundary itself.
Then, Proposition 2.5(iv) implies that T ε maps W 1,p (Ω) into L p (Ω; W 1,p (Y )). For simplicity, we assume that Y =]0, 1[ n . Nevertheless, the results we prove here hold true in the case of a general Y , with minor modifications.
Moreover, the limit function w is 1-periodic, with respect to the y k coordinate.
for every s ∈ [0, 1]. The periodicity with respect to y n results from the following computation with an obvious change of variable: which goes to zero.
Then, there exist a subsequence (still denoted ε) and w ∈ L p (Ω; W 1,p (Y )) such that

Moreover, the limit function
Furthermore, if {w ε } converges strongly to w in L p (Ω), the above convergence is strong (this is the case if, for example, W 1,p (Ω) is compactly embedded in L p (Ω)). Proof. Using (3.1), since {w ε } weakly converges, one has the estimates so that there exist a subsequence (still denoted ε) and w in L p (Ω; W 1,p (Y )) such that and ∇ y w = 0. Consequently, w does not depend on y, and Proposition 2.9(iii) immediately gives w = M Y ( w) = w. Moreover, convergence (3.4) holds for the entire sequence ε. Finally, if the sequence {w ε } converges strongly to w in L p (Ω), so does the sequence {T ε (w ε )}, thanks to Proposition 2.9(ii).
Proposition 3.4. Suppose that p is in [1, +∞[. Let {w ε } be a sequence which converges strongly to some w in W 1,p (Ω). Then, Proof. The first asssertion follows from Proposition 2.9(i). To prove (ii), set which has mean value zero in Y . Since thanks to assertion (i), Then recall the Poincaré-Wirtinger inequality in Y : Applying it to the function Z ε − y c · ∇w (which is of mean value zero) gives and this concludes the proof. Theorem 3.5. Suppose that p is in ]1, +∞[. Let {w ε } be a sequence converging weakly to some w in W 1,p (Ω). Up to a subsequence, there exists some w in L p (Ω; W 1,p per (Y )) such that Proof. Following the same lines as in the previous proof, introduce To prove (ii), note that the sequence {∇ y Z ε } is bounded in L p (Ω × Y ). By (3.6), so that there exists w in L p (Ω; W 1,p (Y )) such that, up to a subsequence, Z ε − y c · ∇w w weakly in L p (Ω; W 1,p (Y )).

Since, by construction, M Y (y c ) vanishes, so does M Y ( w).
It remains to prove the Y -periodicity of w. This is obtained in the same way as in the proof of Proposition 3.1 by using a test function ψ ∈ D(Ω × Y ). One has successively, By Proposition 2.9(ii), {T ε (w ε )} converges strongly to w in L p (Ω × Y ), and by (3.7) (i), it converges weakly to the same w in L p (Ω; W 1,p (Y )). By the trace theorem in Hence, the last integral converges to Similarly, since (y c · ∇w)(y , 1) − (y c · ∇w)(y , 0) = ∂w ∂x n , we obtain Ω×Y (y c · ∇w)(y , 1) − (y c · ∇w)(y , 0)] ψ(x, y ) dx dy This, together with (3.8) and convergence (3.7)(ii), shows that Ω×Y w(x, (y , 1)) − w(x, (y , 0) ψ(x, y ) dx dy = 0, so that w is y n -periodic. The same holds in the directions of all of the other periods. Theorem 3.5 can be generalized to the case of W k,p (Ω)-spaces, with k ≥ 1 and p ∈]1, +∞[ . In order to do so, for r = (r 1 , . . . , r n ) ∈ N n with |r| = r 1 + · · · + r n ≤ k, introduce the notation D r and D r y : Then the following result holds.
Theorem 3.6. Let {w ε } be a sequence converging weakly in W k,p (Ω) to w, k ≥ 1, and p ∈]1, +∞[. There exist a subsequence (still denoted ε) and w in the space L p (Ω; W k,p per (Y )) such that Furthermore, if {w ε } converges strongly to w in W k−1,p (Ω), the above convergences are strong in L p (Ω; W k−l,p (Y )) for |l| ≤ k − 1.
Proof. We give a brief proof for k = 2. The same argument generalizes for k > 2. If |l| = 1, the first convergence in (3.9) follows directly from Corollary 3.3. Set The sequence {w ε } is bounded in W 2,p (Ω), hence proceeding as in the proof of Proposition 2.25(iii), one obtains Moreover, This implies that the sequence {W ε } is bounded in L p (Ω; W 2,p (Y )). Therefore, there exist a subsequence (still denoted ε) and w ∈ L p (Ω; W 2,p (Y )) such that (3.10) W ε w weakly in L p (Ω; W 2,p (Y )), Consequently, Now, apply Theorem 3.5 to each of the derivatives ∂w ε ∂x i , i ∈ {1, . . . , n}. There exist a subsequence (still denoted ε) and w i ∈ L p (Ω; Then (3.10) gives By construction, the function w belongs to L p (Ω; W 2,p (Y )). Furthermore, The last equality implies that w belongs to L p (Ω; W 2,p per (Y )). Finally from (3.12) one gets D l y w = D l w + D l y w, with |l| = 2, which together with (3.11), proves the last convergence of (3.9). Corollary 3.7. Let {w ε } be a sequence converging weakly in W 2,p (Ω) to w, and p ∈]1, +∞[. Then, there exist a subsequence (still denoted ε) and w in the space L p (Ω; W 2,p per (Y )) such that

Macro-micro decomposition:
The scale-splitting operators Q ε and R ε . In this section, we give a different proof of Theorem 3.5, which was the one given originally in [24]. It is based on a scale-separation decomposition which is useful in some specific situations, for example, in the statement of general corrector results (see section 6).
The procedure is based on a splitting of functions φ in W 1,p (Ω) (or in W 1,p 0 (Ω)) for p ∈ [1, +∞], in the form where Q ε (φ) is an approximation of φ having the same behavior as φ, while R ε (φ) is a remainder of order ε. Applied to the sequence {w ε } converging weakly to w in W 1,p (Ω), it shows that, while {∇w ε } , {∇(Q ε (w ε ))} and {T ε (∇Q ε (w ε ))} have the same weak limit ∇w in L p (Ω), respectively, in L p (Ω×Y ), the sequence T ε ∇(R ε (w ε )) converges weakly in L p (Ω × Y ) to ∇ y w for some w in L p (Ω; W 1,p per (Y )). We will distinguish between the case W 1,p 0 (Ω) and the case W 1,p (Ω). For the former, any function φ in W 1,p 0 (Ω) is extended by zero to the whole of R n , and this extension is denoted by φ. In the latter case, we suppose that ∂Ω is smooth enough so that there exists a continuous extension operator P : where C is a constant depending on p and ∂Ω only.
The construction of Q ε is based on the Q 1 interpolate of some discrete approximation, as is customary in FEM. The idea of using these types of interpolate was already present in Griso [40], [41] for the study of truss-like structures. For the purpose of this paper, it is enough to take the average on εξ +εY to construct the discrete approximations, but the average on εξ + εY , where Y is any fixed open subset of Y , or any open subset of a manifold of codimension 1 in Y . The only property which is needed is the Poincaré-Wirtinger inequality, which holds in both of these cases.
Definition 4.1. The operator Q ε : L p (R n ) → W 1,∞ (R n ), for p ∈ [1, +∞], is defined as follows: and for any x ∈ R n , we set In the case of the space W 1,p 0 (Ω), the operator Q ε : given by (4.1).
In the case of the space W 1,p (Ω), the operator Q ε : where Q ε (P(φ)) is given by (4.1).
We start with the following estimates.
, there exists a constant C depending on n and Y only, such that Furthermore, for any ψ in L p (Y ), Proof. By definition, the Q 1 interpolate is Lipschitz-continuous and reaches its maximum at some εξ. So, to estimate the L ∞ norm of Q ε (φ), it suffices to estimate the Q ε (φ)(εξ) s. By (4.1), The space Q 1 (Y ) is of dimension 2 n , hence all of the norms are equivalent. So, there are constants c 1 , c 2 , and c 3 (depending only upon p and Y ) such that, for every Φ ∈ Q 1 (Y ),

The constant C depends on Y (via its diameter and its Poincaré-Wirtinger constant) only, and depends neither on Ω nor on ε.
