Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 52, pp. 1–32. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu 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 JESÚS ILDEFONSO DÍAZ, TATIANA A. SHAPOSHNIKOVA, MARIA N. ZUBOVA Abstract. We characterize the homogenization limit of the solution of a Pois- son equation in a bounded domain, either periodically perforated or containing a set of asymmetric periodical small particles and on the boundaries of these particles a nonlinear dynamic boundary condition holds involving a Hölder nonlinear σ(u). We consider the case in which the diameter of the perfora- tions (or the diameter of particles) is critical in terms of the period of the structure. As in many other cases concerning critical size, a “strange” non- linear term arises in the homogenized equation. For this case of asymmetric critical particles we prove that the effective equation is a semilinear elliptic equation in which the time arises as a parameter and the nonlinear expression is given in terms of a nonlocal operator H which is monotone and Lipschitz continuous on L2(0, T ), independently of the regularity of σ. 1. Introduction The main goal of this article is to extend previous papers in the literature dealing with the homogenization of a Poisson equation in a bounded domain, which we can assume either periodically perforated or containing a set of asymmetric periodical small particles, and on the internal boundaries a nonlinear dynamic boundary con- dition holds involving a Hölder continuous nonlinearity and some small parameters. In contrast to the case in which the diameter of the perforation (or the diameter of the particles) is equal to the period of the structure (see, e.g., [1, 29]) when the involved parameters are in a suitable balance with a “ critical value” of the diameters then a new (and thus “strange” in the spirit of [8, 22, 23]) nonlocal term arises in the homogenized equation. That was shown in our previous paper [15] but merely for the case of a linear boundary condition and for symmetrical balls as perforations or particles (see also [33] for the case of Lipschitz nonlinear terms and nonhomogeneous boundary conditions). The more general case (Hölder continuous nonlinear terms and, which is more important, cavities, or particles, of arbitrary shape) leads to new difficulties which require a different framework: the “strange term” is now given by a nonlocal monotone operator which, curiously enough, 2020 Mathematics Subject Classification. 35B27, 35K57, 35K91, 35R01, 47B44. Key words and phrases. Critically scaled homogenization; asymmetric particles; dynamic boundary conditions; Hölder continuous reactions; strange term; nonlocal monotone operator. ©2022. This work is licensed under a CC BY 4.0 license. Submitted May 8, 2022. Published July 18, 2022. 1 2 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 regularizes the nonlinearity (for instance, even if the nonlinearity involved in the dynamic boundary condition is merely Hölder continuous the strange operator is a L2(0, T )-Lipschitz continuous operator. Our results also extend the treatment made in [11] for asymmetric particles but with Robin type boundary conditions (a problem which can be understood as the associate stationary problem associ- ated with the evolution problem considered in the present paper). We point out that, in some sense, the assumption on critical values in the relation between size and distance, giving rise to a different reaction behavior is typical of many pro- cesses in Nanotechnology. New materials, in particular the so-called “Mechanical meta-materials” , are built as artificial structures which have mechanical properties defined by their geometric structure rather than their chemical composition. The occurrence of “strange terms” in the homogenized equation can be understood as a similar process to the design of new materials having properties outside the scope found in Nature. 