Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 59, pp. 1–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu T-COERCIVITY FOR THE ASYMPTOTIC ANALYSIS OF SCALAR PROBLEMS WITH SIGN-CHANGING COEFFICIENTS IN THIN PERIODIC DOMAINS RENATA BUNOIU, KARIM RAMDANI, CLAUDIA TIMOFTE Abstract. We study a scalar problem in thin periodic composite media formed by two materials, a classical one and a metamaterial (also known as negative material). By applying T-coercivity methods and homogenization techniques specific to the thin periodic domains under consideration, for two geomet- ric settings, we derive the homogenized limit problems, which both exhibit dimension-reduction effects. 1. Introduction Metamaterials (also known as negative or left-handed materials) are artificial composite materials exhibiting negative dielectric permittivity and magnetic per- meability for some frequencies, and hence behaving as negative refractive index ma- terials (see the review papers by Shamonina and Solymar [50] and Smith, Pendry and Wiltshire [51]). This property leads to new super lens effects and explains the growing interest for them in the last decade. From a mathematical point a view, Bouchitté et al. proposed a rigorous derivation and a mathematical justification of negative materials in electromagnetics using homogenization techniques [12, 13, 14] and similar results have been established by Ávila et al. [5] in phononics. The well-posedness of scalar problems involving both classical dielectric materials and metamaterials (and, hence, leading to differential operators with sign-changing co- efficients) has been studied by Bonnet-Ben Dhia et al. [8, 10], Chesnel and Ciarlet Jr. [24], Chung and Ciarlet Jr. [25], Nicaise and Venel [47], Nguyen [44, 45]. The case of Maxwell’s system with sign changing coefficients has also been investigated by Bonnet-Ben Dhia, Chesnel and Ciarlet Jr. [7], Fernandes and Raffetto [30], Oliveri and Raffetto [48] and Nguyen and Sil [46]. More recently, homogenization for composite materials involving both positive and negative materials has been considered in [11, 16, 17]. Our goal in this article is to study a scalar problem in specific periodic thin domains, namely composite materials formed by two constituents, a classical one and a metamaterial, occupying a thin three-dimensional region denoted Ωε. The results are valid in Rd (d > 2), but the three-dimensional case is physically more relevant, if we think, for instance, to the practical applications mentioned below. 2010 Mathematics Subject Classification. 35B27, 78M40, 35Q60, 35J20. Key words and phrases. Sign-changing coefficients; T-coercivity; thin domains; asymptotic analysis. c©2021. This work is licensed under a CC BY 4.0 license. Submitted October 29, 2020. Published June 26, 2021. 1 2 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 The domain Ωε is divided in two open subdomains, denoted by Ωε1 and Ωε2. The subdomain Ωε2, assumed to be disconnected, is formed by ε-periodically distributed inclusions which do not touch the boundary of Ωε. We shall consider two distinct geometrical settings, which both correspond to important physical applications, like demultiplexer devices and antennas [49]. In the first geometrical setting (see Figure 1), the inclusions are ε-periodically distributed in an horizontal layer of height ε, while in the second one (see Figure 2), the inclusions are ε-periodically distributed at the interior of a thin vertical rod with thickness of order ε. For both of these geometries, we analyze the well-posedness of the correspond- ing microscopic problems (see (2.4) and (5.2)) and the asymptotic behavior of their solutions as the small parameter ε tends to zero. The main difference with respect to the classical scalar case is the presence of two materials with conductivities of different sign. Consequently, classical methods based on ellipticity arguments to prove well-posedness and obtain energy estimates do not apply for the problems under study. To cope with this difficulty, we apply T-coercivity technics (see Def- inition 3.1; we refer the reader to the papers of Bonnet-Ben Dhia et al. [8, 9] for more details), which have been already successfully used in the context of homog- enization by two of the authors in [17, 16]. The first difference from these last two references is the degenerate geometry considered in this work. The second differ- ence concerns the construction of the T-coercive operators. More precisely, like in [17], the T-coercive operators involved here are constructed using suitably chosen extension operators from one subdomain (the one occupied by the positive or neg- ative material) into the other. Defining the contrast κ between the two materials as in (2.3), we need to distinguish, throughout the whole paper, between the cases of large contrasts and small contrasts, each one involving a particular extension operator (and correspondingly a particular T-coercive operator). We propose in this paper a similar and quite direct proof for both cases, whereas [17] was limited to the case of large contrasts, while the proof proposed in [16] was more involved and indirect. The use of alternative approaches to study homogenization of these problems with sign-changing coefficients, like the analysis of the spectrum of the