Similarly, for the case W 1,p (Ω), one has Moreover, in both cases, where C does not depend on ε. Proof. We start with φ in W 1,p (R n ). From Proposition 2.5(i) and inequality (3.5), we get On the other hand, for any ψ ∈ W 1,p (interior(Y ∪ (Y + e i ))), i ∈ {1, . . . , n}, we have By a scaling argument and using Definition 4.1, this gives for all ξ ∈ εZ n . Let x ∈ ε ξ + Y , and set for every κ = (κ 1 , . . . , κ n ) ∈ {0, 1} n , If ξ ∈ εZ n , for every κ ∈ {0, 1} n , by definition we have . . x (κ n ) n , and so, for example, and a same expression for the other derivatives. This last formula and (4.7)-(4.9) imply estimate (i) written in R n . Now, from (4.9), we get and (ii) (in R n ) follows by using estimate (4.7). Estimate (iii) (again in R n ) is straightforward from the previous ones. If φ is in W 1,p 0 (Ω), let φ be its extension to the whole of R n . To derive (i)-(iii), it suffices to write down the estimates in R n obtained above. Similarly, applying them to P(φ) for φ in W 1,p (Ω) gives (iv)-(vi).
To finish the proof, it remains to show estimate (4.6) . To do so, it is enough to take the derivative with respect to any x k , with k = 1 in the formula of ∂Q ε (φ) ∂x 1 above, and use estimate (4.8).
Now, by Proposition 3.1, there exist a subsequence (still denoted ε) and where w j is y i -periodic for every i = j. Moreover, from Remark 4.6, the function w j does not depend on y j , hence it is Y -periodic. But, by Remark 4.6 again, w j is also piecewise linear, with respect to any variable y i . Consequently, w j is independent of y. On the other hand, from (ii) above we have Now Proposition 2.9(iii) gives w j = ∂w ∂x j , and convergence (iii) holds for the whole sequence ε.
Proposition 4.8 (Theorem 3.5 revisited). Let {w ε } be a sequence converging weakly in W 1,p 0 (Ω) (resp. in W 1,p (Ω)) to w. Then, up to a subsequence there exists some w in the space L p (Ω; W 1,p per (Y )) such that the following convergences hold: Actually, the connection with the w of Theorem 3.5 is given by Proof. Due to estimates of Proposition 4.5, up to a subsequence, there exists w in L p (Ω; W 1,p per (Y )) such that Combining with convergence (iii) of Proposition 4.7 shows that Remark 4.9. In the previous proposition, one can actually compute the average of w . One can check that M Y ( w ) = −M Y (y) · ∇w, and consequently,

Periodic unfolding and the standard homogenization problem.
Definition 5.1. Let α, β ∈ R, such that 0 < α < β and O be an open subset of R n . Denote by M (α, β, O) the set of the n × n matrices A = (a ij ) 1≤i,j≤n ∈ (L ∞ (O)) n×n such that, for any λ ∈ R n and a.e. on O, Let A ε = (a ε ij ) 1≤i,j≤n be a sequence of matrices in M (α, β, Ω). For f given in H −1 (Ω), consider the Dirichlet problem By the Lax-Milgram theorem, there exists a unique u ε ∈ H 1 0 (Ω) satisfying which is the variational formulation of (5.1). Moreover, one has the apriori estimate Consequently, there exist u 0 in H 1 0 (Ω) and a subsequence, still denoted ε, such that We are now interested to give a limit problem, the "homogenized" problem, satisfied by u 0 . This is called standard homogenization, and the answer, for some classes of A ε , can be found in many works, starting with the classical book by Bensoussan, Lions, and Papanicolaou [11] (see, for instance, Cioranescu and Donato [30] and the references herein). We now recall it.