2. Problem statement Let Ω be a bounded domain in Rn, n ≥ 3, with Lipchsiz boundary ∂Ω. In the cube Y = (−1/2, 1/2)n consider a subdomain G0, G0 ⊂ Y , which, for simplicity, we assume that is star shaped with respect to a ball T 0 ρ ⊂ G0 of radius ρ with the center at the origin. Out treatment remains valid if G0 has a finite number of disjoint connect components satisfying the same geometric property (see Remark 5). Let δB = {x : δ−1x ∈ B}, δ > 0. For ε > 0 let 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. Consider the set Gε = ∪j∈Υε(aεG0 + εj) = ∪j∈ΥεG j ε, where Υε = {j ∈ Zn : Gjε ⊂ Y jε = εY + εj,Gjε ∩ Ω̃ε 6= ∅}. We assume that aε = C0ε γ , for some γ > 1 and C0 > 0. (2.1) It is easy to see that |Υε| ∼= dε−n, d = const > 0. Note that Gjε ⊂ T jCaε ⊂ T j ε/4 ⊂ Y j ε , where T jr is the 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 on ε. We introduce the sets Ωε = Ω \Gε, Sε = ∂Gε, ∂Ωε = Sε ∪ ∂Ω, QTε = Ωε × (0, T ), STε = Sε × (0, T ), ΓT = ∂Ω× (0, T ). The main problem considered in this paper deals with the following case of nonlinear dynamic boundary conditions −∆xuε = f(x, t), (x, t) ∈ QTε , ε−γ∂tuε + ∂νuε + ε−γσ(uε) = ε−γg(x, t), (x, t) ∈ STε , uε(x, t) = 0, (x, t) ∈ ΓT , uε(x, 0) = 0, x ∈ Sε, (2.2) EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 3 where γ = n n− 2 , (2.3) ν is the unit outward normal to the boundary STε , ∂νuε is the normal derivative of uε, and, for simplicity in the exposition we assume f ∈ H1(0, T ;L2(Ω)), (2.4) g ∈ L2(0, T ;C(Ω)). (2.5) Problem (2.2) arises in many different contexts (see, e.g., the exposition made in [3, 15, 20] and their many references). The nonlinear term σ(uε) represents, in some models, the chemical reaction on the boundary of the particles. It is well known that a relevant choice of this term is the function given by σ(u) = σ0u α, for some positive constant σ0 and where α ∈ [0, 1] represents the “order of the chemical reaction” (see, e.g. [10]). Motivated by this, we assume that σ is a Hölder continuous function (with a Lipschiz behavior for large values of u), σ : R → R is nondecreasing, σ(0) = 0, and |σ(s)− σ(t)| ≤ K1|s− t|α +K2|s− t| ∀s, t ∈ R and for some 0 < α ≤ 1, (2.6) Here K1, K2 are positive constants. We recall that a function uε ∈ C([0, T ] : L2(Sε)) is a strong solution of (2.2) if uε ∈ L2(0, T ;H1(Ωε, ∂Ω)), ∂tuε ∈ L2(0, T ;L2(Sε)) and σ(uε) ∈ L2(0, T ;L2(Sε)), such that ε−γ ∫ STε ∂tuεv ds dt+ ∫ QTε ∇uε∇v dx dt+ ε−γ ∫ STε σ(uε)v ds dt = ε−γ ∫ STε gv ds dt+ ∫ QTε fv dx dt, (2.7) for all v ∈ L2(0, T ;H1(Ωε, ∂Ω)), and the initial condition uε(x, 0) = 0 holds for x ∈ Sε. Here H1(Ωε, ∂Ω) denotes the Hilbert space obtained as the closure, with the norm H1(Ωε), of the set of all φ ∈ C∞(Ω) such that φ = 0 in a neighborhood of ∂Ω. Notice that if we denote by H1/2(Sε, ∂Ω) to the space of traces on Sε of a function from H1(Ωε, ∂Ω), with the norm ‖v‖H1/2(Sε,∂Ω) = inf w∈H1(Ωε,∂Ω) {‖w‖H1(Ωε,∂Ω) : w ∣∣ Sε = v}, then strong solutions are more regular than other type of weak solutions satisfying merely that ∂tuε ∈ H−1/2(Sε, ∂Ω), where H−1/2(Sε, ∂Ω) is the dual of the space H1/2(Sε, ∂Ω). Our