Periodic homogenization for inner boundary conditions with equi-valued surfaces: the unfolding approach

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1 Periodic homogenization for inner boundary conditions with equi-valued surfaces: the unfolding approach Alain Damlamian Université Paris-Est, Laboratoire d Analyse et de Mathématiques Appliquées, CNRS UMR 8050 Centre Multidisciplinaire de Créteil, Créteil, Cedex, France, damla@u-pec.fr Joint work with Doina Cioranescu and Li Tatsien Our paper will appear in the issue of the Chinese Annals of Mathematics dedicated to this conference.

2 Jacques-Louis Lions and Homogenization: Famous book Asymptotic Analysis for Periodic Structures in 1978 by A. Bensoussan, J.-L. Lions and G. Papanicolaou. Jacques-Louis Lions and the ISFMA: 2-5 February 1998, inauguration of ISFMA. The periodic unfolding method was introduced in 2002 in a joint work with Doina Cioranescu and Georges Griso ([2], [3]). It gives an elementary proof and insight for the classical periodic homogenization problems (see details in Doina s lecture just before this one). In view of the unfolding method, we revisit the homogenization of two problems which are particular cases of the general problem of equi-valued surfaces with corresponding assigned total fluxes.

3 The first example is the linear elastic torsion problem of a 3-dimensional cylindric rod with multiply-connected cross section (see Lanchon [8](1974) and Li Ta-tsien, Zheng Songmu, Tan ongji and Shen Weixi [10](1998) for the setting of the problem): Torsion of an elastic bar.

4 Assuming the material properties are uniform with respect to the vertical variable, the equations are set in the 2-dimensional cross-section Ω obtained from a bounded open set Ω perforated by a finite number of regular closed subsets (which have a non empty interior), S 1, S 2,.... After normalization, the stress function ϕ of the elastic material is shown to be the solution of the following problem: ϕ H 1 0(Ω) with f S j an unknown constant, for each j, S j ϕ = 1 in Ω ϕ n dσ(x) = Sj (the measure of S j ) for each j. (0.1)

5 The electric conductivity problem arising in resistivity well-logging is set in any dimension (usually in 3-D) with the same type of geometry (Ω, Ω and S j s). The conductivity tensor A can vary with the position in Ω, the right hand side is an L 2 function f defined on Ω, and the total fluxes on the S j are given numbers g j : ϕ H 1 0(Ω) with ϕ S j an unknown constant, for each j, div(a(x) ϕ(x)) = f in Ω ϕ dσ(x) = g j for each j. ν A S j Here we refer to the two works of Li Tatsien et al on the subject ([9] (1995)and [10]). Some variants of this problem consider the case of codimension-1 electrodes (cracks or fissures instead of inclusions), and the global conductor (the electrodes are electrically connected) in which case the unknown constant is the same for all inclusions. (0.2)

6 We want to consider the periodic homogenization for these problems. The first proof for the homogenization of the elastic torsion problem (via extension operators and oscillating test functions) was obtained by Cioranescu and Saint Jean Paulin [4](1979), where regularity assumptions are made for the boundary of the inclusions. In Briane [1](1997), the restriction of regularity of the boundaries is lifted, but still with some geometric conditions and by the same use of oscillating test functions. One advantage of the unfolding method is that it requires no regularity for the boundary of the inclusions whatsoever. Actually, there is no need to introduce surface integrals, except if one wants to see the usual strong formulation, valid for Lipschitz boundaries. The other advantage of the method is that an immediate consequence is a corrector result which is completely general (without the need for extra regularity).

7 Plan: 1- The geometric setting 2- The disconnected case (where the constants values on the inclusions are independent). We state the homogenization result and give the main ingedients of the proof. 3- Variants : cracks; two connected cases. General notations In this work, ε indicates the generic element of a bounded subset of R + in the closure of which 0 lies. Convergence of ε to 0 is understood in this set. Also, c and C denote generic constants which do not depend upon ε. As usual, 1 D denotes the characteristic function of the set D. For a measurable set D in R n, D denotes its Lebesgue measure. For simplicity, the notation L p (O) is used both for scalar and vector-valued functions defined on the set O, since no ambiguity will arise.

