Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 03, pp. 1–29. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.03 STRANGE NON-LOCAL OPERATORS HOMOGENIZING THE POISSON EQUATION WITH DYNAMICAL UNILATERAL BOUNDARY CONDITIONS: ASYMMETRIC PARTICLES OF CRITICAL SIZE JESÚS ILDEFONSO DÍAZ, TATIANA A. SHAPOSHNIKOVA, ALEXANDER V. PODOLSKIY Abstract. We study the homogenization of a nonlinear problem given by the Poisson equation, in a domain with arbitrarily shaped perforations (or parti- cles) and with a dynamic unilateral boundary condition (of Signorini type), with a large coefficient, on the boundary of these perforations (or particles). This problem arises in the study of chemical reactions of zero order. The con- sideration of a possible asymmetry in the perforations (or particles) is funda- mental for considering some applications in nanotechnology, where symmetry conditions are too restrictive. It is important also to consider perforations (or particles) constituted by small different parts and then with several connected components. We are specially concerned with the so-called critical case in which the relation between the coefficient in the boundary condition, the pe- riod of the basic structure, and the size of the holes (or particles) leads to the appearance of an unexpected new term in the effective homogenized equation. Because of the dynamic nature of the boundary condition this “strange term” becomes now a non-local in time and non-linear operator. We prove a conver- gence theorem and find several properties of the “strange operator” showing that there is a kind of regularization through the homogenization process. 1. Introduction This article studies the asymptotic behavior, as ε → 0, of the solution uε to a problem given by the Poisson equation in a domain Ωε which we can understand as either to be given as an initial bounded regular domain Ω of Rn, n ≥ 3 which is perforated by many periodical cavities of an arbitrary shape, or that Ωε is the part of Ω which is exterior to a periodical distribution of many particles Gε of arbitrary shape. On the boundary Sε of these perforations, or of the particles Gε, we impose a dynamic unilateral boundary condition (of Signorini type) and we obtain the 2020 Mathematics Subject Classification. 35B27, 35K57, 35K91, 35R01, 47B44. Key words and phrases. Critically scaled homogenization; asymmetric perforated domain; asymmetric particles; unilateral dynamic boundary conditions; strange term; nonlocal monotone operator; Signorini problem. ©2024. This work is licensed under a CC BY 4.0 license. Submitted November 29, 2023. Published January 4, 2024. 1 2 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 formulation −∆xuε = f(x, t), (x, t) ∈ QTε , uε ≥ 0, ε−γ∂tuε + ∂νuε ≥ 0, (x, t) ∈ STε , uε(ε −γ∂tuε + ∂νuε) = 0, (x, t) ∈ STε , uε(x, t) = 0, (x, t) ∈ ΓT , uε(x, 0) = 0, x ∈ Sε. (1.1) Here, we assume to be given a time T > 0 and which leads to define the sets QTε = Ωε× (0, T ), STε = Sε× (0, T ), ΓT = ∂Ω× (0, T ). Function f(x, t) is a datum of the problem and we are assuming an identically zero initial datum (uε(x, 0) = 0 for x ∈ Sε) just for simplicity in the formulation (see Remark 5.3 below for a more general case). Our main interest concerns the so-called “critical case”, corresponding to the situation in which Gε is the set of translations of a cell perforation (or particle) aεG0 where the homothety is of the order aε = C0ε γ , γ = n n− 2 , C0 > 0. (1.2) (notice that the big parameter ε−γ appears in the boundary condition in order to get some relevant problems: for a general exposition see [13]). Although the detailed presentation on the geometric aspects of the domains and notations will be presented in the next Section, we point out that the consideration of a possible asymmetry in the perforations (or particles) is fundamental in order to consider some applications in nanotechnology (see, e.g., [28, 6]) were symmetry conditions are too restrictive. For instance, in the case of a reactive flow through particu- late filters of fixed bed at nanosccale it is inevitable to produce particles that have symmetry defects and therefore the mathematical treatment must be justified as- suming that the particles are asymmetric. As a matter of facts, it is also important to consider perforations (or particles) constituted by small different parts and then with several connected components. Several positive properties that only take place at the nanometer scale correspond faithfully to the case of homogenization at the critical scale (see, for instance, [13, Section 4.9.4]). One of the many applications in which such formulation arises is in Chemical Engineering. In that framework uε represents the concentration of some chemi- cal substance which is distributed in a permanent flow of some Newtonian fluid. This explains the use of a linear elliptic equation in the exterior of the granular chemical particles: here the Poisson equation is a simplification of the stationary Navier-Stokes equation (see, e.g. [8, 22, 26] and its references) and with a dynamic chemical reaction on the boundary Sε of the particles Gε (since the concentration on Sε decreases with the time). An usual kinetics reaction is given by the so-called reactions of order p ∈ [0,+∞) ∂νuε + ε−γ(∂tuε + λσ(uε)) = 0 on STε , with σ(uε) = upε . Since uε represents a concentration then we can assume that uε ≥ 0. The case of reactions of order p ∈ [1,+∞) corresponds to the case in which we assume σ(u) Lipschitz continuous and increasing in u (in this setting it is natural to assume that uε is a bounded function). The case of reactions of order p ∈ (0,+1) leads to the more general assumption of σ(u) Hölder continuous and the important case of the so-called zero-order reactions corresponds to the discontinuous function σ(u) = 1 if u > 0 and σ(0) = 0. It is easy to see (see EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 3 Remark 5.4) that the solutions of the zero-order reactions satisfy, with a slight modification, the unilateral formulation (1.1)). For the purely stationary problem the “critical case” is characterized by the appearance of a new non-local term in the effective equation as ε→ 0. We send the reader to the monograph [13] where many references on the pioneering papers are given. In the literature, this term is usually called as “strange” (see [7]) because nothing similar happens when the exponent γ in (1.2) is different. For the case of “big particles” of arbitrary shape the effective diffusion operator depends crucially of the shape of the particle and the “effectiveness” of the chemical reaction can be optimized by a suitable choice of the shape of the particles (see, e.g., [16] and its references). We also point out that dynamic Signorini boundary conditions also appear while studying different physical and chemical processes, see, e.g. [21, 2, 4, 5, 3]. The homogenization of problems with unilateral (dynamical or stationary) boundary conditions attracted the attention of many researchers: here, we refer to [18, 29, 1, 9, 20, 15, 12, 27, 11, 10, 31]. Here we obtain the homogenized problem containing a non-local “strange” be- cause the size aε is critical, and we prove the convergence of the original problem’s solution to the solution of the elliptic homogenized one in which the time plays the role of a parameter. Theorem 1.1. Let n ≥ 3, aε = C0ε γ , γ = n n−2 . Assume f ∈ H1(0, T ;L2(Ω)). Let uε be the strong solution of the problem (1.1), and let Pεuε be a suitable extension of uε to the whole domain Ω. Then Pεuε ⇀ u0 weakly in L2(0, T ;H1 0 (Ω)) and u0 ∈ L2(0, T ;H1 0 (Ω)) is characterized as the unique weak solution to the problem −∆xu0 + Cn−2 0 H[u0] = f(x, t), x ∈ Ω, t ∈ (0, T ), u0 = 0, x ∈ ∂Ω, t ∈ (0, T ), (1.3) where the “strange operator” H[u0] is defined by the expression (4.1) below. The detailed expression of the non-local in time (and non-linear) strange operator H[·] is rather technical