first result concerns the existence and uniqueness of strong solutions of problem (2.2) under the assumption of Hölder continuity on σ. Theorem 2.1. Assume σ Hölder continuos as well the rest of the above assump- tions. Then for any ε > 0, problem (2.2) has a unique strong solution, satisfying the estimates ‖uε‖2L2(0,T ;H1(Ωε,∂Ω)) + ε−γ‖uε‖2L2(0,T ;L2(Sε)) ≤ K(‖f‖2L2(0,T ;L2(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ), ε−γ‖∂tuε‖2L2(0,T ;L2(Sε)) ≤ K ( ‖f‖2H1(0,T ;L2(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) + ‖f‖2αL2(0,T ;L2(Ω)) + ‖g‖2α L2(0,T ;C(Ω)) ) , (2.8) 4 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 where K is a positive constant independent on ε, f and g. The proof of this Theorem will be given below by means of Galerkin expan- sion arguments. We recall that, by [24], it is well-known the existence of 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 K > 0 is independent of ε. Then, by using the estimate from Theorem 2.1, we conclude that ‖Pεuε‖L2(0,T ;H1 0 (Ω)) ≤ K, as ε→ 0. Therefore, for some subsequence (still denoted by uε), we have Pεuε ⇀ u0, weakly in L2(0, T ;H1 0 (Ω)). (2.9) The main goal of our research is to characterize the function u0 in terms of a homogenized boundary value problem on Ω for nonlinear terms σ assumed to be in the same class of the existence of solutions. To do that we will argue without passing by a regularizing of σ followed by a passing to the limit in such a process. This is in contrast with many other papers for related problems (see, e.g. [11] and its references). 3. Statement of main results In the case of pure Robin type boundary conditions, f(x, t) = f(x), and when the diameter of the perforation (or the diameter of the particles) is equal to the period of the structure (γ = 1 in (2.3)), it can be proved (see [10]) that u0 satisfies a semilinear equation of the type −Lu0 +Cσ(u0) = f(x) in Ω, for a suitable second order elliptic linear operator L and a suitable constant C > 0. As mentioned before we are interested in the characterization of u0 when the critical size condition (2.3) holds in presence of dynamic nonlinear boundary conditions. A “strange term” (replacing the above nonlinear term Cσ(x, u0)) arises in the homogenized semilinear equation satisfied by u0, in the spirit of the series of previous works for different type of boundary conditions (see, e.g., [8, 9, 19, 22, 23] and the monograph [14]). When G0 is a symmetric ball, it was shown in [15, 33] that this strange term is nonlocal in time and contains some memory expressions. In this paper we will consider the general asymmetric case on G0 and we will prove that the correspondent strange term is now given by a monotone operator H : L2(0, T )→ L2(0, T ). As a matter of facts, we will apply this operator to functions which are also x−dependent, where x ∈ Ω, but concerning the operator H the variable x plays the role of a parameter, so that we can also understand that the above mentioned monotone operator can be understood as H : L2(0, T ;L2(Ω)) → L2(0, T ;L2(Ω)). To be more precise, for any function φ ∈ L2(0, T ;L2(Ω)), we define the value of H[φ] by its pointwise application, for a.e. (x, t) ∈ Ω× (0, T ): H[φ](x, t) = φ(x, t)λG0 − ∫ ∂G0 ∂νw̃φ(x, y, t)dsy, (3.1) EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 5 where w̃φ(x, y, t) satisfies the G0-capacity type exterior nonlinear auxiliary problem ∆yw̃φ = 0, Rn \G0 × (0, T ), C0∂tw̃φ + ∂νw̃φ + C0σ(w̃φ) = C0g(x, t) + φ(x, t)∂νκ(y), ∂G0 × (0, T ), w̃φ(x, y, 0) = 0, y ∈ ∂G0, w̃φ(x, y, t)→ 0, |y| → +∞, t ∈ (0, T ), (3.2) with κ(y) the standard G0-capacity exterior problem ∆yκ(y) = 0, y ∈ Rn \G0, κ(y) = 1, y ∈ ∂G0, κ(y)→ 0, |y| → +∞, (3.3) and where λG0 is the capacity of the set G0 given by λG0 = ∫ ∂G0 ∂νκ(y)dsy. Notice that we are denoting by y to the variable in the domain Rn \ G0 and that function w̃φ(x, y, t) depends, in a crucial way, not only of the values of φ(t, ·) but also of the data G0, σ and g, so that, a more significative notation for this operator H[φ] would be H[φ] = HG0,σ,g[φ]. Nevertheless, for the sake of simplicity in the notation, we avoid such a sophisticated notation. Notice that in the definition of H[φ] the variable x ∈ Ω plays the role of a parameter (since the partial differential problems (3.2) and (3.3) are formulated in terms of the y ∈ Rn \G0 variable). We point out that although κ(y) = 1 on ∂G0 (and then there is no direct influence of κ on the definition of operator H), the y−decay of κ(y) allows a better manageability of the function w̃φ(x, y, t). We shall prove in later that the problem (3.2) has a unique weak solution and that H[φ] ∈ L2(0, T ;L2(Ω)) for any φ ∈ L2(0, T ;L2(Ω)). As a matter of fact, we shall prove that H is Lipschitz continuous operator, independently of the regularity of σ. Remark 3.1. When G0 is a ball, G0 = {|y| < 1} (which we will also denote as G0 = T 0 1 ), then κ(y) = |y|2−n and the solution of problem (3.2) is explicitly given as w̃φ(x, y, t) = Hφ(x, t) |y|n−2 , where Hφ(x, t) for any x ∈ Ω is the unique solution of nonlinear Cauchy problem ∂tHφ + n− 2 C0 Hφ + σ(Hφ) = n− 2 C0 φ(x, t) + g(x, t), Hφ(x, 0) = 0. (3.4) where Hφ is the unique solution (according to [5]) of the Cauchy problem associated with the maximal monotone graph given through function σ. A similar expression can be also found for the case σ = σ(x, φ), when x ∈ Ω is taken as a parameter. In the linear case σ(φ) = λφ, for some λ > 0, we obtain Hφ(x, t) = ∫ t 0 (n− 2 C0 φ(x, s) + g(x, s) ) exp { −(λ+ n− 2 C0 )(t− s) } ds. 6 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 Therefore, H[φ](x, t) = (n− 2)ωn(φ(x, t)−Hφ(x, t)), with ωn = |∂T 0 1 |, the area of the unit sphere. In Section 7 we shall prove some properties of the “strange operator” H: it is a monotone operator (see Theorem 9), in the sense that for any φ1, φ2 ∈ L2(0, T ),∫ T 0 (H[φ1]−H[φ2])(φ1 − φ2)dt ≥ 0. (3.5) We will prove also that H satisfies the growth relation (Theorem 6.5) ‖H[φ]‖L2(0,T ) ≤ K(‖φ‖L2(0,T ) + ‖g‖L2(0,T )), (3.6) for any φ ∈ L2(0, T ) and that, in addition, H is a Lipschiz continuous operator on L2(0, T ), in the sense that ‖H[φ1]−H[φ2]‖L2(0,T ) ≤ K‖φ1 − φ2‖L2(0,T ). (3.7) for any φ1, φ2 ∈ L2(0, T ) and for a suitable constant K > 0. In addition, H is a Lipschiz continuous operator on L2(0, T ) with respect to g , i.e. ‖H[φg1 ]−H[φg2 ]‖L2(0,T ) ≤ K‖g1 − g2‖L2(0,T ). (3.8) It will be very useful to get a stronger regularity on the operator H[φ] under some more regularity on the datum g on the boundary: g ∈ L2(0, T ;W 1,∞(Ω)), (3.9) i.e, there exists L ∈ L2(0, T ) with L(t) > 0 such that |g(x1, t)− g(x2, t)| ≤ L(t)|x1 − x2|, ∀x1, x2 ∈ Ω, and for a.e. t ∈ (0, T ). (3.10) Remember that by Rademacher’s theorem W 1,∞(Ω) = Lip(Ω). This will be used as an intermediate step of the proof of the main result concerning of the function u0 defined in (2.9). Theorem 3.2. Let n ≥ 3, aε = C0ε γ , γ = n n−2 , and assume (2.6). Let uε be the strong solution of problem (2.2). Then the function u0 ∈ L2(0, T ;H1 0 (Ω)) is the unique weak solution of the family of