Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 85, pp. 1–37. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.85 PERIODIC UNFOLDING METHOD FOR DOMAINS WITH VERY SMALL INCLUSIONS JAKE AVILA, BITUIN CABARRUBIAS Communicated by Jesus Ildefonso Diaz Abstract. This work creates a version of the periodic unfolding method suit- able for domains with very small inclusions in RN for N ≥ 3. In the first part, we explore the properties of the associated operators. The second part involves the application of the method in obtaining the asymptotic behavior of a stationary heat dissipation problem depending on the parameter γ < 0. In particular, we consider the cases when γ ∈ (−1, 0), γ < −1 and γ = −1. We also include here the corresponding corrector results for the solution of the problem, to complete the homogenization process. 1. Introduction A recent and novel approach for homogenization theory is the periodic unfold- ing method originally introduced by Cioranescu, Damlamian, and Griso for fixed domains in [9, 10, 19, 20]. This method is favored because it gives an elementary proof for the classical periodic homogenization problem and due to the nature of technique which maps the oscillating domain to a fixed domain, it does not further require any extension operators. Later on, to take into account materials with pe- riodic perforations, Cioranescu, Donato and Zaki extended the method in [14] to perforated domains, for more details we refer to [13, 15, 16]. Successively, when the size of the holes are smaller than the period, the technique was adapted by Cio- ranescu, Damlamian, Griso, and Onofrei in [12] (for domains with two small holes, see [3, 28]). For a general presentation of unfolding, we refer to the comprehen- sive book [11]. Meanwhile, extensions to time-dependent functions involved in the heat and wave equations in perforated domains are treated by Donato and Yang in [23, 24], and similarly in small holes by Cabarrubias and Donato in [5]. More- over, Donato, Le Nguyen, and Tardieu constructed a variant for domains with two components in [21], and another type for highly oscillating boundary by Aiyappan, Nandakumaran, and Prakash in [1]. Since then, the adaptation of the method suit- able for different domain configurations has been extensively explored and studied. In this article we intend to develop a version of the periodic unfolding method suitable for domains with very small inclusions whose sizes are smaller than its 2020 Mathematics Subject Classification. 35B27, 35M32, 35Q79. Key words and phrases. Homogenization; imperfect interface; small inclusions; unfolding method. ©2023. This work is licensed under a CC BY 4.0 license. Submitted September 13, 2023. Published December 20, 2023. 1 2 J. AVILA, B. CABARRUBIAS EJDE-2023/85 period. Next, we apply this method by studying the asymptotic behavior of an elliptic problem where in the interface of the components, the jump of the solution is proportional to the flux. To complete the homogenization process, we also obtain some corrector results. To this goal, for N ≥ 3 we consider an open and bounded set Ω ⊂ RN with a Lipschitz continuous boundary ∂Ω, where Ω is a union of the open sets Ωδ,ε1 and Ωδ,ε2 with a common boundary Γδ,ε. The component Ωδ,ε2 is a disconnected union of ε-periodic very small inclusions of size δ(ε) � ε in Ω. Moreover, we let the component Ωδ,ε1 = Ω \ Ωδ,ε2 be connected while Γδ,ε := ∂Ωδ,ε2 . For the first part, we introduce two unfolding operators: T δ,ε1 acting on functions defined in Ωδ,ε1 and another operator T δ,ε2 for the functions defined in Ωδ,ε2 . Here, we prove their corresponding properties and establish the relationship of these two operators and the behavior of their traces on the interface. To achieve the second goal of the paper, we apply this method to describe the asymptotic behavior and obtain corrector results for the elliptic problem given by −div(Aε∇uδ,ε1 ) = f in Ωδ,ε1 , −div(Aε∇uδ,ε2 ) = f in Ωδ,ε2 , Aε∇uδ,ε1 · n δ,ε 1 = −Aε∇uδ,ε2 · n δ,ε 2 on Γδ,ε, −Aε∇uδ,ε1 · n δ,ε 1 = εγhδ,ε(uδ,ε1 − u δ,ε 2 ) on Γδ,ε, uδ,ε1 = 0 on ∂Ω, (1.1) for γ < 0 where nδ,εi is the unit outward normal to Ωδ,εi , for i = 1, 2. We assume that f is a square integrable function in Ω, the matrix field Aε is bounded and uniformly elliptic and hδ,ε is a positive and bounded Y -periodic function. The physical motivation of this problem concerns a stationary heat dissipation in a two-component composite with very small inclusions with a thermal barrier on the interface whose influence in the heat propagation varies with εγ . This condition on the interface can be observed in radiation phenomena. As discussed in [6], if there is no conduction due to the continuity of the temperature field when we traverse on the components due to imperfect bonding among the phases, we obtain an interfacial condition which relates the jump of the temperature to the heat flux across the interface. The pioneer work in homogenization for two-component domains can be traced back to Auriault and Ene [2] using the multiple scale method. Meanwhile, for inclusions whose size is the same as the period via Tartar’s method, one may consult the work of Monsurrò [26] for γ ≤ −1, together with Donato [22] for γ > −1. For a related problem in domains with very small inclusions via Tartar’s method, the reader is referred to the article [27] by Monsurrò. The first time where periodic unfolding method was used in a two-component domain for γ ≤ 1 was due to the work of Donato, Le Nguyen, and Tardieu in [21]. In our case, we only consider γ < 0 because when γ ∈ [0, 1] one cannot obtain the necessary trace convergences and when γ > 1 the solution becomes unbounded as investigated by Hummel [25]. Meanwhile, for works concerning the homogenization in domains with small holes, Tartar’s method was used by Cioranescu and Murat in[17] (see also [29, 30] by Tartar) to obtain the limiting behavior of a Poisson equation with Dirichlet boundary condition in perforated domains where the critical size εN/(N−2) gives EJDE-2023/85 PERIODIC UNFOLDING METHOD 3 rise to an additional zero-order “strange term” in the limit problem which depends on the capacity of the set of holes in the limit. Also, a related problem for the case of a nonhomogeneous Neumann problem for the Laplacian in the same geo- metric setting but with critical size of order εN/(N−1) was done in [18] by Conca and Donato. The primary novelty in this work is the introduction of another version of the Periodic Unfolding Method that is suited for domains with very small inclusions. Alongside is its application in finding the asymptotic behavior of a particular elliptic problem as well as in determining the corresponding corrector results. In fact, with the aid of this new version, we effectively reveal the contribution of the small inclusions on the homogenized problems by means of a zero order strange term at the limit. This key feature arising from the small scale in this type of domain has not been previously observed for instance in [27] where the limit is only the classical Dirichlet problem. Also, this is the first time for this class of problem where we obtain an additional corrector as a consequence of the strange term at the limit. The main difficulties addressed in this work are the following: establishing the necessary conditions associated with the parameters ε and δ in order to describe the appropriate trace behaviors, demonstrating the contribution of the small inclusions in the limit problem using an appropriate class of test functions, and analyzing how the small scale manifests in the correctors. This article organized as follows: Section 2 recalls the unfolding operators for two-component domains. Next, Section 3 develops the version of the unfolding method suitable for domains with very small inclusions. We first assume that γ ≤ 1 and provide the properties of the associated operators and then due to some limitations, we shift to γ < 0 for the trace behaviors. In Section 4, we describe the asymptotic behavior of problem (1.1) by starting with the case γ < −1 followed by γ ∈ (−1, 0) where no interface influence can be observed at the limits. We present lastly, the case γ = −1 as the integral term on the common boundary appears at the homogenized problem. At the end of this work is Section 5 which gives the convergence of the energies leading to the corrector results. 2. Unfolding operator for two-component domains We start by recalling the periodic unfolding operator for two-component domains originally developed by Donato, Le Nguyen, and Tardieu in [21]. This operator is one of the key tools in the homogenization results later. Let Ω ⊂ RN for N ≥ 2 be an open bounded set with a Lipschitz continuous boundary ∂Ω. Set Y = ∏N i=1(0, `i) to be a reference cell where each `i > 0. Let Y1 and Y2 be two open connected subsets of Y such that Y 2 ⊂ Y and Y = Y1 ∪ Y 2 and the boundary Γ = ∂Y2 is also Lipschitz continuous. Let i = 1, 2. For any k ∈ ZN , denote by k` = (k1`1, . . . , kN `N ) and define the sets Y k = k` + Y, Y ki = k` + Yi and Kε = { k ∈ ZN | εY k2 ⊂ Ω } , Ωε2 = int ∪k∈Kε εY k2 , Γε = ∂Ωε2 and Ωε1 = Ω \ Ωε2. We also consider the sets for i = 1, 2, K̂ε = { k ∈ ZN | εY k ⊂ Ω } , Ω̂ε = int∪k∈K̂εε(k`+Y ) and Λε = Ω\ Ω̂ε. Ω̂εi = ∪k∈K̂εεY k i ,Λ ε i = Ωεi \ Ω̂εi , and Γ̂ε = ∂Ω̂ε2. 