as to be detailed here but we will devote a subsection to get a series of properties which explains that there is a kind of regularization through the homogenization process. Indeed, the non-linear operator H[·] is bounded, monotone and Lipschitz continuous (see Theorem 4.1), in contrast with the unilateral nature of the original problem. In addition, it allows to get solutions u0 changing sign on Ω when the datum f(t, x) is negative in some part of QTε (something which is impossible for the original solutions uε on the boundary STε of so many particles, when ε→ 0): see Remark 5.2. The study of dynamical boundary conditions under the assumption of particles of critical size was already initiated with the series of papers [14, 15, 18, 20, 31]. What distinguishes this article from the previous ones is that here we address the homogenization of the problem with the dynamic Signorini boundary condition in the critical case for arbitrarily shaped particles or perforations. In the case of symmetric particles the “strange operator” H[u0] is given in terms of the solution of a quite simple unilateral problem (see Remark 5.1 and [18]). The main difficulty of the asymmetric case, as it arises in many applications of nanotechnology, is that the particles Gε have arbitrary shape and then the corrections functions which are used in the proof of the “method of oscillating test functions” must depend of the 4 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 exact nonlinear term arising in the boundary condition. That was carried out in the case of σ(u) Hölder continuous in the paper [17] but, in contrast to the case of the purely stationary problem, the extension to the case in which σ is a general maximal monotone operator (as it is the case of the Signorini boundary conditions) made in [11] does not work when the particles are not symmetric and the boundary conditions are dynamic. So, this paper covers such an important lack. The organization of this paper is the following: Section 2 is devoted to present the geometric aspects of the domains, some useful notations and the derivation of suitable a priori estimates on the solutions implying the weak convergence in some functional space. Two important auxiliary problems are introduced in Section 3: they play a fundamental role for the definition of the “strange operator” H[·] which is carried out in Section 4. Finally, the proof of Theorem 1.1, the comparison with the results for symmetric particles and other remarks are presented in Section 5. 2. Statement of the problem and a priori estimates of solutions Let Ω be a bounded domain in Rn, n ≥ 3, with Lipschitz boundary ∂Ω. In the cube Y = (−1/2, 1/2)n, we consider a subdomain G0, G0 ⊂ Y , which, for simplicity, is star-shaped with respect to a ball T 0 ρ ⊂ G0 of radius ρ with the center at the origin. Our treatment remains valid if G0 consists of a finite number of disjoint connected components satisfying the same geometric property. Let δB = {x : δ−1x ∈ B}, δ > 0. For ε > 0, we define Ω̃ε = {x ∈ Ω : ρ(x, ∂Ω) > 2ε}. Denote by Zn the set of all vectors j = (j1, . . . , jn) with integer coordinates ji, i = 1, . . . , n. We consider a set Gε = ∪j∈Υε(aεG0 + εj) = ∪j∈ΥεG j ε, where Υε = {j ∈ Zn|Gjε ⊂ Y jε = εY + εj,Gjε ∩ Ω̃ε 6= ∅}. Our assumption on the size of the inclusions is aε = C0ε γ , with γ = n n− 2 and C0 > 0. (2.1) It is easy to see that |Υε| ∼= dε−n, with d a positive constant. Note that Gjε ⊂ T jCaε ⊂ T j ε/4 ⊂ Y j ε , where T jr is a ball in Rn of radius r with the center at P jε = εj (the center of the cell Y jε ), C is a positive constant independent of ε. We introduce sets Ωε = Ω \Gε, Sε = ∂Gε, ∂Ωε = Sε ∪ ∂Ω, QTε = Ωε × (0, T ), STε = Sε × (0, T ), ΓT = ∂Ω× (0, T ). We define Kε = {v ∈ H1(Ωε, ∂Ω) : v ≥ 0 a.e. x ∈ Sε}. As usual, we denote by H1(Ωε, ∂Ω) the completion with respect to the norm in H1(Ωε) of the set of infinitely differentiable functions in Ωε, vanishing in a neigh- borhood of ∂Ω. We also introduce the convex closed set Kε = {v ∈ L2(0, T ;H1(Ωε, ∂Ω)) : v(·, t) ∈ Kε for a.e. t ∈ [0, T ]}. EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 5 Given f ∈ H1(0, T ;L2(Ω)) we say that uε ∈ Kε is a strong solution to (1.1) if ∂tuε ∈ L2(0, T ;L2(Sε)), uε(x, 0) = 0 and we have ε−γ ∫ T 0 ∫ Sε ∂tuε(φ− uε) ds dt+ ∫ T 0 ∫ Ωε ∇uε∇(φ− uε) dx dt ≥ ∫ T 0 ∫ Ωε f(φ− uε) dx dt, (2.2) for all φ ∈ Kε. Notice that (2.2) is a variational formulation of the unilateral problem with the Signorini dynamic boundary conditions. Theorem 2.1. For any ε > 0 problem (1.1) has a unique strong solution uε. Moreover uε satisfies the following estimates ‖uε‖L2(0,T ;H1(Ωε)) + ε−γ/2‖uε‖C([0,T ];L2(Sε)) ≤ K‖f‖L2(QT ), ε−γ/2‖∂tuε‖L2(0,T ;L2(Sε)) + ‖∇uε‖C([0,T ],L2(Ωε)) ≤ K‖f‖H1(0,T ;L2(Ω)), (2.3) where K > 0 is a constant independent of ε and f . Proof. We use the penalty method (see, e.g., [24]). Given a parameter δ > 0, the penalized problem associated to the original problem (1.1) has the form −∆xu δ ε = f(x, t), (x, t) ∈ QTε , ε−γ∂tu δ ε + ∂νu δ ε + ε−γδ−1(uδε) − = 0, (x, t) ∈ STε , uδε(x, t) = 0, (x, t) ∈ ΓT , uδε(x, 0) = 0, x ∈ Sε, (2.4) where u+ = sup(0, u(x, t)), u− = u − u+. Note that function σ(u) = u− is a monotone Lipschitz continuous function that satisfies |u− − v−| ≤ |u− v|, ∀u, v ∈ R. We say that a function uδε ∈ C([0, T ];L2(Sε)) is a strong solution to the problem (2.4) if uδε ∈ L2(0, T ;H1(Ωε, ∂Ω)), ∂tu δ ε ∈ L2(0, T ;L2(Sε)), and it satisfies the integral identity ε−γ ∫ T 0 ∫ Sε ∂tu δ εv ds dt+ ∫ T 0 ∫ Ωε ∇uδε∇v dx dt + ε−γδ−1 ∫ T 0 ∫ Sε (uδε) −v ds dt = ∫ QT ε fv dx dt, (2.5) for arbitrary functions v ∈ L2(0, T ;H1(Ωε, ∂Ω)), and the initial condition uδε(x, 0) = 0 holds for a.e. x ∈ Sε. By applying the results from [20], we conclude that for any δ > 0 the problem (2.4) has a unique strong solution and the following estimates hold ‖uδε‖L2(0,T ;H1(Ωε,∂Ω)) + ε−γ/2 max t∈[0,T ] ‖uδε‖L2(Sε) ≤ K‖f‖L2(QT ), ε−γ/2‖(uδε)−‖L2(0,T ;L2(Sε)) ≤ K √ δ‖f‖L2(QT ), ε−γ/2‖∂tuδε‖L2(0,T ;L2(Sε)) + max t∈[0,T ] ‖∇uδε‖L2(Ωε) ≤ K‖f‖H1(0,T ;L2(Ω)), (2.6) 6 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 From (2.6), we derive that there exists a subsequence such that uδε ⇀ uε weakly in L2(0, T ;H1(Ωε, ∂Ω)), ∂tu δ ε ⇀ ∂tuε weakly in L2(0, T ;L2(Sε)), uδε ⇀ uε weakly in L2(0, T ;L2(Ωε)), (uδε) − → 0 in L2(0, T ;L2(Sε)), as δ → 0. Then by the compactness result of [5, Theorem 2.1] we conclude that uδε → uε in C([0, T ];L2(Sε)), as δ → 0. Next, we show that uε is a solution to the variational inequality (2.2). Indeed, we take v = φ− uδε, where φ ∈ Kε, as a test function in (2.5) and obtain ε−γ ∫ T 0 ∫ Sε ∂tu δ ε(φ− uδε) ds dt+ ∫ T 0 ∫ Ωε ∇uδε∇(φ− uδε) dx dt + ε−γδ−1 ∫ T 0 ∫ Sε (uδε) −(φ− uδε) ds dt = ∫ T 0 ∫ Ωε f(φ− uδε) dx dt. Applying the inequality ‖∇uε‖L2(QT ε ) ≤ lim δ→0 ‖∇uδε‖L2(QT ε ), we obtain lim δ→0 ∫ T 0 ∫ Ωε ∇uδε∇(φ− uδε) dx dt ≤ ∫ T 0 ∫ Ω ∇uε∇(φ− uε) dx dt. Then, using ‖uε(x, T )‖2L2(Sε) ≤ lim δ→0 ‖uδε(x, T )‖2L2(Sε), we obtain ε−γ lim δ→0 ∫ T 0 ∫ Sε ∂tu δ ε(φ− uδε) ds dt ≤ ε−γ ∫ T 0 ∫ Sε ∂tuε(φ− uε) ds dt. Taking into account that φ ∈ Kε, we conclude that∫ T 0 ∫ Sε (uδε) −(φ− uδε) ds dt = ∫ T 0 ∫ Sε (uδε) −φdsdt− ∫ T 0 ∫ Sε |(uδε)−|2 ds dt ≤ 0. Combining the derived inequalities, we obtain that uε satisfies (2.2). Finally, the estimates (2.6) imply (2.3). This concludes the proof. � We recall that by [25], there exists a linear extension operator Pε : H1(Ωε, ∂Ω)→ H1 0 (Ω), such that ‖∇(Pεu)‖L2(Ω) ≤ K‖∇u‖L2(Ωε), ‖Pεu‖H1 0 (Ω) ≤ K‖u‖H1(Ωε), where constant K > 0 is independent of ε. Then, the estimate from Theorem 2.1 implies ‖Pεuε‖L2(0,T ;H1 0 (Ω)) ≤ K. Therefore, for some subsequence (still denoted with the subindex ε), we have as ε→ 0 Pεuε ⇀ u0 weakly in L2(0, T ;H1 0 (Ω)), (2.7) EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 7 for some u0 ∈ L2(0, T ;H1 0 (Ω)). 