Neumann-Poincaré operator developed in Bonnetier et al. [11], is also probably possible. Once the well-posedness proved, by using the two-scale convergence method (see Nguetseng [43] and Allaire [1]) adapted to thin periodic domains (see, for instance, Gahn and Neuss-Radu [32] and Jerez-Hanckes et al. [39]), we derive for both geometries depicted in Figures 1 and 2 the associated homogenized problems (see Theorem 4.3 and Theorem 5.3). Let us emphasize that they both exhibit dimension reduction effects, 3D → 2D and 3D → 1D, respectively. Nevertheless, the values of the homogenized coefficients keep track of the lost variables, at a local scale, through the cell problems (see Remarks 4.4 and 5.4). For mathematical studies of diffusion problems in thin periodic media, we refer, for instance, to [6, 18, 19, 28, 29, 31, 32, 33, 34, 38, 39, 40, 42] and the references therein. For elasticity problems in related thin periodic domains, we refer to [20, 21, 22, 27, 36, 37]. For flow problems in thin porous media, we refer, for example, to [2, 3, 4, 15, 35]. This article is organized as follows: in Section 2, we introduce the microscopic problem (2.4) stated in the thin periodic layer and we fix the notation. In Section 3, we prove the well-posedness of problem (2.4), using appropriate extension operators EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 3 and the T-coercivity technics. The homogenization result for the case of a thin periodic layer is stated and proved in Section 4. Finally, in Section 5, we address the case of a three-dimensional thin periodic rod. 2. Setting of the problem in the case of a thin periodic layer Figure 1. Example of a thin periodic layer and the corresponding reference cell Y . We start by describing more precisely the geometry of the domain Ωε, which represents a two-phase thin periodic composite medium (see Figure 1). Let ω be a smooth and bounded domain in R2. We denote the independent variable x ∈ R3 by x = (x1, x2, x3) = (x̄, x3) and we define Ωε = ω × (0, ε) = {x = (x̄, x3) ∈ R3 : x̄ ∈ ω, 0 < x3 < ε}. (2.1) Here, ε ∈ (0, 1) is a sequence of strictly positive numbers such that 1 ε ∈ N∗. This small parameter is related to the characteristic dimension of our domain. Thus, Ωε is a thin heterogeneous layer, its thickness, the periodicity of its heterogeneities and their size being of order ε. More precisely, the microscopic structure of Ωε consists of an exact number of replicated unit cells Y = (0, 1)3, rescaled with ε. The reference cell is given by Y = Y1 ∪ Y 2, where Y1 and Y2 are two non-empty disjoint connected open subsets of Y such that Y 2 ⊂ Y . We assume that the boundary ∂Y2 is Lipschitz continuous. For each k ∈ Z3, we define the shifted cells Y k 1 = k + Y1 and Y k 2 = k + Y2. We also define, for each ε, the set of indexes Zε = { k ∈ Z3 : εY k 2 ⊂ Ωε } . Finally, we set Ωε2 = ∪k∈Zε ( εY k 2 ) and Ωε1 = Ω \ Ω ε 2. We denote by nε the unit outward normal to Ωε2. The boundary of the domain Ωε is split into three parts: the lateral boundary Σε,D = {x ∈ R3 : x̄ ∈ ∂ω, 0 < x3 < ε}, the top boundary Σε,N+ = {x ∈ R3 : x̄ ∈ ω, x3 = ε} and the bottom boundary Σε,N− = {x ∈ R3 : x̄ ∈ ω, x3 = 0}. Given two real constants a1 and a2 such that a1a2 < 0, let a ∈ L∞(R3) denote the 1-periodic function in the variables y1 and y2 defined by a(y) = a11Y1(y) + a21Y2(y). For simplicity and without loss of generality, we assume that a1 > 0, a2 < 0, (2.2) 4 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 and we define the contrast κ as the positive number κ = ∣∣∣a1 a2 ∣∣∣ = a1 |a2| . (2.3) Setting aε(x) = a (x ε ) , ∀x ∈ Ωε, our goal is to analyze the asymptotic behavior, as ε→ 0, of the solution uε of the problem −div(aε∇uε) = f in Ωε, uε = 0 on Σε,D, ∂uε ∂νε± = 0 on Σε,N± , (2.4) where the function f ∈ L2(ω) is given and νε± is the unit outward normal to Σε,N± . Let us emphasize that this particular choice of the function f is classical in the framework of asymptotic analysis in thin domains. Nevertheless, more general right-hand sides can be considered, see, for instance, [41, Section 2]. It is worth noticing that the above scalar problem is nothing but the problem obtained from the homogenization problem −div(aε∇uε) = f in Ωε, uε = 0 on Σε,D, aε∇uε · νε± = 0 on Σε,N± , in the special case where the 3x3 matrix a = (aij)16i,j63 is given by aε(x) = a (x ε ) , a(y) = a(y)I3. (2.5) To write the variational formulation of problem (2.4), for every positive ε < 1, we introduce the Hilbert space V ε = { v ∈ H1(Ωε) : v = 0 on Σε,D } , (2.6) endowed with the norm ‖v‖V ε = ‖∇v‖L2(Ωε), for any v ∈ V ε. The variational formulation of problem (2.4) is as follows: find uε ∈ V ε such that Aε(uε, v) = `ε(v), ∀v ∈ V ε, (2.7) where the bilinear form Aε : V ε × V ε → R and the linear form `ε : V ε → R are given by Aε(u, v) = ∫ Ωε aε∇u · ∇v dx = a1 ∫ Ωε 1 ∇u · ∇v dx+ a2 ∫ Ωε 2D ∇u · ∇v dx, `ε(v) = ∫ Ωε fv dx. EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 5 3. Well-posedness of the microscopic problem in a thin periodic layer via T-coercivity technics Since the bilinear form involved in the above weak formulation is not coercive, the well-posedness of (2.7) is far from being obvious. To overcome this difficulty, an issue is to make use of T-coercivity technics (see, for instance, [8, 9]). In the framework of homogenization, this method has already been successfully used for scalar problems set in a fixed domain with periodic geometry; see [17] for the case of a Dirichlet scalar problem (case of large contrasts only) and [16] for the one of Dirichlet and Neumann scalar problems and vector Maxwell’s equations for extreme contrasts, large or small. It is worth mentioning reference [11], in which homogenization of scalar problems with sign-changing coefficients is achieved using another approach, namely the analysis of the spectrum of the Neumann-Poincaré operator. 