problems (depending on the parameter t ∈ (0, T )), −∆xu0(x, t) + Cn−2 0 H[u0](x, t) = f(x, t), x ∈ Ω, t ∈ (0, T ), u0(x, t) = 0, x ∈ ∂Ω, t ∈ (0, T ), (3.11) with H[u0](x, t) defined by the nonlocal operator (3.1) for a.e. x ∈ Ω. The proof of this theorem follows the master lines of the so called “Tartar’s oscil- lating test functions method” (see, e.g., [28]), nevertheless many quite sophisticated variants must be introduced for its application to the problem under consideration. We send the reader to the general exposition made in the monograph [14] for several different problems. EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 7 4. Proof of Theorem 3.2 As usual, we can prove the uniqueness of the strong solution of (2.2) using the integral identity in the definition of solution and then applying the monotonicity of the function σ. For proving the existence, we consider the auxiliary problem δ∂tu δ ε −∆uδε = f(x, t), (x, t) ∈ QTε , ε−γ∂tu δ ε + ∂νu δ ε + ε−γσ(uδε) = ε−γg(x, t), (x, t) ∈ STε , uδε(x, t) = 0, (x, t) ∈ ΓT , uδε(x, 0) = 0, x ∈ Ωε, uδε(x, 0) = 0, x ∈ Sε, (4.1) where δ > 0 is a small parameter. We use the Galerkin method to prove the exis- tence of weak solution of this problem. By a weak solution we mean a function uδε in L2(0, T ;H1(Ωε, ∂Ω)), with ∂tu δ ε ∈ L2(0, T ;H−1(Ωε, ∂Ω)), ∂tu δ ε ∈ L2(0, T ;H−1/2(Sε, ∂Ω)), uδε(x, 0) = 0 for x ∈ Ωε, u δ ε(x, 0) = 0, x ∈ Sε, and such that the following integral identity holds for any test function v ∈ L2(0, T ;H1(Ωε, ∂Ω)), δ ∫ T 0 〈∂tuδε, v〉Ωεdt+ ε−γ ∫ T 0 〈∂tuδε, v〉Sεdt+ ∫ QTε ∇uδε∇v dx dt + ε−γ ∫ T 0 ∫ Sε σ(uδε)v ds dt = ε−γ ∫ T 0 ∫ Sε g(x, t)v(x, t) ds dt+ ∫ QTε fv dx dt. (4.2) Here, we have denoted by 〈·, ·〉Ωε and 〈·, ·〉Sε the duality relations between the spaces H1(Ωε, ∂Ω) and H−1(Ωε, ∂Ω), and H1/2(Sε, ∂Ω) and H−1/2(Sε, ∂Ω), respectively. In fact, we will prove that the solution of (4.1) is more regular. Namely, uδε is a strong solution in the sense that uδε ∈ C([0, T ];L2(Ωε)), u δ ε ∈ C([0, T ];L2(Sε)), uδε ∈ L2(0, T ;H1(Ωε, ∂Ω)), ∂tu δ ε ∈ L2(0, T ;L2(Ωε)), ∂tu δ ε ∈ L2(0, T ;L2(Sε)), uδε(x, 0) = 0, if x ∈ Ωε ∪ Sε and the integral identity δ ∫ QTε ∂tu δ εv dx dt+ ∫ QTε ∇uδε∇v dx dt+ ε−γ ∫ STε ∂tu δ εv ds dt + ε−γ ∫ STε σ(uδε)v ds dt = ε−γ ∫ STε g(x, t)v(x, t) ds dt+ ∫ QTε fv dx dt (4.3) holds for any function v ∈ L2(0, T ;H1(Ωε, ∂Ω)). We introduce the space Hδ ε = L2(Ωε)× L2(Sε) with the scalar product ((u, ũ), (v, ṽ))Hδε = δ(u, v)L2(Ωε) + ε−γ(ũ, ṽ)L2(Sε). Let Vε = {(u, u|Sε) : u ∈ H1(Ωε, ∂Ω)}, where u|Sε is the trace of function u ∈ H1(Ωε, ∂Ω) on Sε. On Vε we introduce the norm ‖(v, v ∣∣ Sε )‖2Vε = ‖v‖2H1(Ωε,∂Ω) + ‖v ∣∣ Sε ‖2H1/2(Sε,∂Ω). 8 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 It is easy to see that Vε is a reflexive separable Banach space dense in the space Hδ ε , so that the linear span of the basis is dense in Vε. In H1/2(Sε, ∂Ω) we use the following norm ‖v‖H1/2(Sε,∂Ω) = inf{‖g‖H1(Ωε,∂Ω) : g ∣∣ Sε = v}. We denote by {(wmε , wmε ∣∣ Sε )}∞m=1 the orthonormal basis for the space Hδ ε , so that the linear span of the this basis is dense in Vε. Let us apply the Galerkin method to prove the existence of a weak solution to the problem (2.2). Let us start by showing the existence of (uδ,mε , uδ,mε ∣∣ Sε ) ∈ C([0, T ];Vε), such that ∂t(u δ,m ε , uδ,mε ∣∣ Sε ) ∈ L2(0, T ;Hδ ε ) and (∂t(u δ,m ε , uδ,mε ∣∣ Sε ), (wlε, w l ε ∣∣ Sε ))Hδε + (∇uδ,mε ,∇wlε)L2(Ωε,∂Ω) + ε−γ(σ(uδ,mε ∣∣ Sε ), wlε ∣∣ Sε )L2(Sε) = ε−γ(g, wlε ∣∣ Sε )L2(Sε) + (f, wlε)L2(Ωε), (4.4) where l = 1, . . . ,m, and (uδ,mε , uδ,mε ∣∣ Sε )(x, 