3. Adaptation of global test functions 3.1. First auxiliary problem: a different unilateral problem in the macro- scopic variables. Let φ(x, t) = ψ(x)η(t), where ψ ∈ C∞(Ω), η ∈ C1([0, T ]). For every j ∈ Υε, we consider the auxiliary elliptic problem with the dynamic unilateral boundary condition given by ∆xw j ε,φ(x, t) = 0, x ∈ T jε/4 \G j ε, t ∈ (0, T ), wjε,φ ≤ φ(P jε , t), x ∈ ∂Gjε, t ∈ (0, T ), ∂νw j ε,φ ≤ ε −γ∂t(φ(P jε , t)− w j ε,φ), x ∈ ∂Gjε, t ∈ (0, T ), (wjε,φ − φ(P jε , t))(∂νw j ε,φ − ε −γ∂t(φ(P jε , t)− w j ε,φ) = 0, x ∈ ∂Gjε, t ∈ (0, T ), wjε,φ(x, t) = 0, x ∈ ∂T jε/4, t ∈ (0, T ), wjε,φ(x, 0) = φ(P jε , 0), x ∈ ∂Gjε. (3.1) Notice that the above unilateral boundary condition has a different nature to the one in the original problem (1.1). To define the notion of solution, we introduce the convex closed sets: Kj,ε = {g ∈ H1(T jε/4 \G j ε, ∂T j ε/4) : g ≤ φ(P jε , t) a.e. on ∂Gjε}, Kj,ε = {g ∈ L2(0, T ;H1(T jε/4 \G j ε, ∂T j ε/4)) : g ∈ Kj,ε for a.e. t ∈ [0, T ]}. We say that a function wjε,φ ∈ Kj,ε is a strong solution to the problem (3.1), if ∂tw j ε,φ ∈ L2(0, T ;L2(∂Gjε)), w j ε,φ(x, 0) = φ(P jε , 0) for a.e. x ∈ ∂Gjε and if it satisfies the variational inequality∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(v − wjε,φ) dx dt ≥ ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w j ε,φ)(v − wjε,φ) ds dt, (3.2) for an arbitrary function v ∈ Kj,ε. Notice that the inequality (3.2) is equivalent to the two following relations: the first one is ∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(v − φj,ε) dx dt ≥ ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w j ε,φ)(v − φ(P jε , t)) ds dt, (3.3) 8 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 where v ∈ Kj,ε, φj,ε(x, t) = φ(P jε , t)ψj,ε(x), ψj,ε ∈ C∞0 (T jε/4), ψj,ε(x) ≡ 1, x ∈ T jCaε , C > 1, ψj,ε ≡ 0, x ∈ T jε/4 \ T j 2Caε , |∇ψj,ε| ≤ K aε . The second relation is∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(wjε,φ − φj,ε) dx dt = ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w j ε,φ)(wjε,φ − φ(P jε , t)) ds dt. (3.4) Indeed, to show this equivalency, we start by pointing out that by subtracting from (3.3) the equality (3.4), we obtain (3.2). Thus, to get the reverse implication, we set v = φj,ε ∈ Kj,ε in (3.2) and obtain∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(φj,ε − wjε,φ) dx dt ≥ ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w j ε,φ)(φ(P jε , t)− w j ε,φ) ds dt. (3.5) Then, if we take v = 2wjε,φ − φj,ε, since on ∂Gjε, we have v = 2wjε,φ − φ(P jε , t) ≤ 2φ(P jε , t)−φ(P jε , t) = φ(P jε , t), then v ∈ Kj,ε is an admissible test function. Setting v as a test function in (3.2), we obtain∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(wjε,φ − φj,ε) dx dt ≥ ≥ ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w j ε,φ)(wjε,φ − φ(P jε , t)) ds dt. (3.6) From (3.5) and (3.6), we derive equality (3.4). Then, taking in (3.2) v = h+wjε,φ− φj,ε, where h ∈ Kj,ε, we obtain (3.3). Thus, we have proved the equivalence of the two formulations. Using the penalty method we will prove the following result. Theorem 3.1. Let φ(x, t) = ψ(x)η(t), where ψ ∈ C∞(Ω) and η ∈ C1([0, T ]). Then, the problem (3.1) has a unique solution that satisfies the estimates ‖∇wjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) + ε−γ max [0,T ] ‖wjε,φ‖ 2 L2(∂Gj ε) ≤ Kεn‖φ‖2 L2(0,T ;C(Ω)) , ‖wjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) ≤ Kεn+2‖φ‖2 L2(0,T ;C(Ω)) , (3.7) and the estimate ε−γ‖∂twjε,φ‖ 2 L2(0,T ;L2(∂Gj ε)) ≤ Kεn‖φ‖2 L2(0,T ;C(Ω)) . (3.8) Proof. We consider the penalty problem associated with the auxiliary problem (3.1) ∆wj,δε,φ = 0, x ∈ T jε/4 \G j ε, t ∈ (0, T ), ε−γ∂tw j,δ ε,φ + ∂νw j,δ ε,φ − ε −γ∂tφ(P jε , t) − δ−1ε−γ(φ(P jε , t)− w j,δ ε,φ)− = 0, x ∈ ∂Gjε, t ∈ (0, T ), wj,δε,φ(x, t) = 0, x ∈ ∂T jε/4, t ∈ (0, T ), wj,δε,φ(x, 0) = φ(P jε , 0), x ∈ ∂Gjε. (3.9) EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 9 We say that a function wj,δε,φ ∈ L2(0, T ;H1(T jε/4\G j ε, ∂T j ε/4))∩C([0, T ];L2(∂Gjε)) is a strong solution of the penalized problem if ∂tw j,δ ε,φ ∈ L2(0, T ;L2(∂Gjε)), w j,δ ε,φ(x, 0) = φ(P jε , 0) on ∂Gjε, and it satisfies the integral identity∫ T 0 ∫ T j ε/4 \Gj ε ∇wj,δε,φ∇v dx dt+ ε−γ ∫ T 0 ∫ ∂Gj ε ∂tw j,δ ε,φv ds dt − ε−γδ−1 ∫ T 0 ∫ ∂Gj ε (φ(P jε , t)− w j,δ ε,φ)−v ds dt = ε−γ ∫ T 0 ∫ ∂Gj ε ∂tφ(P jε , t)v ds dt, (3.10) for all arbitrary functions v ∈ L2(0, T ;H1(T jε/4 \G j ε, ∂T j ε/4)). This problem was investigated in [17, Theorem 5.1]. It was proved that there exists a unique solution and it satisfies the estimates ε−γ‖wj,δε,φ‖ 2 C([0,T ];L2(∂Gj ε)) + ‖wj,δε,φ‖ 2 L2(0,T ;H1(T j ε/4 \Gj ε,∂T j ε/4 )) + δ−1ε−γ‖(φ(P jε , t)− w j,δ ε,φ)−‖2 L2(0,T ;L2(∂Gj ε)) ≤ Kεn max Ω ‖φ‖2L2(0,T ), ε−γ‖∂twj,δε,φ‖ 2 L2(0,T ;L2(∂Gj ε)) + max [0,T ] ‖∇wj,δε,φ‖ 2 L2(T j ε/4 \Gj ε) + δ−1ε−γ max [0,T ] ‖(φ(P jε , t)− w j,δ ε,φ)−‖2 L2(∂Gj ε) ≤ Kεn(max QT |φ|2 + max Ω ‖∂tφ‖2L2(0,T )). From these estimates, we have that there exists a subsequence (still denoted as the original one) such that, as δ → 0, we have wj,δε,φ ⇀ wjε,φ weakly in L2(0, T ;H1(T jε/4 \G j ε, ∂T j ε/4)), ∂tw j,δ ε,φ ⇀ ∂tw j ε,φ weakly in L2(0, T ;L2(∂Gjε)), (φ(P jε , t)− w j,δ ε,φ)− → 0 in L2(0, T ;L2(∂Gjε)), wj,δε,φ ⇀ wjε,φ weakly in L2(0, T ;L2(∂Gjε)), wj,δε,φ ⇀ wjε,φ weakly in L2(0, T ;L2(T jε/4 \G j ε)). Then, by the compactness result of [5, Theorem 2.1], we conclude that wj,δε,φ → wjε,φ in C([0, T ];L2(∂Gjε)), as δ → 0. From the above convergences we conclude that the limit function wjε,φ satisfies the analogous estimates as wj,δε,φ. Now we show that wjε,φ is the strong solution to the problem (3.1). Indeed, using the convergence of (φ(P jε , t) − w j,δ ε,φ)− to zero, it is easy to see that wjε,φ ∈ Kj,ε. Next, we take v − wj,δε,φ, where v ∈ Kj,ε as a test function in the integral identity 10 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 for wj,δε,φ, we obtain ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j,δ ε,φ − φ(P jε , t))(v − w j,δ ε,φ) ds dt + ∫ T 0 ∫ T j ε/4 \Gj ε ∇wj,δε,φ∇(v − wj,δε,φ) dx dt = δ−1ε−γ ∫ T 0 ∫ ∂Gj ε (φ(P jε , t)− w j,δ ε,φ)−(v − wj,δε,φ) ds dt ≥ 0. From the above convergences we derive that ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w j ε,φ)(φ(P jε , t)− w j ε,φ)dsdt = ε−γ 2 ‖φ(P jε , T )− wjε,φ(x, T )‖2 L2(∂Gj ε) ≤ ε−γ 2 lim δ→0 ‖φ(P jε , T )− wj,δε,φ(x, T )‖2 L2(∂Gj ε) = lim δ→0 ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w j,δ ε,φ)(φ(P jε , t)− w j,δ ε,φ) ds dt. From this and the above convergences we conclude that lim δ→0 ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j,δ ε,φ − φ(P jε , t))(v − w j,δ ε,φ) ds dt = lim δ→0 ( ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j,δ ε,φ − φ(P jε , t))(φ(P jε , t)− w j,δ ε,φ) ds dt + ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j,δ ε,φ − φ(P jε , t))(v − φ(P jε , t)) ds dt ) ≤ ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j ε,φ − φ(P jε , t))(φ(P jε , t)− w j ε,φ) ds dt + ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j ε,φ − φ(P jε , t))(v − φ(P jε , t)) ds dt = ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j ε,φ − φ(P jε , t))(v − w j ε,φ) ds dt. Thus, we have proved that lim δ→0 ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j,δ ε,φ − φ(P jε , t))(v − w j,δ ε,φ) ds dt ≤ ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j ε,φ − φ(P jε , t))(v − w j ε,φ) ds dt. (3.11) The above convergences also imply that ‖∇wjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) ≤ lim δ→0 ‖∇wj,δε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) . EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 11 From this inequality we deduce that lim δ→0 ∫ T 0 ∫ T j ε/4 \Gj ε ∇wj,δε,φ∇(v − wj,δε,φ) dx dt ≤ ∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(v − wjε,φ) dx dt. Combining the above results we derive that wjε,φ satisfies ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j ε,φ − φ(P jε , t))(v − w j ε,φ) ds dt + ∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(v − wjε,φ) dx dt ≥ 0, for any arbitrary function v ∈ Kj,ε. Hence, wjε,φ is a strong solution of the problem (3.1). The uniqueness of the strong solution is consequence of the