8 1 The geometric setting for periodic homogenization We use the general framework of the periodic unfolding method [3] and the notations therein. Let b = (b 1,..., b n ) be a basis of R n. We denote by G = { ξ R n ξ = n } k i b i, (k 1,..., k n ) ZZ n i=1 (1.1) the group of macroscopic periods for the problem. Let be the open parallelotope generated by the basis b, i.e., { y R n y = n y i b i, (y 1,..., y n ) (0, 1 ) }. n (1.2) i=1 More generally, can be any bounded connected subset of R n with Lipschitz boundary and having the paving property with respect to the group G. For z R n, [z] denotes the unique (up to a set of measure zero) integer combination n j=1 k jb j of the periods such that z [z] belongs to. Set now {z} = z [z] a.e. for z R n.

9 Definition of [z] and {z}

10 Let S be a given compact subset in, which is the finite disjoint union of S j for j = 1,..., J (with the same property). The only condition on the S j s is that they are pair-wise separated and with strictly positive measure. For the two examples we consider, the sets S j are naturally assumed to be connected (although this is not necessary for the treatment given here). The set \ S is denoted : The sets and

11 Let now Ω be an open bounded subset of R n. We define the holes in Ω as Definition 1.1. S ε S j ε } {. x = {x Ω, { ε. x } = {x Ω, ε Ω ε. = Ω \ Sε. S}, S j }, (1.3) The sets Ω ε (in green) and S ε (in yellow)

12 2 The disconnected case We consider a boundary value problem which generalizes both cases above and does not require the consideration of surface integrals. It applies as long as the inclusions Sε j have strictly positive measure. We make no regularity assumption regarding the sets S j. We only make the natural assumption that they are well separated of each other and of in (if some are not well separated, then they should be merged). We introduce the following two families of subspaces, where G ε denotes the elements ξ G such that the corresponding cell εξ + ε intersects Ω. Definition = { v H0(Ω); 1 ξ G ε and j {1,..., J} v εξ+εs j is a constant function, W ε 0 the value of which depends on (ξ, j)}. (2.1) L ε. = { w L 2 (Ω); ξ G ε and j {1,..., J} w εξ+εs j is a constant function, the value of which depends on (ξ, j)}. (2.2) Note that a condition such as v εξ+εs j is a constant function is taken in the sense of a.e., which makes sense because each εξ + εs j is of positive measure. It also follows that W0 ε is a closed subspace of H0(Ω). 1 The space L ε is a finite dimensional subspace of L 2 (Ω). Note that L ε is the image of L 2 (Ω) under the local average map M ε.

13 The conductivity matrix field A ε is assumed to belong to the set M(α, β, Ω ε ) defined as Definition 2.2. Let α, β R, such that 0 < α < β. M(α, β, O) denotes the set of the n n matrices B = B(x), B = (b ij ) 1 i,j n (L (O)) n n such that for any λ R n and a.e. on O, (B(x)λ, λ) α λ 2, B(x)λ β λ. Let f ε be given in L 2 (Ω ε ) and g j ε in L ε for j = 1,..., J. The problem we consider is given in variational form as (P ε ) Using the obvious formula Find u ε in W0 ε such that w W0 ε, A ε u ε w dx = f ε w dx + Ω ε Ω ε ξ G g j ε εξ+ε w εξ+εs j = 1 ε n S j S j ε ξ G ε, j=1,...,j g j ε εξ+ε w εξ+εs j. (2.3) g j ε(x) w(x) dx, (2.4) this problem can be written as Find u ε in W0 ε such that w W0 ε, (P ε ) A ε u ε w dx = F ε w dx, Ω ε Ω (2.5)

14 where F ε. = fε 1 Ωε + ξ G ε, j=1,...,j 1 ε n S j gj ε 1 S j ε This problem has a unique solution by the Lax-Milgram theorem applied in the space W ε 0 because of Poincaré s inequality in H 1 0(Ω). Furthermore, there is a constant independent of ε such that u ε H 1 0 (Ω) C F ε L 2 (Ω ε) = C ( f ε 2 L 2 (Ω ε) + ξ G ε, j=1,...,j 1 ε n S j gj ε(x) S j ε 2) 1/2. (2.6) Assuming at least Lipschitz regularity for the boundaries involved, the strong formulation of problem (P ε ) is: Find u ε W0 ε such that div(a ε u ε ) = f ε in Ω ε (2.7) ξ G ε, j = 1,... J, A ε u ε n, 1 H 1/2, H 1/2 (εξ+ε S j ) = gε j εξ+ε, where n(x) is the outward unit normal to εξ + ε S j (note that since f ε belongs to L 2 (Ω ε ), A ε u ε n is an element of H 1/2 (εξ + ε S j )). The duality pairing is often (somewhat incorrectly) written as the integral εξ+ε S j A ε u ε n(x) dσ(x).