4 J. AVILA, B. CABARRUBIAS EJDE-2023/85 In the sequel, we let ε take on values from a positive real sequence tending to zero and for g ∈ L1(O), where O is an open set in RN , we use the notations: (i) θi = |Yi| |Y | and (ii) MO(g) = 1 |O| ∫ O g dy. (2.1) We also denote by ϕ̃ the zero extension of the function ϕ defined on Ωεi to the whole of Ω. Definition 2.1. Let i = 1, 2. For any Lebesgue measurable function ϕ in Ωεi the periodic unfolding operator T εi is given by T εi (ϕ)(x, y) = { ϕ ( ε[xε ]Y + εy ) for a.e. (x, y) ∈ Ω̂ε × Yi, 0 for a.e. (x, y) ∈ Λε × Yi. Remark 2.2. Notice that if we define the operator Tε in Ω× Y to be Tε(ϕ) = { T ε1 (ϕ) in Ω× Y1, T ε2 (ϕ) in Ω× Y2, we obtain the unfolding operator for the fixed domain Ω given in [9]. Next, let us recall some properties of this unfolding operators. We only state here the necessary properties for this work. Theorem 2.3. Let p ∈ [1,+∞) and i = 1, 2. The operators T εi (ϕ) are linear and continuous from Lp(Ωεi ) to Lp(Ω× Y ). Moreover, (1) T εi (ϕψ) = T εi (ϕ)T εi (ψ) for every Lebesgue measurable functions ϕ,ψ on Ωεi ; (2) for every ϕ ∈ L1(Ωεi ), 1 |Y | ∫ Ω×Yi T εi (ϕ)(x, y) dxdy = ∫ Ω̂εi ϕ(x) dx = ∫ Ωεi ϕ(x) dx− ∫ Λεi ϕ(x) dx; (3) T εi (ϕ)→ ϕ strongly in Lp(Ω× Yi), for ϕ ∈ Lp(Ω); (4) if ϕε ∈ Lp(Ωεi )) satisfies ‖ϕε‖Lp(Ωεi ) ≤ C and T εi (ϕε) ⇀ ϕ̂ weakly in Lp(Ω× Yi), then ϕ̃ε ⇀ θiMYi(ϕ̂) weakly in Lp(Ω). We now have the following adjoints of these unfolding operators together with properties that we will need later. Definition 2.4. For p ∈ [1,+∞], the averaging operators Uεi : Lp(Ω × Yi) → Lp(Ω̂εi ), i = 1, 2, are defined as follows: Uεi (ϕ)(x) = { 1 |Y | ∫ Y ϕ ( ε [ x ε ] Y + εz, { x ε } Y ) dz for a.e. x ∈ Ω̂εi , 0 for a.e. x ∈ Λεi . Theorem 2.5. Let p ∈ [1,+∞) and i = 1, 2. The averaging operators Uεi are linear and continuous. Moreover, (1) ‖Uεi (ϕ)− ϕ‖Lp(Ωεi ) → 0 for every ϕ ∈ Lp(Ω); (2) if ϕε belongs to Lp(Ωεi ), then the following assertions are equivalent: (a) T εi (ϕε)→ ϕ̂ strongly in Lp(ω × Yi) and ∫ Λεi |ϕε|pdx→ 0, (b) ‖ϕε − Uεi (ϕ̂)‖Lp(Ωεi ) → 0. EJDE-2023/85 PERIODIC UNFOLDING METHOD 5 Now, for γ ≤ 1 we define the space Hε γ (see [21] for the details regarding this space) as, Hε γ = { u = (u1, u2) | u1 ∈ V ε, u2 ∈ H1(Ωε2) } , where V ε = {v ∈ H1(Ωε1) | v = 0 on ∂Ω}. Theorem 2.6. Let uε = (uε1, u ε 2) ∈ Hε γ . Then 1 ε|Y | ∫ Ω×Γ |T ε1 (uε1)− T ε2 (uε2)|2 dx dσy ≤ ∫ Γε |uε1 − uε2|2 dσx. Theorem 2.7. If ϕ ∈ D(Ω) and uε = (uε1, u ε 2) ∈ Hε γ , then for ε small enough, ε ∫ Γε hε(uε1 − uε2)ϕdσx = 1 |Y | ∫ Ω×Γ h(y) (T ε1 (uε1)− T ε2 (uε2))ϕdx dσy. Theorem 2.8. If u = (uε1, u ε 2) is a bounded sequence in Hε γ , then ‖T ε1 (∇uε1)‖Lp(Ω×Y1) ≤ C, ‖T ε2 (∇uε2)‖Lp(Ω×Y2) ≤ C, ‖T ε1 (uε1)− T ε2 (uε2)‖Lp(Ω×Γ) ≤ Cε 1−γ 2 . Theorem 2.9. Let uε = (uε1, u ε 2) be a bounded sequence in Hε γ . Then there exists a subsequence (still denoted by ε), u1 ∈ H1 0 (Ω) and û1 ∈ L2(Ω;H1 per(Y1)) such that T ε1 (uε1)→ u1 strongly in L2(Ω;H1(Y1)), (2.2) T ε1 (∇uε1) ⇀ ∇u1 +∇yû1 weakly in L2(Ω× Y1), (2.3) with MΓ(û1) = 0 for almost every x ∈ Ω. Furthermore, Zε1 = 1 ε (T ε1 (uε1)−MΓ(T ε1 (uε1))) ⇀ yΓ∇u1 + û1 weakly in L2(Ω;H1(Y1)), where yΓ = y −MΓ(y). (2.4) Theorem 2.10. Let γ ≤ 1 and uε = (uε1, u ε 2) be a bounded sequence in Hε γ . Then there exists a subsequence (still denoted by ε) and u2 ∈ L2(Ω) such that T ε2 (uε2) ⇀ u2 weakly in L2(Ω;H1(Y2)), εT ε2 (∇uε2)→ 0 strongly in L2(Ω× Y2)). (2.5) Moreover, if γ < 1 and (2.2) and (2.3) hold for a subsequence, then u2 = u1, i.e., T ε2 (uε2) ⇀ u1 weakly in L2(Ω;H1(Y2)). Theorem 2.11. Let uε be a bounded sequence in Hε γ . Then there exists a subse- quence (still denoted by ε) and û2 ∈ L2(Ω;H1(Y2)) such that Zε2 = 1 ε (T ε2 (uε2)−MΓ(T ε2 (uε2))) ⇀ û2 weakly in L2(Ω;H1(Y2)), (2.6) T ε2 (∇uε2) ⇀ ∇yû2 weakly in L2(Ω× Y2), (2.7) where MΓ(û2) = 0 for almost every x ∈ Ω. 6 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Theorem 2.12. If γ ≤ 1, and uε = (uε1, u ε 2) is a bounded sequence in Hε γ , then there exist a subsequence (still denoted by ε), u1 ∈ H1 0 (Ω), u2 ∈ L2(Ω), û1 ∈ L2(Ω;H1 per(Y1)), and û2 ∈ L2(Ω;H1(Y2)) such that (2.2), (2.3), and (2.5)–(2.7) hold. Furthermore, if γ < 1, then u1 = u2 and (i) if γ < −1 then û1 = û2 − yΓ∇u1 on Ω× Γ, (ii) if γ = −1, then for some function ξΓ ∈ L2(Ω), T ε1 (uε1)− T ε2 (uε2) ε ⇀ û1 − û2 + yΓ∇u1 + ξΓ weakly in L2(Ω× Γ). 3. Unfolding operator for domains with very small inclusions Let Ω ⊂ RN for N ≥ 3 be an open bounded set with a Lipschitz continuous boundary ∂Ω. Let ε be a positive real sequence that approaches zero and let δ = δ(ε) < 1 be such that δ → 0 as ε→ 0. Let Y = ∏N i=1(0, `i) be a reference cell where each `i > 0. Let B ⊂ Y and set Y2 = δB, the δ-scaled version of B, with Lipschitz continuous boundary Γ = ∂Y2. Moreover, we set Y1 = Y \ Y2. From this construction, Y is the disjoint union Y = Y1 ∪Y2 ∪Γ. We assume that both Y1 and Y2 are connected For any ξ ∈ ZN , denote by ξ` = (ξ1`1, . . . , ξN`N ) and define the set Kε = { ξ ∈ ZN | ε(ξ` + Y2) ∩ Ω 6= ∅ } . From here, define the sets Ωδ,ε2 = int ∪ξ∈Kε ε(ξ` + Y2), Γδ,ε = ∂Ωδ,ε2 and Ωδ,ε1 = Ω \ Ωδ,ε2 . Thus, we have ∂Ωδ,ε1 = ∂Ω ∪ Γδ,ε and Ω is the disjoint union Ω = Ωδ,ε1 ∪ Ωδ,ε2 ∪Γδ,ε. We mention that Ωδ,ε1 is connected and assume that ∂Ω∩Γδ,ε = ∅ so that Ωδ,ε2 is a collection of pairwise disjoint translated sets Y2 distributed with period ε. ∂Ω Ωδ,ε1 Ωδ,ε2 Γδ,ε Y1 Y Γ Y2 = δB B Figure 1. Two-component domain with very small inclusions Furthermore, we define the sets K̂ε = { ξ ∈ ZN | ε(ξ` + Y ) ⊂ Ω } , Ω̂ε = int∪ξ∈K̂εε(ξ` + Y ), Λε = Ω \ Ω̂ε, (3.1) Ω̂δ,εi = int ∪ξ∈K̂ε ε(ξ` + Yi), Λδ,εi = Ωδ,εi \ Ω̂δ,εi , Γ̂δ,ε = ∂Ω̂δ,ε2 . (3.2) Again, we use the notation ·̃ to denote the zero extension as defined in the previous section. Let p ∈ [1,∞). We define the functional space V δ,εp = {v ∈W 1,p(Ωδ,ε1 ) | v = 0 on ∂Ω}, (3.3) EJDE-2023/85 PERIODIC UNFOLDING METHOD 7 endowed with the norm ‖v‖V δ,εp = ‖∇v‖Lp(Ωδ,ε1 ) for every v ∈ V δ,εp . Remark 3.1. A Poincaré inequality holds in the space V δ,εp . Consequently, the norms in V δ,εp and W 1,p(Ωδ,ε1 ) are equivalent. For each real number γ, we define the function space Hδ,ε γ,p = { uδ,ε = (uδ,ε1 , uδ,ε2 ) | uδ,ε1 ∈ V δ,εp , uδ,ε2 ∈W 1,p(Ωδ,ε2 ) } , (3.4) equipped with the norm, ‖uδ,ε‖p Hδ,εγ,p = ‖∇uδ,ε1 ‖ p Lp(Ωδ,ε1 ) + ‖∇uδ,ε2 ‖ p Lp(Ωδ,ε2 ) + εγ‖uδ,ε1 − u δ,ε 2 ‖ p Lp(Γδ,ε) . (3.5) Remark 3.2. The norms in Hδ,ε γ,p and V δ,εp ×W 1,p(Ωδ,ε2 ) are equivalent (see, for instance, [27] for analogous developments on this equivalence). Theorem 3.3. If u = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ,p is bounded, then there is a constant C > 0 independent of ε such that ‖uδ,ε1 ‖W 1,p(Ωδ,ε1 ) ≤ C, (3.6) ‖∇uδ,ε2 ‖Lp(Ωδ,ε2 ) ≤ C, (3.7) ‖uδ,ε1 − u δ,ε 2 ‖Lp(Γδ,ε) ≤ Cε−γ/p. (3.8) Moreover, if γ ≤ 1, then ‖uδ,ε2 ‖W 1,p(Ωδ,ε2 ) ≤ C. (3.9) Proof. Suppose that u in Hδ,ε γ,p is bounded. Then (3.6) is immediate from Remark 3.1 and (3.5), while estimates (3.7) and (3.8) follow from (3.5). Finally, estimate (3.9) follows from Remark 3.2 and after minor computations when γ ≤ 1. � 3.1. Unfolding operator. Let us now introduce the unfolding operator suitable for our geometric setting and provide some properties. We also describe here the relationship of the operators and prove the behavior of its traces on the interface. From now on, we let i = 1, 2 unless otherwise stated. Definition 3.4. Let p ∈ [1,+∞). For ϕ ∈ Lp(Ωδ,εi ), the unfolding operator T δ,εi from Lp(Ωδ,εi ) to Lp(Ω× RN ), is defined by T δ,εi (ϕ)(x, z) = { ϕ ( ε [ x ε ] Y + εδz ) for a.e. (x, z) ∈ Ω̂ε × 1 δYi, 0 for a.e. (x, z) ∈ Λε × 1 δYi. For ease of presentation, if ϕ is a function defined in Ω, we denote T δ,εi (ϕ) = T δ,εi ( ϕ|Ωδ,εi ) . Remark 3.5. The operator T δ,ε1 is the unfolding operator “Tε,δ” in [11, 12]. When δ = 1, the operators T δ,εi are the unfolding operators “T εi ” in [21]. Moreover, we also recover the unfolding operator “Tε” for fixed domains given in [9] if we set Tε(ϕ) = { T δ,ε1 (ϕ) in Ω× 1 δY1, T δ,ε2 (ϕ) in Ω× 1 δY2. Before we proceed, let us first introduce the mean value and local average oper- ators. 