Homogenization of the Poisson equation in a non-periodically perforated domain
aa r X i v : . [ m a t h . A P ] S e p Homogenization of the Poisson equation in a non-periodicallyperforated domain
X. Blanc and S. Wolf October 1, 2020
Abstract
We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundaryconditions. The size of the perforations is denoted by ε >
0, and is proportional to the distancebetween neighbouring perforations. In the periodic case, the homogenized problem (obtained inthe limit ε →
0) is well understood (see [20]). We extend these results to a non-periodic case whichis defined as a localized deformation of the periodic setting. We propose geometric assumptionsthat make precise this setting, and we prove results which extend those of the periodic case:existence of a corrector, convergence to the homogenized problem, and two-scale expansion.
Contents H convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3.2 L ∞ convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A Proof of technical lemmas 20 Introduction
In this article, we study the following problem: ( − ∆ u ε = f in Ω ε u ε = 0 on ∂ Ω ε , (1.1)where f is a given smooth, compactly-supported function (this assumption may be relaxed, as wewill see below in Remarks 1.4 and 2.4), and Ω ε is a perforated domain that we make precise in thefollowing. Our aim is to study the asymptotic behaviour of u ε as ε →
0, deriving a two-scale expansionand proving convergence estimates. In [20], these results were obtained in the periodic case (that is, ifthe perforations are a periodic array of period ε ). Here, we adapt this work to a non-periodic setting.Using Assumptions (A1) and (A2) below, which are inspired from the setting developed in [2, 3, 4], wefirst prove the existence of a corrector (Theorem 2.1 below). While this result is trivial in the periodiccase, it is not in the present setting. Then, we prove the convergence result stated in Theorem 2.2,which is a generalization of [20, Theorem 3.1] to the present setting. We also prove such a convergencein L ∞ norm (Theorem 2.3 below), a result which was not proved in [20]. The crucial point in order toprove such results is a Poincar´e inequality with an explicit scaling in ε , for functions vanishing in theperforations, as in the periodic case (see Lemma 1.1 below in the periodic case, and Theorem 3.2 inthe non-periodic case).To our knowledge, the first contribution on the homogenization of elliptic problems in perforateddomains is [14]. The setting is periodic, the equation is elliptic in divergence form, and the Dirichletcondition on the boundary of the holes is not 0. This implies that the limit is not trivial, in contrastto [20], where, as we will see below, u ε ( x ) ≈ ε f ( x ) w ( x/ε ), for some periodic function w . The caseof Neumann boundary conditions was studied in [12, 13] and [9], where the geometry is periodic, butthe holes are assumed to be asymptotically small compared to the period. In this case, an importanttool to study the problem is the so-called extension operator, which is studied in details in [1]. In [15],sufficient conditions on periodic holes are given which allow for homogenization. In [10, 11], the caseof Robin boundary conditions is addressed, with the help of the periodic unfolding method [8, 7]. Thecase of eigenvalue problems was considered in [23].In [6], a formalization in link with the H-convergence was proposed under general assumptionson the perforations. However, the computations are less explicit than in our setting. A general(non-periodic) perforated domain was also considered in [21]: this setting requires that, among otherassumptions, the same perforation is reproduced in some cells of a periodic grid (but not necessarilyall of them).In the following subsection, we recall the results proved in the periodic setting in [20]. Then, inSubsection 1.2, we study the case of a locally perturbed periodic geometry. We give conditions onthe perforations (inspired from [2, 3, 4]), which imply that, away from the defect, the perforationsbecome periodic, and which allow to prove convergence results similar to those of the periodic case. InSection 2, we give the main results of the article, together with some remarks and comments. Section 3is devoted to a Poincar´e-type inequality which is crucial in our proof. Finally, Section 4 is devoted tothe proofs of the results stated in Section 2. We start with some notations. We consider the d − dimensional unit cube Q =]0 , d with d ≥
2. Let O per0 be an open subset of Q such that O per0 ⊂⊂ Q and Q \ O per0 is connected. For simplicity, we chooseto impose that O per0 cannot intersect the boundary of Q . Suppose, for elliptic regularity, that O per0 isa C ,γ domain, for some 0 < γ < k ∈ Z d , O per k = O per0 + k and O per = [ k ∈ Z d O per k . (1.2)2 ε ε ε Figure 1: The periodic set for two choices of ε , ε and ε = ε / . k ∈ Z d , we have O per k ⊂⊂ Q k where Q k := Q + k .Given ε >
0, denote by O per ε the set of ε − perforations : O per ε = [ k ∈ Z d ε O per k = [ k ∈ Z d ε ( O per0 + k ) = ε O per . (1.3)We now define some useful functional spaces : H , per ( Q \ O per0 ) := (cid:8) u ∈ H ( R d \ O per ) s . t . u and ∂ i u are Q − periodic for all i ∈ { , ..., d } (cid:9) (1.4)and H , per0 ( Q \ O per0 ) := n u ∈ H , per ( Q \ O per0 ) s . t . u | ∂ O per0 = 0 o . (1.5)The two spaces defined by (1.4) and (1.5) are Hilbert spaces for the norm k u k H , per ( Q \O per0 ) := Z Q \O per0 | u | + Z Q \O per0 |∇ u | ! / . In the sequel, a function of H , per ( Q \ O per0 ) or of H , per0 ( Q \ O per0 ) will naturally be extended to R d \ O per by periodicity.All along the paper, we will denote the H − semi-norm on a set V by | · | H ( V ) : | u | H ( V ) := (cid:18)Z V |∇ u | (cid:19) / . Let Ω be a bounded, open and connected domain of R d . For ε >
0, denote by Ω ε := Ω \ O per ε . Notethat Ω ε is open and bounded but may not be connected.One has Ω ε = Ω ∩ [ k ∈ Z d ε ( Q k \ O per k ) . (1.6)Figure 1 shows the set Ω ε for two values ε and ε satisfying ε > ε . The set Ω ε is colored in lightgrey. We are interested in the Poisson problem (1.1). As we already mentionned, the source term f is supposed, as in [20], smooth and compactly supported in Ω. In fact, (see Remark 3.3 of [20]), itis sufficient to suppose that f ∈ C m − (Ω) and that D p f | ∂ Ω = 0 for all | p | ≤ m −
2, where m is theorder of the two scale expansion of u ε . As pointed out in [19], the assumptions on f can be weakenedfurther (see Remark 1.4 below). 3y a simple application of the Lax-Milgram Lemma, we have existence and uniqueness of a solution u ε to (1.1). In order to study the dependance of u ε on ε , we will need the following Lemma which isa Poincar´e-type inequality in H (Ω ε ). It is proved in [22, Lemma 1] (see also [5, Proposition 3.1]). Acrucial point in the non-periodic case will be to have a similar result, with the same scaling in ε . Thisis why we use Assumption (A2) , which allows to prove Lemma 3.2 below. Lemma 1.1 (Lemma 1 of [22]) . There exists a constant C > independent of ε such that ∀ u ∈ H (Ω ε ) , k u k L (Ω ε ) ≤ C ε k∇ u k L (Ω ε ) . This allows to prove Lemma 2 of [22]:
Lemma 1.2 (Lemma 2 of [22]) . The solution u ε of Problem (1.1) satisfies the estimates k u ε k L (Ω ε ) ≤ Cε and k u ε k H (Ω ε ) ≤ Cε, (1.7) where C is a constant independent of ε . Using a two-scale expansion of the form (see [20, Section 2]) u ε ( x ) = u per0 (cid:16) x, xε (cid:17) + εu per1 (cid:16) x, xε (cid:17) + ε u per2 (cid:16) x, xε (cid:17) + ε u per3 (cid:16) x, xε (cid:17) + · · · , (1.8)where the functions u per i are defined on Ω × ( Q \ O per0 ), smooth and Q − periodic in the second variable,one proves that, at least formally, u ε ( x ) = ε w per (cid:16) xε (cid:17) f ( x ) + · · · , (1.9)where w per is the periodic solution of ( − ∆ w per = 1 ,w per ∈ H , per0 ( Q \ O per0 ) . (1.10)We note that Problem (1.10) is well-posed. Indeed, it suffices to apply Lax-Milgram’s Lemma to thefollowing variational form ∀ v ∈ H , per0 ( Q \ O per0 ) , Z Q \O per0 ∇ w per · ∇ v = Z Q \O per0 v. The following convergence result is proved in [20, Theorem 3.1] (take m = 2 there). Theorem 1.3 (Consequence of Theorem 3.1 of [20]) . Assume that O per0 is an open subset of Q suchthat O per0 ⊂⊂ Q . Let f ∈ D (Ω) and u ε be the solution to (1.1) . Then there exists a constant C independent of ε such that ε − (cid:13)(cid:13) u ε − ε w per ( · /ε ) f (cid:13)(cid:13) L (Ω ε ) + (cid:12)(cid:12) u ε − ε w per ( · /ε ) f (cid:12)(cid:12) H (Ω ε ) ≤ Cε , (1.11) where w per ∈ H , per0 ( Q \ O per0 ) is the unique function satisfying − ∆ w per = 1 . Remark 1.4.