15 The homogenization problem is to investigate the weak convergence of the sequence u ε and the possible problem satisfied by its limit when F ε is bounded in L 2 (Ω), under suitable assumptions on the data. Remark: The sequence {u ε } in H 1 0(Ω) does not usually converge strongly in H 1 0(Ω) because, when it does, its limit is the zero function. More generally, if v ε is in W ε 0 and converges strongly to v 0 in H 1 0(Ω), then v 0 = 0. The proof is elementary: going to the limit for the product 0 1 Sε v ε, which, as a consequence of the assumptions, converges weakly to S v 0, implies that v 0 0, hence the result.

16 The convergences In order to use the unfolding operator T ε, in S ε we extend A ε by αi without changing the notation. This implies that T ε (A ε ) Ωε S αi. We make the following assumptions concerning the data, for ε converging to 0: (H) : { Tε (A ε ) converges in measure (or a.e.) in Ω to A 0, T ε (F ε ) converges weakly to F 0 in L 2 (Ω ). (2.8) Since A ε belongs to M(α, β, Ω ε ), it follows that A 0 belongs to M(α, β, Ω ). Proposition 2.3. Under hypothesis (H), up to a subsequence (which we still denote by {ε}), there exist u 0 H 1 0(Ω) and û L 2 (Ω; H 1 per( )) such that T ε (u ε ) u 0 weakly in L 2 (Ω; H 1 ( )), T ε ( u ε ) u 0 + y û weakly in L 2 (Ω ), 1 ε (T ε(u ε ) M ε (u ε )) converges weakly in L 2 (Ω; H 1 ( )) to y M u 0 + û, η ε. = Tε (A ε )T ε ( u ε ) η 0. = A 0 ( u 0 + y û) weakly in L 2 (Ω ), η 0 vanishes almost everywhere in Ω S, y M u 0 + û is independent of y on each Ω S j, j = 1,..., J. (2.9)

17 Proof. The existence of a subsequence, of u 0 and û satisfying the first three conditions follows from the standard unfolding theory. The next convergence follows from the fact that convergence in measure (or a.e.) is a multiplyier for strong as well as for weak convergence in L 2 (Ω ). The last property follows from the fact that u ε εξ+εs j is independent of x, which implies that ε 1 (T ε (u ε ) M ε (u ε )) Ω S j is a function only of x (i.e. independent of y!). This property is preserved by weak limit, and holds for y M u 0 + û. A similar proof implies that η 0 vanishes a.e on Ω S. Consequently, the pair (u 0, û) belongs to the space W as defined below: Definition 2.4. Let W be the following space: W = {(w, ŵ); w H 1 0(Ω), ŵ H 1 per( ), M (ŵ) = 0, ŵ + (y M w) S j is a constant (depending on j) for j = 1,..., J}. (2.10) It is a closed subspace of H 1 0(Ω), H 1 per( ), hence a Hilbert space.

18 The Main theorem Theorem 2.5. Under Hypothesis (H), the whole sequence {u ε } converges weakly in H0(Ω) 1 to a function u 0. There also exists û in L 2 (Ω; Hper( 1 )) such that (u 0, û) is the unique solution of the following problem: Find (u 0, û) W, such that (w, ŵ) W 1 A 0 ( u 0 + y û)( w + y ŵ) dx dy = Ω (2.11) 1 F 0 (x, y) w(x) dxdy. Ω The proof is obtained through several propositions. As usual in homogenization, the choice of the proper test functions is crucial (and a posteriori often obvious!) Definition 2.6. Let H S per denote the subspace of Hper( 1 ) consisting of functions which are constant on each S j (with independent constants for each j). We use two different test functions, where w is fixed in D(Ω):