8 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Definition 3.6. For p ∈ [1,+∞), the mean value operator M 1 δYi from Lp(Ω× 1 δYi) to Lp(Ω) is defined as M 1 δYi (ϕ)(x) = δN |Yi| ∫ 1 δYi ϕ(x, z) dz, ∀ϕ ∈ Lp(Ω× 1 δ Yi). An immediate consequence of this definition is the following proposition. Proposition 3.7. If ϕ ∈ Lp(Ω× 1 δYi), then ‖M 1 δYi (ϕ)‖Lp(Ω) ≤ δN |Yi| ‖ϕ‖Lp(Ω× 1 δYi) . Definition 3.8. For p ∈ [1,+∞), the local average operator Mδ,ε 1 δYi from Lp(Ωδ,εi ) to Lp(Ω) is defined as Mδ,ε 1 δYi (ϕ)(x) = δN |Yi| ∫ 1 δYi T δ,εi (ϕ)(x, z) dz, ∀ϕ ∈ Lp(Ωδ,εi ). We are in a position to give some properties of T δ,εi . In what follows, for p ∈ [1,∞) and N ≥ 3, set p∗ to be the associated Sobolev exponent to p given by p∗ = pN N − p . (3.10) Theorem 3.9. Let p ∈ [1,+∞). The unfolding operator T δ,εi is linear and contin- uous. Moreover, it has the following properties. (i) For every vδ,εi , wδ,εi ∈ Lp(Ω δ,ε i ), T δ,εi (vδ,εi wδ,εi ) = T δ,εi (vδ,εi )T δ,εi (wδ,εi ). (ii) For every uδ,εi ∈ Lp(Ω δ,ε i ), δN |Y | ∫ Ω× 1 δYi T δ,εi (uδ,εi ) dx dz = ∫ Ω̂δ,εi uδ,εi dx = ∫ Ωδ,εi uδ,εi dx− ∫ Λδ,εi uδ,εi dx. (iii) For every uδ,εi ∈ Lp(Ω δ,ε i ), ‖T δ,εi (uδ,εi )‖Lp(Ω×RN ) ≤ ( |Y | δN )1/p‖uδ,εi ‖Lp(Ωδ,εi ). (iv) For every uδ,εi ∈ L1(Ωδ,εi ),∣∣∣ ∫ Ωδ,εi uδ,εi dx− δN |Y | ∫ Ω× 1 δYi T δ,εi (uδ,εi ) dx dz ∣∣∣ ≤ ∫ Λδ,εi |uδ,εi | dx. (v) Let uδ,εi ∈W 1,p(Ωδ,εi ). Then T δ,εi (∇xuδ,εi ) = 1 εδ ∇zT δ,εi (uδ,εi ) in Ω× 1 δ Yi, (3.11) ‖∇zT δ,εi (uδ,εi )‖Lp(Ω×RN ) ≤ ε|Y |1/p δ N p −1 ‖∇uδ,εi ‖Lp(Ωδ,εi ). (3.12) (vi) If {wδ,εi } is a sequence in Lp(Ωδ,εi ) such that w̃δ,εi → wi strongly in Lp(Ω), then we have the convergence T δ,εi (wδ,εi )→ wi strongly in Lp(Ω× RN ). (vii) Let ωi be a bounded open set in RN . For every uδ,εi ∈ W 1,p(Ωδ,εi ), the following estimates hold for i = 1, 2 :∥∥T δ,εi [ uδ,εi −M δ,ε 1 δYi (uδ,εi ) ]∥∥ Lp(Ω;Lp∗ (RN )) ≤ C ε|Y | 1/p δ N p −1 ‖∇uδ,εi ‖Lp(Ωδ,εi ), (3.13) EJDE-2023/85 PERIODIC UNFOLDING METHOD 9 ‖T δ,εi (uδ,εi )‖Lp(Ω×ωi) ≤ 2C ε|Y |1/p δ N p −1 ‖∇uδ,εi ‖Lp(Ωδ,εi ) + 2|ωi|1/p |Y |1/p |Yi| δN(1− 1 p )‖uδ,εi ‖Lp(Ωδ,εi ), (3.14) where C is the Sobolev-Poincaré-Wirtinger constant for W 1,p(Yi) and p∗ as in (3.10). (viii) Let {wδ,εi } be a sequence in W 1,p(Ωδ,εi ) which is uniformly bounded when both ε and δ approach zero. Then, up to a subsequence, there is Wi ∈ Lp(Ω;Lp ∗ (RN )) with ∇zWi ∈ Lp(Ω× RN )N such that δ N p −1 ε T δ,εi [ wδ,εi −M δ,ε 1 δYi (wδ,εi ) ] 1 1 δYi ⇀Wi weakly in Lp(Ω;Lp ∗ (RN )), (3.15) δ N p −1 ε ∇zT δ,εi (wδ,εi )1 1 δYi ⇀ ∇zWi weakly in Lp(Ω× RN )N , (3.16) where p∗ is given by (3.10). Assuming furthermore that lim sup ε→0 δ N p −1 ε < +∞, (3.17) then we can choose the subsequence above and some Ui ∈ Lp(Ω;Lploc(RN )) such that δ N p −1 ε T δ,εi (wδ,εi ) ⇀ Ui weakly in Lp(Ω;Lploc(RN )). (3.18) Proof of (i)–(vi). The corresponding properties for T δ,ε1 follow by a change of vari- able y = δz as similarly shown in [12] for the case p = 2 and in [11] for p ∈ [1,∞) (for time-dependent functions, see [5]). The proofs for T δ,ε2 are analogously ob- tained. � To prove Theorem 3.9 (vii) and (viii), we require the next result which describes the interplay between the mean value and local average operators. Proposition 3.10. Let p ∈ [1,∞). (i) For uδ,εi ∈ Lp(Ω δ,ε i ), we have T δ,εi [ Mδ,ε 1 δYi (uδ,εi ) ] =M 1 δYi [ T δ,εi (uδ,εi ) ] =Mδ,ε 1 δYi (uδ,εi ). (ii) If {wδ,εi } is a sequence in Lp(Ωδ,εi ) such that w̃δ,εi → wi strongly in Lp(Ω), then Mδ,ε 1 δYi (wδ,εi )→M 1 δYi (wi) = wi strongly in Lp(Ω). (iii) If uδ,εi ∈ Lp(Ω δ,ε i ), then ‖Mδ,ε 1 δYi (uδ,εi )‖Lp(Ω) ≤ |Y | 1/p |Yi| δ N(1− 1 p )‖uδ,εi ‖Lp(Ωδ,εi ). Proof. Let us show the case i = 1; when i = 2 the proofs are similar. The identity in (i) uses Definitions 3.4, 3.6, 3.8, and the fact that y = δz. (ii) If {wδ,ε1 } is a sequence in Lp(Ωδ,ε1 ) such that w̃δ,ε1 → w1 strongly in Lp(Ω), then by (i), linearity of M 1 δY1 , and using Proposition 3.7 and Theorem 3.9 (vi) we have ‖Mδ,ε 1 δY1 (wδ,ε1 )− w1‖Lp(Ω) = ‖M 1 δY1 [ T δ,ε1 (wδ,ε1 ) ] −M 1 δY1 (w1)‖Lp(Ω) 10 J. AVILA, B. CABARRUBIAS EJDE-2023/85 = ‖M 1 δY1 [ T δ,ε1 (wδ,ε1 )− w1 ] ‖Lp(Ω) ≤ δN |Y1| ‖T δ,ε1 (wδ,ε1 )− w1‖Lp(Ω×RN ) → 0. Finally, (iii) follows from Definition 3.8 and Theorem 3.9 (iii). � Proof of Theorem 3.9 (vii) and (viii). We only show here the case i = 1; when i = 2 the proofs are essentially the same. Estimate (3.13) in (vii) is a direct consequence of the Sobolev-Poincaré-Wirtinger inequality. Meanwhile, the second estimate in (vii) follows from the fact that by Proposition 3.10 (i) one has |T δ,ε1 (uδ,ε1 )|p = |T δ,ε1 (uδ,ε1 )− T δ,ε1 (Mδ,ε 1 δY1 (uδ,ε1 )) + T δ,ε1 (Mδ,ε 1 δY1 (uδ,ε1 ))|p ≤ 2p ( |T δ,ε1 [ uδ,ε1 −M δ,ε 1 δY1 (uδ,ε1 ) ] |p + |Mδ,ε 1 δY1 (uδ,ε1 )|p ) . (3.19) This, since ω1 ⊂ RN , by (3.13) and Proposition 3.10 (iii), we obtain the desired estimate. Let us now prove (viii). Using estimate (3.13), there exists W1 ∈ Lp(Ω;Lp ∗RN )) such that (3.15) holds. Next, we show (3.16). From (3.12), there exists S1 ∈ Lp(Ω× RN )N such that δ N p −1 ε ∇zT δ,ε1 (wδ,ε1 )1 1 δY1 ⇀ S1 weakly in Lp(Ω× RN )N . (3.20) Meanwhile,from Proposition 3.10 (i) and Definition 3.8 we have ∇zT δ,ε1 [ wδ,ε1 − Mδ,ε 1 δY1 (wδ,ε1 ) ] = ∇zT δ,ε1 (wδ,ε1 ). Using this, for ϕ ∈ D ( Ω× RN ) we have∫ Ω× 1 δY1 δ N p −1 ε ∇zT δ,ε1 (wδ,ε1 )ϕdx dz = − ∫ Ω× 1 δY1 δ N p −1 ε T δ,ε1 [ wδ,ε1 −M δ,ε 1 δY1 (wδ,ε1 ) ] ∇ϕdx dz. Passing to the limit in this equation using (3.20) and (3.15), we obtain∫ Ω×RN S1ϕdx dz = − ∫ Ω×RN W1∇ϕdx dz = ∫ Ω×RN ∇zW1ϕdx dz, and so S1 = ∇zW1. In view of (3.20), then (3.16) holds. The last convergence in (3.18) is immediate from (3.14) with the aid of (3.17). � Also, an unfolding criterion for integrals holds for this operator. This is an immediate consequence of Theorem 3.9 (iv). Proposition 3.11. If a sequence {wδ,εi } in L1(Ωδ,εi ) satisfies∫ Λδ,εi |wδ,εi (x)| dx→ 0, then ∫ Ωδ,εi wδ,εi (x) dx− δN |Y | ∫ Ω× 1 δYi T δ,εi (wδ,εi )(x, z) dx dz → 0. Moreover, we write∫ Ωδ,εi wδ,εi (x) dx T δ,εi' δN |Y | ∫ Ω× 1 δYi T δ,εi (wδ,εi )(x, z) dx dz. EJDE-2023/85 PERIODIC UNFOLDING METHOD 11 Corollary 3.12. Let {uδ,εi } be a bounded sequence in Lp(Ωδ,εi ) and v ∈ Lq(Ωδ,εi ) such that 1 p + 1 q = 1. Then∫ Ωδ,εi uδ,εi v dx T δ,εi' δN |Y | ∫ Ω× 1 δYi T δ,εi (uδ,εi )(x, z) T δ,εi (v)(x, z) dx dz. (3.21) Furthermore, if {vδ,εi } is a bounded sequence in Lp0(Ωδ,εi ) such that 1 p + 1 p0 < 1, then∫ Ωδ,εi uδ,εi vδ,εi dx T δ,εi' δN |Y | ∫ Ω× 1 δYi T δ,εi (uδ,εi )(x, z) T δ,εi (vδ,εi )(x, z) dx dz. (3.22) Proof. For the relation in (3.21), we employ the Lebesgue Dominated Convergence Theorem. Meanwhile, (3.22) is straightforward from the boundedness of the bound- ary. � 3.2. Trace behaviors. Let us now proceed to some results concerning the jump on the interface. We investigate the relationship between the two operators and establish the behavior of their traces on the common boundary. Lemma 3.13. Let p ∈ [1,∞). If uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ,p, then δN−1 ε|Y | ∫ Ω×Γ ∣∣∣T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 ) ∣∣∣p dx dσz ≤ ∫ Γδ,ε |uδ,ε1 − u δ,ε 2 |p dσx. Proof. In view of (3.11), all traces are well-defined. By the definition of Ω̂ε given in (3.1), and Definition 3.4 of the unfolding operator, δN−1 ε|Y | ∫ Ω×Γ ∣∣∣T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 ) ∣∣∣p dx dσz = δN−1 ε|Y | ∑ ξ∈K̂ε ∫ ε(ξ`+Y )×Γ ∣∣∣uδ,ε1 ( ε [x ε ] Y + εδz ) − uδ,ε2 ( ε [x ε ] Y + εδz )∣∣∣p dx dσz. Note that if x ∈ ε(ξ`+Y ), then x = ε(ξ`+y1) for some y1 ∈ Y . This, the periodicity in Y , and by a change of variable x = εξ` + εδz, along with the definition of Γ̂δ,ε in (3.2), the above equation becomes δN−1 ε|Y | ∫ Ω×Γ ∣∣∣T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 ) ∣∣∣p dx dσz = δN−1 ε|Y | ∑ ξ∈K̂ε ∫ ε(ξ`+Y )×Γ ∣∣∣uδ,ε1 (εξ` + εδz)− uδ,ε2 (εξ` + εδz) ∣∣∣p dx dσz = δN−1 ε|Y | εN |Y | ∑ ξ∈K̂ε ∫ Γ ∣∣∣uδ,ε1 (εξ` + εδz)− uδ,ε2 (εξ` + εδz) ∣∣∣p dσz = εN−1δN−1 ∑ ξ∈K̂ε ∫ Γ ∣∣∣uδ,ε1 (εξ` + εδz)− uδ,ε2 (εξ` + εδz) ∣∣∣p dσz = ∑ ξ∈K̂ε ∫ ∂[ε(ξ`+Y2)] ∣∣∣uδ,ε1 (x)− uδ,ε2 (x) ∣∣∣p dσx = ∫ Γ̂δ,ε ∣∣∣uδ,ε1 − u δ,ε 2 ∣∣∣p dσx 12 J. AVILA, B. CABARRUBIAS EJDE-2023/85 ≤ ∫ Γδ,ε ∣∣∣uδ,ε1 − u δ,ε 2 ∣∣∣p dσx, which yields the desired inequality. � Now, we prove some estimates for the unfolding of the gradients and the jump on the unfolded functions along the boundary. Theorem 3.14. Let p ∈ [1,∞) and γ ≤ 1. If u = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ,p is bounded, then there is a C > 0 independent of ε and δ such that ‖T δ,ε1 (∇uδ,ε1 )‖Lp(Ω×RN ) ≤ Cδ− N p , (3.23) ‖T δ,ε2 (∇uδ,ε2 )‖Lp(Ω×RN ) ≤ Cδ− N p , (3.24) ‖T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 )‖Lp(Ω×Γ) ≤ Cε 1−γ p δ 1−N p . (3.25) Proof. Estimates (3.23) and (3.24) are immediate from the boundedness of u in Hδ,ε γ,p along with Theorem 3.9 (iii), (3.6), and (3.7). Meanwhile, one obtains (3.25) using Lemma 3.13, (3.8), and the boundedness hypothesis. � To accurately describe the trace behaviors, let us introduce the following mean value operator acting on the interface. Definition 3.15. For p ∈ [1,+∞), the mean value operator MΓ is a function from Lp(Ω× Γ) to Lp(Ω) and is defined as MΓ(ϕ)(x) = 1 |Γ| ∫ Γ ϕ(x, z) dσz, ∀ϕ ∈ Lp(Ω× Γ). Remark 3.16. The properties obtained under Theorem 3.9 (vii) and (viii) also hold when formulated for this operator. In the sequel, we suppose that (3.17) holds and we further assume that lim ε→0 δN−1 ε exists in R+. (3.26) We are now ready to give the trace behaviors. In what follows, we assume that γ < 0 since some of the properties do not hold when γ ∈ [0, 1]. Theorem 3.17. Let p ∈ [1,∞) and γ < 0. If uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ,p is bounded, then up to a subsequence, there exist U1 ∈ Lp(Ω;Lploc(RN )) and W1 ∈ Lp(Ω;W 1,p(RN )) such that δ N p −1 ε T δ,ε1 (uδ,ε1 ) ⇀ U1 weakly in Lp(Ω;Lploc(RN )), (3.27) δ N p T δ,ε1 (∇uδ,ε1 ) ⇀ ∇zW1 weakly in Lp(Ω× RN )N . (3.28) Moreover, we have Zδ,ε1 := δ N p −1 ε [ T δ,ε1 (wδ,ε1 )−MΓ(T δ,ε1 (wδ,ε1 )) ] 1 1 δY1 ⇀W1 (3.29) weakly in Lp(Ω;Lp ∗ (RN )), where MΓ(W1) = 0. EJDE-2023/85 PERIODIC UNFOLDING METHOD 13 Proof. The existence of U1 ∈ Lp(Ω;Lploc(RN )) and W1 ∈ Lp(Ω;W 1,p(RN )) such that convergences (3.27) and (3.29) hold are immediate consequences of (3.18), (3.15) and Remark 3.16, respectively. Furthermore, by the linearity and Definition 3.15 of MΓ, in view of (3.29), we have MΓ(W1) = 0 since MΓ(Zδ,ε1 ) = δ N p −1 ε ( MΓ(T δ,ε1 (wδ,ε1 ))− ( 1 |Γ| ) |Γ|MΓ(T δ,ε1 (wδ,ε1 )) ) = 0. (3.30) On the other hand, (3.28) follows from (3.11) and (3.16) with Remark 3.16. � Theorem 3.18. Let p ∈ [1,∞) and γ < 0. If uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ,p is uniformly bounded, then up to a subsequence, there exists U2 ∈ Lp(Ω;Lploc(RN )) such that δ N p −1 ε T δ,ε2 (uδ,ε2 ) ⇀ U2 weakly in Lp(Ω;Lploc(RN )), (3.31) εδT δ,ε2 (∇uδ,ε2 )→ 0 strongly in Lp(Ω× RN )N . (3.32) If we further assume that (3.26), and (3.27) and (3.28) hold up to subsequences, then U2 = U1. That is, δ N p −1 ε T δ,ε2 (uδ,ε2 ) ⇀ U1 weakly in Lp(Ω;Lploc(RN )). (3.33) Proof. Using (3.18), we have the existence of U2 ∈ Lp(Ω;Lploc( 1 δY2)) such that (3.31) holds. Moreover, (3.32) follows from using (3.11) and (3.24) which yield εδT δ,ε2 (∇uδ,ε2 ) = ∇zT δ,ε2 (uδ,ε2 )→ 0 strongly in Lp(Ω× RN )N . By triangle inequality and (3.25),∥∥δNp −1 ε T δ,ε2 (uδ,ε2 )− U1 ∥∥ Lp(Ω×Γ) ≤ δ N p −1 ε ‖T δ,ε2 (uδ,ε2 )− T δ,ε1 (uδ,ε1 )‖Lp(Ω×Γ) + ∥∥δNp −1 ε T δ,ε1 (uδ,ε1 )− U1 ∥∥ Lp(Ω×Γ) ≤ (δNp −1 ε ) Cε 1−γ p δ 1−N p + ∥∥δNp −1 ε T δ,ε1 (uδ,ε1 )− U1 ∥∥ Lp(Ω×Γ) = C (δNp −1 ε )(δN−1 ε )−1/p ε−γ/p + ∥∥δNp −1 ε T δ,ε1 (uδ,ε1 )− U1 ∥∥ Lp(Ω×Γ) . (3.34) The first term in the right-hand side of (3.34) approaches zero by (3.17), (3.26) and since γ < 0. Meanwhile, for the second term, by the trace theorem and (3.27),∥∥δNp −1 ε T δ,ε1 (uδ,ε1 )− U1 ∥∥ Lp(Ω×Γ) → 0. From these observations and (3.34), we obtain δ N p −1 ε T δ,ε2 (uδ,ε2 )→ U1 strongly in Lp(Ω× Γ). From (3.31), uniqueness of the limit implies that U2 = U1 a.e in Ω× Γ, (3.35) which gives (3.33). � 14 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Remark 3.19. It is important to notice that unlike the trace behaviors in [21] which hold for γ ≤ 1, Theorem 3.18 only permits the case γ < 0. This is due to the fact that when γ ∈ [0, 1], one cannot have convergence (3.33). Theorem 3.20. Let uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ,p be bounded. Then up to subsequences, there exists W2 ∈ Lp(Ω;W 1,p(RN )) such that Zδ,ε2 := δ N p −1 ε [ T δ,ε2 (wδ,ε2 )−MΓ(T δ,ε2 (wδ,ε2 )) ] 1 1 δY2 ⇀W2 (3.36) weakly in Lp(Ω;Lp ∗ (RN )), where MΓ(W2) = 0, and δ N p T δ,ε2 (∇uδ,ε2 ) ⇀ ∇zW2 weakly in Lp(Ω× RN )N . (3.37) Proof. Convergence (3.36) is an immediate consequence of (3.15), where W2 ∈ Lp(Ω;W 1,p(RN )). A similar computation to (3.30) shows thatMΓ(Zδ,ε2 ) = 0 which by (3.36) yields MΓ(W2) = 0. To see (3.37), we just use (3.11) and (3.16) so that δ N p T δ,ε2 (∇uδ,ε2 ) = δ N p [ 1 εδ ∇zT δ,ε2 (uδ,ε2 ) ] = δ N p −1 ε ∇zT δ,ε2 (uδ,ε2 )1 1 δY2 ⇀ ∇zW2, weakly in Lp(Ω× RN )N . � We end this section with a theorem that summarizes our results so far. Theorem 3.21. Let p ∈ [1,∞), γ < 0, and uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ,p be bounded. Then, up to subsequences, there exist Ui ∈ Lp(Ω;Lploc(RN )) and Wi ∈ Lp(Ω;W 1,p(RN )), such that (3.27), (3.28), (3.31), (3.32), (3.36), and (3.37) hold. Furthermore, if (3.26) holds, then U1 = U2 and W1 = W2 on Ω× Γ. (3.38) Proof. Convergences (3.27), (3.28), (3.31), (3.32), (3.36), and (3.37) hold by Theo- rems 3.17–3.20. Moreover, as in (3.35) we have U1 = U2. Now, notice that using the convergences in (3.29) and (3.36) together with the trace theorem for W 1,p( 1 δYi) for i = 1, 2, we obtain Zδ,ε1 − Z δ,ε 2 ⇀W1 −W2 weakly in Lp(Ω× Γ). (3.39) Meanwhile, by Definition 3.15, we have ‖MΓ(ϕ)‖Lp(Ω×Γ) ≤ 1 |Γ|1− 1 p (∫ Ω×Γ |ϕ|p dx dσy )1/p ≤ ‖ϕ‖Lp(Ω×Γ). This, along with the definitions of Zδ,ε1 and Zδ,ε2 in (3.29) and (3.36), triangle inequality, and (3.25), we obtain ‖Zδ,ε1 − Z δ,ε 2 ‖Lp(Ω×Γ) ≤ δ N p −1 ε [ ‖T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 )‖Lp(Ω×Γ) + ∥∥MΓ ( T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 ) )∥∥ Lp(Ω×Γ) ] ≤ (δNp −1 ε ) 2‖T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 )‖Lp(Ω×Γ) ≤ (δNp −1 ε ) 2Cε 1−γ p δ 1−N p EJDE-2023/85 PERIODIC UNFOLDING METHOD 15 = 2C (δNp −1 ε )(δN−1 ε )−1/p ε−γ/p. Then (3.17), (3.26) and since γ < 0 imply that Zδ,ε1 − Z δ,ε 2 → 0 strongly in Lp(Ω× Γ). Comparing this with (3.39), we obtain (3.38). � 4. Homogenization results Let us now obtain the asymptotic behavior of our dissipation problem given in (1.1) for γ < 0 as (ε, δ)→ (0, 0). First, we denote by M(α, β,O) the set of matrix fields A ∈ L∞(O)N×N satisfying (A(y)ξ, ξ) ≥ α|ξ|2 and |A(y)ξ| ≤ β|ξ|, ∀ξ ∈ RN ,∀y ∈ O, where O is an open set in RN and α, β ∈ R such that 0 < α < β. We define the functions Aε(x) = A (x ε ) and hδ,ε(x) = h (1 δ {x ε }) , (4.1) and consider the following assumptions on our data: (A1) A ∈M(α, β, Y ) and f ∈ L2(Ω); (A2) h ∈ L∞(Γ) be periodic in Y2 and there exists h0 ∈ R such that 0 < h0 < h(z) a.e. in Γ; (A3) Suppose that δ = δ(ε) is such that (3.26) holds and k1 = lim ε→0 δ N 2 −1 ε exists in R+. Remark 4.1. The number k1 corresponds to the critical size εN/(N−2) of Dirichlet small holes first observed in [17] (this is also (3.17) for the case p = 2). Meanwhile, (3.26) corresponds to the critical size εN/(N−1) of the Neumann small holes from [18]. From (3.3) and (3.4), when p = 2, we write V δ,ε := V δ,ε2 and Hδ,ε γ := Hδ,ε γ,2. The variational formulation of (1.1) is: Find uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ such that for all v = (v1, v2) ∈ Hδ,ε γ , ∫ Ωδ,ε1 Aε∇uδ,ε1 ∇v1 dx+ ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇v2 dx + εγ ∫ Γδ,ε hδ,ε(uδ,ε1 − u δ,ε 2 )(v1 − v2) dσx = 2∑ i=1 ∫ Ωδ,εi fvi dx. (4.2) Theorem 4.2. Under assumptions (A1) and (A2), problem (4.2) admits a unique solution uδ,ε ∈ Hδ,ε γ such that for some constant C > 0, ‖uδ,ε‖Hδ,εγ ≤ C. (4.3) 16 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Proof. The existence and uniqueness of the solution of (4.2) follows from invoking the Lax-Milgram Theorem together with (A1) and (A2). On the other hand, taking the unique solution uδ,ε as a test function in (4.2) and then applying (A1), (A2), triangle and Cauchy-Schwarz inequalities, and Remark 3.2, one obtains (4.3). � All throughout, let N ≥ 3 and p∗ as in (3.10). Let us start by recalling some spaces and results from [11] for homogenization in domains with small holes via the periodic unfolding method. For p ∈ [1, N), we recall the homogeneous space Ẇ 1,p(RN ) = {ϕ ∈ Lp∗(RN ) | ∇ϕ ∈ Lp(RN )N}. In what follows, for p = 2, we write Ḣ1(RN ) := Ẇ 1,2(RN ). Moreover, we define the functional space W1,p(RN ) := { W 1,p loc (RN ) | ∇ϕ ∈ Lp(RN )N } equipped with the norm ‖ϕ‖pW1,p(RN ) = ‖∇ϕ‖p Lp(RN ) + |ϕ(∞)|p, ∀ϕ ∈ W1,p(RN ). Proposition 4.3 ([11]). Let p ∈ [1, N). (i) The space W1,p(RN ) is isomorphic to Ẇ 1,p(RN ) ⊕ R. That is, for every ϕ ∈ W1,p(RN ), there exists a real number ϕ(∞) (called the weak limit of ϕ at infinity) such that ϕ− ϕ(∞) ∈ Ẇ 1,p(RN ). (ii) We have the estimate ‖ϕ−ϕ(∞)‖Lp∗(RN ) ≤ C‖∇ϕ‖Lp(RN ) ∀ϕ ∈ W1,p(RN ), where C > 0 is the constant for the Sobolev embedding of Ẇ 1,p(RN ) to Lp ∗ (RN ). For an open set B ⊂ RN , we define the subspace KB of Ḣ1(RN )⊕ R by KB = {ϕ ∈ H1 loc(RN ) | ∇ϕ ∈ L2(RN )N and ϕ = 0 on B}, equipped with the norm ‖ϕ‖KB = ‖∇ϕ‖L2(RN ), ∀ϕ ∈ KB . Furthermore, we define the space LB = {V ∈ L2(Ω; Ḣ1(RN ))⊕H1 0 (Ω) | V = 0 a.e. in Ω×B}, equipped with the norm ‖V ‖2LB = ‖∇V (·,∞)‖2L2(Ω) + ‖∇yV ‖2L2(Ω×RN ), ∀V ∈ LB . Next we recall some density results from [11] that are essential in reaching our goal. Lemma 4.4 ([11]). (i) For p ∈ [1,∞), the set ∪δ∈(0,δ0]{ϕ ∈W 1,p(Y ) | ϕ constant on ∂B} is dense in W 1,p(Y ). (ii) For p ∈ [1, N), the set ∪δ∈(0,δ0]{ϕ ∈ W 1,p(Y ) | ϕ = 0 on ∂B} is dense in W 1,p(Y ). Remark 4.5 ([11]). Lemma 4.4 holds also true in the space W 1,p per(Y ) (in place of W 1,p(Y )). We now introduce the important class of test functions which will aid us in revealing the contribution of the small scale of the inclusions to the homogenized problems. EJDE-2023/85 PERIODIC UNFOLDING METHOD 17 Lemma 4.6 ([11]). Let p ∈ [1,∞) and v ∈ W 1,p loc (RN ) such that ∇zv ∈ Lp(RN )N and has compact support. Set vδ,ε(x) = v (1 δ {x ε } Y ) for a.e. x ∈ Ω. (4.4) By Proposition 4.3(i), v has a limit at infinity denoted by v(∞). If δ is small enough, the function vδ,ε belongs to W 1,p(Ω) and vδ,ε → v(∞) strongly in Lp(Ω). (4.5) Moreover, if δ N p −1 ε is uniformly bounded, then vδ,ε ⇀ v(∞) weakly in W 1,p(Ω). Remark 