If we assume in addition that O per0 is of class C ,γ for some < γ < , then Theorem 1.3still holds true under the weaker hypotheses f ∈ H ∩ L ∞ (Ω) and f | ∂ Ω = 0 in the trace sense (see [19,Appendix A.2]). If we do not suppose that f vanishes on ∂ Ω , u ε − ε w ( · /ε ) f does not vanish on ∂ Ω either and we have the weaker estimate k u ε − ε w ( · /ε ) f k H (Ω ε ) ≤ Cε / N ( f ) , where N ( f ) = k f k L ∞ + k∇ f k L + k ∆ f k L . .2 The non-periodic case We aim at extending the previous results to non-periodically perforated medium, in the special caseof local perturbations of the periodic structure. More precisely, we define a reference periodic config-uration by (1.2)-(1.3)-(1.6), and, for each k ∈ Z d , we denote by O k the (non-periodic) perforation inthe cell k . We assume that O per0 is of class C ,γ for some 0 < γ <
1. Our first assumption is thatperforations should be sufficiently regular: (A1) For all k ∈ Z d , O k is a C ,γ open set such that O k ⊂⊂ Q k and Q k \ O k is connected. We now introduce geometric tools. For α >
0, define the Minkowski-content of ∂ O per0 (i.e a widenedboundary of O per0 ) by the set U per0 ( α ) := { x ∈ R d s . t . dist( x, ∂ O per0 ) < α } . Similarly, if k ∈ Z d and α >
0, denote the set U per k ( α ) := { x ∈ R d s . t . dist( x, ∂ O per k ) < α } = U per0 ( α ) + k. Now (see Figure 2 left), we define the reduction and the enlargement of O per k by O per , − k ( α ) := O per k \ U per k ( α ) and O per , + k ( α ) := O per k ∪ U per k ( α ) . One has O per , − k ( α ) ⊂ O per , + k ( α ) and U per k ( α ) = O per , + k ( α ) \ O per , − k ( α ) . We clearly have O per , + k ( α ) = { x ∈ R d s . t . dist( x, O per k ) < α } , (1.12)and O per , − k ( α ) = { x ∈ O per k s . t . dist( x, ∂ O per k ) > α } . (1.13)Assumption (A2) reads : (A2) There exists a sequence ( α k ) k ∈ Z d such that α k ≥ , ( α k ) k ∈ Z d ∈ ℓ ( Z d ) and ∀ k ∈ Z d , O per , − k ( α k ) ⊂ O k ⊂ O per , + k ( α k ) (1.14) i.e O k is between the enlargement and the reduction of O per k .Remark 1.5. Assumption (A2) is a way to impose that the defect is localized. In [2, 3, 4], such anassumption is written as a = a per + e a , with e a ∈ L q ( R d ) , and a per is periodic, where a is the coefficientof the considered elliptic equation. Here, writing a similar condition, we impose that the characteristicfunction of the perforations is a perturbation (i.e, a function in L q ( R d ) ) of the periodic case. For acharacteristic function, being in L q ( R d ) is equivalent to being in L ( R d ) , hence the condition (A2) . Note that if α k is sufficiently large, O per , − k ( α k ) = ∅ and Q k ⊂ O per , + k ( α k ). Thus, there is potentiallyno restriction on (a finite number of) O k . Figure 2 (right) explains Assumption (A2) .We define O := [ k ∈ Z d O k . (1.15)We split the domain R d \ O into two subdomains: R d \ ( O ∪ O per ) and O per \ O . Note that these domains are not necessarily connected.5 per , − k αα O per k O per , + k U per k ( α ) α k α k O k Figure 2: On the left, illustration of O per k (red), its widened boundary U per k , its enlargement O per , + k and its reduction O per , − k (grey). On the right, O k .We split the boundary of the domain O per \ O into two parts (the one surrounding O per and theone surrounding O ). For k ∈ Z d , we defineΓ k = ∂ O per k \ O k and Γ k = ∂ O k ∩ O per k s . t ∂ ( O per k \ O k ) = Γ k ∪ Γ k . (1.16)We denote by Γ (resp. Γ ) the union of the Γ k (resp. Γ k ), k ∈ Z d :Γ = [ k ∈ Z d Γ k , Γ = [ k ∈ Z d Γ k . (1.17)We also split the boundary of R d \ ( O ∪ O per ) into two parts. Write ∂ ( R d \ ( O ∪ O per )) = ∂ ( O∪O per ) , and define for k ∈ Z d Γ k = ∂ O k \ O per k s . t ∂ ( O k ∪ O per k ) = Γ k ∪ Γ k . (1.18)Note that ∂ O k = Γ k ∪ Γ k . Γ denotes the union of the Γ k . Note that Γ is in fact the complement of Γ in ∂ O . Figure 3 explainsthe above definitions. O per k \ O k O k Γ k Γ k Γ k O per k \ O k O k Γ k Γ k Figure 3: Pictures of perforated cells divided into two subdomains (white and light grey) with boundaryΓ i , i = 1 , ,
3. Left: O k ∩ O per k = ∅ . Right: O k ∩ O per k = ∅ We deduce from Assumption (A2)
Lemma A.1, A.2 and A.3, which are stated and proved inAppendix A. 6
Results
In order to state our main result, we first need to prove that a corrector exists:
Theorem 2.1 (Existence and uniqueness of the corrector) . Let ( O k ) k ∈ Z d be a sequence of open setssatisfying Assumptions (A1)-(A2) . Let O be defined by (1.15) , and g = R d \O per + e g, with e g ∈ L ( R d ) . There exists a unique e w ∈ H ( R d \ O ) such that w := w per + e w ∈ H , per ( Q ) + H ( R d \ O ) is solutionin the sense of distributions of ( − ∆ w = g in R d \ O w | ∂ O = 0 , (2.1) where w per ∈ H , per0 ( Q \ O per0 ) is the unique solution of the periodic corrector problem (1.10) extendedby zero to R d . Using Theorem 2.1 and a two-scale expansion, as it is done in the periodic case, we have thefollowing result, which is the generalization of Theorem 1.3 to the present setting
