Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 60, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.?? ASYMPTOTIC ANALYSIS OF SIGN-CHANGING TRANSMISSION PROBLEMS WITH RAPIDLY OSCILLATING INTERFACE RENATA BUNOIU, KARIM RAMDANI, CLAUDIA TIMOFTE Abstract. We study the asymptotic behavior of a sign-changing transmission problem, stated in a symmetric oscillating domain obtained by gluing together a positive and a negative material, separated by an imperfect and rapidly oscillating interface. The interface separating the two heterogeneous materials has a periodic microstructure and is a small perturbation of a flat interface. The solution of the transmission problem is continuous and its flux has a jump on the oscillating interface. Under certain conditions on the properties of the two materials, we derive the limit problem and we prove the convergence result. The T-coercivity method is used to handle the lack of coercivity for both the microscopic and the macroscopic limit problems. 1. Introduction Metamaterials are artificial composite materials with unusual properties. For instance, electromagnetic metamaterials can exhibit negative dielectric permittiv- ity and magnetic permeability, leading to a negative index of refraction [31, 30]. Among the applications of these materials in optics, one can mention sub-diffraction imaging or sensing and detection technologies. In practice, these negative materials are usually in contact with classical positive materials and this destroys the coer- civity of the underlying operators governing the physics of the problem. This leads to several difficulties from both the mathematical (well-posedness) and numerical (convergence analysis) viewpoints. Reference [7] is one of the first papers dealing with the well-posedness issue and this was done using the T-coercivity approach. Special interest has been devoted to the particular case of a symmetric indefinite scalar transmission problem through a smooth interface (see [7, Section 3.4]). In- spired by this geometry, our goal in this paper is to investigate a more general situation by considering three modifications with respect to the problem studied in [7, Section 3.4]: • we consider an interface rapidly oscillating at a speed of ε−1, where ε is a small parameter, and with a small amplitude of order εk+1, k ⩾ 0 being a positive real number (see Figure 1); 2020 Mathematics Subject Classification. 35B40, 35Q60, 78M35. Key words and phrases. Positive and negative materials; transmission problem; asymptotic analysis; oscillating interface; imperfect interfaces; flux jump. ©2024. This work is licensed under a CC BY 4.0 license. Submitted September 24, 2024. Published October 11, 2024. 1 2 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? • we assume that the two materials separated by the rapidly oscillating in- terface, namely the positive and the negative one, are both anisotropic and strongly heterogeneous; • we impose imperfect transmission conditions across the oscillating interface, with a continuous solution and a discontinuous flux (see problem (2.7)). Because of the rapid oscillations of the upper and lower boundaries and of the interface, the microscopic problem is difficult to handle numerically. It is natural then to perform an asymptotic analysis, in order to derive an equivalent macroscopic problem, set in a domain with flat boundaries and interface (see Figure 1). We prove that, as ε → 0, the solution of the microscopic problem (2.7) converges to the unique solution of a well-posed indefinite limit problem (4.6), involving the homogenized matrices associated with each sub-domain. Owing to the indefinite character of the problem, the well-posedness of the mi- croscopic problem (2.7) needs a careful analysis. This is done by using the T- coercivity method (see [7]), which allows us to obtain a well-posedness result and uniform energy estimates, under certain conditions on the coefficients describing the properties of the two heterogeneous materials. Let us emphasize that these well-posedness conditions involve a real number κ (see (3.14)), which can be seen as a generalized contrast between the positive and negative materials. With the uniform estimates in hand, we can pass to the limit in the bulk terms by adapting techniques from [24]. For the term on the interface, we combine results from [18] with the periodic unfolding