3.1. Background on T-coercivity. We start by recalling the definition of T- coercivity. Definition 3.1. Let V be a Hilbert space endowed with the norm ‖ · ‖ and let T ∈ L(V ) be a bounded linear operators on V . A bilinear form a(·, ·) defined on V × V is called T-coercive if there exists γ > 0 such that |a(u,Tu)| > γ‖u‖2, ∀u ∈ V. The T-coercivity result used throughout the paper is detailed below (Theorem 3.2). Note that this result is stated in the special case of symmetric bilinear forms (only such bilinear forms are involved in this work) depending on a small parameter. In particular, it is slightly different from the result given in [17, Theorem 2.1]) which holds for arbitrary bilinear forms. The T-coercivity operators needed to prove well-posedness for the three problems with sign -changing coefficients involved in our study (namely the microscopic problem, the cell problems and the two- scale limit problem) are constructed using suitable extension operators (from the background medium into the inclusions or vice-versa). It is worth noticing that such extension operators are generally used in homogenization theory to obtain compactness results, while they are used here to obtain well-posedness and energy estimates for the solutions of our problems. We also emphasize that the construction of suitable extension operators from the inclusions into the exterior domain is more technical than the construction of extension operators from the outer domain into the inclusions. Indeed, this requires to adapt an idea used in [23] to obtain uniform energy estimates of gradient type for these operators (see (3.13) and (3.21)). Theorem 3.2. Let V be a Hilbert space equipped with the norm ‖ ·‖ and let Aε(·, ·) be a bilinear form on V satisfying the following conditions: (1) Aε(·, ·) is symmetric: Aε(u, v) = Aε(v, u). (2) Aε(·, ·) is uniformly continuous: there exists M > 0 such that |Aε(u, v)| 6M‖u‖‖v‖, ∀u, v ∈ V. (3.1) (3) There exists a family (Tε)ε>0 of uniformly bounded linear operators on V and γ > 0 such that |Aε(u,Tεu)| > γ‖u‖2, ∀u ∈ V. (3.2) 6 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 Then, given a family (`ε)ε>0 in V ′, the space of linear forms on V , the variational problem: Find uε ∈ V such that Aε(uε, v) = `ε(v) for all v ∈ V (3.3) admits a unique solution uε ∈ V for all ε > 0 and there exists C > 0 independent of ε such that ‖uε‖ 6 C‖`ε‖V ′ . (3.4) Proof. First of all, we prove that (Tε)ε>0 is a family of uniformly boundedly invert- ible operators, i.e. that Tε ∈ L(V ) is an isomorphism and there exists a constant C > 0 independent of ε such that ‖(Tε)−1‖ 6 C. Let us introduce the bounded operator Aε ∈ L(V ) associated to the bilinear form Aε(·, ·), (Aεu, v) = Aε(u, v), ∀u, v ∈ V. Equation (3.1) implies that ‖Aεu‖ = sup v 6=0 (Aεu, v) ‖v‖ = sup v 6=0 Aε(u, v) ‖v‖ 6M‖u‖. Consequently, (Aε)ε>0 is a family of uniformly bounded operators (with ‖Aε‖ 6 M). Since ‖(Tε)∗Aε‖ 6 ‖(Tε)∗‖‖Aε‖ = ‖Tε‖‖Aε‖, the family ((Tε)∗Aε)ε>0 is also uniformly bounded (recall that (Tε)ε>0 is uniformly bounded by assumption). Moreover, thanks to (3.2), the Lax-Milgram theorem implies that (Tε)∗Aε is an isomorphism, and the estimates γ‖u‖2 6 |Aε(u,Tεu)| = | (Aεu,Tεu) | = | ((Tε)∗Aεu, u) | 6 ‖(Tε)∗Aεu‖‖u‖, show that ‖ ((Tε)∗Aε) −1 ‖ 6 1/γ. The family ((Tε)∗Aε))ε>0 is thus uniformly boundedly invertible. As a consequence, Aε is injective and has closed range. On the other hand, since Aε(·, ·) is symmetric, (3.2) shows that AεTε is uniformly boundedly invertible, as well. Hence, Aε is onto. Consequently, Aε and Tε define two isomorphisms on V and are uniformly boundedly invertible, as can it be seen from the identities (Aε)−1 = Tε(AεTε)−1 and (Tε)−1 = (AεTε)−1Aε. Let us prove now that the variational problem (3.3) is well-posed. Since Tε is invertible, (3.3) is equivalent to the variational problem Find uε ∈ V such that Aε(uε,Tεv) = `ε(Tεv) for all v ∈ V . (3.5) By assumption, the bilinear form Ãε(·, ·) := Aε(·,Tε·) satisfies Ãε(u, u)| > γ‖u‖2, ∀u ∈ V. Applying the Lax-Milgram theorem to the coercive bilinear form Ãε(·, ·) and the linear form ˜̀ε(·) := `ε(Tε·), we obtain the existence of a unique solution uε ∈ V for (3.5). Moreover, since (Tε)ε>0 is uniformly bounded, the estimate (3.4) holds. � The result above is used in the next two sections to establish the well-posedness of a scalar problem set in the reference cell (namely the cell problem (4.10) involved in the definition of the homogenized matrix) and of problem (2.4) (or equivalently (2.7)) for extreme contrasts. EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 7 3.2. Well-posedness in the reference cell. Let H1 per(Y ) denote the subset of H1(Y ) constituted of functions being 1-periodic with respect to the variables y1 and y2: H1 per(Y ) = {v ∈ H1(Y ) | v is 1-periodic in y1 and y2}. Setting M(u) = 1 |Y | ∫ Y udy = ∫ Y udy, let Wper(Y ) be the Hilbert space Wper(Y ) = { u ∈ H1 per(Y ) :M(u) = 0 } , (3.6) endowed with the norm ‖u‖Wper(Y ) := ‖∇u‖L2(Y ). Given f ∈ L2(Y ), we investigate in this section the well-posedness of the follow- ing cell-problem. Find u ∈Wper(Y ) such that − div(a∇u) = f , in Y , (3.7) whose variational formulation reads Find u ∈Wper(Y ) such that A(u, v) = ∫ Y fv dy for all v ∈Wper(Y ), (3.8) where A(·, ·) denotes the symmetric bilinear form defined on H1(Y ) by A(u, v) = ∫ Y a(y)∇u · ∇v dy = a1 ∫ Y1 ∇u · ∇v dy + a2 ∫ Y2 ∇u · ∇v dy, (3.9) for all u, v ∈ H1(Y ). In the sequel, we denote the restrictions to Y1 and Y2 of a given function u ∈ H1(Y ) by u1 := u|Y1 and u2 := u|Y2 . To prove the well-posedness of (3.7) (or, equivalently, (3.8)), we need to distin- guish between the case of large contrasts (the positive number κ defined by (2.3) is large enough) and the one of small contrasts (κ is small enough), as each one is treated using a specific T-coercivity operator. Large contrasts. Let P denote the harmonic extension operator from Y1 to Y2. In other words, given u ∈ H1(Y1), Pu ∈ H1(Y ) is defined by setting Pu = u in Y1, while in Y2, Pu ∈ H1(Y2) is the unique solution of the Dirichlet boundary value problem −∆(Pu) = 0 in Y2, Pu = u on ∂Y2. Clearly, there exists κY > 0 such that ‖∇(Pu)‖2L2(Y ) 6 κY ‖∇u‖ 2 L2(Y1), ∀u ∈ H1(Y1). (3.10) Proposition 3.3. For u ∈ H1(Y ), let TY u := T̃Y u−M ( T̃Y u ) , (3.11) with T̃Y u = { u1 in Y1, −u2 + 2Pu1 in Y2. Then, the following assertions hold: (1) TY ∈ L(Wper(Y )). 8 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 (2) TY y = y −M(y). (3) For κ > κY , where κ is defined in (2.3), there exists γ > 0 such that for all u ∈Wper(Y ): A(u,TY u) = ∫ Y a(y)∇u(y) · ∇(TY u)(y) dy > γ‖∇u‖2L2(Y ). (3.12) Proof. 1. For every u ∈ Wper(Y ), TY u clearly defines a function of H1(Y ) (since it is continuous across the interface ∂Y2) satisfying M ( TY u ) = 0. Moreover, we have TY u = u −M ( T̃Y u ) on the boundary ∂Y , and hence TY u is 1-periodic in y1 and y2. Consequently, TY u ∈ Wper(Y ) for every u ∈ Wper(Y ). The continuity of the map TY is straightforward. 2. Since affine functions are harmonic, they are invariant by P and hence T̃Y y = y. 3. For all u ∈Wper(Y ) and for all η > 0, we have by Young’s inequality A(u,TY u) = a1 ∫ Y1 |∇u|2 dy + |a2| ∫ Y2 |∇u|2 dy + 2a2 ∫ Y2 ∇u · ∇(Pu) dy > a1‖∇u‖2L2(Y1) + |a2|‖∇u‖2L2(Y2) − |a2|η‖∇u‖2L2(Y2) − |a2| η ‖∇(Pu)‖2L2(Y2). Consequently, using (3.10), we obtain that A(u,TY u) > |a2| {( κ− κY η ) ‖∇u‖2L2(Y1) + (1− η)‖∇u‖2L2(Y2) } . Thus, if the contrast satisfies κ > κY , we can choose η ∈]κY /κ, 1[ and get the existence of a constant γ > 0 such that for all u ∈Wper(Y ), A(u,TY u) > γ‖∇u‖2L2(Y ) = γ‖u‖2Wper(Y ). � Remark 3.4. Let us emphasize that the results of the above proposition have been already obtained in [17, Theorem 2.5]. However, the authors omitted to subtract the average term (the validity of the result was not affected). Small contrasts. We denote by H1 0,Σ(Y ) the space of functions in H1(Y ) which are zero on the lateral boundary ΣD of the domain Y (the top and bottom boundaries are denoted ΣN± ). Let Q denote the harmonic extension operator from H1(Y2) onto H1 0,Σ(Y ). In other words, given u ∈ H1(Y2), Qu ∈ H1 0,Σ(Y ) is defined by setting Qu = u in Y2, while in Y1, Qu is the unique solution of the boundary value problem −∆(Qu) = 0 in Y1, Qu = u on ∂Y2, Qu = 0 on ΣD, ∂(Qu) ∂ν = 0 on ΣN± . Setting M2(u) = 1 |Y2| ∫ Y2 udy, EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 9 let H1 mean(Y2) = {u ∈ H1(Y2) | M2(u) = 0}. It can be easily seen that there exists κ′Y > 0 such that ‖∇(Qu)‖2L2(Y ) 6 κ ′ Y ‖∇u‖2L2(Y2), ∀u ∈ H1 mean(Y2). (3.13) Proposition 3.5. For u ∈ H1(Y ), set TY u := T̃Y u−M ( T̃Y u ) , (3.14) where T̃Y u = { u1 − 2Q(u2 −M2(u2)) in Y1, −u2 + 2M2(u2) in Y2. (3.15) Then, the following assertions hold: (1) TY ∈ L(Wper(Y )). (2) For κ < 1/κ′Y , where κ′Y is defined in (3.13) there exists γ′ > 0 such that for all u ∈Wper(Y ): A(u,TY u) = ∫ Y a(y)∇u(y) · ∇(TY u)(y) dy > γ′‖∇u‖2L2(Y ). (3.16) Proof. 1. For every u ∈ Wper(Y ), TY u clearly defines a function of H1(Y ) (since it is continuous across the interface ∂Y2) satisfying by construction M(TY u) = 0. Moreover, we have TY u = u −M ( T̃Y u ) on the boundary ∂Y , and hence TY u is 1-periodic in y1 and y2. Consequently, TY u ∈ Wper(Y ) for every u ∈ Wper(Y ). The continuity of the map TY is straightforward. 2. For all u ∈Wper(Y ) and for all η > 0, we have by Young’s inequality A(u,TY u) = A(u, T̃Y u) = a1 ∫ Y1 |∇u|2 dy + |a2| ∫ Y2 |∇u|2 dy − 2a1 ∫ Y2 ∇u · ∇(Q(u2 −M2(u2))) dy > a1‖∇u‖2L2(Y1) + |a2|‖∇u‖2L2(Y2) − a1η‖∇u‖2L2(Y1) − a1 η ‖∇(Q(u2 −M2(u2)))‖2L2(Y1). Consequently, using (3.13), we obtain that A(u,TY u) > a1‖∇u‖2L2(Y1) + |a2|‖∇u‖2L2(Y2) − a1η‖∇u‖2L2(Y1) − a1κ ′ Y η ‖∇u‖2L2(Y2), that is, A(u,TY u) > a1 { (1− η)‖∇u‖2L2(Y1) + ( 1 κ − κ′Y η ) ‖∇u‖2L2(Y1) } . Thus, if the contrast satisfies κ < 1/κ′Y , we can choose η ∈]κκ′Y , 1[ and get the existence of a constant γ′ > 0 such that for all u ∈Wper(Y ): A(u,TY u) > γ‖∇u‖2L2(Y ) = γ′‖u‖2Wper(Y ). � We can now state a well-posedness result for the cell problem (3.7) for extreme contrasts. 