Theorem 2.2 (Convergence theorem in H − norm) . Let ( O k ) k ∈ Z d be a sequence of open sets satisfyingAssumptions (A1)-(A2) , and assume that O is defined by (1.15) .Let Ω ⊂ R d be a bounded domain and define for ε > the perforated set Ω ε := Ω \ ε O .Let f ∈ D (Ω) and u ε be the solution of Problem (1.1) . Then there exists a constant C > independent of ε such that ε − (cid:13)(cid:13) u ε − ε w ( · /ε ) f (cid:13)(cid:13) L (Ω ε ) + (cid:12)(cid:12) u ε − ε w ( · /ε ) f (cid:12)(cid:12) H (Ω ε ) ≤ Cε , (2.2) where w = w per + e w ∈ H , per ( Q ) + H ( R d \ O ) is the unique solution of the corrector Problem (2.1) with g = 1 . We note that the constant C appearing in Theorem 2.2 is independent of ε but depends on f , onthe non-periodic corrector constructed in Theorem 2.1 and on the Poincar´e-Friedrichs constant of Ω ε (denoted C in Lemma 3.2 below).Theorem 2.2 provides an error estimate of u ε − ε w ( · /ε ) f in H (Ω ε ) − norm. However, for thischoice of norm, the use of a non-periodic corrector appears to be irrelevant, which means that wecould also have used the periodic corrector w per in (2.2) without changing the rate of convergence.Indeed, we have (cid:13)(cid:13)(cid:13) ε e w (cid:16) · ε (cid:17) f (cid:13)(cid:13)(cid:13) H (Ω ε ) = O (cid:0) ε (cid:1) . (2.3)In order to prove (2.3), we only deal with the leading order term of the above quantity, that is, the L − norm of the gradient. One has Z Ω ε (cid:12)(cid:12)(cid:12) ∇ h ε e w (cid:16) · ε (cid:17) f i ( x ) (cid:12)(cid:12)(cid:12) ≤ ε Z Ω ε (cid:12)(cid:12)(cid:12) ∇ e w (cid:16) xε (cid:17)(cid:12)(cid:12)(cid:12) | f ( x ) | + 2 ε Z Ω ε e w (cid:16) xε (cid:17) |∇ f ( x ) | ≤ Cε Z Ω ε (cid:12)(cid:12)(cid:12) ∇ e w (cid:16) xε (cid:17)(cid:12)(cid:12)(cid:12) d x + Cε Z Ω ε (cid:12)(cid:12)(cid:12) e w (cid:16) xε (cid:17)(cid:12)(cid:12)(cid:12) d x. Thus, after the change of variable y = x/ε , Z Ω ε (cid:12)(cid:12)(cid:12) ∇ h ε e w (cid:16) · ε (cid:17) f i ( x ) (cid:12)(cid:12)(cid:12) d x ≤ Cε d +2 Z R d \O |∇ e w ( y ) | d y + Cε d +4 Z R d \O | e w ( y ) | d y. We thus have (2.3), which implies (since d ≥ (cid:13)(cid:13)(cid:13) u ε − ε w per (cid:16) · ε (cid:17) f (cid:13)(cid:13)(cid:13) H (Ω ε ) ≤ (cid:13)(cid:13)(cid:13) u ε − ε w (cid:16) · ε (cid:17) f (cid:13)(cid:13)(cid:13) H (Ω ε ) + (cid:13)(cid:13)(cid:13) ε e w (cid:16) · ε (cid:17) f (cid:13)(cid:13)(cid:13) H (Ω ε ) = O (cid:0) ε (cid:1) . w per instead of w in convergence Theorem 2.2 does not change the order O ( ε ) of the error.The following Theorem states that the use of w instead of w per improves the rate of convergencein L ∞ − norm for a non-periodic domain. Theorem 2.3 (Convergence Theorem in L ∞ − norm) . Let ( O k ) k ∈ Z d be a sequence of open sets satis-fying Assumptions (A1)-(A2) , and assume that O is defined by (1.15) . Assume that the C ,γ normsof the charts that flatten ∂ O k are uniformly bounded in k .Let Ω ⊂ R d be a bounded domain and define for ε > the perforated set Ω ε := Ω \ ε O .Let f ∈ D (Ω) and u ε be the solution of (1.1) . Then there exists a constant C > independent of ε such that (cid:13)(cid:13) u ε − ε w ( · /ε ) f (cid:13)(cid:13) L ∞ (Ω ε ) ≤ Cε , where w = w per + e w ∈ H ( Q ) + H ( R d \ O ) is the unique solution of (2.1) with g = 1 . Note that k ε e w ( · /ε ) f k L ∞ (Ω ε ) is generally of order ε exactly.Fix K ⊂ Ω. One has (cid:13)(cid:13)(cid:13)h ε e w (cid:16) · ε (cid:17) f i ( ε · ) (cid:13)(cid:13)(cid:13) L ∞ ( K ) ∼ ε → ε f (0) k e w k L ∞ ( K ) . Besides, Theorem 2.3 implies (cid:13)(cid:13)(cid:13)h u ε − ε w (cid:16) · ε (cid:17) f i ( ε · ) (cid:13)(cid:13)(cid:13) L ∞ ( K ) ≤ Cε . Thus, (cid:13)(cid:13)(cid:13)h u ε − ε w per (cid:16) · ε (cid:17) f i ( ε · ) (cid:13)(cid:13)(cid:13) L ∞ ( K ) ∼ Cε . We have the same results for L ∞ ( K ) − norm replaced by L ( K ) − norm. This proves that convergenceof u ε /ε − w ( · /ε ) f holds at the microscale in L − norm when we use w . This is not the case when weuse the periodic corrector w per . Remark 2.4.
This Remark is analogous to Remark 1.4 in the present non-periodic setting. Thecondition f ∈ D (Ω) can be weakened in Theorem 2.2 provided that we use Lemma 4.11 proved below.Under H¨older regularity conditions on the perforations, one has thanks to Lemma 4.11 that w ∈ L ∞ ( R d \ O ) and ∇ w ∈ L ∞ ( R d \ O ) . Thus, if we suppose that f ∈ H (Ω) and f | ∂ Ω = 0 in the tracesense, we obtain (see (4.24) ), k g ε k L (Ω ε ) ≤ k∇ w k L ∞ k∇ f k L + k w k L ∞ k ∆ f k L ≤ C k f k H (Ω) for ε < . We deduce by integration by parts that k u ε − ε w ( · /ε ) f k H (Ω ε ) ≤ Cε .If f does not vanish on ∂ Ω , we can prove that there exists a constant C independent of ε such that k u ε − ε w ( · /ε ) f k H (Ω ε ) ≤ Cε / N ( f ) . The proof is analogous to [19, Appendix A.2] provided we use Lemma 4.11 below. This requires f ∈ H ∩ L ∞ (Ω) . The main ingredient of the proof of Theorem 2.1 is the following Poincar´e-type inequality.
Theorem 3.1.