4.7. For vδ,ε defined in (4.4), in view of (3.11) for i = 1, 2, T δ,εi (∇vδ,ε) = 1 εδ∇zv. Theorem 4.8. Let p ∈ [1,∞), uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ,p and hδ,ε be as in (4.1) such that h satisfies (A2). Then, for ϕ ∈ D(Ω) we have δN−1 ε|Y | ∫ Ω×Γ h(z) ( T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 ) ) T δ,ε1 (ϕ) dx dσz = ∫ Γδ,ε hδ,ε(uδ,ε1 − u δ,ε 2 )ϕdσx. Proof. By similar arguments as in the proof of Lemma 3.13, we have δN−1 ε|Y | ∫ Ω×Γ h(z) ( T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 ) ) T δ,ε1 (ϕ) dx dσz = εN−1δN−1 ∑ ξ∈K̂ε ∫ Γ h(z) ( uδ,ε1 (εξ` + εδz)− uδ,ε2 (εξ` + εδz) ) ϕ (εξ` + εδz) dσz. By the change of variable x = εξ` + εδz, (A2), and since the support of ϕ is a compact subset of Ω, the integral on Γδ,ε is the same over Γ̂δ,ε so that the above equation is transformed into δN−1 ε|Y | ∫ Ω×Γ h(z) ( T δ,ε1 (uδ,ε1 )− T δ,ε2 (uδ,ε2 ) ) T δ,ε1 (ϕ) dx dσz = εN−1δN−1 ∑ ξ∈K̂ε ∫ ∂[ε(ξ`+Y2)] h ( x εδ − ξ` δ ) (uδ,ε1 (x)− uδ,ε2 (x))ϕ(x) dσx εN−1δN−1 = ∑ ξ∈K̂ε ∫ ∂[ε(ξ`+Y2)] hδ,ε(uδ,ε1 (x)− uδ,ε2 (x))ϕ(x) dσx = ∫ ∂[∪ξ∈K̂εε(ξ`+Y2)] hδ,ε(uδ,ε1 − u δ,ε 2 )ϕdσx = ∫ Γ̂δ,ε hδ,ε(uδ,ε1 − u δ,ε 2 )ϕdσx = ∫ Γδ,ε hδ,ε(uδ,ε1 − u δ,ε 2 )ϕdσx, which completes the proof. � We are now in a position to describe the asymptotic behavior of (4.2) for different values of γ < 0. To aid us in passing to the limit, we further assume that, 18 J. AVILA, B. CABARRUBIAS EJDE-2023/85 (A4) there exists a matrix A ∈ M(α, β,Ω × Y ) such that T εi (Aε)(x, y) → A(x, y) for a.e. (x, y) ∈ Ω× Y ; and (A5) there exists a matrix F ∈ M(α, β,Ω × RN ) such that T δ,εi (Aε)(x, z) → F (x, z) for a.e. (x, z) ∈ Ω× RN . 4.1. Case γ < −1. Theorem 4.9. Let γ < −1. Under assumptions (A1)–(A3), let uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ be the solution of (4.2). Then there exist u1 ∈ H1 0 (Ω) and G1 ∈ LB such that ũδ,εi ⇀ θiu1 weakly in L2(Ω), (4.6) T δ,εi (uδ,εi ) ⇀ G1 weakly in L2(Ω;L2 loc(RN )) with G1(·,∞) = u1, (4.7) ∇zT δ,εi (uδ,εi ) ⇀ ∇zG1 weakly in L2(Ω× RN )N , (4.8) where θi is given by (2.1) (i), and ûi ∈ L2(Ω;H1 per,0(Yi)) such that T ε1 (∇uδ,ε1 ) ⇀ ∇u1 +∇yû1 weakly in L2(Ω× Y1)N , (4.9) T ε2 (∇uδ,ε2 ) ⇀ ∇yû2 weakly in L2(Ω× Y2)N , (4.10) for i = 1, 2. Moreover, under assumptions (A4) and (A5), the pair (G1, û) is the unique solution of the unfolded limit problem k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zV dx dz + 1 |Y | ∫ Ω×Y A(x, y)(∇G1(·,∞) +∇yû)(∇V (·,∞) +∇yΨ) dx dy = ∫ Ω fV (·,∞) dx, ∀Ψ ∈ L2(Ω;H1 per(Y )), ∀V ∈ LB , (4.11) where û ∈ L2(Ω;H1 per,0(Y )) is the extension by periodicity of the function û(·, y) = { û1(·, y) y ∈ Y1, û2(·, y)− yγ∇u1 y ∈ Y2, (4.12) where yΓ = y −MΓ(y). (4.13) Proof. We divide the proof in four steps. Step 1. Using (4.3), (2.2) and (2.5) in Theorem 2.3 (iv), we obtain ũδ,εi ⇀ θiui weakly in L2(Ω), (4.14) proving (4.6) for i = 1. We will show later that u2 = u1 so that u2 ∈ H1 0 (Ω) which will prove the case i = 2 in (4.6). Furthermore, the existence of ûi ∈ L2(Ω;H1 per,0(Yi)) for i = 1, 2 such that convergences (4.9) and (4.10) are true, come from (2.3) and (2.7). We also delay the proof of (4.7) and (4.8). Step 2. To capture the effect of the periodic oscillation of the coefficients in (4.2), for ϕ ∈ D(Ω) and ψ ∈ H1 per(Y ) vanishing in a neighborhood of the origin, we let EJDE-2023/85 PERIODIC UNFOLDING METHOD 19 v1 = v2 = ϕ = εϕ(x)ψ ( x ε ) be the test functions in (4.2). Since v1 = v2, the third term in (4.2) becomes zero and so we obtain∫ Ωδ,ε1 Aε∇uδ,ε1 ∇ϕdx+ ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇ϕdx = 2∑ i=1 ∫ Ωδ,εi fϕ dx. (4.15) Concerning the test functions, direct computations show that T εi (ϕ)→ 0 strongly in L2(Ω× Yi), (4.16) T εi (∇ϕ)→ ∇yΨ strongly in L2(Ω× Yi)N , (4.17) where Ψ(x, y) = ϕ(x)ψ(y). Unfolding the right-hand side of (4.15) using T ε1 and T ε2 , respectively, using Theorem 2.3 (i) (iii), and passing to the limit using (4.16) yield lim ε→0 2∑ i=1 ∫ Ωδ,εi fϕ dx T ε1' lim ε→0 2∑ i=1 1 |Y | ∫ Ω×Yi T εi (f)T εi (ϕ) dx dy = 0. (4.18) For the first term on the left-hand side of (4.15), we apply T ε1 and Theorem 2.3 (i), then pass to the limit using (A4), (4.9), and (4.17) yield lim ε→0 ∫ Ωδ,ε1 Aε∇uδ,ε1 ∇ϕdx = 1 |Y | ∫ Ω×Y1 A(x, y)(∇u1 +∇yû1)∇yΨ dx dy. (4.19) Now we consider the second term on the left-hand side of (4.15). By using T ε2 and Theorem 2.3 (i), passing to the limit with (A4), (4.10), and (4.17), we obtain lim ε→0 ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇ϕdx = 1 |Y | ∫ Ω×Y2 A(x, y)∇yû2∇yΨ dx dy. (4.20) Therefore, via (4.18), (4.19), and (4.20), we obtain the limit equation 1 |Y | ∫ Ω×Y1 A(x, y)(∇u1 +∇yû1)∇yΨ dx dy + 1 |Y | ∫ Ω×Y2 A(x, y)∇yû2∇yΨ dx dy = 0. (4.21) Finally, observe that using Theorem ?? (i) as γ < −1, we can define a function û (extended by periodicity) in L2(Ω;H1 per(Y )) given by (4.12), where yΓ is simi- larly defined as in (2.4). Using this along with the density of D(Ω) × H1 per(Y ) in L2(Ω;H1 per(Y )), then equation (4.21) becomes∫ Ω×Y A(x, y)(∇u1 +∇yû)∇yΨ dx dy = 0, ∀Ψ ∈ L2(Ω;H1 per(Y )). (4.22) Step 3. Let us now show the existence of G1 ∈ LB such that (4.7) and (4.8) hold, and that u2 = u1 as postponed in Step 1. For i = 1, 2, let us have the following results. From (A3) and (3.18) with Remark 3.16, there exists Gi ∈ L2(Ω;L2 loc(RN )) such that up to subsequences, T δ,εi (uδ,εi ) ⇀ Gi weakly in L2(Ω;L2 loc(RN )). (4.23) Theorem 3.21 then implies that G1 = G2 so that the above becomes T δ,εi (uδ,εi ) ⇀ G1 weakly in L2(Ω;L2 loc(RN )). (4.24) 20 J. AVILA, B. CABARRUBIAS EJDE-2023/85 From estimates (3.6) and (3.9) and Theorem 3.10 (ii), we have Mδ,ε 1 δYi (uδ,εi )1 1 δYi → ui strongly in L2(Ω;L2 loc(RN )). (4.25) Meanwhile, by (3.13), there exists Ji ∈ L2(Ω;L2∗(RN )) with ∇zJi ∈ L2(Ω × RN )N such that T δ,εi [ uδ,εi −M δ,ε 1 δYi (uδ,εi ) ] 1 1 δYi ⇀ Ji weakly in L2(Ω;L2∗(RN )). (4.26) Now, the linearity of T δ,εi together with (4.24), (4.25), and (4.26) imply G1 − ui = Ji and ∇zG1 = ∇zJi. By (3.38), J1 = J2 and since ui is independent of z, we obtain u1 = u2 in Ω concluding the proof of (4.6) from Step 1. Meanwhile, by (A3) and (3.16) with Remark 3.16, ∇zT δ,εi (uδ,εi ) ⇀ ∇zG1 weakly in L2(Ω× RN )N . (4.27) For the case i = 1, Definition 3.4 gives T δ,ε1 (uδ,ε1 ) = 0 in Ω×B, (4.28) and with (4.24), we obtain G1 = 0 in Ω×B. (4.29) Thanks to (4.26) - (4.29), we have G1 ∈ LB and G1(·,∞) = u1. (4.30) This along with (4.24) and (4.27) prove (4.7) and (4.8). Step 4. In this part, we will prove the rest of the limit equations. To see the effect of the very small inclusions, let ϕ ∈ D(Ω) and v ∈ KB such that ∇zv has compact support. Take v1 = v2 = vδ,ε(x)ϕ(x) as a test function in (4.2), where vδ,ε is defined in (4.4). Since v1 = v2, the third term in (4.2) vanishes and we obtain∫ Ωδ,ε1 Aε∇uδ,ε1 ∇(vδ,εϕ) dx+ ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇(vδ,εϕ) dx = 2∑ i=1 ∫ Ωδ,εi fvδ,εϕdx. (4.31) Before we proceed, observe that T δ,ε1 (ϕ)∇zv → ϕ∇zv strongly in L2(Ω× RN ). (4.32) The first term in the left-hand side of (4.31) becomes∫ Ωδ,ε1 Aε∇uδ,ε1 ∇(vδ,εϕ) dx = ∫ Ωδ,ε1 Aε∇uδ,ε1 ∇vδ,εϕdx+ ∫ Ωδ,ε1 Aε∇uδ,ε1 vδ,ε∇ϕdx. (4.33) For the first term on the right-hand side of (4.33), we unfold using T δ,ε1 due to the factor ∇vδ,ε. Using Theorem 3.11, (3.11), Remark 4.7, and passing to the limit using (A3), (A5), (4.8), and (4.32), lim ε→0 ∫ Ωδ,ε1 Aε∇uδ,ε1 ∇vδ,εϕdx = k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)ϕ∇zv dx dz. (4.34) EJDE-2023/85 PERIODIC UNFOLDING METHOD 21 Unfolding the second integral in (4.33) using T ε1 and passing to the limit using (A4), (4.9), (4.5), Theorem 2.3 (iii) we obtain lim ε→0 ∫ Ωδ,ε1 Aε∇uδ,ε1 vδ,ε∇ϕdx = v(∞) |Y | ∫ Ω×Y1 A(x, y)(∇u1 +∇yû1)∇ϕdx dy. (4.35) Similarly, for the second term in the left-hand side of (4.31),∫ Ωδ,ε2 Aε∇uδ,ε2 ∇(vδ,εϕ) dx = ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇vδ,εϕdx+ ∫ Ωδ,ε2 Aε∇uδ,ε2 vδ,ε∇ϕdx. (4.36) For the first term in the right-hand side of (4.36), we unfold using T δ,ε2 in a similar fashion in getting (4.34). Using Theorem 3.11, (3.11), and Remark 4.7, and then passing to the limit using (A3), (A5), (4.8), and (4.32), because of (4.29), we obtain lim ε→0 ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇vδ,εϕdx = k2 1 |Y | ∫ Ω×B F (x, z)∇zG1(x, z)ϕ∇zv dx dz = 0. (4.37) Unfolding the second term in the right-hand side of (4.36) using T ε2 and invoking (A4), (2.7), (4.5), and Theorem 2.3 (iii), we obtain lim ε→0 ∫ Ωδ,ε2 Aε∇uδ,ε2 vδ,ε∇ϕdx = v(∞) |Y | ∫ Ω×Y2 A(x, y)∇yû2∇ϕdx dy. (4.38) For the right-hand side of (4.31), we unfold the terms using T ε1 and T ε2 , respec- tively, 2∑ i=1 ∫ Ωδ,εi fvδ,εϕdx T εi' 2∑ i=1 1 |Y | ∫ Ω×Yi T εi (f)T εi (vδ,ε)T εi (ϕ) dx dy, and so by (4.5) and Theorem 2.3 (iii), and since f and ϕ are independent of y, lim ε→0 2∑ i=1 ∫ Ωδ,εi fvδ,εϕdx = v(∞) ∫ Ω fϕ dx. (4.39) Therefore, (4.35), (4.34), (4.38), (4.37), and (4.39) allow us to pass to the limit as ε → 0 in (4.31), by (4.30), and the density of D(Ω) ×KB in LB , using (4.12), we obtain k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zV dx dz + 1 |Y | ∫ Ω×Y A(x, y)(∇G1(·,∞) +∇yû)∇V (·,∞) dx dy = ∫ Ω fV (·,∞) dx, ∀V ∈ LB . (4.40) Taking into account (4.22) into (4.40) gives (4.11). Furthermore, the existence, uniqueness, and stability of the solution (G1, û) to problem (4.11) follows from Lax- Milgram Theorem, and so the convergences mentioned in the theorem holds for the whole sequence. � The next result gives the classical form of the homogenized system given in (4.11). 