method [19]. For obtaining the boundary conditions of the limit solution, we use the zero extension of the microscopic solution and results from [18]. For definite problems in a similar geometrical configuration, presenting an oscil- lating interface, we refer to [10, 24, 22, 23, 29, 5]. In all these studies, imperfect transmission conditions across the oscillating interface were considered, but in con- trast to our case, the flux of the solution was supposed there to be continuous and proportional to the jump of the solution. For the asymptotic analysis of definite problems in two-component periodic composites involving flux jump, we refer, for instance, to [28, 25, 26, 16, 17, 21]. For similar problems stated in domains with oscillating boundaries, we refer to [6, 2, 20, 3, 27, 4]. For the asymptotic analysis of indefinite problems in a different geometrical setting, namely a two-composite medium with periodically distributed negative inclusions, we refer to [12, 9, 13, 11, 14, 15]. This article is organized as follows. In Section 2, we state the problem under study and the notation. In Section 3, we use the T-coercivity method to prove the well-posedness of the microscopic problem, under certain conditions on the properties of the materials. Theorem 3.6 provides the main result of this section, namely the energy estimate for the unique solution of the microscopic problem. In Section 4, we pass to the limit in the weak formulation of the microscopic problem and obtain the limit macroscopic problem. This indefinite limit problem is showed to be well-posed and the convergence is proved in Theorem 4.2. Finally, we collect in the Appendix some results on the unfolding method used to prove certain of the convergence results. EJDE-2024/?? SIGN-CHANGING TRANSMISSION PROBLEMS 3 Figure 1. Description of the geometry: microscopic case ε > 0 (left) and limit case ε = 0 (right). 2. Problem setting Given ε ∈ (0, 1), we consider a two-dimensional bounded domain Ωε constituted by two sub-domains Ωε 1 and Ωε 2 separated by an oscillating interface Σε (see Figure 1). More precisely, given L, l > 0, we set Ωε = {x = (x1, x2) : x1 ∈ (0, l), −L+Hε(x1) < x2 < L+Hε(x1)} , (2.1) Ωε 1 = {x = (x1, x2) ∈ Ωε : x2 > Hε(x1)} , Ωε 2 = {x = (x1, x2) ∈ Ωε : x2 < Hε(x1)} , Σε = {x = (x1, x2) ∈ Ωε : x2 = Hε(x1)} . (2.2) Note that Ωε is symmetric with respect to Σε and that Ωε = Ωε 1 ∪ Ωε 2 ∪ Σε. The oscillating interface Σε is described through the one-dimensional function Hε given by Hε(x1) = εk+1h (x1 ε ) , (2.3) where h ∈ C1([0, 1];R+) is a 1-periodic function and k ⩾ 0. From now on, we assume, for simplicity, that ε is a sequence of strictly positive numbers such that l/ε ∈ N∗. We set h = ∥h∥L∞(0,1), h′ = ∥h′∥L∞(0,1), (2.4) which we suppose to be finite, with h < L. We also introduce the oscillating upper and lower boundaries of the domain Ωε, described by Σε 1 = {x = (x1, x2) : x1 ∈ (0, l), x2 = L+Hε(x1)} (2.5) Σε 2 = {x = (x1, x2) : x1 ∈ (0, l), x2 = −L+Hε(x1)} . (2.6) We denote by nε the unit exterior normal to Ωε 1. For any function v defined on Ωε, we denote by v1 and v2 its restrictions to Ωε 1 and to Ωε 2, respectively. We point out that the results of this paper are still valid in the n-dimensional case (n ⩾ 3). Assume that the sub-domains Ωε 1 and Ωε 2 are occupied by a positive and, respec- tively, by a negative material, described by anisotropic matrix-valued coefficients Aε 1(x) and Aε 2(x). Given a volume source term f and a prescribed jump flux gε, our 4 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? goal is to analyze the asymptotic behavior, as ε → 0, of the solution uε ∈ H1 0 (Ω ε) of the indefinite transmission problem −div (Aε 1(x)∇uε 1) = f in Ωε 1 −div (Aε 2(x)∇uε 2) = f in Ωε 2 uε = 0 on ∂Ωε uε 1 − uε 2 = 0 on Σε Aε 1(x)∇uε 1 · nε −Aε 2(x)∇uε 2 · nε = gε on Σε. (2.7) On the exterior boundary, we prescribed homogeneous Dirichlet boundary condi- tions. We remark that, across the oscillating interface Σε, the solution uε of problem (2.7) is continuous, while its flux exhibits a jump. Let us now make more precise the hypotheses on the coefficients Aε 1 and Aε 2 and on the data f and gε. Let Ms be the linear space of 2 × 2 symmetric matrices. Following [1, Chapter 1], given two positive constants α, β > 0, with αβ ⩽ 1, we define the subspace Ms α,β of coercive matrices with coercive inverses Ms α,β := { M ∈ Ms : Mξ · ξ ⩾ α|ξ|2, M−1ξ · ξ ⩾ β|ξ|2, ∀ξ ∈ R2 } . (2.8) As pointed out in [1, Remark 1.2.9], it is worth noticing that if M ∈ Ms α,β (and even if M is not symmetric), then one necessarily has α|ξ|2 ⩽ Mξ · ξ ⩽ β−1|ξ|2, ∀ξ ∈ R2. (2.9) The above relation shows, in particular, why we need to impose the condition αβ ⩽ 1, as soon as Ms α,β is not empty. With these notation, we make the following assumptions. (P1) We denote by L∞(Y ;Ms α,β) the set of matrix-valued bounded functions de- fined on Y with values in Ms α,β . Let A1 = (a1ij)1⩽i,j⩽2, A2 = (a2ij)1⩽i,j⩽2 ∈ Ms be 2×2 real symmetric matrix-valued functions defined on Y = (0, 1)2 and extended to the whole plane by Y -periodicity. We assume that A1 ∈ L∞(Y ;Ms α1,β1 ) and −A2 ∈ L∞(Y ;Ms α2,β2 ) for some positive constants α1, β1, α2, β2 > 0. In particular, according to (2.9), we have for every ξ ∈ R2 and for almost every y ∈ Y α1|ξ|2 ⩽ A1(y)ξ · ξ ⩽ β−1 1 |ξ|2, α2|ξ|2 ⩽ −A2(y)ξ · ξ ⩽ β−1 2 |ξ|2. (2.10) For almost every x ∈ R2, we define the ε-periodic functions Aε 1(x) := A1 (x ε ) , Aε 2(x) := A2 (x ε ) . (2.11) Obviously, it follows from (2.10) that for every ξ ∈ R2 and for almost every x ∈ R2 α1|ξ|2 ⩽ Aε 1(x)ξ · ξ ⩽ β−1 1 |ξ|2, α2|ξ|2 ⩽ −Aε 2(x)ξ · ξ ⩽ β−1 2 |ξ|2. (2.12) (P2) The function f belongs to L2(Ω̃), where Ω̃ = (0, l)× (−L, 2L). (P3) We assume that, for any ε > 0, gε(x) = g (x1 ε ) , ∀x = (x1, H ε(x1)) ∈ Σε, (2.13) where g ∈ L∞(0, 1) is a 1-periodic function and Hε is defined in (2.3). EJDE-2024/?? SIGN-CHANGING TRANSMISSION PROBLEMS 5 Let V ε = H1 0 (Ω ε), endowed with the standard gradient norm. The variational formulation of problem (2.7) is the following one: find uε ∈ V ε such that Aε(uε, v) = ℓε(v), ∀v ∈ V ε, (2.14) where the bilinear form Aε : V ε × V ε → R and the linear form ℓε : V ε → R are given by Aε(u, v) = ∫ Ωε 1 Aε 1(x)∇u1(x) · ∇v(x) dx+ ∫ Ωε 2 Aε 2(x)∇u2(x) · ∇v(x) dx, (2.15) ℓε(v) = ∫ Ωε f(x)v(x) dx+ ∫ Σε gε(x1)v(x) dσx. (2.16) Throughout this article, C will denote a positive constant, independent of ε, whose value can change from line to line. 3. Well-posedness Since the bilinear form Aε(·, ·) given by (2.15) is indefinite (because of (2.10) and (2.11)), one cannot use Lax-Milgram lemma to obtain a well-posedness result for the variational problem (2.14). Thus, for obtaining the well-posedness, we apply the T-coercivity method introduced in [8] and used in [7] to study a large class of sign-changing scalar transmission problems. We start by recalling the definition of T-coercivity. Definition 3.1. Let T ∈ L(V ) be a bounded linear operator on a Hilbert space V . A bilinear form a(·, ·) defined on V × V is T-coercive if there exists γ > 0 such that a(u,Tu) ⩾ γ∥u∥2, ∀u ∈ V. For the reader’s convenience, we recall a well-posedness result given in [13, Theo- rem 3.2]) which shows that uniform T-coercivity yields well-posedness and uniform estimates for variational problems involving a parameter. 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), for all u, v ∈ V . (2) Aε(·, ·) is uniformly continuous: there exists M > 0 such that Aε(u, v) ⩽ M∥u∥∥v∥, ∀u, v ∈ V. (3.1) (3) Aε(·, ·) is uniformly T-coercive: 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) Then, given a uniformly bounded family (ℓε)ε>0 in V ′, the space of linear forms on V , the variational problem find uε ∈ V such that Aε(uε, v) = ℓε(v), ∀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∥ ⩽ C. (3.4) 6 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? We are going to use the above abstract result to investigate the well-posedness of the sign-changing transmission problem (2.7) set in Ωε. Our objective is to con- struct two families of uniformly T-coercive operators and this will be done by using suitably chosen lifting (or extension) operators Rε 1 and Rε 2 for one sub-domain to another. More precisely, we first adapt [7, Theorem 2.1] to the anisotropic case studied here (see Proposition 3.3 below). This result shows that T-coercivity holds provided the “maximal contrasts” between the positive and negative mate- rials (measured through the positive numbers α1β2 and α2β1) are large enough compared to the norms of the lifting operators Rε 1 and Rε 2. Next, we obtain upper bounds for these lifting operators, by extending [7, Theorem 3.10] to the case of highly oscillating interface, paying a special attention to the dependence on ε of the involved constants (see Proposition 3.4 below). Proposition 3.3. We