monotonicity of the associate operator (as in [17]). � The next theorem gives a pointwise estimate for wjε,φ. Theorem 3.2. The solution to the problem (3.1) satisfies the estimate sup (T j ε/4 \Gj ε)×(0,T ) |wjε,φ| ≤ 2 max QT |φ(x, t)|. (3.12) Proof. We denote hj,δε,φ(x, t) = φ(P jε , t) − w j,δ ε,φ. It is easy to see that hj,δε,φ satisfies the problem ∆hjδε,φ = 0, x ∈ T jε/4 \G j ε, t ∈ (0, T ), ε−γ∂th j,δ ε,φ + ∂νh j,δ ε,φ + δ−1ε−γ(hj,δε,φ)− = 0, x ∈ ∂Gjε, t ∈ (0, T ), hj,δε,φ(x, t) = φ(P jε , t), x ∈ ∂T jε/4, t ∈ (0, T ), hj,δε,φ(x, 0) = 0, x ∈ ∂Gjε. (3.13) We define K = max QT |φ(x, t)|. Then take v = (K − hj,δε,φ)− ∈ L2(0, T ;H1(T jε/4 \ Gjε, ∂T j ε/4)) as a test function in the integral identity for problem (3.13), − ∫ T 0 ∫ T j ε/4 \Gj ε |∇(K − hj,δε,φ)−|2 dx dt− ε−γ 2 ‖(K − hj,δε,φ)−(T )‖2 L2(∂Gj ε) + δ−1ε−γ ∫ T 0 ∫ ∂Gj ε (hj,δε,φ)−(K − hj,δε,φ)− ds dt = 0. (3.14) Suppose now that hj,δε,φ ≤ 0, then K−hj,δε,φ ≥ 0. So, (hj,δε,φ)−(K−hj,δε,φ)− = 0 on ∂Gjε. Thus, from (3.14), we deduce that (K − hj,δε,φ)− ≡ 0 for x ∈ T jε/4 \ G j ε, t ∈ (0, T ). Hence wj,δε,φ ≥ −K + φ ≥ −2K. 12 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 Similarly, taking v = (K + hj,δε,φ)− ∈ L2(0, T ;H1(T jε/4 \ G j ε, ∂T j ε/4)) as a test function in the integral identity for problem (3.13), we obtain∫ T 0 ∫ T j ε/4 \Gj ε |∇(K + hj,δε,φ)−|2 dx dt+ ε−γ 2 ‖(K + hj,δε,φ)−(T )‖2 L2(∂Gj ε) + δ−1ε−γ ∫ T 0 ∫ ∂Gj ε (hj,δε,φ)−(K + hj,δε,φ)− ds dt = 0. (3.15) All of the terms in (3.15) are non-negative, thus, we conclude that K + hj,δε,φ ≥ 0 and so wj,δε,φ ≤ φ(P jε , t) +K ≤ 2K. Using standard arguments of the penalty method, one can show that wj,δε,φ ⇀ wjε,φ weakly in L2(0, T ;H1(T jε/4 \ G j ε)) as δ → 0. Now, consider a function g ∈ L2(0, T ;H1(T jε/4 \ G j ε)) such that g ≥ 0 and multiply by it in both sides of the inequality wj,δε,φ − 2K ≤ 0. Then, we integrate the obtained expression over (T jε/4 \ Gjε)× (0, T ) and obtain∫ T 0 ∫ T j ε/4 \Gj ε (wj,δε,φ − 2K)g dx dt ≤ 0. Using the weak convergence, we pass to the limit as δ → 0 in the above inequality and derive ∫ T 0 ∫ T j ε/4 \Gj ε (wjε,φ − 2K)g dx dt ≤ 0, where g is an arbitrary non-negative function as before. Taking g = (wjε,φ − 2K)+, we obtain ∫ T 0 ∫ T j ε/4 \Gj ε |(wjε,φ − 2K)+|2 dx dt ≤ 0. Thus, we conclude that wjε,φ − 2K ≤ 0 for a.e. x ∈ Ω, t ∈ (0, T ). Analogously, we can get the opposite estimate and then we arrive to (3.12). � Remark 3.3. From the penalty method we have the weak convergence of wj,δε,φ to wjε,φ in L2(0, T ;H1(T jε/4 \ G j ε)). Nevertheless, we can show that the above con- vergence is in fact strong. Indeed, the penalty method also implies that ∂tw j,δ ε,φ ⇀ ∂tw j ε,φ weakly in L2(0, T ;L2(∂Gjε)). As mentioned before we have wj,δε,φ → wjε,φ in C([0, T ];L2(∂Gjε)) as δ → 0. From the integral equality for problem (3.9), we obtain∫ T 0 ∫ T j ε/4 \Gj ε ∇wj,δε,φ∇(ψ − wj,δε,φ) dx dt + ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j,δ ε,φ − φ(P jε , t))(ψ − w j,δ ε,φ) ds dt = δ−1ε−γ ∫ T 0 ∫ ∂Gj ε (φ(P jε , t)− w j,δ ε,φ)−(ψ − wj,δε,φ) ds dt EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 13 = δ−1ε−γ ∫ T 0 ∫ ∂Gj ε ((φ(P jε , t)− w j,δ ε,φ)− − (φ(P jε , t)− ψ)−)(ψ − wj,δε,φ) ds dt for an arbitrary function ψ ∈ Kj,ε (i.e. (φ(P jε , t)− ψ)− ≡ 0 on ∂Gjε). The function λ → −(C − λ)− is monotone non-decreasing, hence, the right-hand side of the derived expression is non-negative. Thus, we have the same inequality as (3.2) for the wj,δε,ϕ. From here, we obtain ‖∇wj,δε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) ≤ ∫ T 0 ∫ T j ε/4 \Gj ε ∇wj,δε,φ∇ψ dx dt+ ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j,δ ε,φ − φ(P jε , t))(ψ − w j,δ ε,φ) ds dt For the second integral on the right-hand side of the inequality, we use (3.11), and passing to the limit as δ → 0, we obtain lim δ→0 ‖∇wj,δε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) ≤ ∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇ψ dx dt+ ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(w j ε,φ − φ(P jε , t))(ψ − w j ε,φ) ds dt Taking ψ = wjε,φ, we obtain lim δ→0 ‖∇wj,δε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) ≤ ‖∇wjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) Using this estimate and the properties of the weak convergence, we derive lim δ→0 ‖∇wj,δε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) = ‖∇wjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) Thus, wj,δε,φ converges to wjε,φ strongly in L2(0, T ;H1(T jε/4 \G j ε)) as δ → 0. 3.2. Second auxiliary problem: a different unilateral exterior problem on the cell variables. To define the notion of a strong solution to the exterior Signorini problem, we denote by M the set of functions w ∈ C∞(Rn \G0) such that w(y) = 0 for y ∈ Rn \ T oR for some ball T 0 R such that G0 ⊂ T 0 R. We denote by M the closure of M with respect to the norm ‖w‖M = ‖∇w‖L2(Rn\G0). Given φ ∈ H1(0, T ;H1(Ω)) let us consider the exterior problem with dynamic Signorini boundary condition ∆ywφ(x, y, t) = 0, y ∈ Rn \G0, t ∈ (0, T ), wφ ≤ φ(x, t), y ∈ ∂G0, t ∈ (0, T ), ∂νwφ ≤ C0(∂tφ− ∂twφ), y ∈ ∂G0, t ∈ (0, T ), (wφ − φ)(∂νwφ − C0(∂tφ− ∂twφ)) = 0, y ∈ ∂G0, t ∈ (0, T ), wφ(x, y, 0) = φ(x, 0), y ∈ ∂G0, wφ(x, y, t)→ 0, |y| → ∞, (3.16) where now x ∈ Ω is a parameter. For x ∈ Ω and t ∈ [0, T ], we define the closed convex sets Kφ(x) = {v ∈M : v ≤ φ for a.e. y ∈ ∂G0}, Kφ(x) = {v ∈ L2(0, T ;M) : v ∈ Kφ(x) for a.e. t ∈ [0, T ]}. 14 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 By a strong solution to the problem (3.16), we mean a function wφ(x, ·, ·) ∈ Kφ(x) such that wφ ∈ C([0, T ];L2(∂G0)), ∂twφ ∈ L2(0, T ;L2(∂G0)) and wφ(x, y, 0) = φ(x, 0) for a.e. x ∈ Ω, and wφ satisfies the variational inequality∫ T 0 ∫ Rn\G0 ∇wφ∇(v − wφ) dy dt ≥ C0 ∫ T 0 ∫ ∂G0 ∂t(φ− wφ)(v − wφ) ds dt, (3.17) for any arbitrary test function v ∈ Kφ(x), for a.e. x ∈ Ω. We define now the function ŵφ(x, y, t) = φ(x, t)κ(y)− wφ(x, y, t), (3.18) where κ(y) is the solution to the G0 – capacity problem in the cell variables ∆yκ = 0, y ∈ Rn \G0, κ(y) = 1, y ∈ ∂G0, κ(y)→ 0, |y| → ∞. (3.19) Clearly, the function ŵφ(x, y, t) is a solution to the unilateral problem in the cell variables ∆yŵφ(x, y, t) = 0, y ∈ Rn \G0, t ∈ (0, T ), ŵφ ≥ 0, ∂νŵφ + C0∂tŵφ ≥ φ(x, t)∂νκ(y), y ∈ ∂G0, t ∈ (0, T ), ŵφ(∂νŵφ + C0∂tŵφ − φ(x, t)∂νκ(y)) = 0, y ∈ ∂G0, t ∈ (0, T ), ŵφ(x, y, 0) = 0, y ∈ ∂G0, ŵφ → 0, |y| → +∞. (3.20) This reformulation of the auxiliary function wφ reduces the conditions imposed on the function φ as the problem (3.20) doesn’t contain ∂tφ. Thus, here and in the theorems below, we consider φ ∈ L2(0, T ;L2(Ω)) unless otherwise stated explicitly. Theorem 3.4. Problem (3.20) has a unique strong solution and, for a.e. x ∈ Ω, the following estimates hold ‖ŵφ(x, ·, ·)‖C([0,T ];L2(∂G0)) + ‖ŵφ(x, ·, ·)‖C([0,T ];M) ≤ K‖φ(x, ·)‖L2(0,T ), ‖∂tŵφ(x, ·, ·)‖L2(0,T ;L2(∂G0)) ≤ K‖φ(x, ·)‖L2(0,T ). (3.21) Proof. Given δ > 0, let us consider, again, the auxiliary penalty formulation in the cell variables ∆yŵ δ φ(x, y, t) = 0, y ∈ Rn \G0, t ∈ (0, T ), ∂νŵ δ φ + C0∂tŵ δ φ + δ−1(ŵδφ)− = φ∂νκ, y ∈ ∂G0, t ∈ (0, T ), ŵδφ(x, y, 0) = 0, y ∈ ∂G0, ŵδφ → 0, |y| → ∞, (3.22) where x ∈ Ω is taken as a parameter. It is well known (see [20, 24]) that the problem (3.22) has a unique strong solution, and that for a.e. x ∈ Ω, the following estimates hold ‖ŵδφ(x, ·, ·)‖C([0,T ];L2(∂G0)) + ‖ŵδφ(x, ·, ·)‖C([0,T ];M) ≤ K‖φ(x, ·)‖L2(0,T ), ‖∂tŵδφ(x, ·, ·)‖L2(0,T ;L2(∂G0)) ≤ K‖φ(x, ·)‖L2(0,T ). (3.23) Also, it is easy to obtain that ‖(ŵδφ)−(x, ·, ·)‖L2(0,T ;L2(∂G0)) ≤ K √ δ‖φ(x, ·)‖L2(0,T ). (3.24) EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 15 From estimates (3.23), (3.24), we conclude that, for some subsequence, as δ → 0 we have ŵδφ ⇀ ŵφ weakly in L2(0, T ;M), ∂tŵ δ φ ⇀ ∂tŵφ weakly in L2(0, T ;L2(∂G0)), ŵδφ ⇀ ŵφ weakly in L2(0, T ;L2(∂G0)). (3.25) Then, by the compactness result of [5, Theorem 2.1], we conclude that ŵδφ → ŵφ in C([0, T ];L2(∂G0)), as δ → 0. The integral identity for problem (3.22) takes the form∫ T 0 ∫ Rn\G0 ∇ŵδφ∇(ψ − ŵδφ) dy dt+ C0 ∫ T 0 ∫ ∂G0 ∂tŵ δ φ(ψ − ŵδφ)dsydt + δ−1 ∫ T 0 ∫ ∂G0 (ŵδφ)−(ψ − ŵδφ)dsydt = ∫ T 0 φ(x, t) ∫ ∂G0 ∂νκ(y)(ψ − ŵδφ)dsydt, (3.26) where ψ ∈ L2(0, T ;M), ψ ≥ 0 on ∂G0 for a.e. t ∈ [0, T ], x ∈ Ω is a parameter. We rewrite it in the form∫ T 0 ∫ Rn\G0 ∇ŵδφ∇(ψ − ŵδφ) dy dt+ C0 ∫ T 0 ∫ ∂G0 ∂tŵ δ φ(ψ − ŵδφ)dsydt − ∫ T 0 φ(x, t) ∫ ∂G0 ∂νκ(y)(ψ − ŵδφ)dsydt = δ−1 ∫ T 0 ∫ ∂G0 (ψ− − (ŵδφ)−)(ψ − ŵδφ)dsydt ≥ 0, (3.27) where we have used that ψ ≥ 0 on ∂G0 a.e. t ∈ [0, T ] and that the real function λ → λ− is a Lipschitz and monotone function. Now, we are going to pass to the limit as δ → 0 in the inequality (3.27). Taking into account that ‖ŵφ(x, ·, ·)‖L2(0,T ;M) ≤ lim δ→0 ‖ŵδφ(x, ·, ·)‖L2(0,T ;M), we conclude that lim δ→0 ∫ T 0 ∫ Rn\G0 ∇ŵδφ∇(ψ − ŵδφ) dy dt ≤ ∫ T 0 ∫ Rn\G0 ∇ŵφ∇(ψ − ŵφ) dy dt. (3.28) From the convergences (3.25), we have C0 lim δ→0 ∫ T 0 ∫ ∂G0 ∂tŵ δ φ(ψ − ŵδφ)dsydt = C0 lim δ→0 (∫ T 0 ∫ ∂G0 ∂tŵ δ φψdsydt− 1 2 ‖ŵδφ(·, T )‖2L2(∂G0) ) ≤ C0 (∫ T 0 ∫ ∂G0 ∂tŵφψdsydt− 1 2 ‖ŵψ(·, T )‖2L2(∂G0) ) = C0 ∫ T 0 ∫ ∂G0 ∂tŵφ(ψ − ŵφ)dsydt. (3.29) 16 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 From (3.29)-(3.31) we conclude that∫ T 0 ∫ Rn\G0 ∇ŵφ∇(ψ − ŵφ) dy dt+ C0 ∫ T 0 ∫ ∂G0 ∂tŵφ(ψ − ŵφ) ds dt ≥ ∫ T 0 φ(x, t) ∫ ∂G0 ∂νκ(y)(ψ − ŵφ)dsydt, (3.30) where ψ ∈ L2(0, T ;M) and ψ ≥ 0 for a.e. y ∈ ∂G0, t ∈ [0, T ]. The inequality (3.30) is a variational formulation of the problem (3.20), hence, w̃φ is a solution. Then, estimates (3.21) are a consequence of (3.23), (3.24) and the weak conver- gences (3.25). � Remark 3.5. Notice that we can transform the inequality (3.27) into ‖ŵδφ(x, ·, ·)‖2L2(0,T ;M) + C0 2 ‖ŵδφ(x, ·, T )‖2L2(∂G0) ≤ ∫ T 0 ∫ Rn\G0 ∇wδφ∇ψdydt+ C0 ∫ T 0 ∫ ∂G0 ∂tŵ δ φψdsydt − ∫ T 0 φ(x, t) ∫ ∂G0 ∂νκ(y)(ψ − ŵδφ)dsydt. Using Hardy’s inequality, we have ŵδφ(x, ·, T ) is uniformly bounded in δ in H1(T 0 R0 \ G0), where R0 > 0 is large enough to have G0 ⊂ T 0 R0 . From the embedding theorem, we have that there exists a subsequence (for which we preserve the notation of the original) such that ŵδφ(x, ·, T ) converges to ŵφ(x, ·, T ) in L2(∂G0) as δ → 0. Passing to the limit as δ → 0, we obtain lim δ→0 (‖ŵδφ(x, ·, ·)‖2L2(0,T ;M) + C0 2 ‖ŵδφ(x, ·, T )‖2L2(∂G0)) ≤ ∫ T 0 ∫ Rn\G0 ∇ŵφ∇ψdydt+ C0 ∫ T 0 ∫ ∂G0 ∂tŵφψ dsy dt − ∫ T 0 φ(x, t) ∫ ∂G0 ∂νκ(y)(ψ − ŵφ) dsy dt. Taking ψ = ŵφ as a test function in this inequality, we derive lim δ→0 ‖ŵδφ(x, ·, ·)‖2L2(0,T ;M) ≤ ‖ŵφ(x, ·, ·)‖2L2(0,T ;M). Hence, we actually have the strong convergence ŵδφ → ŵφ in L2(0, T ;M), as δ → 0. Theorem 3.6. Let φ1, φ2 ∈ L2(0, T ;L2(Ω)). Then, for a.e. x ∈ Ω, we have ‖(ŵφ1 − ŵφ2)(x, ·, ·)‖C([0,T ];L2(∂G0)) + ‖(ŵφ1 − ŵφ2)(x, ·, ·)‖L2(0,T ;M) ≤ K‖(φ1 − φ2)(x, ·)‖L2(0,T ). (3.31) In addition, we have the following estimate for the time derivative of the functions ‖(∂tŵφ1 − ∂tŵφ2 )(x, ·, ·)‖L2(0,T ;L2(∂G0)) ≤ K‖(φ1 − φ2)(x, ·)‖L2(0,T ), (3.32) for a.e. x ∈ Ω. EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 17 Proof. Let v̂ = ŵφ1 − ŵφ2 . We use again the penalized problem (3.22). If we define v̂δ = ŵδφ1 − ŵδφ2 then, according to (3.25), we have v̂δ ⇀ v̂ weakly in L2(0, T ;M), ∂tv̂ δ ⇀ ∂tv̂ weakly in L2(0, T ;L2(∂G0)), v̂δ ⇀ v̂ weakly in L2(0, T ;L2(∂G0)). (3.33) For the solution of the function v̂δ, we have some estimates similar to the estimates obtained in [20], ‖v̂δ(x, ·, ·)‖C([0,T ];L2(∂G0)) + ‖v̂δ(x, ·, ·)‖L2(0,T ;M) ≤ K‖(φ1 − φ2)(x, ·)‖L2(0,T ), ‖∂tv̂δ(x, ·, ·)‖L2(0,T ;L2(∂G0)) ≤ K‖(φ1 − φ2)(x, ·)‖L2(0,T ), (3.34) where the constant K does not depend on δ neither on φi, i = 1, 2. Then, using the convergence (3.33), we conclude that the estimates (3.31), (3.32) follow from (3.34). � Theorem 3.7. Consider a test function of the form φ(x, t) = ψ(x)η(t), with ψ(x) ∈ C∞(Ω) and η ∈ C1([0, T ]). Then, for a.e. t ∈ [0, T ] and x ∈ Ω, we have |ŵφ(x, y, t)| ≤ K max Q T |φ(x, t)| |y|n−2 for y ∈ Rn \G0, (3.35) |∇ywφ(x, y, t)| ≤ K max Q T |φ(x, t)| |y|n−1 , for y ∈ Rn \G0, (3.36) where the positive constant K is independent of φ, x ∈ Ω and t ∈ [0, T ]. Proof. Once again, we consider the solution ŵδφ of the associate penalized problem (3.22). Then we have the point-wise estimates (see [20]) |ŵδφ(x, y, t)| ≤ K max Q T |φ(x, t)| |y|n−2 , ∀y ∈ Rn \G0, (3.37) |∇yŵδφ(x, y, t)| ≤ K max Q T |φ(x, t)| |y|n−1 , ∀y ∈ Rn \G0, (3.38) where K is independent of φ, δ, x ∈ Ω and t ∈ [0, T ]. By Remark 3.5 we know the strong convergence of ŵδφ to ŵφ in L2(0, T ;M). Thus, we can extract a subsequence converging almost everywhere. Passing to the limit in (3.37) for this subsequence we obtain the first statement of this theorem. Using the same arguments, we can take a subsequence converging almost every- where for the gradient of ŵδφ and this proves the second part of the statement. � Theorem 3.8. Let φ as in Theorem 3.7 and let ŵφ be the solution to the problem (3.20). Then, for a.e. x1, x2 ∈ Ω ‖ŵφ(x1, ·, ·)− ŵφ(x2, ., .)‖L2(0,T ;M) ≤ K‖φ(x1, ·)− φ(x2, ·)‖L2(0,T ). (3.39) Proof. We use the following estimate for the solution of the penalized problem (3.22) proved in [20] ‖ŵδφ(x1, ·, ·)− ŵδφ(x2, ·, ·)‖L2(0,T ;M) ≤ K‖φ(x1, ·)− φ(x2, ·)‖L2(0,T ). 18 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 From here and the weak convergence (3.33), we derive ‖ŵφ(x1, ·, ·)− ŵφ(x2, ·, ·)‖L2(0,T ;M) ≤ ≤ lim δ→0 ‖ŵδφ(x1, ·, ·)− ŵδφ(x2, ·, ·)‖L2(0,T ;M) ≤ K‖φ(x1, ·)− φ(x2, ·)‖L2(0,T ). � 3.3. Asymptotic similarity between the two types of auxiliary functions after a suitable substitution. We use a natural substitution to define the func- tion w̃jε,φ(x, t) = wφ(P jε , x−P j ε aε , t). Then we are in conditions to get some asymp- totic estimates on the difference of the two types of auxiliary functions. We define vjε,φ(x, t) = w̃jε,φ(x, t)− wjε,φ(x, t). Then, we have the following result. Theorem 3.9. Let φ(x, t) = ψ(x)η(t), with ψ ∈ C∞(Ω) and η ∈ C1([0, T ]). Then we have the following estimate for the function vjε,φ expressing the difference of the two auxiliary functions sup (T j ε/4 \Gj ε)×(0,T ) |vjε,φ| ≤ sup ∂T j ε/4 ×(0,T ) |w̃jε,φ|. (3.40) Proof. Given δ > 0, we consider the solutions to the penalized problems (3.9), and (3.22) and we define vj,δε,φ(x, t) = w̃j,δε,φ(x, t)− wj,δε,φ(x, t), where w̃j,δε,φ(x, t) = w̃δφ(P jε , x− P jε aε , t), w̃δφ(x, y, t) = k(y)φ(x, t)− ŵδφ(x, y, t), where ŵδφ(x, y, t) was defined similarly to (3.18). From Remark 3.5, we conclude that w̃j,δε,φ converges to w̃jε,φ in L2(0, T ;H1(T jε/4 \G j ε)) as δ → 0. From Remark 3.3, we have wj,δε,φ → wjε,φ in L2(0, T ;H1(T jε/4 \ G j ε)) as δ → 0. Therefore, vj,δε,φ → vjε,φ in L2(0, T ;H1(T jε/4 \ G j ε)). Hence, we can obtain a sub-sequence (that we denote as the original one) that converges almost everywhere. Moreover, concerning the function vj,δε,φ, we have the estimate sup (T j ε4\G j ε)×(0,T ) |vj,δε,φ| ≤ sup ∂T j ε/4 ×(0,T ) |w̃j,δε,φ|, (see. [20]). From here, we derive the estimate (3.40). � Theorem 3.10. Let φ = ψ(x)η(t), ψ ∈ C∞(Ω), η ∈ C1([0, T ]). The following global estimates hold ε−γ ∑ j∈Υε max t∈[0,T ] ‖vjε,φ‖ 2 L2(∂Gj 0) + ∑ j∈Υε ‖∇vjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) ≤ Kε2 max QT φ2(x, t), ∑ j∈Υε ‖vjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) ≤ Kε4 max QT φ2(x, t). (3.41) Proof. Recall that φj,ε(x, t) = φ(P jε , t)ψj,ε(x), where ψj,ε ∈ C∞0 (T jε/4), with ψj,ε(x) ≡ 1, if x ∈ T jCaε , C > 1 and ψj,ε ≡ 0, if x ∈ T jε/4 \ T j 2Caε , and that |∇ψj,ε| ≤ K aε . EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 19 Taking into account the definition of the w̃jε,φ and using that wjε,φ−φj,ε ∈ H1(T jε/4\ Gjε, ∂T j ε/4), we obtain ∫ T 0 ∫ T j ε/4 \Gj ε ∇w̃jε,φ∇(wjε,φ − φj,ε) dx dt = − ∫ T 0 ∫ ∂Gj ε ∂νw̃ j ε,φ(φ(P jε , t)− w j ε,φ) ds dt ≥ −ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w̃ j ε,φ)(φ(P jε , t)− w j ε,φ) ds dt = ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w̃ j ε,φ)(wjε,φ − φ(P jε , t)) ds dt. Additionally, we have (see (3.4))∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(wjε,φ − φj,ε) dx dt = ε−γ ∫ T 0 ∫ ∂Gj ε ∂t(φ(P jε , t)− w j