10 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 Theorem 3.6. Let κY and κ′Y denote the constants in (3.10) and (3.13), respec- tively. If the contrast κ := ∣∣a1 a2 ∣∣ satisfies κ > κY or κ < 1/κ′Y , (3.17) then, for every given f ∈ L2(Y ), problem (3.7) admits a unique solution and there exists a constant C > 0 such that ‖u‖Wper(Y ) = ‖∇u‖L2(Y ) 6 C‖f‖L2(Y ). (3.18) Proof. The result follows immediately from Theorem 3.2 using the T-coercivity operators of Proposition 3.3 and Proposition 3.5, respectively, for large or small contrasts. � 3.3. Well-posedness of the problem in Ωε. For every function u ∈ H1(Ωε), we set u1 := u|Ωε 1 , u2 := u|Ωε 2 . Let H1 0,Σ(Ωε1) be the space of functions u in H1(Ωε1) and such that u = 0 on the lateral boundary Σε,D of Ωε. Like for the cell problem considered in the previous section, we need to distin- guish the cases of large and small contrasts to study the well-posedness of (2.4) (or equivalently of its variational formulation (2.7)). The case of large contrasts can be easily adapted from [17] (even though the domain considered here is thin) and therefore we only state the results for the sake of completeness. For the case of small contrasts, we propose a new proof which is more elementary than the ones given for the case of a fixed domain in [16] (using the T-coercivity) and in [11] (via the analysis of the spectrum of the Neumann-Poincaré operator). Large contrasts. As in [17, Proposition 2.4], one can prove the following result. Proposition 3.7. There is a family of linear bounded extension operators (Pε)ε>0 from Ωε1 to Ωε such that (1) Pε ∈ L(H1 0,Σ(Ωε1), V ε). (2) For all u ∈ H1 0,Σ(Ωε1), (Pεu)(x) = u(x) for all x ∈ Ωε1. (3) If κY is the constant defined in (3.10), one has for all u ∈ H1 0,Σ(Ωε1), ‖∇(Pεu)‖2L2(Ωε) 6 κY ‖∇u‖ 2 L2(Ωε 1). (3.19) According to [17, Theorem 2.6], we also have the following result. Proposition 3.8. For u ∈ H1(Ωε), set Tεu = { u1 in Ωε1, −u2 + 2Pεu1 in Ωε2. (3.20) Then, the following assertions hold: (1) (Tε)ε>0 is a family of uniformly bounded linear operators of L(V ε). (2) For κ > κY , where κY is defined in (3.10), there exists γ > 0 such that for all ε > 0 and all u ∈ V ε, Aε(u,Tεu) = ∫ Ωε aε(x)∇u(x) · ∇(Tεu)(x) dx > γ‖∇u‖2L2(Ωε). EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 11 Small contrasts. Given u ∈ H1(Ωε2), let Mε 2(u) ∈ H1(Ωε2) denote the piecewise constant function defined on each inclusion εY k 2 ⊂ Ωε, k ∈ Zε, by Mε 2(u)(x) = 1 |εY k 2 | ∫ εY k 2 udx, ∀x ∈ εY k 2 . We set H1 mean(Ωε2) = { u ∈ H1(Ωε2) | Mε 2(u) = 0 } . As in [23, Lemma 2.3], one has the following result. Proposition 3.9. There is a family of linear bounded extension operators (Qε)ε>0 from Ωε2 to Ωε such that (1) Qε ∈ L(H1 mean(Ωε2), V ε). (2) For all u ∈ H1 mean(Ωε2), (Qεu)(x) = u(x) for all x ∈ Ωε2. (3) If κ′Y is the constant defined in (3.13), one has for all u ∈ H1 mean(Ωε2): ‖∇(Qεu)‖2L2(Ωε) 6 κ ′ Y ‖∇u‖2L2(Ωε 2). (3.21) Proposition 3.10. For u ∈ H1(Ωε), let Tεu = { u1 − 2Qε(u2 −Mε 2(u2)) in Ωε1, −u2 + 2Mε 2(u2) in Ωε2. (3.22) Then, the following assertions hold: (1) (Tε)ε>0 is a family of uniformly bounded linear operators of L(V ε). (2) For 0 < κ < 1/κ′Y , where κ′Y is defined in (3.13), there exists γ′ > 0 such that for all ε > 0 and all u ∈ V ε, Aε(u,Tεu) = ∫ Ωε aε(x)∇u(x) · ∇(Tεu)(x) dx > γ′‖∇u‖2L2(Ωε). Proof. 1. Let u be given in V ε. Since u2−Mε 2(u2) ∈ H1 mean(Ωε2), Qε(u2−Mε 2(u2)) is well-defined and so is Tεu. Moreover, Tεu is continuous across the interface ∂Ωε1 ∩ ∂Ωε2 and satisfies Tεu|Σε,D = 0. Hence, Tεu ∈ V ε and one clearly has Tε ∈ L(V ε) and is uniformly bounded. 2. For all u ∈ H1(Ωε) and for all η > 0, we have Aε(u,Tεu) = ∫ Ωε aε(x)∇u(x) · ∇(Tεu)(x) dx = a1 ∫ Ωε 1 |∇u1|2 dx+ |a2| ∫ Ωε 2 |∇u2|2 dx − 2a1 ∫ Ωε 1 ∇u1 · ∇ (Qε(u2 −Mε 2(u2)) dx > a1‖∇u1‖2L2(Ωε 1) + |a2|‖∇u2‖2L2(Ωε 2) − a1η‖∇u1‖2L2(Ωε 1) − a1 η ‖∇ (Qε(u−Mε 2(u)) ‖2L2(Ωε 1), where we have used Young’s inequality for the last estimate. Consequently, using (3.21), we obtain that |Aε(u,Tεu)| > a1 { (1− η)‖∇u1‖2L2(Ωε 1) + ( 1 κ − κ′Y η ) ‖∇u2‖2L2(Ωε 2) } . 12 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 Thus, if the contrast satisfies 0 < κ < 1/κ′Y , we can choose η ∈]κκ′Y , 1[ and get the existence of a constant γ > 0 such that |Aε(u,Tεu)| > γ′‖∇u‖2L2(Ωε), ∀u ∈ H1(Ωε). � We can now state a well-posedness result for problem (2.4) for extreme contrasts. Theorem 3.11. Let κY and κ′Y denote the constants in (3.19) and (3.21), respec- tively. If the contrast κ := |a1a2 | satisfies κ > κY or κ < 1/κ′Y , (3.23) then, for every given f ∈ L2(ω), problem (2.4) (or, equivalently, its variational formulation (2.7)) admits a unique solution uε ∈ V ε and there exists a constant C > 0 independent of ε such that for all ε > 0, ‖∇uε‖L2(Ωε) 6 C √ ε‖f‖L2(ω). (3.24) Proof. The claimed result follows immediately from Theorem 3.2, Proposition 3.8 and Proposition 3.10 and from the hypothesis on f (f does not depend on x3), which yields ‖f‖L2(Ωε) = √ ε‖f‖L2(ω). � 4. Homogenization results in the case of a thin periodic layer Our goal in this section is to pass to the limit, with ε → 0, in the variational formulation (2.7) of problem (2.4). To this end, we use the two-scale convergence method [43, 1], adapted to thin periodic domains (see, for instance, [42]). Let C∞per(Y ) be the space of infinitely differentiable functions in R3 that are 1- periodic in the first two variables y1 and y2. We