Let u ε be the solution of the corresponding problem (5.1), with f in H −1 (Ω). Then the whole sequence {u ε } converges to a limit u 0 , which is the unique solution of the homogenized problem where the constant matrix A 0 = (a 0 ij ) 1≤i,j≤n is elliptic and given by In (5.7), the functions χ j (j = 1, . . . , n), often referred to as correctors, are the solutions of the cell systems As will be seen below, using the periodic unfolding, the proof of this theorem is elementary! Actually, with the same proof, a more general result can be obtained, with matrices A ε . Theorem 5.3 (periodic homogenization via unfolding). Let u ε be the solution of problem (5.1), with f in H −1 (Ω), and A ε = (a ε ij ) 1≤i,j≤n be a sequence of matrices in M (α, β, Ω). Suppose that there exists a matrix B such that Then there exists u 0 ∈ H 1 0 (Ω) and u ∈ L 2 (Ω; H 1 per (Y )) such that and the pair (u 0 , u) is the unique solution of the problem Remark 5.4. System (5.11) is the unfolded formulation of the homogenized limit problem. It is of standard variational form in the space Remark 5.5. Hypothesis (5.9) implies that B ∈ M (α, β, Ω × Y ). Remark 5.6. If A ε is of the form (5.5), then B(x, y) = A(y). In the case where A ε (x) = A 1 (x)A 2 ( x ε ), one has (5.9), with B(x, y) = A 1 (x)A 2 (y). Remark 5.7. Let us point out that every matrix B ∈ M (α, β, Ω × Y ) can be approached by the sequence of matrices A ε in M (α, β, Ω), with A ε defined as follows: Proof of Theorem 5.3. Convergences (5.10) follow from estimate (5.3), Corollary 3.3, and Proposition 4.7, respectively.
Let us choose v = Ψ, with Ψ ∈ H 1 0 (Ω) as test function in (5.2). The integration formula (2.5) from Proposition 2.7 gives We are allowed to pass to the limit in (5.12), due to (5.9), (5.10), and Proposition 2.9(i), to get Now, taking in (5.2), as test function v ε (x) = εΨ(x)ψ( x ε ), Ψ ∈ D(Ω), ψ ∈ H 1 per (Y ), one has, due to (2.5) and Proposition 2.7, Since v ε 0 in H 1 0 (Ω), we get at the limit 1 By the density of the tensor product D(Ω) ⊗ H 1 per (Y ) in L 2 (Ω; H 1 per (Y )), this holds for all Φ in L 2 (Ω; H 1 per (Y )). Remark 5.8. As in the two-scale method, (5.11) gives u in terms of ∇u 0 and yields the standard form of the homogenized equation, i.e., (5.6). In the simple case where A(x, y) = A(y) = (a ij (y)) 1≤i,j≤n , it is easily seen that and that the limit B is precisely the matrix A 0 which was defined in Theorem 5.2 by (5.7) and (5.8). Proposition 5.9 (convergence of the energy). Under the hypotheses of Theorem 5.3, Proof. By standard weak lower-semicontinuity, one successively obtains which gives the first convergence in (5.15), as well as which implies the second convergence in (5.15).
Remark 5.10. From the above proof, we also have Corollary 5.11. The following strong convergence holds: Proof. We have successively Each term in the right-hand side converges, the first one due to Remark 5.10, and the others due to (5.10) and hypothesis (5.9). So, the right-hand side term converges to zero. Then convergence (5.16) is a consequence of the ellipticity of B ε . Remark 5.12. One can consider problem (5.1) with a homogeneous Neumann boundary condition on ∂Ω provided a zero order term is added to the operator. This problem is variational on the space H 1 (Ω) without any regularity condition on the boundary. The exact same method applies and gives the corresponding limit problem. In order for a nonhomogeneous Neumann boundary condition (or Robin condition) on ∂Ω to make sense, a well-behaved trace operator is needed from H 1 (Ω) to L 2 (Ω). In that case, the same method applies.

Some corrector results and error estimates.