19 Type 1 v ε (x) =. ( ε M ε (w)(x)ψ ({ x} ε ) ({x} ) ) + w(x)φ ε where ψ is in D( ) H S per and φ in C per( ) vanishing on S. The function M ε (w) is the local average of w. Clearly, v ε converges to 0 uniformly and it is also straighforward to check that T ε ( v ε ) w(x) y (φ + ψ)(y) uniformly. Note that by a partition of unity argument, every Ψ C per( ) H S per can be written as a sum φ + ψ, so that this gives a dense subset of H S per. Type 2 w ε (x) =. ( w(x) 1 Ψ ({ x} ) ) + M ε ε (w)(x)ψ ({ x} ) ) ε = w(x) ( w(x) M ε (w)(x) ) Ψ ({ x} ) (2.12) ε where Ψ is in Cper( ) with Ψ 1 on S. Clearly, again, w ε converges uniformly to w. It is less obvious that T ε ( v ε ) converges strongly in L 2 (Ω ) to w y ( (ym w)ψ(y) ).

20 Indeed, T ε ( v ε ) = T ε ( w) ( 1 Ψ(y) ) 1 ε (T ε(w) M ε (w)) y Ψ(y), and the result now follows from the fact that 1 ε (T ε(w) M ε (w)) converges strongly to y M w(x) in L 2 (Ω ). Here y M. = y M (y) denotes the normalized variable y. Notation: we use η ε for A ε ( u ε ) and η 0 for A 0 ( u 0 + y û) (so that T ε (η ε ) converges weakly to η 0 ). From now on, Ψ is fixed in C per( ) with Ψ 1 on S. It will simplify things to assume also that M (y M Ψ) = 0, which is easy to achieve.

21 The local problem Using the type 1 test functions, the unfolding formula and a density argument, we get: Proposition 2.7. For almost every x Ω, and for every Φ L 2 (Ω; H S per), the vector field η 0 satisfies Ω η 0 (x, y) y Φ(y) dy = 0. (2.13) The corresponding strong formulation, under regularity assumptions, is div y η 0 = 0 in D ( ), η 0 n, 1 H 1/2,H 1/2 ( S j ) = 0 for j = 1,..., J, and periodicity conditions of the normal flux of η 0 (on opposite faces of ). (2.14) The global problem Using type 2 test functions, the unfolding formula and a density argument, we get Proposition 2.8. For every w H 1 0(Ω): 1 Ω ( η 0 ( w y (ym w)ψ(y) )) dxdy = 1 F 0 (x, y) w(x) dxdy. Ω (2.15)

22 Remark Because η 0 satisfies (2.13), formula (2.15) does not depend upon the choice of Ψ. Indeed, for another such Ψ for a.e. x Ω, by (2.13), one has η 0 y ((y M w) ( Ψ Ψ )) dy = 0. (2.16) 2. If S is assumed regular, and in view of (2.13), the term 1 ) η 0 y ((y M w)ψ dxdy can be interpreted as Ω 1 Ω η 0 n, y M w H 1/2, H 1/2 ( S)dx, because Ψ is identically 1 on S.

23 Proof of the main theorem Combining Propositions 2.7 and 2.8 gives the following: For all w H0(Ω), 1 Φ L 2 (Ω; H S per( )) 1 ( A 0 ( u 0 + y û)( w + y Φ (ym w)ψ ) dx dy = Ω 1 F 0 (x, y) w(x) dxdy. Ω (2.17) This is equivalent to (2.11) by the following change of test functions. For (w, ŵ) in W, set Φ. = ŵ + (y M w)ψ. Then, (w, Φ) in H 1 0(Ω) L 2 (Ω; H S per( )). Conversely, for (w, Φ) in H 1 0(Ω) L 2 (Ω; H S per( )), set (w, ŵ) = (w, Φ (y M w)ψ). Then (w, ŵ) is in W. Recall problem (2.11): Find (u 0, û) W, such that (w, ŵ) W 1 A 0 ( u 0 + y û)( w + y ŵ) dx dy = Ω 1 F 0 (x, y) w(x) dxdy. Ω (2.11) The Lax-Milgram theorem applies to problem (2.11) because its bilinear form is coercive as can be seen as follows:

24 since A 0 belongs to M(α, β, Ω ), 1 A 0 ( u 0 + y û)( u 0 + y û) dx dy α u 0 + y û 2 L 2 (Ω ). Ω But since u 0 is independent of y and û is -periodic, the latter is just α( u 0 2 L 2 (Ω) + 1 yû 2 L 2 (Ω ) ). One concludes using the Poincaré inequality in H 1 0(Ω) and the Poincaré-Wirtinger inequality in H 1 per( ) (since M (û) = 0).