introduce the sub-spaces V ε 1 := { v1 = v|Ωε 1 : v ∈ H1 0 (Ω ε) } , V ε 2 := { v2 = v|Ωε 2 : v ∈ H1 0 (Ω ε) } , endowed with the norms ∥v1∥V ε 1 = ∥∇v1∥L2(Ωε 1) , ∥v2∥V ε 2 = ∥∇v1∥L2(Ωε 2) . Let Rε 1 ∈ L(V ε 1 , V ε 2 ) and Rε 2 ∈ L(V ε 2 , V ε 1 ) be two lifting operators: • Rε 1 ∈ L(V ε 1 , V ε 2 ) such that (Rε 1u1)|Σε = u1|Σε for all u1 ∈ V ε 1 , • Rε 2 ∈ L(V ε 2 , V ε 1 ) such that (Rε 2u2)|Σε = u2|Σε for all u2 ∈ V ε 2 . We associate with these operators the two operators Tε 1,T ε 2 ∈ L ( H1 0 (Ωε) ) defined by: Tε 1u := { u1 in Ωε 1 −u2 + 2Rε 1u1 in Ωε 2, Tε 2u := { u1 − 2Rε 2u2 in Ωε 1 −u2 in Ωε 2. (3.5) Finally, assume that there exist ρ⋆1, ρ ⋆ 2 > 0 such that, for all ε > 0, ∥Rε 1∥2 ⩽ ρ⋆1, ∥Rε 2∥2 ⩽ ρ⋆2. (3.6) Then, under conditions (2.9), the following uniform T-coercivity results for the bilinear form Aε(·, ·) defined by (2.15), hold. • If α1β2 ⩾ ρ⋆1, then Aε(·, ·) is uniformly Tε 1-coercive. • If α2β1 ⩾ ρ⋆2, then Aε(·, ·) is uniformly Tε 2-coercive. Proof. Assume that α1β2 > ρ⋆1 and let us choose η1 such that ρ⋆ 1 α1β2 < η1 < 1. Then, using Cauchy-Schwarz and Young inequalities together with (2.10), (3.5) and (3.6), we have that for every u ∈ H1 0 (Ω ε), Aε(u,Tε 1u) = ∫ Ωε 1 Aε 1∇u1 · ∇u1 dx+ ∫ Ωε 2 ( −Aε 2 ) ∇u2 · ∇u2 dx+ 2 ∫ Ωε 2 Aε 2∇u2 · ∇(Rε 1u1) dx ⩾ ∫ Ωε 1 Aε 1∇u1 · ∇u1 dx+ ∫ Ωε 2 ( −Aε 2 ) ∇u2 · ∇u2 dx− η1 ∫ Ωε 2 ( −Aε 2 ) ∇u2 · ∇u2 dx − 1 η1 ∫ Ωε 2 ( −Aε 2 ) ∇(Rε 1u1) · ∇(Rε 1u1) dx ⩾ α1∥∇u1∥2L2(Ωε 1) + α2(1− η1)∥∇u2∥2L2(Ωε 2) − 1 β2η1 ∥∇(Rε 1u1)∥2L2(Ωε 2) EJDE-2024/?? SIGN-CHANGING TRANSMISSION PROBLEMS 7 ⩾ ( α1 − ∥Rε 1∥2 β2η1 ) ∥∇u1∥2L2(Ωε 1) + α2(1− η1)∥∇u2∥2L2(Ωε 2) ⩾ ( α1 − ρ⋆1 β2η1 ) ∥∇u1∥2L2(Ωε 1) + α2(1− η1)∥∇u2∥2L2(Ωε 2) . Thus, we proved that Aε(u,Tε 1u) ⩾ γ1∥∇u∥2L2(Ωε), with γ1 = min ( α1 − ρ⋆1 β2η1 , α2(1− η1) ) > 0. Similarly, when α2β1 > ρ⋆2, one can prove that Aε(u,Tε 2u) ⩾ γ2∥∇u∥2L2(Ωε), with γ2 = min ( α2 − ρ⋆2 β1η2 , α1(1− η2) ) > 0, for some η2 such that ρ⋆ 2 α2β1 < η2 < 1. We have thus proved the uniform T-coercivity of the bilinear form Aε. □ Taking advantage of the symmetric geometry of our problem, let us now con- struct particular lifting operators whose norms can be explicitly estimated. Proposition 3.4. Let us introduce the two lifting operators Rε 1 ∈ L(V ε 1 , V ε 2 ) and Rε 2 ∈ L(V ε 2 , V ε 1 ) obtained by symmetry with respect to the interface Σε defined by (2.2): ∀u1 ∈ V ε 1 : (Rε 1u1)(x1, x2) = u1 (x1, 2H ε(x1)− x2) ∀u2 ∈ V ε 2 : (Rε 2u2)(x1, x2) = u2 (x1, 2H ε(x1)− x2) . (3.7) Then, we have the estimate ∥Rε 1∥2 = ∥Rε 2∥2 ⩽ ρε := 1 + 2εkh′ + 4ε2kh′2, (3.8) where h′ is defined in (2.4). Moreover, in the particular case where h vanishes identically (i.e. for flat inter- face and flat upper and lower boundaries), we have ∥Rε 1∥ = ∥Rε 2∥ = 1. (3.9) Proof. Following the proof of [7, Theorem 3.10], the change of variables x1 = ξ1, x2 = 2Hε(ξ1)− ξ2 shows that ∥∇(Rε 1u1)∥2L2(Ωε 2) = ∫ Ωε 2 ( |∂(R ε 1u1) ∂ξ1 |2 + |∂(R ε 1u1) ∂ξ2 |2 ) dξ = ∫ Ωε 1 ( |∂u1 ∂x1 + 2 dHε dx1 ∂u1 ∂x2 |2 + |∂u1 ∂x2 |2 ) dx = ∫ Ωε 1 ( |∇u1|2 + 4 dHε dx1 ∂u1 ∂x1 ∂u1 ∂x2 + 4| dH ε dx1 |2|∂u1 ∂x2 |2 ) dx ⩽ ∫ Ωε 1 ( |∇u1|2 + 2 dHε dx1 |∇u1|2 + 4| dH ε dx1 |2|∂u1 ∂x2 |2 ) dx ⩽ ( 1 + 2∥ dHε dx1 ∥L∞(0,l) + 4∥ dHε dx1 ∥2L∞(0,l) ) ∥∇u1∥2L2(Ωε 1) . 8 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? Hence, ∥Rε 1u1∥2V ε 2 ⩽ ( 1 + 2∥ dHε dx1 ∥L∞(0,l) + 4∥ dHε dx1 ∥2L∞(0,l) ) ∥u1∥2V ε 1 . Since dHε dx1 = εkh′(x1 ε ) , we have ∥ dHε dx1 ∥L∞(0,l) = εkh′. The same calculations hold for Rε 2 and this shows that (3.8) holds true. In the particular case of a flat interface, the lifting operators Rε 1 and Rε 2 are simply given by the symmetry with respect to the x1-axis: ∀u1 ∈ V ε 1 : (Rε 1u1)(x1, x2) = u1 (x1,−x2) ∀u2 ∈ V ε 2 : (Rε 2u2)(x1, x2) = u2 (x1,−x2) . Consequently, ∥Rε 1∥ = ∥Rε 2∥ = 1. □ We collect in the next lemma some results needed in the sequel. Lemma 3.5. The following estimates hold true for every v ∈ H1 0 (Ω ε): ∥v∥L2(Ωε) ⩽ C∥∇v∥L2(Ωε), (3.10) | ∫ Σε v(x) dσx| ⩽ C∥∇v∥L2(Ωε). (3.11) Proof. The first inequality states that Poincaré’s inequality holds in Ωε with a constant independent of ε. Indeed, since the zero-extension ṽ of v to Ω̃ = (0, l) × (−L, 2L) belongs to H1 0 (Ω̃), we have by Poincaré’s inequality in the fixed domain Ω̃ ∥v∥L2(Ωε) = ∥ṽ∥L2(Ω̃) ⩽ C∥∇ṽ∥L2(Ω̃) = C∥∇v∥L2(Ωε), where we used for the last equality the identity ∇ṽ = ∇̃v. Hence, (3.10) holds. To prove the second estimate, we first rewrite