ε,φ)(wjε,φ − φ(P jε , t)) ds dt. Subtracting one expression from the other, we obtain∫ T 0 ∫ T j ε/4 \Gj ε ∇vjε,φ∇(wjε,φ − φj,ε) dx dt ≥ ε−γ ∫ ∂Gj ε ∂t(w j ε,φ − w̃ j ε,φ)(wjε,φ − φ(P jε , t)) ds dt = ε−γ ∫ T 0 ∫ ∂Gj ε ∂tv j ε,φ(φ(P jε , t)− w j ε,φ) ds dt. From here, we derive − ∫ T 0 ∫ T j ε/4 \Gj ε |∇vjε,φ| 2 dx dt+ ∫ T 0 ∫ T j ε/4 \Gj ε ∇vjε,φ∇(w̃jε,φ − φj,ε) dx dt ≥ ε−γ ∫ T 0 ∫ ∂Gj ε ∂tv j ε,φ(φ(P jε , t)− w j ε,φ) ds dt. Next, we transform this inequality into∫ T 0 ∫ T j ε/4 \Gj ε |∇vjε,φ| 2 dx dt+ ε−γ ∫ T 0 ∫ ∂Gj ε ∂tv j ε,φv j ε,φ ds dt ≤ ∫ T 0 ∫ T j ε/4 \Gj ε ∇vjε,φ∇(w̃jε,φ − φj,ε) dx dt + ε−γ ∫ T 0 ∫ ∂Gj ε ∂tv j ε,φ(w̃jε,φ − φ(P jε , t)) ds dt = Iε. (3.42) 20 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 Now, we need to estimate the right-hand side of the inequality, i.e. Iε. Using Green’s formula, we obtain Iε = ∫ T 0 ∫ ∂T j ε/4 ∂νv j ε,φw̃ j ε,φ ds dt+ ∫ T 0 ∫ ∂Gj ε ∂νv j ε,φ(w̃jε,φ − φ(P jε , t)) ds dt + ε−γ ∫ T 0 ∫ ∂Gj ε ∂tv j ε,φ(w̃jε,φ − φ(P jε , t)) ds dt = ∫ T 0 ∫ ∂Gj ε (∂νv j ε,φ + ε−γ∂tv j ε,φ)(w̃jε,φ − φ(P jε , t)) ds dt + ∫ T 0 ∫ ∂T j ε/4 ∂νv j ε,φw̃ j ε,φ ds dt ≤ ∫ T 0 ∫ ∂T j ε/4 ∂νv j ε,φw̃ j ε,φ ds dt. (3.43) Here, we used that if w̃jε,φ < φ(P jε , t), then ε−γ∂tw̃ j ε,φ + ∂νw̃ j ε,φ − ε −γ∂tφ(P jε , t) = 0. (3.44) Additionally, on ∂Gjε × (0, T ), we have ε−γ∂tw j ε,φ + ∂νw j ε,φ − ε −γ∂tφ(P jε , t) ≤ 0. (3.45) Subtracting from the equality (3.44) the inequality (3.45), for (x, t) ∈ ∂Gjε × (0, T ) we obtain the inequality ε−γ∂tv j ε,φ + ∂νv j ε,φ ≥ 0. (3.46) From (3.42) and (3.43), we deduce∫ T 0 ∫ T j ε/4 \Gj ε |∇vjε,φ| 2 dx dt+ ε−γ 2 ‖vjε,φ(x, T )‖2 L2(∂Gj ε) ≤ ∫ T 0 ∫ ∂T j ε/4 ∂νv j ε,φw̃ j ε,φ ds dt = ∫ T 0 ∫ T j ε/4 \T j ε/8 ∇vjε,φ∇w̃ j ε,φ dx dt− ∫ T 0 ∫ ∂T j ε/8 ∂νv j ε,φw̃ j ε,φ ds dt. (3.47) Applying Theorems 3.4 and 3.9, we obtain |vjε,φ(x, t)| ≤ Kε2 max QT |φ(x, t)|, (3.48) where K is a constant independent of ε and φ. From here, for any x0 ∈ ∂T jε/8, t ∈ (0, T ), we deduce the estimate ∂xiv j ε,φ(x0, t)| = |T x0 ε/16| −1 ∣∣∫ T x0 ε/16 ∂xiv j ε,φ(x, t)dx ∣∣ = |T x0 ε/16| −1 ∣∣∫ ∂T x0 ε/16 vjε,φνids ∣∣ ≤ Kεmax QT |φ(x, t)|. Consequently, ∣∣∫ T 0 ∫ ∂T j ε/8 ∂νv j ε,φw̃ j ε,φ ds dt ∣∣ ≤ Kεn+2 max QT |φ(x, t)|2. EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 21 From the estimate |∇ywφ(x, y, t)| ≤ K max QT |φ(x, t)| |y|n−1 , we obtain |∇xw̃jε,φ(x, t)| ≤ K max QT |φ(x, t)|ε, if x ∈ T jε/4 \ T j ε/8, t ∈ [0, T ]. Thus, we have ∣∣∫ T 0 ∫ T j ε/4 \T j ε/8 ∇vjε,φ∇w j ε,φ dx dt ∣∣ ≤ 1 2 ∫ T 0 ∫ T j ε/4 \Gj ε |∇vjε,φ| 2 dx dt+Kεn+2 max QT |φ(x, t)|2. From this and (3.47), we conclude that ε−γ max [0,T ] ‖vjε,φ‖ 2 L2(∂Gj ε) + ‖∇vjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gj ε)) ≤ Kεn+2 max QT |φ(x, t)|2. Using Friedrichs inequality, we obtain the second estimate in (3.41). � 4. Definition and properties of the strange non-local operator H[·] We are now in a position to define the important nonlinear nonlocal operator H : L2(0, T ;L2(Ω)) → L2(0, T ;L2(Ω)) which arises in the main Theorem 1.1. We start by defining the operator on a class of smoother functions φ ∈ H1(0, T ;H1(Ω)), but, by density, we can extend it to the general space L2(0, T ;L2(Ω)). We define H[φ](x, t) = ∫ ∂G0 ∂νwφ(x, y, t)dsy, (4.1) where wφ is the solution to the problem (3.16). It is easy to rewrite H[φ] in an equivalent form H[φ](x, t) = φ(x, t)λG0 − ∫ ∂G0 ∂νŵφ(x, y, t)dsy, (4.2) where ŵφ is the solution to problem (3.20) and where it appears the important notion of the capacity of the model set G0 λG0 = ∫ ∂G0 ∂νκ(y)dsy = Cap(G0). Theorem 4.1. Assume φ ∈ L2(0, T ;L2(Ω)). Then ‖H[φ]‖L2(0,T ;L2(Ω)) ≤ K‖φ‖L2(0,T ;L2(Ω)). (4.3) Also we have Lipschitz continuity with respect to φ: for φ1, φ2 ∈ L2(0, T ;L2(Ω)), we have ‖H[φ1]−H[φ2]‖L2(0,T ;L2(Ω)) ≤ K‖φ1 − φ2‖L2(0,T ;L2(Ω)). (4.4) In addition, we have the following monotone time-dependence on φ: given φ1, φ2 ∈ L2(0, T ;L2(Ω)), we have∫ T 0 ∫ Ω ( H[φ1]−H[φ2] ) (φ1 − φ2) dx dt ≥ 0. (4.5) 22 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 Proof. Given δ > 0 we consider the penalized version of the operator H[φ] as the one given by Hδ[φ](x, t) = φ(x, t)λG0 − ∫ ∂G0 ∂νŵ δ φdsy, (4.6) where ŵδφ is the solution to the penalized problem (3.22). The properties of the operator Hδ[φ] were studied in the previous paper [20] and it was shown there that it satisfies properties (4.3)-(4.5). Thus, the estimates (4.3), (4.4) are a direct consequence of (3.21), (3.31), respectively, and the monotonicity property can be derived also by using the integral identity satisfied by ŵδφ1 −ŵδφ2 . Hence, to prove the theorem, we only need to show the convergence of Hδ[φ] to H[φ], in L2(0, T ;L2(Ω)). To do that, we first point out that, by using the definition of κ, we can rewrite (4.6) in the form Hδ[φ](x, t) = φ(x, t)λG0 − ∫ Rn\G0 ∇yŵδφ∇κdy. (4.7) Then, for an arbitrary function ψ ∈ L2(0, T ;L2(Ω)), we have∫ T 0 ∫ Ω Hδ[φ]ψ(t)dxdt = λG0 ∫ T 0 ∫ Ω φψ dx dt− ∫ T 0 ∫ Ω ∫ Rn\G0 ∇yŵδφ∇y(κψ) dy dx dt. Using that ŵδφ ⇀ ŵφ weakly in L2(0, T ;M), as δ → 0, we conclude lim δ→0 ∫ T 0 ∫ Ω Hδ[φ]ψ dx dt = ∫ T 0 ∫ Ω H[φ]ψ dx dt, (4.8) for all arbitrary functions ψ ∈ L2(0, T ;L2(Ω)). Hence, Hδ[φ] ⇀ H[φ] weakly in L2(0, T ;L2(Ω)) and thus ‖H[φ]‖L2(0,T ;L2(Ω)) ≤ lim δ→0 ‖Hδ[φ]‖L2(0,T ;L2(Ω)) ≤ K‖φ‖L2(0,T ;L2(Ω)). (4.9) ‖H[φ1]−H[φ2]‖L2(0,T ;L2(Ω)) ≤ lim δ→0 ‖Hδ[φ1]−Hδ[φ2]‖L2(0,T ;L2(Ω)) ≤ K‖φ1 − φ2‖L2(0,T ;L2(Ω)), (4.10) and we obtain 0 ≤ lim δ→0 ∫ T 0 ∫ Ω ( Hδ[φ1]−Hδ[φ2] ) (φ1 − φ2) dx dt = ∫ T 0 ∫ Ω ( H[φ1]−H[φ2] ) (φ1 − φ2) dx dt. (4.11) � Theorem 4.2. For a.e. x1, x2 ∈ Ω, we have the estimate ‖H[φ](x1, ·)−H[φ](x2, ·)‖L2(0,T ) ≤ K { ‖φ(x1, ·)− φ(x2, ·)‖L2(0,T ) } . (4.12) Proof. We consider again Hδ[φ]. Using Theorem 3.8, we have Hδ[φ](x1, t)−Hδ[φ](x2, t) = (φ(x1, t)− φ(x2, t))λG0 − ∫ Rn\G0 ∇(ŵδφ(x1, y, t)− ŵδφ(x2, y, t))∇κ(y) dy Taking the square of this identity, integrating it in t, from 0 to T , and applying Theorem 3.8, we arrive exactly to the estimate (4.12). � EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 23 Remark 4.3. To bring a connection with some previous works (specially with [18]), we now consider G0 to be given by the unit ball, G0 = {|x| < 1}. Notice that this is a small abuse of notation since G0 * Y , but this is not any problem since for any small ε > 0 we have Gjε ⊂ T jε/4 ⊂ Y jε . Hence, we are under the same conditions as above. Given φ ∈ L2(0, T ). We define the function Hφ as the unique solution to the unilateral problem H ′φ + BnHφ ≥ Bnφ, Hφ ≥ 0, t ∈ (0, T ), Hφ(H ′φ + BnHφ − Bnφ) = 0, t ∈ (0, T ), Hφ(0) = 0, where Bn = (n − 2)C−1 0 . We reformulate this problem as a variational inequality. We search for a function Hφ ∈ H1(0, T ) such that Hφ ≥ 0 on (0, T ) and satisfying the integral inequality∫ T 0 (H ′φ + BnHφ − Bnφ)(v −Hφ)dt ≥ 0, for arbitrary function v ∈ L2(0, T ), v ≥ 0 a.e. t ∈ (0, T ). It is well-known (see, e.g., [24]) that this problem has a unique solution. Moreover, it satisfies ‖Hφ‖L2(0,T ) ≤ ‖φ‖L2(0,T ) , ‖Hφ1 −Hφ2 ‖L2(0,T ) ≤ ‖φ1 − φ2‖L2(0,T ) ,∫ T 0 (Hφ1 −Hφ2)(φ1 − φ2)dt ≥ 0. Now we compare these conclusions with the ones given in Theorem 4.1. Then, if we consider φ ∈ L2(0, T ;L2(Ω)), we can understand x ∈ Ω as a parameter in the problem for Hφ and thus Hφ(x, t) is the unique solution to the problem ∂tHφ + BnHφ ≥ Bnφ(x, t), Hφ ≥ 0, (∂tHφ + Bn(Hφ − φ(x, t))Hφ = 0, Hφ(x, 0) = 0, (4.13) Now, we can use spherical symmetry properties to search for the solution of (3.19) in the form κ(r), where r is the radial coordinate. We get that κ(r) = r2−n. Hence, ∂νκ = d drκ(r) = const on ∂G0. Further, we can search for the function ŵφ, solving the problem (3.20), in the form ŵφ = r2−nHφ(x, t). A direct computation shows that ŵφ satisfies (3.20). The last conclusion is a consequence of that Hφ satisfies (4.13). Actually, we have H[φ] = (n − 2)|∂G0|(φ(x, t) − Hφ). We also notice that, in this case, λG0 = (n− 2)|∂G0| . 