recall the definition of the weak two-scale convergence from [42]. Definition 4.1. A sequence (vε) in L2(Ωε) weakly two-scale converges to v ∈ L2(ω × Y ) if one has 1 ε ∫ Ωε vε(x)ψ ( x̄, x ε ) dx→ ∫ ω×Y v(x̄, y)ψ(x̄, y) dx̄dy, for all ψ ∈ C∞0 (ω, C∞per(Y )). Then, we denote vε 2−s−−⇀ v. Using the a priori estimate (3.24), Poincaré inequality in Ωε (see, for instance, [40], Section 2.1) and [42, Prop. 4.4(i)], it follows that there exist u ∈ H1 0 (ω) and û ∈ L2(ω,Wper(Y )) (recall that Wper(Y ) is defined in (3.6)), such that, up to a subsequence, for ε→ 0, we obtain uε 2−s−−⇀ u, ∇x̄uε 2−s−−⇀ ∇x̄ u+∇ȳû, ∂x3 uε 2−s−−⇀ ∂y3 û. (4.1) The special form of the limits in convergences (4.1)2 and (4.1)3 suggests to intro- duce the following notation: to every w ∈ H1(ω), whose gradient ∇x̄w(x̄) has two components, we associate the tridimensional vector ∇w(x̄) defined by ∇w(x̄) = (∇x̄w(x̄), 0). EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 13 We introduce the space H = H1 0 (ω)× L2 (ω,Wper(Y )) and for all V = (v, v̂) ∈ H we define the norm ‖V‖2H = ‖∇v +∇y v̂‖2L2(ω×Y ), which is equivalent to the norm ‖∇v‖2L2(ω) + ‖∇y v̂‖2L2(ω×Y ) (see, for instance, [26, Lemma 5.4]). Theorem 4.2. Assume that the contrast κ := ∣∣a1 a2 ∣∣ satisfies κ > κY or κ < 1/κ′Y , where κY and κ′Y denote the constants in (3.19) and (3.21), respectively. The unique solution uε of the variational problem (2.7) converges, in the sense of (4.1), to the unique solution (u, û) ∈ H of the well-posed two-scale limit problem∫ ω×Y a(y)(∇u+∇yû) · (∇ϕ+∇yΦ) dx̄dy = ∫ ω f(x̄)ϕ(x̄) dx̄, (4.2) for all ϕ ∈ H1 0 (ω) and Φ ∈ L2(ω,H1 per(Y )). Proof. According to Theorem 3.11, the variational problem (2.7) admits a unique solution uε which satisfies the energy estimate (3.24). Dividing (2.7) by ε and then choosing the test function (recalling that x = (x̄, x3)) v(x̄, x3) = ϕ(x̄) + εψ ( x̄, x ε ) , (4.3) with ϕ ∈ D(ω), ψε ∈ C∞0 (ω, C∞per(Y )), we can pass to the limit as ε→ 0. Using the convergences (4.1), we obtain by standard density arguments the two-scale limit problem (4.2). We prove now the well-posedness of the limit problem (4.2). Define on H the bilinear form B(U ,V) := ∫ ω×Y a(y) ( ∇u(x̄) +∇yû(x̄, y) ) · ( ∇v(x̄) +∇y v̂(x̄, y) ) dx̄dy, for all U = (u, û) ∈ H and all V = (v, v̂) ∈ H. Given F ∈ H′ (H′ denotes here the dual space of H), consider the variational problem Find U = (u, û) ∈ H such that B(U ,V) = 〈F ,V〉H′,H, ∀V = (v, v̂) ∈ H. (4.4) We remark that the two-scale limit problem (4.2) fits in the frame of problem (4.4), for which we shall prove the well-posedness successively for large and small contrasts. (i) Large contrasts. Let T be the operator defined on H by T U = (u,TY û) , ∀U = (u, û) ∈ H, (4.5) where TY ∈ L(Wper(Y )) is defined in (3.11). Since TY ∈ L(Wper(Y )), we clearly have T ∈ L(H). For every U = (u, û) ∈ H, we note that ∇u(x̄) +∇yû(x̄, y) = ∇y ( ∇u(x̄) · yc + û(x̄, y) ) , where we have set yc := y −M(y). Therefore, B(U , T U) 14 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 = ∫ ω×Y a(y)∇y ( ∇u(x̄) · yc + û(x̄, y) ) · ∇y ( ∇u(x̄) · yc + TY û(x̄, y) ) dx̄dy. According to Proposition 3.3, we have TY y c = yc and, thus, ∇y ( ∇u(x̄) · yc + TY û(x̄, y) ) = ∇y ( ∇u(x̄) ·TY y c + TY û(x̄, y) ) = ∇yTY ( ∇u(x̄) · yc + û(x̄, y) ) . Consequently, setting Ux̄(y) := ∇u(x̄) · yc + û(x̄, y), we have B(U , T U) = ∫ ω×Y a(y)∇yUx̄(y) · ∇yTY Ux̄(y) dx̄ dy = ∫ ω A(Ux̄,TY Ux̄) dx̄, where A(·, ·) is defined in (3.9). Using the T-coercivity inequality (3.12) in the above relation, we obtain that, for κ > κY , there exists γ > 0 such that B(U , T U) > γ ∫ ω ‖∇yUx̄‖2L2(Y ) dx̄ = γ‖∇u(x̄) +∇yû(x̄, y)‖2L2(ω×Y ) = γ‖U‖2H. (ii) Small contrasts. As above, the idea is to use the T-coercivity of the reference cell problem given in (3.14) to derive a suitable two-scale T-coercivity operator T . However, since affine functions are not anymore stable by the extension operator Q used to define TY , the proof needs to be adapted. Let T be the operator defined on H by T U = (u,TY û+ TY ψ − ψ) , ∀U = (u, û) ∈ H, (4.6) where TY ∈ L(Wper(Y )) is defined in (3.14) and ψ := ∇u · yc. Let us first check that T ∈ L(H). Obviously, we only need to prove that TY û + TY ψ − ψ ∈Wper(Y ) for all U = (u, û) ∈ H. First of all, we have M (TY û+ TY ψ − ψ) = −M(ψ) = −∇u · M(yc) = 0. Next, by the definition of TY (see (3.14)), TY û is 1-periodic in the variables y1 and y2. Furthermore, it follows from the definition of T̃Y (see (3.15)) that T̃Y ψ = ψ on the lateral boundary ΣD of ∂Y . Hence, TY ψ − ψ = T̃Y ψ −M(T̃Y ψ)− ψ = −M(T̃Y ψ), on ΣD. From the above relation we deduce that TY ψ − ψ is constant on ΣD, and thus, TY û + (TY ψ − ψ) is 1-periodic in y1 and y2. Consequently, TY û + TY ψ − ψ ∈ Wper(Y ), which shows that T ∈ L(H). Moreover, for every U = (u, û) ∈ H, one can easily check that B(U , T U) = ∫ ω×Y a(y)∇y ( ψ(x̄, y) + û(x̄, y) ) · ∇y ( TY ψ(x̄, y) + TY û(x̄, y) ) dx̄dy. Hence, setting Ux̄(y) := ψ(x̄, y) + û(x̄, y), we have B(U , T U) = ∫ ω×Y a(y)∇yUx̄(y) · ∇yTY Ux̄(y) dx̄dy = ∫ ω A(Ux̄,TY Ux̄) dx̄. Using the T-coercivity inequality (3.16) in the above relation, we obtain that, for κ < 1/κ′Y , there exists γ′ > 0 such that B(U , T U) > γ′ ∫ ω ‖∇yUx̄‖2L2(Y ) dx̄ = γ′‖∇u(x̄) +∇yû(x̄, y)‖2L2(ω×Y ) = γ′‖U‖2H. EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 15 From the uniqueness of (u, û) ∈ H, all