the integral in the left-hand side as a one-dimensional integral in the coordinate x1, namely∫ Σε v(x) dσx = ∫ l 0 v ( x1, ε k+1h (x1 ε ))√ 1 + ε2k|h′ (x1 ε ) |2 dx1. Since k ⩾ 0 and h′ satisfies (2.4), one has∣∣ ∫ Σε v(x) dσx ∣∣ ⩽ C ∫ l 0 ∣∣v(x1, ε k+1h (x1 ε ))∣∣dx1 ⩽ C (∫ l 0 ∣∣v(x1, ε k+1h (x1 ε )) − v(x1, 0) ∣∣ dx1 + ∫ l 0 |v(x1, 0)|dx1 ) . (3.12) According to [24, Proposition 3.2] (which is an adaptation of [18, Lemma 1]) written with our notation, one has ∥v ( x1, ε k+1h (x1 ε )) − v(x1, 0)∥L2(0,l) ⩽ C √ εk+1∥∇v∥L2(Ωε). (3.13) Then, using in (3.12) the Cauchy-Schwarz inequality and estimate (3.13) for the first term, and the classical trace inequality and the Poincaré inequality (3.10) for the second one, we obtain (since k ⩾ 0) | ∫ Σε v(x) dσx| ⩽ C∥∇v∥L2(Ωε) EJDE-2024/?? SIGN-CHANGING TRANSMISSION PROBLEMS 9 and the proof is complete. □ We are now in position to prove a well-posedness result for the microscopic problem (2.7). Theorem 3.6. Assume that κ := max (α1β2, α2β1) > κ⋆, (3.14) where κ⋆ := { 1 if k > 0 1 + 2h′ + 4h′2 if k = 0. Then, there exists ε∗ > 0 such that, for all ε ∈ (0, ε∗), the variational formulation (2.14) of problem (2.7) admits a unique solution uε ∈ V ε. Moreover, there exists a positive constant C independent of ε such that the following a priori estimates hold true for all ε ∈ (0, ε∗), ∥uε∥L2(Ωε) ⩽ C, ∥∇uε∥L2(Ωε) ⩽ C. (3.15) Proof. To apply Theorem 3.2, we first prove the uniform T-coercivity of the bilinear form Aε(·, ·). According to (3.8) in Proposition 3.4, we have ∥Rε 1∥2 = ∥Rε 2∥2 ⩽ ρε := 1 + 2εkh′ + 4ε2kh′2. Now, we need to distinguish between the two cases k > 0 and k = 0. Case k > 0. In this case, we have limε→0 ρ ε = 1 and, by assumption, κ := max (α1β2, α2β1) > 1 = κ⋆. Let us choose ε∗ such that 1 ⩽ ρ⋆ := ρε ∗ < κ. Therefore, ∥Rε 1∥2 = ∥Rε 2∥2 ⩽ ρε ⩽ ρε ∗ = ρ⋆ < κ, ∀ε ∈ (0, ε∗). (3.16) Case k = 0. In this case, ρε is independent of ε since ρε = 1 + 2h′ + 4h′ := ρ⋆, for all ε > 0. Hence, for all ε > 0, we have by (3.8) and by using the assumption (3.14) on κ, ∥Rε 1∥2 = ∥Rε 2∥2 ⩽ ρε = ρ⋆ < κ, ∀ε > 0. (3.17) Thanks to (3.16) and (3.17), we can apply Proposition 3.3 with ρ⋆1 = ρ⋆2 = ρ⋆ and we obtain the following alternative to hold. • If κ = α1β2, then we have κ ⩾ ρ⋆, and thus Aε(·, ·) is uniformly Tε 1-coercive (for all ε ∈ (0, ε∗) when k > 0, and for all ε > 0 when k = 0). • If κ = α2β1, then we have κ ⩾ ρ⋆, and thus Aε(·, ·) is uniformly Tε 2-coercive (for all ε ∈ (0, ε∗) when k > 0 and for all ε > 0 when k = 0). To conclude the proof, the only assumption in Theorem 3.2 that needs to be checked is the uniform continuity of the linear form ℓε given by (2.16). Since f ∈ L2(Ω̃), the first term (the volume term) of ℓε is clearly continuous on H1 0 (Ω ε) by Poincaré’s inequality (3.10). For the second term (the surface term) of ℓε, we first use the boundedness of the function g to obtain∣∣ ∫ Σε gε(x)v(x) dσx ∣∣ ⩽ C ∣∣ ∫ Σε v(x) dσx ∣∣. (3.18) By combining (3.18) and (3.11), we obtain the continuity of the second term in the linear form ℓε, and the proof is now complete. □ 10 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? Remark 3.7. Let us consider in (2.10) the particular case of matrices of the form A1(y) = a1(y)Id and A2(y) = a2(y) Id, where a1 and a2 are in L∞(Y ), Y− periodic, taking positive and, respectively, negative values. More precisely, assume that there exist positive constants a−1 , a + 1 , a − 2 , a + 2 such that, for almost every y ∈ Y , 0 < a−1 ⩽ a1(y) ⩽ a+1 , 0 < a−2 ⩽ −a2(y) ⩽ a+2 . In this case, the constant κ = max(α1β2, α2β1), defined in (3.14), becomes κ = max(a−1 /a + 2 , a − 2 /a + 1 ). As expected, this is exactly the constant obtained in [7, Theorem 3.10] for a scalar sign-changing transmission problem through a C1−interface. Moreover, if the func- tions a1 and a2 are constant, then κ = max(a1/|a2|, |a2|/a1). In particular, one has: • if k > 0, then the well-posedness condition κ > 1 = κ⋆ reads a1 ̸= |a2|, i.e. the constants a1 and a2 should not be opposite; • if k = 0, then the well-posedness condition reads a1/|a2| /∈ [1/κ⋆, κ⋆], with κ⋆ = 1 + 2h′ + 4h′2, i.e. the contrasts should be large or small enough. 