Under additional regularity assumptions on the homogenized solution u 0 and the cell-functions χ j , the strong convergence for the gradient of u 0 with a corrector is known (cf. [11] Chapter 1, section 5, [30] Chapter 8, section 3 and references therein). More precisely, suppose that ∇ y χ j ∈ (L r (Y )) n , j = 1, . . . , n and ∇u 0 ∈ L s (Ω), with 1 ≤ r, s < +∞ and such that 1/r + 1/s = 1/2. Then Our next result gives a corrector result without any additional regularity assumption on χ j , and its proof reduces to a few lines. We also include a new type of corrector.
Theorem 6.1. Under the hypotheses of Theorem 5.2, one has
In what follows in this paragraph, we suppose that the open set Ω is a bounded domain in R n , n = 2 or 3, of polygonal (n = 2) or polyhedral (n = 3) boundary. We assume that Ω is on one side only of its boundary, and that Γ 0 is the union of some edges (n = 2) or some faces (n = 3) of ∂Ω. Recall that classical regularity results show that the solution u 0 of the homogenized problem (5.6) belongs to H 1+s (Ω) for s in ]1/2, 1[ (s = 1 if the domain is convex) depending only on ∂Ω, on A 0 , and satisfies the estimate The error estimate for this case is given in the following result.
Theorem 6.4 (see [45]). The solution u ε of problem (5.1) satisfies the estimate The constants depend on n, A, and ∂Ω. Corollary 6.5 (see [45]). Let Ω be an open set strongly included in Ω, then The constant depends on n, A, Ω , and ∂Ω.

Periodic unfolding and multiscales.
As we mentioned in the Introduction, the periodic unfolding method turns out to be particularly well-adapted to multiscales problems. As an example, we treat here a problem with two different small scales.
Consider two periodicity cells Y and Z, both having the properties introduced at the beginning of section 2 (each associated to its set of periods). Suppose that Y is "partitioned" in two nonempty disjoint open subsets Y 1 and Y 2 , i.e., such that Let A εδ be a matrix field defined by where A 1 is in M (α, β, Y 1 ) and A 2 in M (α, β, Z) (cf. Definition 5.1). Here we have two small scales, namely, ε and εδ, associated, respectively, to the cells Y and Z (see Figure 7). Consider the problem So, there is some u 0 such that, up to a subsequence, u εδ u 0 weakly in H 1 0 (Ω). Using the unfolding method for scale ε, as before we have T ε ∇u εδ ∇u 0 + ∇ y u in L 2 (Ω × Y ).
These convergences do not see the oscillations at the scale εδ. In order to capture them, one considers the restrictions to the set Ω × Y 2 defined by v εδ (x, y) .
Now, we apply to v εδ , a similar unfolding operation, denoted T y δ , for the variable y, thus adding a new variable z ∈ Z.
It is essential to remark that all of the estimates and weak convergence properties which were shown for the original unfolding T ε still hold for T y δ , with x being a mere parameter. For example, Proposition 4.7 and Theorem 3.5 adapted to this case imply that T y δ ∇ y v εδ ∇ y u| Ω 2 + ∇ z u weakly in L 2 (Ω × Y 2 × Z), T y δ T ε ∇u εδ ∇u 0 + ∇ y u + ∇ z u weakly in L 2 (Ω × Y 2 × Z).
The proof uses test functions of the form where Ψ, Ψ 1 , Ψ 2 are in D(Ω), Φ 1 in H 1 per (Y ), Φ 2 ∈ D(Y 2 ), and Θ 2 ∈ H 1 per (Z). Remark 7.2. The same theorem holds true for a general A εδ under the hypotheses Proposition 5.9 (convergence of the energy) and Corollary 5.11 extend without any difficulty to the multiscale case.
Remark 7.5. A corrector result, similar to that of Theorem 6.1, can be obtained. Remark 7.6. Theorem 7.1 can be extended to the case of any finite number of distinct scales by a simple reiteration.
8. Further developments. The unfolding method has some interesting properties which make it suitable for more general situations than that presented here. In problems which are set on a domain Ω ε which depends on the parameter ε, it may be difficult to have a good notion of convergence for the sequence of solutions u ε . The traditional way is to extend the solution by 0 outside Ω ε ; however, this precludes the strong convergence of these extended functions in general. For the case of holes of the