5. Proof of the main result Before proving Theorem 1.1 it is useful to make some remarks. Remark 5.1. As in Remark 4.3, if we consider the case in which G0 is a unit ball, G0 = {|x| < 1}, then the homogenized problem corresponding to problem (1.1) is −∆xu0 +An(u0 −Hu0 ) = f, (x, t) ∈ QT , ∂tHu0 + BnHu0 ≥ Bnu0, (x, t) ∈ QT , Hu0 ≥ 0, Hu0(∂tHu0 + Bn(Hu0 − u0)) = 0, (x, t) ∈ QT , Hu0 (x, 0) = 0, x ∈ Ω, 24 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 u0 = 0, (x, t) ∈ ∂Ω× (0, T ), where the constants are given by An = (n − 2)Cn−2 0 ωn, ωn = |∂G0| and Bn = (n − 2)C−1 0 respectively (see [18]). By using the relation between Hφ and H[φ] of Remark 4.3, we obtain exactly problem (1.3). Remark 5.2. As in the case in which the particles are radially symmetric, it is possible to prove that for suitable source negative function f(t, x), the solution of the homogenized problem (1.1) may become negative in some parts of the domain Ω, even if for each ε, the approximate solution uε is non-negative on the many points of the boundary of the particles contained in Ωε. See [18], for a detailed proof for the radial symmetric case. This unexpected property also holds in the asymmetric case but, as it is natural, with some additional difficulties. For instance, the boundedness of the solution u0 (when the datum f(t, x) is assumed to be bounded, or in some Lp(QT ) with p large enough, requires to get, previously, some L∞-estimates on ∂tw j ε,φ. This can be justified by working with Lipschitz solutions of the auxiliary unilateral problems as, for instance, in [23]. The rest of details are very similar to the symmetric case. Proof of Theorem 1.1. Let φ(x, t) = ψ(x)η(t), where ψ ∈ C∞0 (Ω), η ∈ C1([0, T ]). To adapt any test function of the limit equation to the heterogeneous original problem we introduce our last auxiliary function Wε,φ(x, t) =  wjε,φ(x, t)− (φ(P jε , t)− φ(x, t))κjε(x), if x ∈ T jε/4 \G j ε, t ∈ (0, T ), j ∈ Υε, 0, if x ∈ Rn \ ∪j∈Υε T jε/4, t ∈ (0, T ), (5.1) where κjε(x) is the unique solution to the problem ∆κjε = 0, x ∈ T jε/4 \G j ε, κjε = 1, x ∈ ∂Gjε, κjε = 0, x ∈ ∂T jε/4. (5.2) Notice that the function v = φ(x, t) −Wε,φ(x, t) is a good test function since v ∈ Kε. Indeed, if x ∈ ∂Gjε, then v(x, t) = φ(x, t)− wjε,φ(x, t) + φ(P jε , t)− φ(x, t) = φ(P jε , t)− w j ε,φ(x, t) ≥ 0. (5.3) Therefore, we can take the function v as a test function in the integral inequality of the original problem ε−γ ∫ T 0 ∫ Sε ∂tv(v − uε) ds dt+ ∫ T 0 ∫ Ωε ∇v∇(v − uε) dx dt ≥ ≥ ∫ T 0 ∫ Ωε f(v − uε) dx dt− ε−γ 2 ‖v(x, 0)‖2L2(Sε). (5.4) EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 25 Thus, we obtain ε−γ ∑ j∈Υε ∫ T 0 ∫ ∂Gj ε (∂tφ(P jε , t)− ∂tw j ε,φ(x, t))(φ(P jε , t)− w j ε,φ(x, t)− uε) ds dt + ∫ T 0 ∫ Ωε ∇φ∇(φ(x, t)−Wε,φ(x, t)− uε) dx dt − ∑ j∈Υε ∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(φ(x, t)− wjε,φ + κjε(φ(P jε , t)− φ(x, t))− uε) dx dt + ∑ j∈Υε ∫ T 0 ∫ T j ε/4 \Gj ε ∇(κjε(x)(φ(P jε , t)− φ(x, t)))∇(φ(x, t)− wjε,φ + κjε(φ(P jε , t)− φ(x, t))− uε) dx dt ≥ ∫ T 0 ∫ Ωε f(φ(x, t)−Wε,φ − uε) dx dt. (5.5) Here, we have used that v(x, 0) = 0 for x ∈ Sε. Moreover, we have − ∑ j∈Υε ∫ T 0 ∫ T j ε/4 \Gj ε ∇wjε,φ∇(φ(x, t)− wjε,φ + κjε(x)(φ(P jε , t)− φ(x, t))− uε) dx dt = − ∑ j∈Υε ∫ T 0 ∫ ∂Gj ε ∂νw j ε,φ(φ(P jε , t)− w j ε,φ − uε) ds dt − ∑ j∈Υε ∫ T 0 ∫ ∂T j ε/4 ∂νw j ε,φ(φ(x, t)− uε) ds dt. (5.6) Combining all of the integrals over ∂Gjε, j ∈ Υε, we obtain Jε ≡ ∑ j∈Υε ∫ T 0 ∫ ∂Gj ε ( ε−γ(∂tφ(P jε , t)−∂tw j ε,φ)−∂νwjε,φ ) ((φ(P jε , t)−w j ε,φ)−uε) ds dt. (5.7) Using that wjε,φ is a solution to the problem (3.1), we conclude that Jε = − ∑ j∈Υε ∫ T 0 ∫ ∂Gj ε ( ε−γ∂t(φ(P jε , t)− w j ε,φ)− ∂νwjε,φ)uε ds dt ≤ 0. (5.8) Taking into account (5.6)-(5.8), we derive from (5.5) that∫ T 0 ∫ Ωε ∇φ∇(φ−Wε,φ − uε) dx dt− ∑ j∈Υε ∫ T 0 ∫ ∂T j ε/4 ∂νw j ε,φ(φ− uε) ds dt + ∑ j∈Υε ∫ T 0 ∫ T j ε/4 \Gj ε ∇(κjε(x)(φ(P jε , t)− φ(x, t)))∇(φ(x, t)− wjε,φ + κjε(φ(P jε , t)− φ(x, t))− uε) dx dt ≥ ∫ T 0 ∫ Ωε f(φ−Wε,φ − uε) dx dt. (5.9) 26 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 If we define the function κε(x) =  κjε(x), x ∈ T jε/4 \G j ε, j ∈ Υε, 1, x ∈ Gjε, j ∈ Υε, 0, x ∈ Rn \ ∪j∈ΥεT j ε/4, (5.10) we have that κε ⇀ 0 weakly in H1 0 (Ω) and that κε → 0 strongly in L2(Ω). Using this and the estimate |φ(P jε , t)−φ(x, t)| ≤ Kε for x ∈ T jε/4, we derive that the sum of the integrals over T jε/4 \G j ε in (5.9) converges to zero as ε→ 0. Using Theorem 3.10 and results from [14] (see also the “from the surface to the volume integrals” [13, Theorem 4.5]), we obtain the limit of the integrals over the “big” balls lim ε→0 ∑ j∈Υε ∫ T 0 ∫ ∂T j ε/4 ∂νw j ε,φ(φ− uε) ds dt = lim ε→0 ∑ j∈Υε ∫ T 0 ∫ ∂T j ε/4 ∂νwφ(P jε , x− P jε aε , t)(φ− uε) ds dt = −Cn−2 0 ∫ T 0 ∫ Ω H[φ](x, t)(φ− u0) dx dt. (5.11) Thus, from (5.5)- (5.11), we conclude that u0 satisfies∫ QT ∇φ∇(φ− u0) dx dt+ Cn−2 0 ∫ QT H[φ](x, t)(φ− u0) dx dt ≥ ∫ QT f(φ− u0) dx dt (5.12) for any smooth test function φ(x, t) = ψ(x)η(t), ψ ∈ C∞0 (Ω), η ∈ C1([0, T ]). Taking into account that the linear span of functions {ψ(x)η(t) : ψ ∈ C∞0 (Ω), η ∈ C1([0, T ])} is dense in the space L2(0, T ;H1 0 (Ω)), we derive that inequality (5.12) is valid for an arbitrary function φ ∈ L2(0, T ;H1 0 (Ω)). Then we take φ = u0 ± λθ, where λ ≥ 0 and θ ∈ L2(0, T ;H1 0 (Ω)), as a test function in the inequality (5.12) and pass to the limit as λ → 0. Notice that H[u0 ± λθ] → H[u0] in L2(QT ) as λ→ 0. Combining the two limit inequalities, we conclude that u0 satisfies∫ QT ∇u0∇θ dx dt+ Cn−2 0 ∫ QT H[u0]θ dx dt = ∫ QT fθ dx dt for any θ ∈ L2(0, T ;H1 0 (Ω)). Hence, u0 satisfies the problem (1.3). The uniqueness of solutions of problem (1.3) is a trivial consequence of the monotonicity properties of the operator H[u0] (see the fourth step of the proof of [17, Theorem 3.2]). � Remark 5.3. The case of non-zero initial datum in the original problem, uε(x, 0) = u0(x) for x ∈ Sε can be also treated with the arguments of this paper. In fact, that was detailed in [18] for the case of symmetric particles. The important modification is that now the “strange operator” also depends on u0 (since u0 appears in the definition of H[u0]). We leave the details to the interested reader. EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 27 Remark 5.4. Remark that the solutions of the zero-order reactions satisfy, with a slight modification, the unilateral formulation (1.1). Indeed, the boundary con- ditions are now ∂νuε + ε−γ(∂tuε + λσ(uε)) 3 0 on STε , with σ(s) the maximal monotone graph of R2 given by σ(s) =  0 if s < 0, [0, 1] if s = 0, 1 if s > 0. We recall that in this case, automatically uε ≥ 0 on STε since uε represents a concentration. Then, we obtain that, if σS(s) is the maximal monotone graph corresponding to the Signorini type boundary conditions, then ∂νuε + ε−γ(∂tuε + λ(σS(uε) + 1) 3 0 on STε , and thus we arrive that if uε satisfies the zero-order reaction on STε then uε satis- fies also the non-homogeneous unilateral boundary conditions similar to the ones considered in this paper uε ≥ 0, ε−γ∂tuε + ∂νuε ≥ −ε−γλ, (x, t) ∈ STε , uε(ε −γ∂tuε + ∂νuε − ε−γλ) = 0, (x, t) ∈ STε . It is easy to see that the arguments of this paper can be adapted to the case in which there is a non-homogeneous boundary data g(x) = −ε−γλ (see, for instance, the exposition made in [13, Section 4.7.1] for the symmetric case and stationary boundary conditions). We leave the details to the interested reader. Remark 5.5. In the case n = 2 the critical case must be written in a different way (see, the general exposition made in [13, Section 4.7.2]). In fact, it can be shown (see [19]) that in that case it is possible to prove an universal homogenized non- local problem when the particles Gε have different geometrical shapes but having the same perimeter on their boundary. Acknowledgements. J. I. Dı́az was partially supported by the project PID2020- 112517GB-I00 of the Spanish State Research Agency (AEI) and by the Research Group MOMAT (Ref. 910480) of the UCM. References [1] Anguiano, M.; Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media, Mediterranean Journal of Mathe- matics 17 1 (2020), 18 [2] Arrieta, M.; Quittner, P.; Rodriguez-Bernal, A.; Parabolic problems with nonlinear dynamical boundary conditions and singular initial data, Diff. and Int. Eq., 14 12, (2011), 1487-1510. [3] Attouch, H.; Picard, C.; Variational inequalities with varying obstacles: the general form of the limit problem, J. Funct. Anal., 50 3 (1983), 329-386. [4] Bandle, C.; Below, J.; Reichel, W.; Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up, Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni, (2006), 35-67. [5] Bejenaru, I.; Dı́az, J. I.; Vrabie, I. I.; An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamical boundary conditions, Electronic Journal of Differential Equations 2001, 50 (2001), 1-19. 