4. Convergence analysis We remark that the dependence on ε of the domain Ωε is due to the oscillations of its upper and lower boundaries. Since the solution uε of problem (2.7) is defined in Ωε, general compactness results do not apply and, hence, we shall extend uε to a fixed domain. Taking into account the homogeneous Dirichlet boundary conditions, we extend uε by zero to a fixed domain, namely Ω̃ = (0, l)× (−L, 2L). For any function v defined on Ωε, ṽ stands for its extension with zero to Ω̃. We remark that ∇ṽ = ∇̃v. For this particular extension, classical compactness arguments allow us to state the following result. Proposition 4.1. Let uε be the unique solution of problem (2.7), which satisfies the a priori estimates (3.15). Then, its extension ũε satisfies the a priori estimates ∥ũε∥L2(Ω̃) ⩽ C, ∥∇ũε∥L2(Ω̃) ⩽ C. (4.1) These estimates obviously imply that there exists a function ũ ∈ H1 0 (Ω̃) such that, up to a subsequence, one has ũε → ũ strongly in L2(Ω̃), ∇ũε ⇀ ∇ũ weakly in L2(Ω̃). (4.2) To state the main convergence result of the paper, we introduce the limit domain (see Figure 1, right) Ω = Ω1 ∪ Ω2 ∪ Σ0, where Ω1 = (0, l)× (0, L), Ω2 = (0, l)× (−L, 0), Σ0 = (0, l)× {0}. In what follows, we denote by u the restriction of the limit function ũ to the domain Ω: u = ũ|Ω. (4.3) EJDE-2024/?? SIGN-CHANGING TRANSMISSION PROBLEMS 11 Also we associate to any matrix A ∈ L∞(Y ;Ms α,β), Ms α,β being defined in (2.8), a corresponding 2× 2 homogenized matrix defined by Ahom = (Ahom ij )1⩽i,j⩽2 Ahom ij = 1 |Y | ∫ Y A(y)∇(χi + yi) · ∇(χj + yj) dy, (4.4) where the functions χ1, χ2 ∈ H1(Y ) are Y -periodic solutions, defined up to an additive constant, of the cell problems −div [A(y)∇ (χi + yi)] = 0 in Y, i = 1, 2. (4.5) Classically (see, for instance, [1, Remark 1.3.12]), the homogenized matrix also belongs to the set Ms α,β and, hence, α|ξ|2 ⩽ Ahomξ · ξ ⩽ β−1|ξ|2, ∀ξ ∈ R2. Theorem 4.2. Assume that κ := max (α1β2, α2β1) > κ⋆, where κ⋆ := { 1 if k > 0 1 + 2h′ + 4h′2 if k = 0. Then, the unique solution uε of the variational problem (2.14) converges, in the sense of (4.2)-(4.3), to the unique solution u ∈ H1 0 (Ω) of the sign-changing limit transmission problem −div ( Ahom 1 ∇u1 ) = f in Ω1 −div ( Ahom 2 ∇u2 ) = f in Ω2 u = 0 on ∂Ω u1 − u2 = 0 on Σ0 Ahom 1 ∇u1 · n−Ahom 2 ∇u2 · n = G on Σ0, (4.6) where the constant homogenized symmetric matrices Ahom 1 and Ahom 2 , associated to A1 and, respectively, to A2, are defined by (4.4)-(4.5) and G := {∫ 1 0 g(y1) dy1 if k > 0∫ 1 0 g(y1) √ 1 + |h′(y1)|2 dy1 if k = 0. (4.7) Proof. We remark that the homogenized symmetric matrices Ahom 1 and −Ahom 2 are positive definite, due to (2.10). We start by proving the well-posedness of the sign- changing problem (4.6) by using the T-coercivity method. The proof goes along the same lines as those detailed in the proof of Theorem 3.6. Since the interface is flat, the corresponding lifting operators are of norm one in this case. Consequently, adapting the proof of Proposition 3.3 to the flat case and in the limit domain Ω, we obtain the T-coercivity of the bilinear form associated with the limit problem (4.6) for κ > 1. This last inequality holds true since, by assumption, κ > κ⋆ ⩾ 1. We prove now the convergence result for the solution uε of (2.7) to the solution u of (4.6). According to (2.14), the variational formulation of problem (2.7) reads as follows: find uε ∈ V ε such that∫ Ωε 1 Aε 1(x)∇uε 1(x) · ∇v(x) dx+ ∫ Ωε 2 Aε 2(x)∇uε 2(x) · ∇v(x) dx = ∫ Ωε f(x)v(x) dx+ ∫ Σε gε(x)v(x) dσx, (4.8) 12 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? for all v ∈ V ε. From (4.8), we are led to∫ Ω̃ χΩε 1 (x)Aε 1(x)∇ũε 1(x) · ∇v(x) dx+ ∫ Ω̃ χΩε 2 (x)Aε 2(x)∇ũε 2(x) · ∇v(x) dx = ∫ Ω̃ χΩε(x)f(x)v(x) dx+ ∫ Σε gε(x)v(x) dσx, (4.9) for any v ∈ D(Ω̃), where χD denotes the characteristic function of a domain D. Our aim is to pass to the limit with ε → 0 in (4.9). For the passage to the limit in its left-hand side, we adapt to our situation some ideas from [24]. To this end, we set Bε 1 = {x = (x1, x2) ∈ Ωε : L < x2 < L+Hε(x1)} , Bε 2 = {x = (x1, x2) ∈ Ωε : −L < x2 < −L+Hε(x1)} , Bε − = {x = (x1, x2) ∈ Ωε : 0 < x2 < Hε(x1)} , Bε + = { x = (x1, x2) ∈ Ωε : Hε(x1) < x2 < εk+1h } . We first notice that Ωε 1 = (Ω1 \Bε −)∪Bε 1 and Ωε 2 = (Ω2 ∪Bε −) \Bε 2. The oscillating interface Σε, as well as the top and the bottom oscillating boundaries Σε 1 and Σε 2 of the domain Ωε, are contained in the sets Sε = (0, l) × [0, εk+1h] and, respectively, in Sε 1 = (0, l) × [L,L + εk+1h] and Sε 2 = (0, l) × [ −L,−L+ εk+1h ] . An important feature of the sets Sε, Sε 1 , and Sε 2 is that, when ε tends to zero, their measures tend to zero. This is a