0) = 0. (4.5) We look for (uδ,mε , uδ,mε ∣∣ Sε ) in the form (uδ,mε , uδ,mε ∣∣ Sε ) = m∑ l=1 Clε(t)(w l ε, w l ε ∣∣ Sε ). The coefficients Clε, l = 1, . . . ,m, are found as solutions of the system (4.4),(4.5) which is a Cauchy problem for a system of nonlinear ordinary differential equations on Clε. By well known results, this problem has a unique global solution defined on the whole time-interval [0, T ]. Notice that although σ is merely Hölder continuous, the uniqueness of solution is consequence of the monotonicity assumption made on σ. From (4.4) we obtain δ 2 ‖uδ,mε (x, τ)‖2L2(Ωε) + ε−γ 2 ‖uδ,mε (x, τ)‖2L2(Sε) + ∫ τ 0 ‖∇uδ,mε ‖2L2(Ωε) dt + ε−γ ∫ τ 0 ∫ Sε σ(uδ,mε )uδ,mε ds dt = ε−γ ∫ τ 0 (g, uδ,mε )L2(Sε)dt+ ∫ τ 0 (f, uδ,mε )L2(Ωε)dt. Using that σ(u)u ≥ 0, and applying Friedrich’s inequality we obtain δ‖uδ,mε (x, τ)‖2L2(Ωε) + ε−γ‖uδ,mε (x, τ)‖2L2(Sε) + ∫ τ 0 ‖∇uδ,mε ‖2L2(Ωε) dt ≤ K(‖f‖2L2(0,T ;L2(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ) + ε−γ ∫ τ 0 ‖uδ,mε ‖2L2(Sε) dt, (4.6) where the constant K depends only on the domain Ω. From Gronwall’s lemma we deduce the explicit estimate ε−γ ess sup[0,T ] ‖uδ,mε ‖2L2(Sε) ≤ K(‖f‖2L2(0,T ;L2(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ). (4.7) Using (4.6) and (4.7) we have ess sup[0,T ] ‖(uδ,mε , uδ,mε ∣∣ Sε )‖2Hδε + ∫ T 0 ‖∇uδ,mε ‖2L2(Ωε) dt ≤ K(‖f‖2L2(0,T ;L2(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ). (4.8) EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 9 Similarly, by multiplying by ∂tu δ,m ε we obtain δ ∫ τ 0 ‖∂tuδ,mε ‖2L2(Ωε) dt+ ε−γ ∫ τ 0 ‖∂tuδ,mε ‖2L2(Sε) dt+ 1 2 ‖∇uδ,mε (x, τ)‖2L2(Ωε) + ε−γ ∫ τ 0 ∫ Sε σ(uδ,mε )∂tu δ,m ε ds dt = ∫ τ 0 ∫ Ωε f∂tu δ,m ε dx dt+ ε−γ ∫ τ 0 ∫ Sε g(x, t)∂tu δ,m ε ds dt. (4.9) Thanks to (2.4) we have∫ τ 0 ∫ Ωε f∂tu δ,m ε dx dt = − ∫ τ 0 ∫ Ωε ∂tfu δ,m ε dx dt+ ∫ Ωε f(x, τ)uδ,mε (x, τ)dx. (4.10) Using assumption (2.6) on σ, and applying the Hölder inequality, we obtain ε−γ ∫ τ 0 ∫ Sε |σ(uδ,mε )||∂tuδ,mε | ds dt ≤ Cβε−γ ∫ τ 0 ‖σ(uδ,mε )‖2L2(Sε) dt+ βε−γ ∫ τ 0 ‖∂tuδ,mε ‖2L2(Sε) dt ≤ C1,βε −γ ∫ τ 0 ∫ Sε |uδ,mε |2αdsdt+ C2,βε −γ ∫ τ 0 ‖uδ,mε ‖2L2(Sε) dt + βε−γ ∫ τ 0 ‖∂tuδ,mε ‖2L2(Sε) dt. (4.11) Then, applying again Hölder inequality, we have ε−γ ∫ τ 0 ∫ Sε |uδ,mε |2α ds dt ≤ ε−γ (∫ τ 0 ∫ Sε |uδ,mε |2 dx dt )α |Sε|1−α ≤ Kεγ(α−1) ( ε−γ ∫ τ 0 ∫ Sε |uδ,mε |2 ds dt )α εγ(1−α) ≤ K(‖f‖2αL2(0,T ;L2(Ω)) + ‖g‖2α L2(0,T ;C(Ω)) ), (4.12) where the constant K does not depend on f and g, on m, δ, ε. From (4.8), (4.11) and (4.12) we conclude that ε−γ ∫ τ 0 ∫ Sε |σ(uδ,mε )||∂tuδ,mε | ds dt ≤ Kβ ( (‖f‖2αL2(0,T ;L2(Ω)) + ‖f‖2L2(0,T ;L2(Ω))) + (‖g‖2α L2(0,T ;C(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ) ) + βε−γ ∫ τ 0 ‖∂tuδ,mε ‖2L2(Sε) dt, (4.13) where β is an arbitrary positive constant. Setting β = 1/2, from (4.9), (4.10), (4.13), we obtain δ ∫ T 0 ‖∂tuδ,mε ‖2L2(Ωε) dt+ ε−γ ∫ T 0 ‖∂tuδ,mε ‖2L2(Sε) dt+ ess sup[0,T ] ‖∇uδ,mε ‖2L2(Ωε) ≤ K ( ‖f‖2H1(0,T ;L2(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) + ‖f‖2αL2(0,T ;L2(Ω)) + ‖g‖2α L2(0,T ;C(Ω)) ) . (4.14) 10 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 Consequently, from (4.8) and (4.14), we derive that there exists a subsequence such that as m→∞, uδ,mε ⇀ uδε, weakly in L2(0, T ;H1(Ωε, ∂Ω)), ∂tu δ,m ε ⇀ ∂tu δ ε, weakly in L2(0, T ;L2(Sε)), ∂tu δ,m ε ⇀ ∂tu δ ε, weakly in L2(0, T ;L2(Ωε)), uδ,mε → uδε, in L2(0, T ;L2(Sε)). From (4.4) we deduce that uδε satisfies the integral identity δ ∫ T 0 ∫ Ωε ∂tu δ εφdx dt+ ε−γ ∫ T 0 ∫ Sε ∂tu δ εφds dt+ ∫ T 0 ∫ Ωε ∇uδε∇φdx dt + ε−γ ∫ T 0 ∫ Sε σ(uδε)φds dt = ε−γ ∫ T 0 ∫ Sε