28 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, A. V. PODOLSKIY EJDE-2024/03 [6] Cecere, G.; König, Ch. M.; Alleva, J. L.; MacMillan, D. W. C.; Enantioselective direct α-amination of aldehydes via a photoredox mechanism: A strategy for asymmetric amine fragment coupling, J. Am. Chem. Soc. 135 31 (2013), 11521–11524. [7] Cioranescu, D.; Murat, F.; Un terme étrange venu d’ailleurs, Nonlinear Partial Diff. Eq. and their Applications, V. II, College de France Seminar, Paris, France. Ed. by H. Brezis and J. L. Lions, Research Notes in Mathematics. London: Pitman, 60, 1982, 98-138. [8] Conca, C.; Dı́az, J. I.; A. Liñán, A.; Timofte, C.; Homogenization in Chemical Reactive Flows. Electronic Journal of Differential Equations 2004, 40 (2004), 1-22. [9] Conca, C.; Murat F.; Timofte, C.; A generalized strange term Signorini’s type problems, ESAIM:Math. Modeling and Numerical Analysis, 3, 57, (2003), 773-805. [10] Dı́az J. I.; Gómez-Castro, D.; Podolskii, A. V.; Shaposhnikova, T. A.; Homogenization of variational inequalities of Signorini type for the p-Laplacian in perforated domains when p ∈ (1, 2), Dokl. Math., 95 2 (2017), 151-156. [11] Dı́az, J. I.; Gómez-Castro, D.; Podol’skiy, A. V.; Shaposhnikova, T. A.; Characterizing the strange term in critical size homogenization: quasilinear equations with a nonlinear boundary condition involving a general maximal monotone graph. Advances in Nonlinear Analysis, 8 (2019), 679–693. [12] Dı́az J. I., Gómez-Castro D., Podolskiy A. V., and Shaposhniova T. A., Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis nanocomposite membranes, Adv. Nonlinear Anal., 9 (2020), 193-227. [13] Dı́az, J. I.; Gómez-Castro, D.; Shaposhnikova, T. A.; Nonlinear Reaction-Diffusion Processes for Nanocomposites. Anomalous improved homogenization, De Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2021. [14] Dı́az, J. I.; Gómez-Castro, D.; Shaposhnikova, T. A.; Zubova, M. N.; Change of homoge- nized absorption term in diffusion processes with reaction on the boundary of periodically distributed asymmetric particles of critical size, Electronic Journal of Differential Equations, 2017 (2017), No. 178. 1-25. [15] Dı́az, J. I.; Gomez-Castro, D.; Shaposhnikova, T. A.; Zubova, M. N.; A nonlocal memory strange term arising in the critical scale homogenization of a diffusion equation with a dynamic boundary condition, Electronic Journal of Differential Equations, 2019, 77 (2019), 1-13. [16] Dı́az, J. I.; Gómez-Castro, D.; Timofte, C.; The effectiveness factor of reaction-diffusion equations: homogenization and existence of optimal pellet shapes, Journal of Elliptic and Parabolic Equations, 2 1 (2016), 119-129. [17] Dı́az, J. I.; Shaposhnikova, T. A.; Zubova, A.; Strange non-local monotone operator arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape, Electronic Journal of Differential Equations, 2022, 52 (2022),1–32. [18] Dı́az, J .I.; Podolskiy, A. V.; Shaposhnikova, T. A.; Unexpected regionally negative solutions of the homogenized problem associated with the Poisson equation and dynamic unilateral boundary conditions: the case of symmetric particles of critical size. Revista de la Real Academia de Ciencias Exactas, F́ısicas y Naturales. Serie A. Matemáticas, RACSAM, (2023) DOI 10.1007/s13398-023-01503-w. [19] Dı́az, J. I.; Podolskiy, A. V.; Shaposhnikova, T. A.; Aperiodical isoperimetric planar homog- enization with critical diameter: universal non-local strange term for a dynamical unilateral boundary condition. To appear in Dokl. Math. [20] Dı́az, J.I.; Shaposhnikova, T.A.; Zubova, M.N.; A strange non-local monotone operator aris- ing in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape, Electronic Journal of Differential Equations 2022, 52 (2022), 1-32. [21] Escher, J.; Quasiliniar parabolic systems with dynamical boundary conditions, Comm. Par- tial Differential Equations 18 (1993), 1309-1364. [22] Gómez, D.; Lobo, M.; Pérez, E.; Sánchez-Palencia, E.; Homogenization in perforated do- mains: a Stokes grill and an adsorption process, Applicable Analysis, 97, 16 (2018), 2893– 2919. [23] Kinderlehrer, D.; Stampacchia, G.; An Introduction to Variational Inequalities and Their Applications, Society for Industrial and Applied Mathematics, 2000. [24] Lions, J.-L.; Quelques méthodes de ré solution des problèmes aux limites non linéaires, Dunod, Paris, 1968. EJDE-2024/03 STRANGE NONLOCAL ASYMMETRIC HOMOGENIZATION 29 [25] Oleinik, O. A.; Shaposhnikova, T. A.; On homogenization problem for the Laplace operator in partially perforated domain with Neumann condition on the boundary of cavities, Rend. Mat. Acc. Lincei, 6, 9 (1995), 133-142. [26] Iliev, O.; Mikelic, A.; Prill, T.; Sherly. A.; Homogenization approach to the upscaling of a reactive flow through particulate filters with wall integrated catalyst, Adv. Water Resour, 146 (2020), 103779. [27] Podolskii, A. V.; Shaposhnikova, T. A.; Homogenization of the boundary-value problem for the Laplace operator in a domain perforated along (n − 1)-dimensional manifold with nonlinear Robin type boundary condition on the boundary of arbitrary shaped holes: critical case, Dokl. Math., 96, 3 (2017), 601-606. [28] Schimpf, S.; Lucas, M.; Mohr, C.; Rodemerck, U.; Brückner, A.; Radnik, J.; Hofmeister, H.; Claus, P.; Supported gold nanoparticles: in-depth catalyst characterization and application in hydrogenation and oxidation reactions, Catal. Today, 72, 1 (2002), 63–78. [29] Timofte, C.; Parabolic problems with dynamical boundary conditions in perforated media, Math. Modelling and Analysis, 8 (2003), 337-350. [30] Zubova M.N., Shaposhnikova T.A., Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem, Diff. Eq., 47 1 (2011), 1-13. [31] Zubova, M .N.; Shaposhnikova, T. A.; Homogenization of a boundary value problem in a domain perforated by cavities of arbitrary shape with nonlinear boundary condition on their boundaries: the case of critical values of the parameters, Journal of Mathematical Sciences, 244, 2 (2020), 235-253. [32] Zubova, M. N.; Shaposhnikova, T. A.; Appearance of a nonlocal monotone operator in trans- mission conditions when homogenizing the Poisson equation in a domain preforated along an n−1-dimensional manifold by sets of arbitrary shape and critical size with nonlinear dynamic boundary conditions on the boundary of perforations, Journal of Mathematical Sciences, 255, 4 (2021), 423-443. Jesús Ildefonso D́ıaz Instituto de Mathematica Interdisciplinar, Universidad Complutense de Madrid, Spain Email address: jidiaz@ucm.es Tatiana A. Shaposhnikova Lomonosov Moscow State University, Moscow, Russia Email address: shaposh.tan@mail.ru Alexander V. Podolskiy Lomonosov Moscow State University, Moscow, Russia Email address: avpodolskiy@yandex.ru 1. Introduction 2. Statement of the problem and a priori estimates of solutions 3. Adaptation of global test functions 3.1. First auxiliary problem: a different unilateral problem in the macroscopic variables 3.2. Second auxiliary problem: a different unilateral exterior problem on the cell variables 3.3. Asymptotic similarity between the two types of auxiliary functions after a suitable substitution 4. Definition and properties of the strange non-local operator H[] 5. Proof of the main result Acknowledgements References