the above convergences hold for the whole sequence. � Theorem 4.3. Under the hypotheses of Theorem 4.2, the unique solution uε of the variational problem (2.7) converges, in the sense of (4.1), to (u, û) ∈ H, where u is the unique solution of the homogenized problem −divx̄ ( ahom∇x̄u(x̄) ) = f(x̄) in ω, u = 0 on ∂ω (4.7) and û(x̄, y) = − 2∑ j=1 ∂u ∂xj (x̄)χj(y) ∀(x̄, y) ∈ ω × Y. (4.8) Here, ahom is the constant homogenized 2× 2 matrix whose entries are defined, for i, j ∈ {1, 2}, by ahom ij = ∫ Y ( aij − 3∑ k=1 aik ∂χj ∂yk ) dy, (4.9) where the coefficients (aij)16i,j63 are those of the (diagonal) matrix a defined in (2.5). The function χ = (χ1, χ2) ∈ (H1 per(Y ))2 is the weak solution of the cell problem (j = 1, 2), −divy ( a(y)(∇yχj − ej) ) = 0, in Y, (∇yχj − ej) · ν± = 0, on ΣN± , MY (χj) = 0, (4.10) where ΣN+ = (0, 1)2 × {1}, ΣN− = (0, 1)2 × {0}, and ν± = (0, 0,±1). Proof. By choosing ϕ = 0 in the two-scale limit problem (4.2), we obtain∫ ω×Y a(y)(∇u+∇yû) · ∇yΦ dx̄dy = 0, (4.11) for all Φ ∈ L2(ω,H1 per(Y )). Choosing Φ ∈ L2(ω,H1 per(Y )) such that Φ = 0 on ω × ∂Y and integrating by parts with respect to y, we are formally led to −divy (a(y)∇yû) = divy(a(y)∇u) in ω × Y. Taking in (4.11) a test function Φ ∈ L2(ω,H1 per(Y )) which is zero on ΣN− (and, respectively, which is zero on ΣN+ ), we formally obtain ∇yû · ν+ = −∇u · ν+ on ω × ΣN+ , ∇yû · ν− = −∇u · ν− on ω × ΣN− . The linearity of the problem suggests us to search û(x̄, y) = −∇u(x̄) · χ(y), where the vector χ(y) = (χ1(y), χ2(y), χ3(y)) belonging to (H1 per(Y ))3 has to be determined. Recalling that ∇u(x̄) = (∇x̄u(x̄), 0), we notice that only the first two components of χ will play a role in our analysis. Inserting this factorization into the equation, we therefore obtain the two local problems (4.10). By choosing now Φ = 0 in (4.2), we obtain∫ ω×Y a(y) ( ∇u+∇yû ) · ∇ϕdx̄dy = ∫ ω f(x̄)ϕ(x̄) dx̄. (4.12) 16 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 We integrate this last equality by parts with respect to x̄ and, by using (4.8) and (4.10), we are led to the homogenized problem (4.7) with the homogenized coefficients given by (4.9). � Remark 4.4. The expected dimension reduction effect can be seen in the fact that the 2 × 2 homogenized matrix ahom is lower dimensional with respect to the initial matrix. However, information coming from the vertical direction of the initial problem is preserved. Indeed, the value of the coefficients (4.9) is influenced by the vertical local variable y3, through the solution of the cell problem (4.10). Consequently, despite the fact that the derivative in the third direction does not explicitly appear in the homogenized problem (4.7), its solution u is implicitly influenced by the third local variable y3. Due to the diagonal form of the matrix a = (aij)16i,j63 (see (2.5)), the homog- enized coefficients (4.9) can be written as ahom 11 = ∫ Y a(y) ( 1− ∂χ1 ∂y1 ) dy, ahom 12 = ahom 21 = − ∫ Y a(y) ∂χ2 ∂y1 dy, ahom 22 = ∫ Y a(y) ( 1− ∂χ2 ∂y2 ) dy, The above coefficients can be further simplified in the particular, but relevant, situation in which the reference cell Y is invariant under rotation of angle π/2 (see [17, Proposition 4.2]). 5. The case of a thin periodic rod Figure 2. Example of a thin periodic rod and the corresponding reference cell Y . In this section, we study the case of a thin rod. We start by describing the geometry of the domain Ωε, which represents now a two-phase thin rod (see Figure EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 17 2). We denote the independent variable x ∈ R3 by x = (x1, x2, x3) and we define Ωε = εY ′ × (0, L), (5.1) where Y ′ = (0, 1)2, L > 0 and ε ∈ (0, 1) is a sequence of strictly positive numbers such that L ε ∈ N∗. This small parameter is related to the characteristic dimension of our domain. Thus, Ωε is a thin heterogeneous rod, its thickness, the periodicity of its heterogeneities and their size being of order ε. More precisely, the microscopic structure of Ωε consists of an exact number of replicated unit cells Y = Y ′×(0, 1) = (0, 1)3, rescaled with ε. One has Y = Y1 ∪ Y 2, where Y1 and Y2 are two non-empty disjoint connected open subsets of Y such that Y 2 ⊂ Y . We assume that ∂Y2 is Lipschitz continuous. For each k ∈ Z3, we denote Y k α = k + Yα, for α ∈ {1, 2}. We also define, for each ε, Zε = { k ∈ Z3 : εY k 2 ⊂ Ωε } ; we set Ωε2 = ∪k∈Zε ( εY k 2 ) and Ωε1 = Ω \ Ω ε 2. The boundary of the domain Ωε is split into three parts: ΣεN , the lateral boundary of the domain Ωε, Σε,N = {x ∈ R3 : (x1, x2) ∈ ∂εY ′, 0 < x3 < L}, and Σε,D+ , Σε,D− , the top and the bottom boundaries, respectively, Σε,D+ = {x ∈ R3 : (x1, x2) ∈ εY ′, x3 = L}, Σε,D− = {x ∈ R3 : (x1, x2) ∈ εY ′, x3 = 0}. The goal in this section is to analyze the asymptotic behavior, as ε → 0, of the solution uε of the problem −div(aε∇uε) = f in Ωε, uε = 0 on Σε,D± , ∂uε ∂νε = 0 on Σε,N , (5.2) where the function f ∈ L2((0, L)) is given and νε is the unit outward normal to the lateral boundary Σε,N . Here, we have set as before aε(x) = a (x ε ) , ∀x ∈ Ωε, where a(y) denotes the 1-periodic in the variable y3 function defined on Y by a(y) = a11Y1(y) + a21Y2(y), with a1, a2 ∈ R and a1a2 < 0. To write the variational formulation of problem (5.2), we introduce, for every positive ε < 1, the Hilbert space V ε = { v ∈ H1(Ωε) : v = 0 on