Let Q be the unit cube of R d and let U be an open subset of Q containing a box R = Q di =1 [ a i , b i ] . Then ∀ v ∈ H ( Q \ U ) s . t v | ∂U = 0 , Z Q \ U | v | ≤ d |R| Z Q \ U |∇ v | . (3.1) Similarly, ∀ v ∈ H ( Q ) s . t v | U = 0 , Z Q | v | ≤ d |R| Z Q |∇ v | .
8n important point in (3.1), is that the constant is explicit and depends only on R . This crucialpoint will allow us, with the help of Assumption (A2) , to prove Lemma 3.2 below, in which thefundamental point is that the constant does not depend on ε . We thus have an explicit scaling withrespect to ε , similarly to the periodic case. This allows us to adapt the proofs of [20]. Proof.
By density, it is enough to show the result for v ∈ C ( Q ) satisfying v = 0 on U . Fix x ∈ Q andwrite v ( x ) − v (ˆ x ) = Z ∇ v ((1 − t )ˆ x + tx ) · ( x − ˆ x )d t, where ˆ x = ( a i + x i ( b i − a i )) ≤ i ≤ d ∈ R . Note that v (ˆ x ) = 0 and | x − ˆ x | ≤ d . Thus by the Cauchy-Schwarz inequality | v ( x ) | ≤ d Z |∇ v ((1 − t )ˆ x + tx ) | d t. Integrating with respect to x ∈ Q andexchanging the two integrals yields Z Q | v ( x ) | d x ≤ d Z (cid:18)Z Q |∇ v ((1 − t )ˆ x + tx ) | d x (cid:19) d t Fix t ∈ [0 ,
1] and define the diffeomorphism φ t : Q ∋ x (1 − t )ˆ x + tx . Note that φ t ( Q ) ⊂ Q and that | det J ( φ t ) | = d Y i =1 [(1 − t )( b i − a i ) + t ] ≥ d Y i =1 ( b i − a i ) . Thus by a change of variables, Z Q |∇ v ((1 − t )ˆ x + tx ) | d x ≤ Q di =1 ( b i − a i ) Z Q |∇ v | . Integrating with respect to t concludes the proof.Theorem 3.1 and Assumption (A2) allow to prove the following, which is a generalization to thepresent setting of Lemma 1.1. Lemma 3.2 (Poincar´e-type inequality in H (Ω ε )) . Let ( O k ) k ∈ Z d be a sequence of open sets such that O k ⊂⊂ Q k . Suppose that the sequence ( O k ) k ∈ Z d satisfies Assumption (A2) . Let Ω be an open subsetof R d . Define for ε > , Ω ε = Ω \ ε O = Ω ∩ [ k ∈ Z d ε ( Q k \ O k ) . There exists a constant
C > independent of ε such that ∀ u ∈ H (Ω ε ) , Z Ω ε u ≤ Cε Z Ω ε |∇ u | . Proof.
We first recall (see Lemma A.3 in the appendix) that K := { k ∈ Z d , O k ∩ O per k = ∅} is finite.We show that there exists e ρ > k ∈ Z d , there exists a box R k ⊂ O k satisfying |R k | ≥ e ρ . Fix k ∈ Z d , there are two cases : • Case 1 : k ∈ K (see Lemma A.1). The open set O k contains a ball and thus a box R k . • Case 2 : k / ∈ K . By Lemma A.2, there exists a ball B k ⊂ O k such that | B k | ≥ ρ with ρ independent of k . Thus, there exists a box R k ⊂ O k such that |R k | ≥ C ( d ) ρ where C ( d ) is aconstant depending only on d .We define e ρ := min (cid:18) C ( d ) ρ, min k ∈K |R k | (cid:19) > ∀ k ∈ Z d , ∀ w ∈ H (cid:0) R d \ O (cid:1) , Z Q k \O k w ≤ d e ρ Z Q k \O k |∇ w | . k ∈ Z d each inequality and defining C := d/ e ρ yields ∀ w ∈ H ( R d \ O ) , Z R d \O w ≤ C Z R d \O |∇ w | . (3.2)Now, fix u ∈ H (Ω ε ). We extend u by zero to R d \ ε O and define v := u ( ε · ). It is clear that v ∈ H ( R d \ O ) and that ∀ y ∈ R d \ O , ∇ v ( y ) = ε ∇ u ( εy ) . (3.3)Applying (3.2) with w = v ∈ H ( R d \ O ) and using (3.3) yields Z ε Ω ε u ( εy )d y = Z R d \O u ( εy ) d y ≤ Cε Z R d \O |∇ u | ( εy ) d y = Cε Z ε Ω ε |∇ u | ( εy )d y. Making the change of variables x = εy in each integral finally concludes the proof. The aim of this section is to find an asymptotic equivalent of u ε as ε goes to zero. We begin by thetwo scale expansion of u ε . Write u ε ( x ) = u (cid:16) x, xε (cid:17) + εu (cid:16) x, xε (cid:17) + ε u (cid:16) x, xε (cid:17) + ε u (cid:16) x, xε (cid:17) + · · · , where the functions u i are now defined on Ω × ( R d \ O ), and are of the form u per i + e u i . Suppose that e u i ( x, · ) ∈ H ( R d \ O ) and use the u per i ’s defined in Section 1.1 and extended by zero to R d . Becauseof the Dirichlet Boundary conditions on u ε , we impose that u i ( x, y ) = 0 for y ∈ ∂ O and any x ∈ Ω.The calculations leading to (1.9) (see [20, Section 2]) are still valid, so we have: − ∆ y u = 0 − ∆ y u − ∇ x · ∇ y ) u = 0 − ∆ y u − ∇ x · ∇ y ) u − ∆ x u = f − ∆ y u − ∇ x · ∇ y ) u − ∆ x u = 0 · · · , (4.1)where all these equations are posed on Ω × ( R d \ O ). These equations imply that u and u areconstantly equal to zero. Indeed, fix x ∈ Ω. Since u per0 ≡
0, we get that f u ( x, · ) satisfies the PDE ( − ∆ y f u = 0 in R d \ O , f u | ∂ O = 0 . Multiplying by f u ( x, · ) ∈ H ( R d \ O ) and integrating by parts yields f u ( x, · ) ≡
0. Thus u ≡ u ≡
0. We are now left with the following equation on u : ( − ∆ y u ( x, y ) = f ( x ) in R d \ O u ( x, y ) = 0 , x ∈ Ω , y ∈ ∂ O . (4.2)According to (4.2), u ( x, y ) = f ( x ) w ( y ), where w is a solution to the corrector equation (2.1) with g ≡
1. This is why we introduced the corrector equation.10 .2 Proof of the existence of a corrector
The aim of this section is to prove Theorem 2.1. The difficulty of this theorem is that equation (2.1)is posed on an unbounded domain.We search for w in the form w per + e w , where we impose that e w ∈ H ( R d \ O ). We write theequation on e w and prove by energy minimization that there is a solution. The equation we want to solve for e w is − ∆ e w = R d \O per + e g + ∆ w per , (4.3)where e g ∈ L ( R d ) and w per ∈ H , per ( Q ) is the solution to (1.10) defined in Section 1. We recall that w per is extended by zero in O per . We impose that e w = − w per on ∂ O .It is worth noticing that w per / ∈ H ( Q ), and thus the right-hand side of (4.3) cannot be in L ( R d \O ).Thus the linear form of the weak formulation of (4.3) is not of the form v R f v . In fact, we willhave to deal with boundary terms along ∂ O per . These terms express the fact that ∆ w per is a Diracmeasure on ∂ O per (or that w per has normal derivative jumps along ∂ O per ). Notation.