g(x, t)φds dt+ ∫ T 0 ∫ Ωε fφ dx dt, (4.15) where φ is an arbitrary test function in L2(0, T ;H1(Ωε, ∂Ω)). Let us estimate the limit value of the term ε−γ ∫ T 0 ∫ Sε σ(uδ,mε )φds dt, as m→∞. By using the Hölder continuity of σ (2.6) we have ε−γ ∫ T 0 ∫ Sε |σ(uδ,mε )− σ(uδε)||φ| ds dt ≤ K ( ε−γ ∫ T 0 ∫ Sε |uδ,mε − uδε||φ| ds dt+ ε−γ ∫ T 0 ∫ Sε |uδ,mε − uδε|α|φ| ds dt ) ≤ K ( ε− γ(1+α) 2 ‖uδ,mε − uδε‖αL2(0,T ;L2(Sε)) + ε−γ‖uδ,mε − uδε‖L2(0,T ;L2(Sε)) ) ‖φ‖L2(0,T ;L2(Sε)). (4.16) Thus as m→∞ the right hand side of (4.16) tends to zero. For the solution uδε we have similar estimates to the ones given in (4.8) and (4.14) ess sup[0,T ] ‖(uδε, uδε ∣∣ Sε )‖2Hδε + ∫ T 0 ‖∇uδε‖2L2(Ωε) dt ≤ K(‖f‖2L2(0,T ;L2(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ), (4.17) δ ∫ T 0 ‖∂tuδε‖2L2(Ωε) dt+ ε−γ ∫ T 0 ‖∂tuδε‖2L2(Sε) dt+ ess sup[0,T ] ‖∇uδε‖2L2(Ωε) ≤ K ( ‖f‖2H1(0,T ;L2(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) + ‖f‖2αL2(0,T ;L2(Ω)) + ‖g‖2α L2(0,T ;C(Ω)) ) . (4.18) Thus, for every ε > 0 there is exist some subsequence δ → 0 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 in L2(0, T ;L2(Sε)). EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 11 Taking into account (4.18) we find that δ ∫ T 0 ∫ Ωε ∂tu δ εφdx dt→ 0, δ → 0. Hence, passing to the limit in (4.2) we obtain that uε is a strong solution to the problem (2.2) and the estimate (2.8) holds. 5. Auxiliary Gjε-capacity problems to adapt the global test function . The first step of our adaptation of the master lines of the “oscillating test func- tions method” consists in to replace the weak formulation of the problem by a kind of “very weak formulation” thanks to the monotonicity in L2(Sε) of the operator given, formally by uε → ∂νuε (see, e.g. [17, 21] and, more precisely, [3, Lemma 4.1]). By applying [6] (see also [21, Théorème 8.4]) we know that uε ∈ L2(0, T ;H1(Ωε, ∂Ω)) is a weak solution of problem (2.2) if and only if for any regular test function, for instance, φ ∈ L2(0, T ;H1 0 (Ω) ∩ C(Ω)) ∩ H1(0, T ;C(Ω)), we have ε−γ ∫ T 0 ∫ Sε ∂tφ(φ− uε) ds dt+ ∫ QTε ∇φ∇(φ− uε) dx dt + ε−γ ∫ T 0 ∫ Sε σ(φ)(φ− uε) ds dt ≥ ∫ QTε f(φ− uε) dx dt+ ε−γ ∫ T 0 ∫ Sε g(φ− uε) ds dt− 1 2 ε−γ‖φ(x, 0)‖2L2(Sε) . (5.1) The second step of our adaptation of this very general set of ideas constituting the “oscillating test functions method” consists in modifying, in a suitable way, any test function (which will be used to check the homogenized equation satisfied by the weak limit function u0(x, t) obtained in Theorem 3.2) to a different family of test functions better adapted to problem (2.2): the new set of “oscillating test functions” . This step is rather involved and should be carried out in a very sharp way accordingly the problem under consideration. To this purpose, it is useful to simplify the properties satisfied by the departing test function. In our case, it will be enough to consider the given potential test function of the form φ(x, t) = ϕ(x)η(t), where ϕ ∈ C∞(Ω) and η ∈ H1(0, T ). (5.2) As mentioned in the introduction, the identification of the “strange term” oper- ator H[φ] defined by (3.1) will use the G0-capacity type exterior nonlinear problem given by (3.2) which we will consider in the next Section. It turns out that it is convenient to approximate the problem (3.2) by a set of auxiliary capacity problems which are sharply adapted to the starting spatial domain Ωε = Ω \ Gε: thus, this other set of auxiliary problems will depend on the parameter ε and are built in terms of the cell reference set Y jε . We recall that we