22 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Corollary 4.10. Under the assumptions of Theorem 4.9, the limit function u1 ∈ H1 0 (Ω) is the unique solution of the homogenized problem −div(Ahom,1∇u1) + k2 1Θu1 = f in Ω, u1 = 0 in ∂Ω, (4.41) where the homogenized matrix Ahom,1 = (ahom,1 ij )N×N ∈M(α, β,Ω) has entries ahom,1 ij =MY ( aij + N∑ k=1 aik ∂χ̂j ∂yk ) (4.42) such that the correctors χ̂j for j = 1, . . . , N solve the cell problems −div(A(x, y)∇(χ̂j + yj)) = 0 in Y, χ̂j is Y -periodic, MY (χ̂j) = 0, (4.43) and where for a.e. x ∈ Ω, Θ(x) := 1 |Y | ∫ RN\B F (x, z)∇zθ(x, z)∇zθ(x, z) dz. (4.44) Proof. We first introduce the classical correctors χ̂j for j = 1, . . . , N for the ho- mogenization in fixed domains (for more details, see [4]) that solve (4.43) which is equivalently given by χ̂j ∈ L∞(Ω;H1 per(Y )),∫ Y A(x, y)∇(χ̂j + yj)∇ϕdy = 0 a.e. in Ω, ∀ϕ ∈ H1 per(Y ). From (4.30), we also have that ∇G1(·,∞) = ∇u1. (4.45) Using this along with (4.22) implies that û can be expressed as û(x, y) = N∑ j=1 ∂u1 ∂xj (x)χ̂j(x, y) = N∑ j=1 ∂G1 ∂xj (x,∞)χ̂j(x, y) a.e. in Ω× Y. (4.46) Using this expression for û in (4.40), we have for all V ∈ LB ,∫ Ω Ahom,1∇G1(·,∞)∇V (·,∞) dx dy + k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zV dx dz = ∫ Ω fV (·,∞) dx, (4.47) where for a.e. x ∈ Ω, and by (4.45), Ahom,1(x)∇G1(x,∞) := 1 |Y | ∫ Y A(x, y) [ ∇G1(x,∞) +∇yû(x, y) ] dy = 1 |Y | ∫ Y A(x, y) [ ∇u1 +∇yû(x, y) ] dy, EJDE-2023/85 PERIODIC UNFOLDING METHOD 23 from which Ahom,1 given by (4.42) follows (see, for instance, [11]). Now, let us intro- duce θ to be the solution of the following cell problem corresponding to component B given by θ ∈ L∞(Ω; KB), θ(x,∞) ≡ 1,∫ RN\B F (x, z)∇zθ(x, z)∇zΨ(z) dz = 0 a.e. in Ω, ∀Ψ ∈ Ḣ1(RN ) with Ψ = 0 on B. (4.48) Multiplying the equation in (4.48) by u1(x) which is independent of z and setting Ψ(z) = V , we have a.e. in Ω that∫ RN\B F (x, z)∇z ( u1(x)θ(x, z) ) ∇zV dz = 0. (4.49) On the other hand, in view of (4.47), setting V (·,∞) = 0 we have∫ RN\B F (x, z)∇zG1(x, z)∇zV dz = 0 a.e. in Ω. (4.50) By the Lax-Milgram Theorem, (4.48) admits a unique solution. Therefore, in view of (4.49) and (4.50), since G1(x, ·) and u1(x)θ(x, ·) are both inKB and are solutions of the same problem which admits a unique solution, we have that G1(x, ·) = u1(x)θ(x, ·). Indeed, this coincides with (4.30) since G1(·,∞) = u1(x)θ(x,∞) = u1(x)(1) = u1(x). Now, if we set V (x, z) = Υ(x)θ(x, z) a.e. in Ω× RN in (4.47), where Υ ∈ H1 0 (Ω) and θ as in (4.48), then (4.47) becomes∫ Ω Ahom,1∇G1(·,∞)∇Υ dx dy + k2 1 ∫ Ω Θu1Υ dx dz = ∫ Ω fΥ dx, ∀Υ ∈ H1 0 (Ω), (4.51) where Θ(x) is given by (4.44), a nonnegative function and can be interpreted as the local capacity of the set B. Finally, we have (4.51) as the variational formulation of (4.41). � 4.2. Case γ ∈ (−1, 0). Theorem 4.11. Let γ ∈ (−1, 0). Under assumptions (A1)–(A3), let uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ be the solution of (4.2). Then there exist u1 ∈ H1 0 (Ω) and G1 ∈ LB such that ũδ,εi ⇀ θiu1 weakly in L2(Ω), (4.52) T δ,εi (uδ,ε1 ) ⇀ G1 weakly in L2(Ω;L2 loc(RN )), with G1(·,∞) = u1 (4.53) ∇zT δ,ε1 (uδ,ε1 ) ⇀ ∇zG1 weakly in L2(Ω× RN )N , (4.54) ∇zT δ,ε2 (uδ,ε2 ) ⇀ 0 weakly in L2(Ω× RN )N . (4.55) and û1 ∈ L2(Ω;H1 per,0(Y1)) such that T ε1 (∇uδ,ε1 ) ⇀ ∇u1 +∇yû1 weakly in L2(Ω× Y1)N , (4.56) T ε2 (∇uδ,ε2 ) ⇀ 0 weakly in L2(Ω× Y2)N . (4.57) 24 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Moreover, under assumptions (A4) and (A5), the pair (G1, û1) satisfies the unfolded limit problem k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zV dx dz + 1 |Y | ∫ Ω×Y1 A(x, y)(∇G1(·,∞) +∇yû1)(∇V (·,∞) +∇yΨ1) dx dy = ∫ Ω fV (·,∞) dx, ∀Ψ1 ∈ L2(Ω;H1 per(Y )), ∀V ∈ LB . (4.58) Proof. We divide the proof in five steps. Step 1. As in Step 1 from the proof of Theorem 4.9, by a Poincaré inequality and using the boundedness of the solution from Theorem 4.2, there exists u1 ∈ H1 0 (Ω) such that (4.52) holds for i = 1. Also, as in (4.14), we have ũδ,ε2 ⇀ θ2u2 weakly in L2(Ω). (4.59) Moreover, the existence of û1 ∈ L2(Ω;H1 per,0(Y1)) such that convergence (4.56) is true, come from (2.3). Step 2. To capture the effect of the periodic oscillation of the coefficients in (4.2), for ϕi ∈ D(Ω) and ψi ∈ H1 per(Y ) vanishing in a neighborhood of the origin, we let vi = ϕi = εϕi(x)ψi ( x ε ) be a test function in (4.2) for i = 1, 2. Following the arguments in Step 2 in obtaining (4.21), we obtain the limit equation 1 |Y | ∫ Ω×Y1 A(x, y)(∇u1 +∇yû1)∇yΨ1 dx dy + 1 |Y | ∫ Ω×Y2 A(x, y)∇yû2∇yΨ2 dx dy = 0. (4.60) This along with the density of D(Ω) × H1 per(Y ) in L2(Ω;H1 per(Y )), then equation (4.60) holds for every Ψi ∈ L2(Ω;H1 per(Yi)). Step 3. As argued in Step 3 from the proof of Theorem 4.9, we have the existence of Gi ∈ L2(Ω;L2 loc(RN )) such that T δ,εi (uδ,εi ) ⇀ Gi weakly in L2(Ω;L2 loc(RN )). (4.61) In view of (4.52) and (4.59), we obtain Gi − ui = Ji, ∇zGi = ∇zJi, (4.62) ∇zT δ,εi (uδ,εi ) ⇀ ∇zGi weakly in L2(Ω× RN )N , (4.63) for i = 1, 2. We also have here that G1 ∈ LB and G1(·,∞) = u1. (4.64) From these statements, when i = 1 in (4.62) and (4.63) prove (4.53) and (4.54). Step 4. Let us now show the contribution of the inclusions to the limit equations. As in Step 4 of the proof of Theorem 4.9, to capture the effect of the very small inclusions, let ϕ ∈ D(Ω) and v ∈ KB such that ∇zv has compact support. Take v1 = v2 = vδ,ε(x)ϕ(x) as test functions in (4.2) where vδ,ε is defined in (4.4). EJDE-2023/85 PERIODIC UNFOLDING METHOD 25 Since v1 = v2, the third term in (4.2) vanishes and we obtain∫ Ωδ,ε1 Aε∇uδ,ε1 ∇(vδ,εϕ) dx+ ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇(vδ,εϕ) dx = 2∑ i=1 ∫ Ωδ,εi fvδ,εϕdx. (4.65) For the first and second terms in the left-hand side of (4.65), similar computations used in (4.34)–(4.38) yield lim ε→0 ∫ Ωδ,ε1 Aε∇uδ,ε1 ∇(vδ,εϕ) dx = k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)ϕ∇zv dx dz + v(∞) |Y | ∫ Ω×Y1 A(x, y)(∇u1 +∇yû1)∇ϕdx dy, (4.66) and lim ε→0 ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇(vδ,εϕ) dx = k2 1 |Y | ∫ Ω×B F (x, z)∇zG2(x, z)ϕ∇zv dx dz + v(∞) |Y | ∫ Ω×Y2 A(x, y)∇yû2∇ϕdx dy. (4.67) Meanwhile, for the right-hand side of (4.65), computations to those used in (4.39) give lim ε→0 2∑ i=1 ∫ Ωδ,εi fvδ,εi ϕi dx = v(∞) ∫ Ω fϕ dx. (4.68) Using equations (4.66)–(4.68) when passing to the limit as ε → 0 in (4.65), by (4.64) and the density of D(Ω) × KB in LB , and taking into account (4.60), we obtain the limit equation k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zU(x, z)∇zV dx dz + 1 |Y | ∫ Ω×Y1 A(x, y)(∇G1(·,∞) +∇yû1)(∇V (·,∞) +∇yΨ1) dx dy + k2 1 |Y | ∫ Ω×B F (x, z)∇zG2(x, z)∇zV dx dz + 1 |Y | ∫ Ω×Y2 A(x, y)∇yû2(∇V (·,∞) +∇yΨ2) dx dy = ∫ Ω fV (·,∞) dx, ∀Ψi ∈ L2(Ω;H1 per(Yi)) for i = 1, 2, ∀V ∈ LB . (4.69) Let us now have an insight about the explicit form of ∇zG2 and ∇yû2. To this aim, choose V ≡ 0, Ψ1 ≡ 0, and Ψ2 ≡ û2, then (4.69) becomes 1 |Y | ∫ Ω×Y2 A(x, y)∇yû2∇yû2 = 0. (4.70) However, as A(x, y) ∈M(α, β,Ω×Y ), we have by ellipticity and (4.70) that∇yû2 ≡ 0, which by (2.7) gives (4.57). Furthermore, in view of (3.11) for i = 2, and (3.32), we have ∇zT δ,ε2 (uδ,ε2 ) = εδT δ,ε2 (∇uδ,ε2 )→ 0. 26 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Comparing this with (4.63) for i = 2, we have ∇zG2 ≡ 0 giving (4.55). Therefore, since ∇yû2 ≡ 0 and ∇zG2 ≡ 0, equation (4.69) simplifies to (4.58). Furthermore, the existence, uniqueness, and stability of the solution (G1, û1) to problem (4.58) follows from Lax-Milgram Theorem, and so the convergences men- tioned in the theorem holds for the whole sequence. Step 5. Finally, since γ < 0, the second part of Theorem 3.21 is applicable; and consequently, G1 = G2. In view of (4.61), we obtain (4.53) for i = 2. Moreover, using (3.38) in (4.62), we have J1 = J2 and since ui is independent of z, we have u1 = u2. This along with (4.59) yield (4.52) for i = 2. � Let us now have the classical form of the homogenized system given in (4.58). The arguments of the proof are similar to that of Corollary 4.10, however instead of integrating over Y at some parts, here the integrals are over Y1 only. Corollary 4.12. Under the assumptions of Theorem 4.11, the limit function u1 ∈ H1 0 (Ω) is the unique solution of the homogenized problem −div(Ahom,2∇u1) + k2 1Θu1 = f in Ω, u1 = 0 in ∂Ω, (4.71) where the homogenized matrix Ahom,2 = (ahom,2 ij )N×N ∈M(α, β,Ω) has entries ahom,2 ij =MY1 ( aij + N∑ k=1 aik ∂χj ∂yk ) (4.72) such that the correctors χj for j = 1, . . . , N solve the cell problems −div(A(x, y)∇(χj + yj)) = 0 in Y1, A(x, y)∇(χj + yj) · n1 = 0 on Γ, χj is Y -periodic, MY (χj) = 0, and where Θ is given by (4.44). 