We denote by ∂u∂n (cid:12)(cid:12) ext (resp. ∂u∂n (cid:12)(cid:12) int ) the exterior normal derivative of u on the outside(resp. inside) of a piecewise smooth closed surface Γ (when it is defined i.e u is H on each side of theboundary). Definition 4.1.
We say that e w ∈ H ( R d \ O ) is a weak solution of (4.3) if ∀ v ∈ C c ( R d \ O ) , Z R d \O ∇ e w · ∇ v + Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext v − Z R d \O e gv = 0 , (4.4) and e w | ∂ O = − w per in the trace sense. Remark 4.2.
We could also have written equation (4.3) as a system of PDEs coupled by transmissionconditions: − ∆ e w = e g in R d \ ( O ∪ O per ) − ∆ e w = e g in O per \ O e w = − w per on Γ ∪ Γ ∂ e w∂n (cid:12)(cid:12) ext + ∂ e w∂n (cid:12)(cid:12) int = ∂w per ∂n (cid:12)(cid:12) ext on Γ (4.5) The three first equations are obviously necessary. The last equation is necessary to guarantee that w = w per + e w ∈ H ( R d \ O ) . Using standard tools of the calculus of variations, one easily proves the following:
Lemma 4.3.
Assume that e w ∈ H (cid:0) R d \ O (cid:1) . It is a weak solution of (4.3) in the sense of Defini-tion 4.1, if and only if it is a solution to the following minimization problem: inf e w ∈ V ( Z R d \O |∇ e w | + Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext e w − Z R d \O e g e w ) , (4.6) where the minimization space V is defined by V := (cid:8) e w ∈ H ( R d \ O ) s . t e w | ∂ O = − w per (cid:9) . (4.7) Definition 4.4.
Let e w ∈ V . We denote by f W its extension to R d defined by e w = − w per in O . per k \ O k O k ∩ O per k O k Γ k Γ k Γ k f W = 0 f W = 0 f W = − w per f W = 0 O per k O k f W = − w per Γ k Γ k Figure 4: Function e w (its extension f W ) on a perforated cell with and without overlappingThe extension f W of e w satisfies f W ∈ H ( R d ) under Assumptions (A1)-(A2) on the sequence( O k ) k ∈ Z d . Figure 4 shows a function e w ∈ V (extended to O by − w per ).In order to study the minimization problem (4.6), we will need the following Poincar´e type inequalityon V . Lemma 4.5 (Poincar´e-type inequality in V ) . Let ( O k ) k ∈ Z d be a sequence of sets satisfying Assumptions (A1)-(A2) . Define O = S k ∈ Z d O k . Let w per be the periodic corrector solution to (1.10) . There existconstants C > and C > such that for any e w ∈ V , Z R d \O e w ≤ C Z R d \O |∇ e w | + C . (4.8) Denoting by f W the extension of e w (see Definition 4.4), we also have Z R d f W ≤ C Z R d |∇ f W | + C . Proof.
Fix e w ∈ V and extend e w by − w per in O . This gives a function f W ∈ H ( R d ). Note that ∀ k ∈ Z d , f W = 0 in O k ∩ O per k . Fix k ∈ Z d , there are two cases : Case 1 : k ∈ K , that is O k ∩ O per k = ∅ . Then e w + w per = 0 on ∂ O k . Thus classical Poincar´einequality gives the existence of C k = C ( Q k \ O k , ∂ O k ) satisfying C k ≥ Z Q k \O k ( e w + w per ) ≤ C k Z Q k \O k |∇ e w + ∇ w per | . We get Z Q k \O k e w ≤ C k Z Q k \O k |∇ e w | + 2 C k k w per k H ( Q k \O k ) ≤ C k Z Q k \O k |∇ e w | + 2 C k k w per k W , ∞ ( Q ) . (4.9)Now, the fact that f W = − w per on O k implies Z Q k f W ≤ C k Z Q k |∇ f W | + 2 C k k w per k H ( Q k ) ≤ C k Z Q k |∇ f W | + 2 C k k w per k W , ∞ ( Q ) . (4.10)12 ase 2 : k / ∈ K so that O k ∩ O per k = ∅ . Note that f W = 0 on O k ∩ O per k . We now use Lemma A.2:there exists a ball B k ⊂ O k ∩O per k such | B k | ≥ ρ and thus a box R k ⊂ O k ∩O per k such that |R k | ≥ C ( d ) ρ where C ( d ) depends only on the dimension.Theorem 3.1 gives the existence of a constant C = C ( d ) /ρ chosen ≥ Z Q k f W ≤ C Z Q k |∇ f W | . (4.11)Recall that Z O k |∇ f W | ≤ k∇ w per k L ∞ ( Q ) |O k \ O per k | ≤ k w per k W , ∞ ( Q ) |O k \ O per k | . We thus have Z Q k \O k e w ≤ C Z Q k \O k |∇ e w | + C k w per k W , ∞ ( Q ) |O k \ O per k | . (4.12)Define C = max (cid:18) k ∈K C k , C (cid:19) and C = C k w per k W , ∞ ( Q ) |K| + X k ∈ Z d |O k \ O per k | < + ∞ . We have proved (see equations (4.9) and (4.12)) that ∀ k ∈ Z d , Z Q k \O k e w ≤ C Z Q k \O k |∇ e w | + C k w per k W , ∞ ( Q ) δ k , where δ k = 1 if k ∈ K and δ k = |O k \ O per k | if k / ∈ K .Summing over k gives the desired results for e w . Equations (4.10) and (4.11) give the analogousresult for f W .Using Lemma 4.5, we prove the following: Lemma 4.6.
Suppose that the sequence ( O k ) k ∈ Z d satisfies Assumption (A2) . Let e w ∈ V and denoteby f W ∈ H ( R d ) its extension (see Definition 4.4). Then, one has the following estimates: (cid:12)(cid:12)(cid:12)(cid:12)Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext f W (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + 14 k∇ f W k L ( R d ) , (4.13) where C is a constant independent of e w , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R d \O e g f W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k e g k L ( R d ) k f W k L ( R d ) (4.14) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z O per \O f W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ |O per \ O| k f W k L ( R d ) . (4.15) Proof.