25 The homogenized limit problem The homogenized limit problem is given by the next proposition. Proposition There exists a matrix field A hom which is bounded and coercive such that the limit u 0 is the unique solution of the homogenized problem Find u 0 H0(Ω) 1 such that w H0(Ω), 1 A hom (2.18) ( u 0 ) w dx = M (F 0 ) w dx. Ω Ω To make A hom explicit, one needs the so-called cell-problem, obtained by freezing u 0 as a constant vector λ. For a given vector λ R n, find χ λ Hper( 1 ) such that M (χ λ ) = 0, (χ λ + y M.λ) S j is independent of y for j = 1,..., J and (2.19) A 0 (x, y)( y χ λ (y) + λ) y ϕ(y) dy = 0 ϕ H S per. This problem itself is not variational, but it easy to show that it has at most one solution: the difference V of two solutions of (2.19) belongs to H S per and satisfies in particular A 0 y V y V dy = 0, which by ellipticity, implies y V 0 so that V = 0 (since M (V ) = 0).

26 To prove existence, look for χ λ in the form χ λ. = Uλ (y λ)ψ for U λ in H S per with M (U λ ) = 0, using the same fixed function Ψ in D( ) with Ψ S 1 and M (y M Ψ) = 0. This U λ is actually the unique solution of the following variational problem, for which the Lax-Milgram theorem applies. Find U λ H S per such that M (U λ ) = 0, A 0 (x, y)( y U λ (y)) y ϕ(y) dy = A 0 ((Ψ 1)λ + (y M λ) y Ψ) y ϕ(y) dy ϕ H S per. (2.20) Using U λ as a test function in (2.20) for a.e. x Ω, it is straightforward to see that, y χ λ L 2 ( ) C(α, β, ) λ, (2.21) with a explicit constant C(α, β, ) which depends only upon α, β and. For simplicity, we write χ(λ) for χ λ. Going back to (2.11) with solution (u 0, û), it follows that û(x, y) = χ( u 0 ) = n i=1 u 0 x i (x) χ ei (x, y). (2.22)

27 Then, (2.11) becomes Find u 0 H0(Ω) 1 such that w H 1 0(Ω), 1 ( ( A 0 ( u 0 + y χ( u 0 )) w y (ym w)ψ(y) )) dxdy = Ω 1 F 0 (x, y) w(x) dxdy. Ω (2.23) This leads to the definition Definition A hom (λ, µ) =. 1 A 0 (λ + y χ λ )(µ + y χ µ ) dy. (2.24) The matrix field A hom belongs to M(α, β(1 + C(α, β, )), Ω).

28 Proof. From (2.23), the homogenized problem (2.18) holds with A hom (λ, µ) =. 1 A 0 (λ + y χ λ ) ( µ y ((y M µ)ψ) ) dy (2.25) for λ, µ R n and for a.e. x Ω. However, by (2.19), since χ µ + (y M µ)ψ is in H S per, so that A 0 (λ + y χ λ )) y χ µ dy = A 0 (λ + y χ λ )) y ((y M µ)ψ) dy A hom (λ, µ) =. 1 A 0 (λ + y χ λ )(µ + y χ µ ) dy, (2.26) which is formula (2.24). From the coerciveness of A 0, we get for a.e. x Ω A hom (x)(λ)µ α λ + yχ λ ) 2 L 2 ( ). As before, since χ λ is -periodic, λ + y χ λ ) 2 L 2 ( ) = λ 2 + y χ λ ) 2 L 2 ( ), which shows the α-coerciveness of A hom. Finally, by (2.21), it follows that A hom (λ) µ (β(1 + C(α, β, )) 2 λ µ, which completes the proof.