crucial argument in the convergence process. Let us notice that, for the passage to the limit in the left-hand side of (4.9), we also use [24, Remark 2.2], which ensures that the convergence results of the paper remain valid for the case of distinct diffusion matrices in the upper and lower part of the domain. Passing to the limit in the first integral in the left-hand side of (4.9), we obtain∫ Ω̃ χΩε 1 (x)Aε 1(x)∇ũε 1(x) · ∇v(x) dx → ∫ Ω1 Ahom 1 ∇ũ1 · ∇v dx. (4.10) Indeed, one has∫ Ω̃ χΩε 1 (x)Aε 1(x)∇ũε 1(x) · ∇v(x) dx = ∫ Ω̃ χΩ1\Bε − (x)Aε 1(x)∇ũε 1(x) · ∇v(x) dx+ ∫ Ω̃ χBε 1 (x)Aε 1(x)∇ũε 1(x) · ∇v(x) dx. By using the hypothesis on the matrix Aε 1 and the estimates (4.1), we obtain | ∫ Bε 1 Aε 1(x)∇ũε 1(x) · ∇v(x) dx| ⩽ C∥∇v∥L2(Bε 1) → 0, since Bε 1 is included in Sε 1 , whose measure tends to zero. We then use [24, Propo- sition 3.1], to obtain (4.10). For the second integral in the left-hand side of (4.9), we obtain∫ Ω̃ χΩε 2 (x)Aε 2(x)∇ũε 2(x) · ∇v(x) dx → ∫ Ω2 Ahom 2 ∇ũ2 · ∇v dx. (4.11) Indeed, one has∫ Ω̃ χΩε 2 (x)Aε 2(x)∇ũε 2(x) · ∇v(x) dx EJDE-2024/?? SIGN-CHANGING TRANSMISSION PROBLEMS 13 = ∫ Ω̃ χΩ2 Aε 2(x)∇ũε 2(x) · ∇v(x) dx+ ∫ Ω̃ χBε − (x)Aε 2(x)∇ũε 2(x) · ∇v(x) dx − ∫ Ω̃ χBε 2 (x)Aε 2(x)∇ũε 2(x) · ∇v(x) dx = J1 + J2 − J3. Classical convergence results in the theory of homogenization give us J1 = ∫ Ω̃ χΩ2 (x)Aε 2(x)∇ũε 2(x) · ∇v(x) dx → ∫ Ω2 Ahom 2 ∇ũ2 · ∇v dx. For J2, by using the hypothesis on the matrix Aε 2 and estimates (4.1), one has J2 = | ∫ Bε − Aε 2(x)∇ũε 2(x) · ∇v(x) dx| ⩽ C∥∇v∥L2(Bε −) → 0, since Bε − is included in Sε 2 , whose measure goes to zero. The value of J3 is zero by the construction of the extension ũε. Hence, we arrive at (4.11). The first term in the right-hand side of (4.9) obviously gives∫ Ω̃ χΩεf(x)v(x) dx → ∫ Ω fv dx. (4.12) For dealing with the second term in the right-hand side of (4.9), namely the surface term, we first express it as a one-dimensional integral in the coordinate x1. Then, due to the C1 regularity of the function h on the interval [0, 1], we can use the one-dimensional version of periodic unfolding operator on fixed domains in [19, Chapter 1], whose definition and main properties are briefly recalled in the Appendix. More precisely, we use the results in the Appendix for the particular values m = 1, ω = (0, l), and Y = (0, 1). Applying Propositions 5.3 and 5.4, we obtain, since g is 1-periodic,∫ Σε gε(x)v(x) dσx = ∫ l 0 g (x1 ε ) v ( x1, ε k+1h (x1 ε ))√ 1 + |εkh′ (x1 ε ) |2 dx1 = ∫ l 0 ∫ 1 0 T ε(g)(x1, y1)T ε(V ε)(x1, y1)T ε(W ε)(x1, y1) dx1 dy1 = ∫ l 0 ∫ 1 0 g(y1)T ε(V ε)(x1, y1)T ε(W ε)(x1, y1) dx1 dy1. Here, we denoted V ε(x1) = v ( x1, ε k+1h (x1 ε )) , W ε(x1) = √ 1 + |εkh′ (x1 ε ) |2. According to (3.13), one has v ( ·, εk+1h ( · ε )) → v(·, 0) strongly in L2(0, l). This, together with Proposition 5.3(6) and (8), leads to T ε(V ε)(x1, y1) → v(x1, 0) strongly in L2((0, l)× (0, 1)). By Proposition 5.4 applied to h′, we have T ε(h′)(x1, y1) → h′(y1) strongly in L2((0, l)× (0, 1)). 14 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? For k > 0, one has T ε(εkh′)(x1, y1) = εk T ε(h′)(x1, y1) → 0 strongly in L2((0, l)× (0, 1)). Using Proposition 5.3(6) and the properties of Nemytskii’s operator, we obtain T ε(W ε)(x1, y1) ⇀ 1 weakly in L2((0, l)× (0, 1)). Therefore, for k > 0, for the term involving the flux jump, we obtain∫ Σε gε(x1)v(x1, x2) dσx → ∫ l 0 ∫ 1 0 g(y1)v(x1, 0) dx1 dy1 = ∫ 1 0 g(y1) dy1 ∫ l 0 v(x1, 0) dx1 = ∫ 1 0 g(y1) dy1 ∫ Σ0 v(x1, x2) dσ. (4.13) For k = 0, the term involving the flux jump reads∫ Σε gε(x1)v(x1, x2) dσx = ∫ l 0 g (x1 ε ) v ( x1, εh (x1 ε ))√ 1 + |h′ (x1 ε ) |2 dx1. By applying again the one-dimensional unfolding operator and using similar argu- ments as above, we obtain∫ Σε gε(x1)v(x1, x2) dσx → ∫ 1 0 g(y1) √ 1 + |h′(y1)|2 dy1 ∫ Σ0 v(x1, x2) dσ. (4.14) It remains now to obtain the boundary conditions on ∂Ω for the limit function u = ũ|Ω. On the lateral boundaries, the homogeneous Dirichlet condition is obvi- ously kept at the limit. For the bottom and the upper boundaries of Ω, we shall prove that the prescribed homogeneous Dirichlet boundary condition will be also preserved at the limit. More precisely, we shall prove that u = 0 on (0, l) × {−L} and on (0, l)× {L}. We first prove that u = 0 on Σ2 = (0, l)× {−L}. Since, by construction, ũε 2 = 0 on Σ2, one has ∥ũ2∥L2(Σ2) = ∥ũε 2 − ũ2∥L2(Σ2) ⩽ C∥∇ũε 2 −∇ũ2∥L2(Ω2), where we use the