Fix e w ∈ V . Let us first show that f W ∈ H ( R d ). Write Z O f W + Z O |∇ f W | = Z O\O per ( w per ) + Z O\O per |∇ w per | ≤ k w per k W , ∞ ( Q ) X k ∈ Z d |O k \ O per k | . (4.16)By Lemma A.1, we conclude that f W ∈ H ( O ). This proves that f W ∈ H ( R d ).13e now prove estimate (4.13). Standard elliptic regularity implies ∂w per ∂n (cid:12)(cid:12) ext ∈ L ∞ ( ∂ O per ). Weapply the trace theorem [16, Theorem 1, p 272] for p = 1 to the open subset O per0 (and thus to O per k by periodicity with the same constant): (cid:12)(cid:12)(cid:12)(cid:12)Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext e w (cid:12)(cid:12)(cid:12)(cid:12) ≤ X k ∈ Z d Z Γ k (cid:12)(cid:12)(cid:12)(cid:12) ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext e w (cid:12)(cid:12)(cid:12)(cid:12) ≤ X k ∈ Z d Z ∂ O per k (cid:12)(cid:12)(cid:12)(cid:12) ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext f W (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ∂ O per0 ) X k ∈ Z d Z ∂ O per k | f W | ≤ C ( w per , O per ) X k ∈ Z d Z O per k | f W | + Z O per k |∇ f W | ! . (4.17)Now, recall that f W = 0 in O k ∩ O per k , so that using successively the Cauchy-Schwarz inequality andtrace continuity (see [16, Theorem 1, p 272] with p = 2), we have (cid:12)(cid:12)(cid:12)(cid:12)Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext f W (cid:12)(cid:12)(cid:12)(cid:12) ≤ C X k ∈ Z d |O per k \ O k | / (cid:16) k f W k L ( O per k ) + k∇ f W k L ( O per k ) (cid:17) . We use the inequality ab ≤ D a + b D with D to be chosen later: (cid:12)(cid:12)(cid:12)(cid:12)Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext f W (cid:12)(cid:12)(cid:12)(cid:12) ≤ CD X k ∈ Z d |O per k \ O k | + CD X k ∈ Z d (cid:16) k f W k L ( O per k ) + k∇ f W k L ( O per k ) (cid:17) . (4.18)Thus, (cid:12)(cid:12)(cid:12)(cid:12)Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext f W (cid:12)(cid:12)(cid:12)(cid:12) ≤ CD X k ∈ Z d |O per k \ O k | + CD (cid:16) k f W k L ( R d ) + k∇ f W k L ( R d ) (cid:17) . Lemma 4.5 implies CD (cid:16) k f W k L ( R d ) + k∇ f W k L ( R d ) (cid:17) ≤ CC D k∇ f W k L ( R d ) + CC D .
Choosing D = 8 CC yields finally (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ O per ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext f W (cid:12)(cid:12)(cid:12)(cid:12) ≤ C X k ∈ Z d |O per k \ O k | + C + 14 k∇ f W k L ( R d ) , (4.19)with C being a constant independent of e w . We infer (4.13) thanks to Lemma A.1.The two last estimates (4.14) and (4.15) are consequences of the Cauchy-Schwarz inequality. Remark 4.7.
Let v ∈ H ( R d \ O ) . Computations (4.17) - (4.18) with e w replaced by v and D = 1 arevalid and give (cid:12)(cid:12)(cid:12)(cid:12)Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext v (cid:12)(cid:12)(cid:12)(cid:12) ≤ C |O per \ O| + C k v k H ( R d \O ) . Thus, the linear form v R Γ ∂w per ∂n (cid:12)(cid:12) ext v is continuous on H ( R d \ O ) . First, we prove below that the minimization space V is not empty: Lemma 4.8.
Let ( O k ) k ∈ Z d satisfy Assumption (A1) and Assumption (A2) . Then V defined by (4.7) is not empty.Proof. We want to build a function φ ∈ H ( R d \ O ) satisfying the boundary conditions φ = − w per on ∂ O . We will first build φ on each cell Q k .Let k ∈ Z d . Recall that δ per0 = dist( O per k , ∂Q k ) and that δ is defined in Lemma A.3 of theAppendix. Set ε per k = min(2 α k , δ per0 /
2) and ε k = min( α k , δ / α k −→ | k |→ + ∞
0, there exists k such that ∀| k | ≥ k , ε per k = 2 α k and ε k = α k . Define U per k ( ε per k ) (resp. U k ( ε k )) to be the ε per k (resp. ε k ) Minkowski content of ∂ O per k (resp. ∂ O k )that is U per k ( ε per k ) = (cid:8) x ∈ R d s . t dist( x, ∂ O per k ) < ε per k (cid:9) ⊂ Q k and U k ( ε k ) = (cid:8) x ∈ R d s . t dist( x, ∂ O k ) < ε k (cid:9) ⊂ Q k . Denote O per , + k ( ε per k ) = O per k ∪ U per k ( ε per k ) = (cid:8) x ∈ R d s . t dist( x, O per k ) < ε per k (cid:9) ⊂ Q k and O + k ( ε k ) = O k ∪ U k ( ε k ) = (cid:8) x ∈ R d s . t dist( x, O k ) < ε k (cid:9) ⊂ Q k . Now, let χ k ∈ C ∞ c ( Q k ) be a cut-off function satisfying ≤ χ k ≤ χ k ≡ O k supp( χ k ) ⊂ O + k , supp( ∇ χ k ) ⊂ U k ( ε k ) |∇ χ k | ≤ C/ε k . We define φ k = − χ k w per . It is clear that φ k ∈ H ( R d ) and that φ k = − w per on ∂ O k .One defines φ ( x ) = X k ∈ Z d φ k ( x ) = X k ∈ Z d φ k ( x )1 Q k ( x ) . Note that since supp( φ k ) ⊂ Q k , all terms but one (which depends on x ) vanish in the above sum.Thus φ = − w per on ∂ O .Our goal is to prove that φ ∈ H ( R d \ O ) to conclude the proof. By Lemma 4.5, it is sufficient toshow that ∇ φ ∈ L ( R d \ O ). Showing this is equivalent to prove that X k ∈ Z d k∇ φ k k L ( U k ( ε k )) < + ∞ . We are thus left to estimate each term k∇ φ k k L ( U ( ε k )) where k ∈ Z d . We study these terms only when | k | ≥ k and k / ∈ K where K is defined in Lemma A.1 of the Appendix (there are only a finite numberof terms k such that k ∈ K and | k | < k ).Let k ∈ Z d such that | k | ≥ k and k / ∈ K that is O k ∩ O per k = ∅ . One has - using Assumption (A2) - the inclusions, O k ⊂ O + k ( α k ) ⊂ O per , + k (2 α k ) and U k ( α k ) ⊂ U per k (2 α k ) . (4.20)We write Z U k ( α k ) |∇ ( χ k w per ) | ≤ Z U k ( α k ) |∇ w per | | χ k | + 2 Z U k ( α k ) | w per | |∇ χ k | ≤ k∇ w per k L ∞ ( U k ( α k )) |U k ( α k ) | + 2 k w per k L ∞ ( U k ( α k )) C α k |U k ( α k ) | . Using that ∇ w per ∈ L ∞ ( R d ), that d ( U k ( α k ) , O per k ) ≤ α k and that w per = 0 in O per k , we infer k w per k L ∞ ( U k ( α k )) ≤ α k k∇ w per k L ∞ ( Q ) . We conclude that Z U k ( α k ) |∇ φ k | ≤ C |U k ( α k ) | + C |U k ( α k ) | α k /α k ≤ C |U k ( α k ) | . Z U k ( α k ) |∇ φ k | ≤ C |U per k (2 α k ) | . We deduce that for k large enough, R U k ( α k ) |∇ φ k | ≤ Cα k (see (A.3)). Since ( α k ) k ∈ Z d ∈ ℓ ( Z d ), oneconcludes that φ ∈ H ( R d \ O ). Proposition 4.9.
Under the assumptions (A1) and (A2) , the minimization Problem (4.6) has asolution.Proof.