Σε,D± } , endowed with the norm ‖v‖V ε = ‖∇v‖L2(Ωε), for any v ∈ V ε. The variational formulation of problem (5.2) is the following: find uε ∈ V ε such that Aε(uε, v) = `ε(v), ∀v ∈ V ε, (5.3) where the bilinear form Aε : V ε × V ε → R and the linear form `ε : V ε → R are given by Aε(u, v) = ∫ Ωε aε∇u · ∇v dx, `ε(v) = ∫ Ωε fv dx . As in the case of the thin periodic layer, the aim is to prove the well-posedness of the variational problem (5.3) and then to pass to the limit with ε → 0 in this 18 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 problem. The proofs follow the same steps as in the previous case, with the use of an appropriate weakly two-scale convergence (see Definition 5.2 below) and with a slight modification in the definition of the extension operator Q, as follows. In this case, we denote by H1 0,Σ(Y ) the space of functions in H1(Y ) which are zero on the upper and lower boundaries ΣD± of the domain Y (the lateral boundary is denoted ΣN ). Let Q denote the harmonic extension operator from H1(Y2) onto H1 0,Σ(Y ). In other words, given u ∈ H1(Y2), Qu ∈ H1 0,Σ(Y ) is defined by setting Qu = u in Y2, while, in Y1, Qu is the unique solution of the boundary value problem −∆(Qu) = 0 in Y1, Qu = u on ∂Y2, Qu = 0 on ΣD± , ∂(Qu) ∂ν = 0 on ΣN . With the notation M2(u) = 1 |Y2| ∫ Y2 udy and setting H1 mean(Y2) = { u ∈ H1(Y2) | M2(u) = 0 } , it can be easily seen that there exists κ′Y > 0 such that ‖∇(Qu)‖2L2(Y ) 6 κ ′ Y ‖∇u‖2L2(Y2), ∀u ∈ H1 mean(Y2). (5.4) Next we have a well-posedness result. Theorem 5.1. Assume that the contrast κ := ∣∣a1 a2 ∣∣ satisfies κ > κY or κ < 1/κ′Y , where κY and κ′Y denote the constants in (3.19) and (5.4) respectively. Then, for any ε ∈ (0, 1), the variational problem (5.3) has a unique solution uε ∈ V ε. Moreover, there exists a constant C > 0, independent of ε, such that ‖∇uε‖L2(Ωε) ≤ Cε‖f‖L2(ω). (5.5) To pass to the limit, with ε→ 0, in the variational formulation (5.3) of problem (5.2), we use again the two-scale convergence method [43, 1], adapted to the thin periodic rod case. Let C∞per(Y ) be the space of infinitely differentiable functions in R3 that are 1-periodic in the variable y3. We recall the definition of the weakly two-scale convergence (see, for instance, [39]). Definition 5.2. A sequence (vε) in L2(Ωε) weakly two-scale converges to v ∈ L2((0, L)× Y ) if one has 1 ε2 ∫ Ωε vε(x)ψ ( x, x ε ) dx→ ∫ (0,L)×Y v(x3, y)ψ(x3, y) dx3 dy, for all ψ ∈ C∞0 ((0, L), C∞per(Y )). Then, we denote vε 2−s−−⇀ v. EJDE-2021/59 SIGN-CHANGING PROBLEMS IN THIN PERIODIC DOMAINS 19 Using the a priori estimate (5.5) and well-adapted compactness results (see, for instance, [39]), it follows that there exist u ∈ H1 0 (0, L) and û ∈ L2((0, L), H1 per(Y )), with MY (û) = 0, such that, up to a subsequence, for ε→ 0, we obtain uε 2−s−−⇀ u, ∇x′uε 2−s−−⇀ ∇y′ û, ∂x3 uε 2−s−−⇀ du dx3 + ∂y3 û, (5.6) where x′ = (x1, x2) and y′ = (y1, y2). Here, the space H1 per(Y ) is defined by H1 per(Y ) = {v ∈ H1(Y ) : v is 1-periodic in y3}. The special form of the limits in convergences (5.6)2 and (5.6)3 suggests the introduction of the following notation: to every w ∈ H1(0, L), we associate the three-dimensional vector ∇w(x3) defined by ∇w(x3) = ( 0, 0, dw dx3 ) . Let Wper(Y ) = {v ∈ H1 per(Y ) :MY (v) = 0}. We introduce the space H = H1 0 (0, L)× L2 ((0, L),Wper(Y )) and for all V = (v, v̂) ∈ H we define the norm ‖V‖2H = ∥∥∇v +∇y v̂ ∥∥2 L2((0,L)×Y ) . Theorem 5.3. Assume that the contrast κ := ∣∣∣a1a2 ∣∣∣ satisfies κ > κY or κ < 1/κ′Y , where κY and κ′Y denote the constants in (3.19) and (5.4), respectively. Then, the unique solution uε of the variational problem (5.3) converges, in the sense of (5.6), to (u, û) ∈ H, where u is the unique solution of the homogenized problem −ahom d2u dx2 3 = f on (0, L), u(0) = u(L) = 0 (5.7) and û(x3, y) = − du dx3 (x3)χ3(y), ∀(x3, y) ∈ (0, L)× Y. (5.8) Here, ahom is the scalar defined by ahom = ∫ Y a(y) ( 1− ∂χ3 ∂y3 ) dy. (5.9) The function χ3 ∈ H1 per(Y ) is the weak solution of the cell problem −divy(a(y)(∇yχ3 − e3)) = 0 in Y, (∇yχ3 − e3) · ν = 0 on ΣN , M(χ3) = 0, (5.10) where ΣN denotes the lateral boundary of the reference cell Y and ν is the unit outward normal to ΣN . 20 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2021/59 Remark 5.4. 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Solymar; Metamaterials: How the subject started, Metamaterials 1 (2007), no. 1, 12–18. [51] D. R. Smith, J. B. Pendry, M. C. K. Wiltshire; Metamaterials and negative refractive index, Science 305 (2004), no. 5, 788–792. Renata Bunoiu Université de Lorraine, CNRS, IECL, F-57000 Metz, France Email address: renata.bunoiu@univ-lorraine.fr Karim Ramdani Université de Lorraine, CNRS, INRIA, IECL, F-54000 Nancy, France Email address: karim.ramdani@inria.fr Claudia Timofte University of Bucharest, Faculty of Physics, Bucharest-Magurele, P.O. Box MG-11, Romania Email address: claudia.timofte@g.unibuc.ro 1. Introduction 2. Setting of the problem in the case of a thin periodic layer 3. Well-posedness of the microscopic problem in a thin periodic layer via T-coercivity technics 3.1. Background on T-coercivity 3.2. Well-posedness in the reference cell 3.3. Well-posedness of the problem in 4. Homogenization results in the case of a thin periodic layer 5. The case of a thin periodic rod Acknowledgments References