4.3. Case γ = −1. As presented in the next theorem, the asymptotic behavior for this case is more delicate as the limit problem contains a jump term on the common boundary. Theorem 4.13. Let γ = −1. Under assumptions (A1)–(A3), let uδ,ε = (uδ,ε1 , uδ,ε2 ) ∈ Hδ,ε γ be the solution of (4.2). Then there exist u1 ∈ H1 0 (Ω) and G1 ∈ LB such that ũδ,εi ⇀ θiu1 weakly in L2(Ω), (4.73) T δ,εi (uδ,εi ) ⇀ G1 weakly in L2(Ω;L2 loc(RN )), (4.74) ∇zT δ,εi (uδ,εi ) ⇀ ∇zG1 weakly in L2(Ω× RN )N , (4.75) and ûi ∈ L2(Ω;H1 per,0(Yi)) such that T ε1 (∇uδ,ε1 ) ⇀ ∇u1 +∇yû1 weakly in L2(Ω× Y1)N , (4.76) T ε2 (∇uε2) ⇀ ∇yû2 weakly in L2(Ω× Y2)N , (4.77) EJDE-2023/85 PERIODIC UNFOLDING METHOD 27 for i = 1, 2, Moreover, under assumptions (A4) and (A5), the triple (G1, û1, ū2) is the unique solution of the unfolded limit problem k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zV dx dz + 1 |Y | ∫ Ω×Y1 A(x, y)(∇G1(·,∞) +∇yû1)(∇V (·,∞) +∇yΨ1) dx dy + 1 |Y | ∫ Ω×Y2 A(x, y)(∇G1(·,∞) +∇yū2)(∇V (·,∞) +∇yΨ2) dx dy + 1 |Y | ∫ Ω×Γ h(y)(û1 − ū2)(Ψ1 −Ψ2) dx dσy = ∫ Ω fV (·,∞) dx, ∀Ψi ∈ L2(Ω;H1 per(Y )) for i = 1, 2, ∀V ∈ LB , (4.78) where ū2 ∈ L2(Ω;H1(Y2)) is the extension by periodicity of the function ū2 = û2 − yΓ∇u1 − ξΓ, (4.79) for some ξΓ ∈ L2(Ω) and where yγ = y −MΓ(y). Proof. We divide the proof in three steps. Step 1. The existence of u1 ∈ H1 0 (Ω), G1 ∈ LB , and ûi ∈ L2(Ω;H1 per,0(Yi)) such that convergences (4.73)–(4.77) hold follow from the same arguments as in the proofs of (4.6)–(4.10) from Theorem 4.9. Step 2. To capture the effect of the periodic oscillation of the coefficients, for ϕi ∈ D(Ω) and ψi ∈ H1 per(Y ) vanishing in a neighborhood of the origin, we let vi = ϕi = εϕi(x)ψi ( x ε ) be a test function in (4.2) for i = 1, 2. Proceeding as in Step 2 of Theorem 4.11 and employing similar arguments in [21] for the interfacial term, we obtain the following limit equation∫ Ω×Y1 A(x, y)(∇u1 +∇yû1)∇yΨ1 dx dy + ∫ Ω×Y2 A(x, y)∇yû2∇yΨ2 dx dy + ∫ Γ h(y)(û1 − ū2)(Ψ1 −Ψ2) dx dσy = 0, (4.80) for all Ψi ∈ L2(Ω;H1 per(Y )) and i = 1, 2. Step 3. Let us now capture the effect of the very small inclusions in (4.2). To this goal, for ϕ ∈ D(Ω) and v ∈ KB such that ∇zv has compact support, we use again the function vδ,ε as given in (4.4) and let v1 = v2 = vδ,ε(x)ϕ(x) be test functions in (4.2). Following the arguments in Step 4 of Theorem 4.9, we obtain lim ε→0 ∫ Ωδ,ε1 Aε∇uδ,ε1 ∇(vδ,ε)ϕdx = k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)ϕ∇zv dx dz + v(∞) |Y | ∫ Ω×Y1 A(x, y)(∇u1 +∇yû1)∇ϕdx dy, (4.81) lim ε→0 ∫ Ωδ,ε2 Aε∇uδ,ε2 ∇(vδ,εϕ) dx = v(∞) |Y | ∫ Ω×Y2 A(x, y)∇yû2∇ϕdx dy, (4.82) 28 J. AVILA, B. CABARRUBIAS EJDE-2023/85 lim ε→0 2∑ i=1 ∫ Ωδ,εi fvδ,εi ϕi dx. = v(∞) ∫ Ω fϕ dx. (4.83) Therefore, in view of (4.81) - (4.83), passing to the limit, by (4.30) and the density of D(Ω)×KB in LB , we obtain k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zV dx dz + 1 |Y | ∫ Ω×Y1 A(x, y)(∇G1(·,∞) +∇yû1)∇V (·,∞) dx dy + 1 |Y | ∫ Ω×Y2 A(x, y)∇yû2∇V (·,∞) dx dy = ∫ Ω fV (·,∞) dx, ∀V ∈ LB . (4.84) Finally, taking into account (4.80) in (4.84) gives (4.78). Furthermore, the existence, uniqueness, and stability of the solution (G1, û1, ū2) to problem (4.78) follows from Lax-Milgram Theorem, and so the convergences mentioned in the theorem holds for the whole sequence. � For the classical form of the homogenized system given in (4.78), the arguments are similar to the the proof of Corollary 4.10 with additional straightforward com- putations on the interface. Corollary 4.14. Under the assumptions of Theorem 4.13, the limit function u1 ∈ H1 0 (Ω) is the unique solution of the homogenized problem −div(Ahom,3∇u1) + k2 1Θu1 = f in Ω, u1 = 0 in ∂Ω, (4.85) where the homogenized matrix Ahom,3 = (ahom,3 ij )N×N ∈M(α, β,Ω) has entries ahom,3 ij =MY1 ( aij + N∑ k=1 aik ∂χ̂1 j ∂yk ) +MY2 ( aij + N∑ k=1 aik ∂χ̂2 j ∂yk ) (4.86) such that the correctors (χ̂1 j , χ̂ 2 j ) for j = 1, . . . , N solve the cell problems −div(A(x, y)∇(χ̂1 j + yj)) = 0 in Y1, −div(A(x, y)∇(χ̂2 j + yj)) = 0 in Y2, A(x, y)∇(χ̂1 j + yj) · n1 = −A(x, y)∇(χ̂2 j + yj) · n2 on Γ, −A(x, y)∇(χ̂1 j + yj) · n1 = h(χ̂1 j − χ̂2 j ) on Γ, χ̂1 j is Y -periodic, MY (χ̂1 j ) = 0. and where Θ is still of the form (4.44). Remark 4.15. Let us have the following observations: (1) From Theorems 4.10, 4.12, and 4.14, the contribution of the coefficient matrix Aε in the corresponding components Y1 and Y2 is taken into account by the term Ahom,j , for j = 1, 2, 3, which also recovers the homogenized matrices in [21]. EJDE-2023/85 PERIODIC UNFOLDING METHOD 29 (2) The appearance of the zero order strange term k2 1Θu1 in (4.41), (4.71), and (4.85) is brought about by the effect of the very small inclusions. (3) In contrast to the limit problem in [27] being the classical Dirichlet problem which does not account for the presence of a strange term, the results of this paper provide an information on the contribution of the small scale in the homogenized problem. To end this section, let us further investigate what happens to the limit function and the contribution of the very small inclusions to the limit problem if instead of (A3), we have the assumptions k1 = +∞ or k1 = 0. 4.4. Case k1 = +∞. If k1 in (A3) is infinite, together with (3.13) for p = 2 as ε→ 0, we obtain∥∥T δ,εi [ uδ,εi −M δ,ε 1 δYi (uδ,εi ) ]∥∥ L2(Ω;L2∗ (RN )) ≤ C ε|Y | 1/2 δ N 2 −1 ‖∇uδ,εi ‖L2(Ωδ,εi )N , → 0, (4.87) for i = 1, 2. Meanwhile, by (3.6) and (3.9), we have the convergence uδ,εi → ui strongly in L2(Ω). This along with a similar proof of (4.25), we have Mδ,ε 1 δYi (uδ,εi )1 1 δYi → ui strongly in L2(Ω;L2 loc(RN )). (4.88) Now for γ < 0, as shown in Theorems 4.9, 4.11, and 4.13, we have u1 = u2. Therefore, by linearity of T δ,εi , Theorem 3.10 (i), and (4.88), in view of (4.87), we obtain T δ,εi (uδ,εi )→ u1 strongly in L2(Ω;L2 loc(RN )). (4.89) However, as seen in the proof of (4.28), T δ,ε1 (uδ,ε1 ) = 0 in Ω× B. Finally, this and (4.89) imply that u1 = 0. Remark 4.16. When k1 = +∞ and γ < 0, then u1 vanishes in Ω. This means that the size of the very small inclusions is too big that it forces the limit function u1 to be zero. 4.5. Case k1 = 0. When k1 in (A3) is zero, then k2 1 = 0 so that the limit problems in Theorems 4.10, 4.12, and (4.14) become −div(Ahom∇u1) = f in Ω, u1 = 0 in ∂Ω, with their corresponding homogenized matrices given in (4.42), (4.72) and (4.86), respectively. Remark 4.17. When k1 = 0 and γ < 0, then the very small inclusions do not contribute in the limit problem. This means that the size of the very small inclusions are too small to influence the limit problem and so we do not have the appearance of a zero order strange term. Furthermore, let us mention that the proof of the homogenization results for k1 = 0 requires slight modifications. For instance, let us consider the case when 30 J. AVILA, B. CABARRUBIAS EJDE-2023/85 γ < −1. Let us focus on Step 2 on the proof of Theorem (4.9). To resolve this, we will use Theorem 3.9 (viii). In particular, in place of (4.23), we will use δ N 2 −1 ε T δ,εi (uδ,εi ) ⇀ Ui weakly in L2(Ω;L2 loc(RN )). The rest of the arguments are similar. 5. Corrector results In this section, we prove some convergence for the associated energies to prob- lem (1.1). As a consequence, corrector results will be obtained using the periodic unfolding method for problem (1.1) where γ < 0. We first recall a classical result which is essential in this part (see e.g. [11]). Theorem 5.1. Let {Dε}ε be a sequence of N ×N matrix fields in M(α, β,O) for some open set O such that Dε → D almost everywhere on O (or more generally, in measure in O). If the sequence of vector fields {ζε}ε converges weakly to ζ in L2(O)N , then lim inf ε→0 ∫ O Dεζεζε dx ≥ ∫ O Dζζ dx. Furthermore if lim sup ε→0 ∫ O Dεζεζε dx ≤ ∫ O Dζζ dx, then lim ε→0 ∫ O Dεζεζε dx = ∫ O Dζζ dx and ζε → ζ strongly in L2(O)N . 