Let ( f w n ) n ∈ N ⊂ V be a minimizing sequence of Problem (4.6) which exists by Lemma 4.8, thatis 12 Z R d \O |∇ f w n | + Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext f w n − Z R d \O e g f w n −→ n → + ∞ inf u ∈ V J ( u ) . We extend each f w n by − w per in the perforations and denote by g W n the extension (see Definition 4.4).The sequence 12 Z R d |∇ g W n | + Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext g W n − Z R d \O e g g W n admits an upper bound independent of n . We first prove that k∇ g W n k L ( R d ) is bounded independentlyof n . We use Lemma 4.5 and Lemma 4.6 to bound each term: (cid:12)(cid:12)(cid:12)(cid:12)Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext g W n (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + 14 k∇ g W n k L ( R d ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R d \O e g g W n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k g W n k L ( R d ) ≤ Lemma . C + C k∇ g W n k L ( R d ) , where C denotes various constants independent of n . Hence, one gets C ≥ Z R d |∇ g W n | + Z ∂ O per ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext g W n − Z R d \O e g g W n ≥ k∇ g W n k L ( R d ) − C k∇ g W n k L ( R d ) − C, and thus k∇ g W n k L ( R d ) ≤ C k∇ g W n k L ( R d ) + C. This proves that k∇ g W n k L ( R d ) is bounded independently of n . With Lemma 4.5, one deduces that k g W n k H ( R d ) is also bounded independently of n .Thus, by weak compactness, there exists a weak limit f W ∈ H ( R d ) such that g W n − ⇀ H f W and g W n −→ L f W .
Denote e w = f W | R d \O . We first show that e w ∈ V .Strong convergence in L and f W n = − w per in O k imply f W = − w per in O k . For the boundary ∂ O k ,recall that the trace operator T k (see [16, Theorem 1, p 272] ) is weakly continuous from H ( O k ) to L ( ∂ O k ). Thus e w | ∂ O k = T k f W = − T k w per = − w per | ∂ O k . Since this is true for all k ∈ Z d , we have proved that e w | ∂ O = − w per . Moreover, e w ∈ H ( R d \ O ). Thus e w ∈ V .We can now pass to the limit. Since w ∋ H ( R d \ O ) R |∇ w | is convex and continuous (in thestrong norm), it is weakly lower semi-continuous and thus Z R d \O |∇ e w | ≤ lim inf n → + ∞ Z R d \O |∇ f w n | . (4.21)16y weak H − convergence, since e g ∈ L ( R d \ O ), Z R d \O e g f w n −→ n → + ∞ Z R d \O e g e w. (4.22)Let us treat the remaining term. We first recall (see Remark 4.7) that the linear form v R Γ ∂w per ∂n (cid:12)(cid:12) ext v is strongly and thus weakly continous on H ( R d \ O ). We apply this continuity to v n = e w n − φ , where φ was defined in the proof of Lemma 4.8. Since Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext φ = 0 , we deduce Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext f w n −→ n → + ∞ Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext e w. (4.23)Finally, collecting (4.21), (4.22)and (4.23) and letting n → + ∞ , we conclude that12 Z R d |∇ e w | + Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext e w − Z R d \O e g e w ≤ inf u ∈ V J ( u ) . This finishes the proof of existence.To conclude the proof, we prove uniqueness: let f w and f w be two weak solutions of (4.6) (in thesense of Definition 4.1). We have that ∀ v ∈ H ( R d \ O ) , Z R d \O ∇ f w i · ∇ v + Z Γ ∂w per ∂n (cid:12)(cid:12)(cid:12)(cid:12) ext v − Z R d \O e gv = 0 , for i = 1 ,
2. Substracting the two equations yields ∀ v ∈ H ( R d \ O ) , Z R d \O ∇ ( f w − f w ) · ∇ v = 0Since f w − f w ∈ H ( R d \ O ), we may choose v = f w − f w in the previous expression. The Poincar´einequality on Q \ O with Γ = ∂ O implies f w − f w = 0. Remark 4.10.
We could also have applied Lax-Milgram’s lemma to show that Problem (4.3) admitsa weak solution. The ingredients are basically the same. Coercivity of the bilinear form is a directconsequence of Lemma 3.1 (see (3.2) ). Continuity is proved using the same method as in the proof ofProposition 4.9, when passing to the limit in the minimizing sequence. H convergence Proof of Theorem 2.2.
We first define the second order approximation of u ε . Let g = R d \O . Withthis choice of g , one has e g = O per \O − O\O per = X k ∈ Z d (cid:16) O per k \O k − O k \O per k (cid:17) . Moreover, Lemma A.1 implies that e g ∈ L ( R d ). Thus we can apply Theorem 2.1 and get the existenceof a unique function e w ∈ H ( R d \ O ) such that w := w per + e w satisfies ( − ∆ w = 1 in R d \ O w | ∂ O = 0 . in sense of distribution. Note that w ∈ H ( R d \ O ).17ow, set φ ε := u ε − ε w ( · /ε ) f. Since f ∈ D (Ω), w = 0 on ∂ O and w ∈ H ( R d \ O ), one gets that φ ε ∈ H (Ω ε ).We have, in the sense of distributions, − ∆ φ ε = f + ∆ w (cid:16) · ε (cid:17) f + 2 ε ∇ w (cid:16) · ε (cid:17) · ∇ f + ε w (cid:16) · ε (cid:17) ∆ f = f − f + εg ε = εg ε , (4.24)where g ε = 2 ∇ w (cid:16) · ε (cid:17) · ∇ f + εw (cid:16) · ε (cid:17) ∆ f. Note that k g ε k L (Ω ε ) is bounded independently of ε .Next, we multiply (4.24) by φ ε , integrate by parts and apply the Cauchy-Schwarz inequality: Z Ω ε |∇ φ ε | = ε Z Ω ε g ε φ ε ≤ Cε (cid:18)Z Ω ε φ ε (cid:19) / . Thanks to Lemma 3.2, one concludes that (cid:18)Z Ω ε |∇ φ ε | (cid:19) / ≤ Cε and (cid:18)Z Ω ε φ ε (cid:19) / ≤ Cε , which concludes the proof. L ∞ convergence We first prove the following Lemma:
Lemma 4.11.