29 Convergence of the energy and correctors This comes standard with the unfolding method! Proposition Under the hypotheses of the preceding sections, the following holds: lim ε 0 A ε ( u ε ) u ε dx = Proof. By definition of problem (P ε ), A ε ( u ε ) u ε dx = A hom ( u 0 ) u 0 dx. (2.27) Ω F ε u ε dx. Using the established convergences, it is straighforward to see that the right-hand side, once unfolded, converges to M (F 0 ) u 0 dx. We conclude by comparing with (2.18). Ω Corollary The following strong convergence holds: T ε ( u ε ) u 0 + y û strongly in L 2 (Ω ), u ε 2 dx 0. Λ ε (2.28) Classically in the unfolding method, the convergences of Corollary 2.13 imply the existence of a corrector, making use of the averaging operator U ε (defined as the adjoint of T ε ). Corollary Under the hypotheses of the preceding sections, as ε 0, u ε u 0 U ε ( y û) L 2 (Ω) 0. (2.29)

30 3 Other related problems Cracks and fissures One can consider the case when some of the sets S j are cracks or fissures, i.e. are Lipschitz submanifolds of codimension one in. The case of a submanifold without boundary corresponds to the case of the boundary of a compact subset j in. The case of a submanifold with a boundary corresponds to a crack in. A combination of the two can also occur. Each of these cases can be seen as limits of thick inclusions. The various cases of cracks and fissures: S 1, S 2 and S 3 as limits of thick inclusions

31 The corresponding conditions for regular cracks (which have two sides!!) in the strong formulation are The solution u is an unknown constant on each fissure εξ + εs j and [ u ] dσ(x) = g j ν A εξ+εs j ε εξ+ε, a given number, (3.1) εξ+εs j [ u ] where denotes the sum of the two outward conormal derivatives from both sides ν A εξ+εs j of the crack (it can be considered as the jump of the conormal derivative across the fissure εξ + εs j, hence the notation). In the definition of the space W0 ε there is the requirement that the functions be constant almost everywhere with respect to the surface measure on the fissures (equivalently the (n-1)- dimensional Hausdorff measure). In the variational formulation, the term associated with the fissure εξ + εs j is ε n 1 gε εξ+ε j w εξ+εs j = 1 g S j ε j w dσ(x). εξ+εs j From there on, the proofs are the same and the statements of the results are modified in an obvious way: the homogenized matrix field is given by the same definition where the ξ λ are given as solutions of the cell problem which here has fissures. The homogenized problem is unchanged. The only modification is in the definition of the space H S per, where the conditions on the fissures are taken in the sense of traces (i.e. almost everywhere for the corresponding surface measures).

32 The Global Conductor 1 In the Global conductor case, the situation is the same as in the previous cases, but all the conductors are somehow connected so that the solution takes the same unknown constant value on all of S ε. The problem is therefore set in the smaller subspace. = {w H0(Ω); 1 w Sε is constant}. (3.2) W ε 0c and is defined for given A ε and f ε as before and for a given real number g ε as ( P Find u ε W0c ε such that for all w W0c ε ε ) A ε u ε w dx = f ε w dx + g ε w Sε. Ω Let F ε denote f ε 1 Ωε + g ε 1 Sε, and we denote by C ε the constant value taken by u ε on S ε. There is an instability in the problem insofar as C ε has to vanish as soon a S ε intersects the outer boundary Ω on a set of non zero capacity. We make the following hypotheses on the data: T ε (A ε ) converges in measure (or a.e.) in Ω to A 0, (Ĥ) : T ε (f ε ) converges weakly to f 0 in L 2 (Ω ) g ε converges to g 0 in R. Furthermore, for the values of ε such that C ε 0, we assume that (H 0 ) F ε dx is a O(ε 1/2 1 ) (which implies that S g 0 = 1 f 0 (x, y) dxdy). Ω Ω Ω Ω (3.3)