5.1. Case γ < −1. Let us now have the convergence of the energy for this case. Theorem 5.2. Let γ < −1. Under the assumptions of Theorems 4.9 and 4.10, one has lim ε→0 2∑ i=1 ∫ Ωδ,εi Aε∇uδ,εi ∇u δ,ε i dx = ∫ Ω Ahom∇u1∇u1 dx+ k2 1 ∫ Ω Θu2 1 dx = 1 |Y | ∫ Ω×Y A(x, y)(∇G1(·,∞) +∇yû)(∇G1(·,∞) +∇yû) dx dy + k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zG1(x, z) dx dz, (5.1) and lim ε→0 2∑ i=1 ∫ Λδ,εi |∇uδ,εi | 2dx = 0. (5.2) Moreover, we have the strong convergences T ε1 (∇uδ,ε1 )1Ω×Y δ1 → ∇u1 +∇yû1 strongly in L2(Ω× Y1)N , (5.3) T ε2 (∇uδ,ε2 )1Ω×Y δ2 → ∇yû2 strongly in L2(Ω× Y2)N , (5.4) and for i = 1, 2, ∇zT δ,εi (uδ,εi ))1 1 δY δ i → ∇zG1 strongly in L2(Ω× RN )N . (5.5) EJDE-2023/85 PERIODIC UNFOLDING METHOD 31 Proof. Let vi = uδ,εi be test functions in (4.2) for i = 1, 2. Then 2∑ i=1 ∫ Ωδ,εi Aε∇uδ,εi ∇u δ,ε i dx = 2∑ i=1 [ ∫ Ωδ,εi fuδ,εi dx ] − εγ ∫ Γδ,ε hδ,ε(uδ,ε1 − u δ,ε 2 )(uδ,ε1 − u δ,ε 2 ) dσx. (5.6) Unfolding the left-hand side of (5.6) using Tε, in view of Remark 2.2 and Theorem 2.3 (ii), we obtain 2∑ i=1 [ 1 |Y | ∫ Ω×Yi T εi (Aε)T εi (∇uδ,εi )T εi (∇uδ,εi ) dx+ ∫ Λδ,εi Aε∇uδ,εi ∇u δ,ε i dx ] = 2∑ i=1 [ ∫ Ωδ,εi fuδ,εi dx ] − εγ ∫ Γδ,ε hδ,ε(uδ,ε1 − u δ,ε 2 )(uδ,ε1 − u δ,ε 2 ) dσx. (5.7) Now, to investigate the convergence of the energy, we set Y δ2 = √ δ B and Y δ1 = Y \ Y δ2 . From here, we note that 1 δY δ 2 = 1√ δ B. With a change of variable y = δz in Y δ2 , transforming the first and third term in the left-hand side of (5.7), in view of Remark 3.5 (2), and (3.11), gives 2∑ i=1 [ 1 |Y | ∫ Ω×Y δi T εi (Aε)T εi (∇uδ,εi )T εi (∇uδ,εi ) dx dy + (δN2 −1 ε )2 1 |Y | ∫ Ω× 1 δY δ i T δ,εi (Aε)∇zT δ,εi (uδ,εi )∇zT δ,εi (uδ,εi ) dx dz + ∫ Λδ,εi Aε∇uδ,εi ∇u δ,ε i dx ] = 2∑ i=1 [ ∫ Ωδ,εi fuδ,εi dx ] − εγ ∫ Γδ,ε hδ,ε(uδ,ε1 − u δ,ε 2 )(uδ,ε1 − u δ,ε 2 ) dσx. (5.8) For conciseness in (5.8), we set Aδ,ε = 2∑ i=1 1 |Y | ∫ Ω×Y δi T εi (Aε)T εi (∇uδ,εi )T εi (∇uδ,εi ) dx dy, Bδ,ε = 2∑ i=1 (δN2 −1 ε )2 1 |Y | ∫ Ω× 1 δY δ i T δ,εi (Aε)∇zT δ,εi (uδ,εi )∇zT δ,εi (uδ,εi ) dx dz, Cδ,ε = 2∑ i=1 ∫ Λδ,εi Aε∇uδ,εi ∇u δ,ε i dx. (5.9) From (4.8)–(4.10), we have T ε1 (∇uδ,ε1 )1Ω×Y δ1 ⇀ ∇u1 +∇yû1 weakly in L2(Ω× Y1)N , (5.10) T ε2 (∇uε2)1Ω×Y δ2 ⇀ ∇yû2 weakly in L2(Ω× Y2)N , (5.11) ∇zT δ,εi (uδ,εi )1 1 δY δ i ⇀ ∇zG1 weakly in L2(Ω× RN )N . (5.12) 32 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Therefore, with (4.12) and in view of (4.11) with V = G1 and Ψ = û, same arguments as in the proof of Theorem 4.9, Theorem 5.1 together with (5.6), (5.7), (5.9) - (5.12), and Theorem 3.9 (ii) yield∫ Ω fG1(·,∞) dx = 1 |Y | ∫ Ω×Y A(x, y)(∇G1(·,∞) +∇yû)(∇G1(·,∞) +∇yû) dx dy + k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zG1(x, z) dx dz ≤ lim inf ε→0 Aδ,ε + lim inf ε→0 Bδ,ε ≤ lim inf ε→0 (Aδ,ε + Bδ,ε) = lim inf ε→0 [ 2∑ i=1 (∫ Ωδ,εi Aε∇uδ,εi ∇u δ,ε i dx ) − Cδ,ε ] ≤ lim sup ε→0 [ 2∑ i=1 (∫ Ωδ,εi Aε∇uδ,εi ∇u δ,ε i dx ) − Cδ,ε ] ≤ lim sup ε→0 2∑ i=1 ∫ Ωδ,εi Aε∇uδ,εi ∇u δ,ε i dx = lim ε→0 [ 2∑ i=1 (∫ Ωδ,εi fuδ,εi dx ) − εγ ∫ Γδ,ε hδ,ε(uδ,ε1 − u δ,ε 2 )(uδ,ε1 − u δ,ε 2 ) dσx ] = ∫ Ω fG1(·,∞) dx, (5.13) which implies that these inequalities are actually equalities. Hence, equations (5.1) and (5.2) hold true. Finally, the convergences in (5.3)–(5.5) follows from (5.13) and with the application of some properties of limits as well as lim sup and lim inf. � Corollary 5.3. Under the assumptions of Theorem 5.2, we have the corrector results for i = 1, 2, ‖∇uδ,εi 1 Ω √ δ,ε i −∇u1− N∑ j=1 Uεi (∂G1 ∂xj (x,∞) ) Uεi ( ∇yχ̂j(x, y) ∣∣ Yi ) ‖L2(Ωδ,εi ) → 0, (5.14) and ‖∇uδ,εi ‖L2(Ωδ,εi \Ω √ δ,ε i ) → k1 |Y |1/2 ‖∇zG1‖L2(Ω×RN ). (5.15) Furthermore, for i = 1, 2, T δ,εi (uδ,εi )→ G1 strongly in L2(Ω;L2 loc(RN )). (5.16) Proof. First, we note that from (4.12) and (4.46), we have û1 = û ∣∣ Ω×Y1 = N∑ j=1 ∂G1 ∂xj (x,∞)χ̂j(x, y) ∣∣ Y1 , which implies ∇yû1 = ∇y N∑ j=1 ∂G1 ∂xj (x,∞)χ̂j(x, y) ∣∣ Y1 = N∑ j=1 ∂G1 ∂xj (x,∞)∇yχ̂j(x, y) ∣∣ Y1 . EJDE-2023/85 PERIODIC UNFOLDING METHOD 33 This along with (5.2), (5.3), Theorem 2.5 (ii), linearity of Uε1 , triangle inequality, and Theorem 2.5 (i), yield ‖∇uδ,ε1 1 Ω √ δ,ε 1 −∇u1 − N∑ j=1 Uε1 (∂G1 ∂xj (x,∞) ) Uε1 ( ∇yχ̂j(x, y) ∣∣ Y1 ) ‖L2(Ωδ,ε1 ) = ‖∇uδ,ε1 1 Ω √ δ,ε 1 −∇u1 − Uε1 (∇yû1)‖L2(Ωδ,ε1 ) ≤ ‖∇uδ,ε1 1 Ω √ δ,ε 1 − Uε1 (∇u1)− Uε1 (∇yû1)‖L2(Ωδ,ε1 ) + ‖Uε1 (∇u1)−∇u1‖L2(Ωδ,ε1 ) → 0, which implies (5.14) for i = 1. Meanwhile, from (4.12) and (4.46), we have û2 = û ∣∣ Ω×Y2 + yγ∇u1 = N∑ j=1 ∂G1 ∂xj (x,∞)χ̂j(x, y) ∣∣ Y2 + yΓ∇u1, where yΓ is given in (4.13). Moreover, a number of computations yield∇y(yγ∇u1) = ∇u1. This implies ∇yû2 = ∇y ( N∑ j=1 ∂G1 ∂xj (x,∞)χ̂j(x, y) ∣∣ Y2 + yΓ∇u1 ) = N∑ j=1 ∂G1 ∂xj (x,∞)∇yχ̂j(x, y) ∣∣ Y2 +∇u1. This along with (5.2), (5.4), Theorem 2.5 (ii), linearity of Uε2 , triangle inequality, and Theorem 2.5 (i) yield ‖∇uδ,ε2 1 Ω √ δ,ε 2 −∇u1 − N∑ j=1 Uε2 (∂G1 ∂xj (x,∞) ) Uε2 ( ∇yχ̂j(x, y) ∣∣ Y2 ) ‖L2(Ωδ,ε2 ) = ‖∇uδ,ε2 1 Ω √ δ,ε 2 −∇u1 − Uε2 (∇yû2 −∇u1)‖L2(Ωδ,ε2 ) ≤ ‖∇uδ,ε2 1 Ω √ δ,ε 2 − Uε2 (∇yû2)‖L2(Ωδ,ε2 ) + ‖Uε2 (∇u1)−∇u1‖L2(Ωδ,ε2 ) → 0, which shows (5.14) for i = 2. Let us prove (5.15). Indeed, from (3.11), (A3), and (5.5) we have by unfolding lim ε→0 ‖∇uδ,εi ‖ 2 L2(Ωδ,εi \Ω √ δ,ε i ) = lim ε→0 (δN2 −1 ε )2 1 |Y | ∫ Ω× 1 δY δ i ∇zT δ,εi (uδ,εi )∇zT δ,εi (uδ,εi ) dx dz = k2 1 |Y | ∫ Ω×RN |∇zG1|2 dx dz = k2 1 |Y | ‖∇zG1‖2L2(Ω×RN ). Finally, we prove (5.16). Let ω be an open and bounded set and choose R > 0 such that ω∪B ⊂ B(O,R), the ball in RN with center at O of radius R. In view of (4.28) and (4.29), a Poincaré inequality holds on the space B(O,R). By Definition 3.4 and since Ω̂δ,εi = Ωδ,εi \ Λδ,εi , we obtain ‖T δ,εi (uδ,εi )−G1‖2L2(Ω̂δ,εi ×B(O,R)) 34 J. AVILA, B. CABARRUBIAS EJDE-2023/85 = ‖T δ,εi (uδ,εi )−G1‖2L2(Ωδ,εi ×B(O,R)) − ‖T δ,εi (uδ,εi )−G1‖2L2(Λδ,εi ×B(O,R)) ≤ C‖∇zT δ,εi (uδ,εi )−∇zG1‖2L2(Ωδ,εi ×B(O,R)) − C‖∇zT δ,εi (uδ,εi ) −∇zG1‖2L2(Λδ,εi ×B(O,R)) + ‖T δ,εi (uδ,εi )−G1‖2L2(Λδ,εi ×B(O,R)) ≤ C ( ‖∇zT δ,εi (uδ,εi )−∇zG1‖2L2(Ωδ,εi ×B(O,R)) + ‖∇zG1‖2L2(Λδ,εi ×B(O,R)) + ‖G1‖2L2(Λδ,εi ×B(O,R)) ) , where C is a generic constant. For δ small enough, ω ⊂ B(O,R) ⊂ RN . This and when (5.5) is applied to the first term in the right-hand side, and (4.29) to the remaining two terms, then the left-hand side above approaches zero and so we obtain T δ,εi (uδ,εi )→ G1 strongly in L2(Ω× ω), which yields (5.16). � 5.2. Case γ ∈ (−1, 0). For this case, we start by giving the energy convergence. Theorem 5.4. Let γ ∈ (−1, 0). Under the assumptions of Theorems 4.11 and 4.12, one has lim ε→0 2∑ i=1 ∫ Ωδ,εi Aε∇uδ,εi ∇u δ,ε i dx = ∫ Ω Ahom∇u1∇u1 dx+ k2 1 ∫ Ω Θu2 1 dx = 1 |Y | ∫ Ω×Y A(x, y)(∇G1(·,∞) +∇yû)(∇G1(·,∞) +∇yû) dx dy + k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zG1(x, z) dx dz, (5.17) lim ε→0 2∑ i=1 ∫ Λδ,εi |∇uδ,εi | 2dx = 0. (5.18) Moreover, we have the following strong convergences hold T ε1 (∇uδ,ε1 )1Ω×Y δ1 → ∇u1 +∇yû1 strongly in L2(Ω× Y1)N , (5.19) T ε2 (∇uδ,ε2 )1Ω×Y δ2 → 0 strongly in L2(Ω× Y2)N , (5.20) ∇zT δ,ε1 (uδ,ε1 ))1 1 δY δ 1 → ∇zG1 strongly in L2(Ω× RN )N , (5.21) ∇zT δ,ε2 (uδ,ε2 ))1 1 δY δ 2 → 0 strongly in L2(Ω× RN )N . (5.22) Proof. For this case, the proof of (5.19)–(5.22) is similar to that of Theorem 5.2, the difference being convergences (5.20) and (5.22) which are immediate from (4.55) and (4.57). � The following result is proved similarly to the one in Corollary 5.3. EJDE-2023/85 PERIODIC UNFOLDING METHOD 35 Corollary 5.5. Under the assumptions of Theorem 5.4, we have the following corrector results: ‖∇uδ,ε1 1 Ω √ δ,ε i −∇u1 − N∑ j=1 Uε1 (∂G1 ∂xj (x,∞) ) Uε1 ( ∇yχ̂j(x, y) ∣∣ Yi ) ‖L2(Ωδ,ε1 ) → 0, ‖∇uδ,ε1 ‖L2(Ωδ,εi \Ω √ δ,ε i ) → k1 |Y |1/2 ‖∇zG1‖L2(Ω×RN ), ‖∇uδ,ε2 ‖L2(Ωδ,ε2 ) → 0. Moreover, T δ,ε1 (uδ,ε1 )→ G1 strongly in L2(Ω;L2 loc(RN )), T ε2 (uδ,ε2 )→ u1 strongly in L2(Ω, H1(Y2)). 5.3. Case γ = −1. The proofs for the next results are similar to the previous cases with appropriate modifications. Theorem 5.6. Let γ = −1. Under the assumptions of Theorems 4.13 and 4.14, one has lim ε→0 2∑ i=1 ∫ Ωδ,εi Aε∇uδ,εi ∇u δ,ε i dx = ∫ Ω Ahom∇u1∇u1 dx+ k2 1 ∫ Ω Θu2 1 dx = 1 |Y | ∫ Ω×Y A(x, y)(∇G1(·,∞) +∇yû)(∇G1(·,∞) +∇yû) dx dy + k2 1 |Y | ∫ Ω×(RN\B) F (x, z)∇zG1(x, z)∇zG1(x, z) dx dz, lim ε→0 2∑ i=1 ∫ Λδ,εi |∇uδ,εi | 2dx = 0. Moreover, we have the strong convergences for i = 1, 2, T ε1 (∇uδ,ε1 )1Ω×Y δ1 → ∇u1 +∇yû1 strongly in L2(Ω× Y1)N , T ε2 (∇uδ,ε2 )1Ω×Y δ2 → ∇yû2 strongly in L2(Ω× Y2)N , ∇zT δ,εi (uδ,εi ))1 1 δY δ i → ∇zG1 strongly in L2(Ω× RN )N . Corollary 5.7. Under the assumptions of Theorem 5.6, we have the corrector results for i = 1, 2, ‖∇uδ,εi 1 Ω √ δ,ε i −∇u1 − N∑ j=1 Uεi (∂G1 ∂xj (x,∞) ) Uεi ( ∇yχ̂j(x, y) ∣∣ Yi ) ‖L2(Ωδ,εi ) → 0, ‖∇uδ,εi ‖L2(Ωδ,εi \Ω √ δ,ε i ) → k1 |Y |1/2 ‖∇zG1‖L2(Ω×RN ). 36 J. AVILA, B. CABARRUBIAS EJDE-2023/85 Furthermore, for i = 1, 2, T δ,εi (uδ,εi )→ G1 strongly in L2(Ω;L2 loc(RN )). Acknowledgments. The authors would like to express their gratitude to the Com- putational Research Laboratory Program of the Institute of Mathematics, Univer- sity of the Philippines Diliman for funding this research and to Prof. Sara Monsurrò for her helpful insights to make the paper better. J. 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Jake Avila Institute of Mathematics, University of the Philippines Diliman, 1101 Diliman, Quezon City, Philippines Email address: javila@math.upd.edu.ph Bituin Cabarrubias Institute of Mathematics, University of the Philippines Diliman, 1101 Diliman, Quezon City, Philippines Email address: bituin@math.upd.edu.ph 1. Introduction 2. Unfolding operator for two-component domains 3. Unfolding operator for domains with very small inclusions 3.1. Unfolding operator 3.2. Trace behaviors 4. Homogenization results 4.1. Case bold0mu mumu <-1<-1<-1<-1<-1<-1 4.2. Case bold0mu mumu (-1, 0)(-1, 0)(-1, 0)(-1, 0)(-1, 0)(-1, 0) 4.3. Case bold0mu mumu =-1=-1=-1=-1=-1=-1 4.4. Case bold0mu mumu k1=+k1=+k1=+k1=+k1=+k1=+ 4.5. Case bold0mu mumu k1=0k1=0k1=0k1=0k1=0k1=0 5. Corrector results 5.1. Case bold0mu mumu < -1< -1< -1< -1< -1< -1 5.2. Case bold0mu mumu (-1,0)(-1,0)(-1,0)(-1,0)(-1,0)(-1,0) 5.3. Case bold0mu mumu =-1=-1=-1=-1=-1=-1 Acknowledgments References