compactness of the trace operator. By using the convergence (4.2), we pass to the limit and we obtain ∥ũ2∥L2(Σ2) = 0, which implies that u2 = 0 on Σ2. To obtain the boundary condition for u on Σ1 = (0, l) × {L}, we notice that, since by construction, ũε 1(·, L+Hε(x1)) = 0, we have ∥ũ1(·, L)∥L2(0,l) = ∥ũε 1(·, L+Hε(x1))− ũ1(·, L)∥L2(0,l) ⩽ ∥ũε 1(·, L+Hε(x1))− ũε 1(·, L)∥L2(0,l) + ∥ũε 1 − ũ1∥L2(Σ1) ⩽ C √ εk+1 + C∥∇ũε 1 −∇ũ1∥L2(Ω1) where we use an adaptation of (3.13), estimate (4.1) and the compactness of the trace operator. By using the convergence (4.2), we pass to the limit and we obtain ∥ũ1(·, L)∥L2(0,l) = 0, which implies ũ1 = 0 on Σ1. Hence, u1 = 0 on Σ1. Collecting convergences (4.10), (4.11), (4.12), (4.13), and (4.14), we are led to∫ Ω1 Ahom 1 ∇u1 · ∇v dx+ ∫ Ω2 Ahom 2 ∇u2 · ∇v dx = ∫ Ω fv dx+G ∫ Σ0 v dσx, (4.15) EJDE-2024/?? SIGN-CHANGING TRANSMISSION PROBLEMS 15 which is equivalent to the limit transmission problem (4.6), where G is given by (4.7). Since the limit problem (4.6) admits a unique solution u, the convergences (4.2) hold for the whole sequence. □ Remark 4.3. If k > 0, the microscopic and the homogenized problems are well- posed for the same range of values of the generalized contrasts, namely κ > 1. On the contrary, for k = 0, the well-posedness of the homogenized problem is ensured as soon as κ > 1, while the one of the microscopic problem holds only for κ > 1 + 2h′ + 4h′2. Remark 4.4. The upper bounds obtained in (3.8) for the norms of the lifting operators are still valid in the case k < 0. Hence, they provide a well -posedness result for the microscopic problem for fixed ε for k < 0. However, as these bounds blow-up as ε → 0, the uniform T-coercivity fails and the macroscopic problem (2.7) might be ill-posed as ε → 0. 5. Appendix: background on the unfolding method In this Appendix, we collect some useful results from [19, Chapter 1] on the periodic unfolding method. Consider, for simplicity, the bounded domain ω = (0, l)m ⊂ Rm. Let ε be a sequence of strictly positive numbers such that l/ε ∈ N∗ and let Y = (0, 1)m. Thus, the domain ω ⊂ Rm is obtained as the union of an entire number of ε−shrinked and translated cells Y . Let us notice that for x ∈ Rm, by denoting [x] the entire part of x in Zm, then x− [x] ∈ Y . Set {x} = x− [x], for x ∈ Rm. In particular, for any x ∈ Rm and any ε > 0, one has x = ε [x ε ] + ε {x ε } . (5.1) Definition 5.1 ([19, Definition 1.2]). For each function φ Lebesgue–measurable on ω, the periodic unfolding operator T ε is defined by T ε(φ)(x, y) = φ ( ε [x ε ] + εy ) for a.e. (x, y) ∈ Ω× Y. The function T ε(φ) is Lebesgue–measurable on ω × Y . One has the following property. Proposition 5.2 ([19, Prop. 1.12]). Let p ∈ (1,+∞) and let {vε} be a bounded sequence in Lp(ω). Then, the sequence {T ε(vε)} is bounded in Lp(ω × Y ) and, if T ε(vε) ⇀ v weakly in Lp(ω × Y ), then vε ⇀ ∫ Y v(x, y) dy weakly in Lp(ω). The unfolding operator T ε has the following properties. Proposition 5.3 ([19, Chapter 1]). Properties of operator T ε : Lp(ω) → Lp(ω×Y ), p ∈ (1,∞): (1) The operator T ε is linear and continuous. (2) One has T ε(vw) = T ε(v)T ε(w), for v, w Lebesgue–measurable functions. 16 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? (3) (Integration formula) If v ∈ L1(ω), then∫ ω v(x) dx = ∫ ω×Y T ε(v)(x, y) dx dy. (4) If {vε} is a sequence of functions in L1(ω), then lim ε→0 ∫ ω vε(x) dx = lim ε→0 ∫ ω×Y T ε(vε)(x, y) dx dy. (5) We have the estimate ∥T ε(v)∥Lp(ω×Y ) ⩽ ∥v∥Lp(ω). (6) If N is a real-valued continuous function and v is a Lebesgue–measurable function, then T ε(N(v)) = N(T ε(v)). Convergence results. (7) If v ∈ Lp(ω), then T ε(v) → v strongly in Lp(ω × Y ). (8) If vε → v strongly in Lp(ω), then T ε(vε) → v strongly in Lp(ω × Y ). Proposition 5.4 ([19, Proposition 1.5]). For a Lebesgue–measurable function h on Y , extended by Y -periodicity to the whole of Rm, we define the sequence {hε} by hε(x) = h (x ε ) for a.e. x ∈ Rm. Then T ε(hε)(x, y) = h(y), for a.e. x ∈ Rm. 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[31] D. R. Smith, J. B. Pendry, M. C. K. Wiltshire; Metamaterials and negative refractive index, Science, 305 (2004), pp. 788–792. 18 R. BUNOIU, K. RAMDANI, C. TIMOFTE EJDE-2024/?? 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. Problem setting 3. Well-posedness 4. Convergence analysis 5. Appendix: background on the unfolding method Acknowledgments References