33 1 ) Theorem 3.1. Under assumptions (H 0 ) and ε( (Ĥ), the sequence T ε (u 2 ε C ε ) weakly in L 2 (Ω; H S ( )) to the unique solution Û of the following variational problem: Find Û L2 (Ω; Hper( S )); Ψ L 2 (Ω; H per( S )), A 0 (x, y) y Û(x, y) y Ψ(x, y) dxdy = f 0 (x, y)ψ(x, y) dxdy. Ω Ω converges This result gives a kind of second order expansion of the solution in terms of ε, but it is not complete in the sense that, in general, C ε is not of order ε 2 itself (only of order ε 3/2 ). Remark 3.2. If one assumes a stronger condition than (H 0 ), namely F ε dx is a o(ε 1/2 ), then one can show the convergence of the energy, which implies that ( uε ) T ε converges strongly in L 2 (Ω ) to y Û. (3.5) ε In turn, this gives the following corrector result u ε = ε U ε ( y Û) + o L 2 (Ω)(ε). Remark 3.3. In the case where S ε intersects Ω in a set of non zero capacity, it follows that C ε is 0. Assuming this is the case for all ε s of the sequence, this problem reduces to a that of a homogeneous Dirichlet condition in S ε. The corresponding problem was originally studied for the Stokes system by L.Tartar in the appendix of Sanchez-Palencia [13](1980) and for the Laplace equation by J.-L. Lions in [11] (1981). The nonlinear case was later studied in Donato-Picard [6] (2000). Ω (3.4)

34 The Global Conductor 2 The presentation of the previous section is not necessarily realistic because of the unpredictability of the intersection of S ε with the boundary of Ω. It may be more realistic to consider that the conductor is restricted to a compact subset Ω 0 of Ω, i.e. is Sε 0. = S ε Ω 0. We make this hypothesis in this section. We shall also assume that Ω 0 is a null set for the Lebesgue measure and denote Ω \ Ω 0 by Ω 1.

35 In this case, the variational space for the original problem is W 0 (S ε ). = { w H 1 0(Ω); w S 0 ε is a constant. } (3.6) as The variational formulation is for given A ε and f ε as before and for a given real number g ε ( P Find u ε W Sε 0c such that for all w W 0 (S ε ) ε ) (3.7) A ε u ε w dx = f ε w dx + g ε w S 0 ε. Ω Ω Proposition 3.4. Suppose that f ε is bounded in L 2 (Ω) and g ε is bounded in R. Assume that the sequence of matrix fields A ε H-converges to A hom in Ω 1, then sequence u ε converges weakly in H0(Ω) 1 to the solution u 0 of the homogenized limit problem given by Find u 0 H such that w H, A hom (3.8) ( u 0 ) w dx = M (F 0 ) w dx. Ω 1 Ω The strong formulation can be given here if Ω 0 is Lipschitz as a problem on Ω 1 : div(a hom u 0 ) = M (f 0 ) in Ω 1, u 0 Ω0 is an unknown constant, u 0 dσ(x) = M (F 0 )dx. Ω 0 ν A hom Ω 0 (3.9)

36 The notion of H-convergence (see Murat-Tartar [12])(1997) is more abstract than the unfolding method. The convergence almost everywhere of the unfolding of A ε implies the latter H- convergence of A ε. References THANK OU! [1] M.Briane, Homogenization of the torsion problem and the Neumann problem in nonregular periodically perforated domains. Math. Models Methods Appl. Sci. 7, No. 6, (1997). [2] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, Série 1, 335 (2002), [3] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. of Math. Anal. Vol. 40, 4 (2008), [4] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes, J. Mathematical Analysis and Applications, 71 (1979), [5] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Applied Mathematical Sciences 136, Springer-Verlag New ork, 1999.

37 [6] P. Donato and C. Picard, Convergence of Dirichlet problems for monotone operators in a class of porous media, Ricerche mat., 49 (2000), [7] G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), [8] Hélène Lanchon, Torsion élastoplastique d un arbre cylindrique de section simplement ou multiplement connexe, J. Mécanique 13 (1974), [9] Li Ta-tsien and Tan ongji, Mathematical problems and methods in resistivity wellloggings. Surveys Math. Indust. 5 (1995), no. 3, [10] Li Tatsien, Zheng Songmu, Tan ongji and Shen Weixi, Boundary value problems with equivalued surface and resistivity well-logging, Pitman Research Notes in Mathematics Series, 382, Longman, Harlow, xiv+210 pp. ISBN: [11] J.-L. Lions, Some methods in the mathematical analysis of systems and their control, Beijing, Science Press; New ork, Gordon and Breach, Science Publishers, Inc., [12] F. Murat and L. Tartar, H - Convergence, Topics in the mathematical modelling of composite materials, R.V. Kohn ed., Progr. Nonlinear Differential Equations Appl., 31. Birkhäuser, Boston, 1997, [13] E. Sanchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, 127, Springer-Verlag, 1980, Appendix by Luc Tartar. THANK OU AGAIN!!