Let ( O k ) k ∈ Z d be a sequence of open sets satisfying Assumptions (A1)-(A2) . Let w bethe solution to (2.1) with g = 1 . Then w ∈ L ∞ (cid:0) R d \ O (cid:1) . Moreover, if the C ,γ norms of the chartsthat flatten ∂ O k are uniformly bounded in k , we have that ∇ w ∈ L ∞ (cid:0) R d \ O (cid:1) .Proof. Let us first prove that w ∈ L ∞ ( R d \ O ). Fix k ∈ Z d and recall that ( − ∆ w = 1 in Q k \ O k w | ∂ O k = 0 . (4.25)There exists a constant C independent of k such that k w k L ∞ ( ∂Q k ) ≤ C. (4.26)Proving (4.26) is equivalent to prove that k e w k L ∞ ( ∂Q k ) ≤ C . Lemma A.3 implies that there exists δ > k ∈ Z d , O k ∪ O per k ⊂ [ k + δ, k + 1 − δ ] d . By translation invariance and since ∂Q iscompact, there exists x , x , ..., x ℓ ∈ ∂Q such that ∀ k ∈ Z d , ∂Q k ⊂ ℓ [ i =1 B ( x i + k, δ/ . (4.27)On each ball B ( x i + k, δ ), e w satisfies − ∆ e w = 0. De Giorgi-Nash-Moser Theory (see [24], Theorem4.22, p. 155) implies that there exists a constant C = C ( d, δ ) independent of x i and k such thatsup B ( x i + k,δ/ | e w | ≤ C ( d, δ ) Z B ( x i + k,δ ) | e w ( x ) | d x ! ≤ C k e w k L ( R d \O ) . (4.28)18he inclusion (4.27) together with (4.28) proves (4.26). We now apply the Maximum principle on w for each domain Q k \ O k . Let R be such that Q k ⊂ B ( k, R ). The functions w + ( x ) := w ( x ) + | x − k | d + k w k L ∞ ( ∂Q k ) and w − ( x ) = w ( x ) + | x − k | − R d − k w k L ∞ ( ∂Q k ) are respectively supersolution and subsolution of (4.25). Thus, thanks to (4.26), k w k L ∞ ( Q k \O k ) isbounded independently of k . Hence w ∈ L ∞ (cid:0) R d \ O (cid:1) .For ∇ w , we use H¨older Regularity results for the first derivatives. First recall that Assumption (A1) implies that R d \ O is connected. For all x ∈ R d \ O such that dist( x, ∂ O ) > δ/
2, there existsa ball B x centered at x such that dist( B x , ∂ O ) = δ/
2. Interior estimates (see [18], Theorem 8.32, p.210) give the existence of a constant C = C ( δ, d ) independent of x such that k w k C ,γ ( B x ) ≤ C (cid:0) k w k L ∞ ( R d \O ) + 1 (cid:1) ≤ C. We have proved that ∇ w is bounded at a distance δ/ ∂ O .For the proof up to the boundary ∂ O , we use Corollary 8.36 p. 212 of [18] with the sets Ω k = { x s . t dist( x, ∂ O k ) < δ } \ O k , Ω ′ k = { x s . t dist( x, ∂ O k ) < δ/ } \ O k and T k = ∂ O k . We have d ′ = δ/ k and thus k w k C ,γ (Ω ′ k ) ≤ C ( T k , δ, d ) (cid:0) k w k L ∞ ( R d \O ) + 1 (cid:1) where the dependence on T k appears through the C ,γ − norms of the charts that flatten T k (see [18],p.210). By hypothesis, we get that C ( T k ) ≤ C . This concludes the proof. Proof of Theorem 2.3.
Fix ε > v ε = u ε ( ε · ). Then v ε ∈ H ( ε Ω ε ) and satisfies − ∆ v ε = ε f ( ε · ) in 1 ε Ω ε v ε = 0 on ∂ (cid:18) ε Ω ε (cid:19) . (4.29)Define ψ ε := v ε − ε wf ( ε · ) ∈ H ( 1 ε Ω ε ) , and note that − ∆ ψ ε = ε [2 ∇ w · ∇ f ( ε · ) + εw ∆ f ( ε · )] =: ε h ε . Lemma 4.11 and the fact that f ∈ D (Ω) imply that k h ε k L ∞ ( ε Ω ε ) ≤ C for all 0 < ε <
1. Define ψ + ε = ψ ε + ε k h ε k L ∞ ( ε Ω ε ) ( w + k w k L ∞ ) . Then ψ + ε is a supersolution of Problem (4.29). Thus, by the weak maximum principle (see [18]Theorem 8.1, p.179), one gets that ψ + ε ≥ ε Ω ε . Similarly, ψ − ε = ψ ε − ε k h ε k L ∞ ( ε Ω ε ) ( w + k w k L ∞ )is a subsolution of (4.29) and thus ψ − ε ≤ ε Ω ε . Finally, − ε k h ε k L ∞ ( ε Ω ε ) ( w + k w k L ∞ ) ≤ ψ ε ≤ ε k h ε k L ∞ ( ε Ω ε ) ( w + k w k L ∞ ) . The bound k h ε k L ∞ ( ε Ω ε ) ≤ C and Lemma 4.11 imply k ψ ε k L ∞ ( ε Ω ε ) ≤ Cε . Rescaling back concludesthe proof. 19 Proof of technical lemmas
Lemma A.1.
Let ( O k ) k ∈ Z d be a sequence of open sets satisfying Assumption (A2) . Then, X k ∈ Z d |O k ∆ O per k | < + ∞ , (A.1) where A ∆ B = ( A ∪ B ) \ ( A ∩ B ) = ( A \ B ) ∪ ( B \ A ) stands for the symmetric subset difference.Moreover, if K := { k ∈ Z d s . t O k ∩ O per k = ∅} , then |K| < + ∞ .Proof. First note that, using (1.14), O k \ O per k ⊂ U per k ( α k ) and O per k \ O k ⊂ U per k ( α k ) . (A.2)We now use [17, Theorem 3.2.39] to control the measure of U per k ( α k ): there exists α > ∀ α < α, |U per0 ( α ) | ≤ C | ∂ O per0 | α. (A.3)By translation invariance, the above assertion is true for U per0 ( α ) replaced by U per k ( α ) : ∀ k ∈ Z d , ∀ α < α, |U per k ( α ) | ≤ C | ∂ O per0 | α. For k large enough such that α k < α , one thus have |U per k ( α k ) | ≤ e Cα k where e C is a constant. This,together with (A.2), proves the (A.1).The fact that |K| < + ∞ is a direct consequence of (A.1) and of the fact that for all k ∈ K , |O k ∆ O per k | ≥ |O per k \ O k | = |O per0 | . Lemma A.2.
Let ( O k ) k ∈ Z d be a sequence of open sets satisfying Assumption (A2) . There exists ρ > such that ∀ k ∈ K c , ∃ B k s . t | B k | ≥ ρ and B k ⊂ O k ∩ O per k , where B k denotes an open ball and K is defined in Lemma A.1.Proof. Since O per0 is open, it contains a ball B ⊂ B ⊂ O per0 . One has δ := dist( B, ∂ O per0 ) > k ∈ Z d , B k := B + k satisfies B k ⊂ B k ⊂ O per k and δ = dist( B k , ∂ O per k ) . Since ( α k ) k ∈ Z d ∈ ℓ ( Z d ), there exists k such that for all | k | ≥ k , α k ≤ δ/
2. Equation (1.13)implies B k ⊂ O per , − k ( α k ) for | k | ≥ k . This proves that ∀| k | ≥ k , B k ⊂ O per k ∩ O k . If | k | < k and O per k ∩ O k = ∅ , there exists a ball B k such that B k ⊂ O per k ∩ O k . Defining ρ = min (cid:18) min | k |
Let ( O k ) k ∈ Z d be a sequence of open sets satisfying Assumptions (A1)-(A2) . Thereexists δ > such that ∀ k ∈ Z d , dist( O k , ∂Q k ) ≥ δ . roof. Recall that for all k ∈ Z d , O per k ⊂⊂ Q k . Thus, by translation invariance, there exists a constant δ per0 > k such that ∀ k ∈ Z d , dist( O per k , ∂Q k ) = δ per0 . One has, using Assumption (A2) and in particular the inclusion O k ⊂ O per , + k ( α k ), δ k := dist( O k , ∂Q k ) ≥ δ per0 − α k ≥ δ per0 / k large enough, say | k | ≥ k .Since for all | k | < k , Assumption (A1) gives δ k = dist( O k , ∂Q k ) > , the Lemma is proved by defining δ := min (cid:18) δ per0 , min | k | We thank C. Le Bris for comments and suggestions that greatly improved the manuscript. References [1] Emilio Acerbi, Valeria ChiadPiat, Gianni Dal Maso, and Danilo Percivale. An extension theo-rem from connected sets, and homogenization in general periodic domains. 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