assumed thatGjε ⊂ T jCaε ⊂ T j ε/4 ⊂ Y j ε , where T jr is the 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 on ε. Then, for every j ∈ Υε we 12 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 consider the auxiliary problem ∆wjε,φ = 0, x ∈ T jε/4 \G j ε, t ∈ (0, T ), ε−γ∂tw j ε,φ + ∂νw j ε,φ − ε −γσ(φ(P jε , t)− w j ε,φ) = ε−γ(∂tφ(P jε , t)− g(P jε , t)), 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. (5.3) Notice that, again, a more exact notation would be wj ε,φ = wj ε,φ,g,G0 , to indicate the important dependence of wj ε,φ with respect also to the datum g and G0, but we drop such a sophisticated notation for the sake of simplicity. In the next Section we will study the asymptotic resemblance among the solutions wj ε,φ of this new problem and the solutions w̃φ(x, y, t) of the G0-capacity type exterior nonlinear problem given by (3.2). To study problem (5.3) we start by introducing the type of solutions we will consider in this paper: A function wjε,φ is a strong solution of problem (5.3) if wjε,φ ∈ C([0, T ];L2(∂Gjε)), w j ε,φ ∈ L2(0, T ;H1(T jε/4 \G j ε)), ∂tw j ε,φ ∈ L2(0, T ;L2(∂Gjε)) and the following integral identity holds for any test function ψ ∈ L2(0, T ;H1(T j ε/4 \ Gj ε, ∂T j ε/4)), ε−γ ∫ T 0 ∫ ∂Gjε ∂tw j ε,φψ ds dt+ ∫ T 0 ∫ T j ε/4 \Gjε ∇wjε,φ∇ψ dx dt − ε−γ ∫ T 0 ∫ ∂Gjε σ(φ(P jε , t)− w j ε,φ)ψ ds dt = ε−γ ∫ T 0 ∫ ∂Gjε (∂tφ(P jε , t)− g(P jε , t))ψ ds dt. (5.4) Theorem 5.1. Assume σ Hölder continuous satisfying (2.6). Then, for any given test function φ(x, t) of the form (5.2), problem (5.3) has a unique strong solution wjε,φ and the following estimates hold ‖wjε,φ‖ 2 L2(0,T ;H1(T j ε/4 \Gjε,∂T jε/4)) + ε−γ‖wjε,φ‖ 2 L2(0,T ;L2(∂Gjε)) ≤ K0ε n(‖φ‖2 L2(0,T ;C(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ), ε−γ‖∂twjε,φ‖ 2 L2(0,T ;L2(∂Gjε)) ≤ K0ε n ( ‖φ‖2 L2(0,T ;C(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) + ‖∂tφ‖2L2(0,T ;C(Ω)) + ‖φ‖2α L2(0,T,C(Ω)) + ‖g‖2α L2(0,T ;C(Ω)) + max Ω×[0,T ] |φ(x, t)|2 ) , ‖wjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gjε)) ≤ K0ε n+2 ( ‖φ‖2 L2(0,T ;C(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ) , where the positive constant K0 is independent on φ, g and ε. EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 13 Proof. As usual, the uniqueness of the strong solution of the problem (5.3) can be derived from the integral identity by using the monotonicity of the function σ. In order to prove the existence of solutions satisfying the above mentioned estimates, let us introduce the functions ψ̃jε = φ(P jε , t)ψ j ε(x), where ψjε ∈ C∞(T jε/4), ψjε(x) = 1 if x ∈ T jCaε and ψjε(x) = 0 if x ∈ T jε/4 \ T j 2Caε , |∇ψjε| ≤ C1a −1 ε , C1 is a positive constant. Then, we make the change of unknowns wjε,φ = vjε,φ + ψ̃jε, and thus, it is easy to see that wjε,φ is a strong solution of the problem (5.3) if and only if vjε,φ is a strong solution to the new problem ∆x(vjε,φ + ψ̃jε) = 0, x ∈ T jε/4 \G j ε, t ∈ (0, T ), ε−γ∂tv j ε,φ + ∂ν(vjε,φ + ψ̃jε)− ε−γσ(−vjε,φ) = −ε−γg(P jε , t), x ∈ ∂Gjε, t ∈ (0, T ), vjε,φ(x, t) = 0, x ∈ ∂T jε/4, t ∈ (0, T ), vjε,φ(x, 0) = 0, x ∈ ∂Gjε. (5.5) To prove that problem (5.5) has a strong solution, we consider the approximate auxiliary problem δ∂tv δ,j ε,φ −∆vδ,jε,φ = ∆ψ̃jε,φ, x ∈ T jε/4 \G j ε, t ∈ (0, T ), ε−γ∂tv δ,j ε,φ + ∂ν(vδ,jε,φ + ψ̃jε,φ)− ε−γσ(−vδ,jε,φ) = −ε−γg(P jε , t), x ∈ ∂Gjε, t ∈ (0, T ), vδ,jε,φ(x, t) = 0, x ∈ ∂T jε/4, t ∈ (0, T ), vδ,jε,φ(x, 0) = 0, x ∈ T jε/4 \G j ε, vδ,jε,φ(x, 0) = 0, x ∈ ∂Gjε. (5.6) The proof of the existence of solution of problem (5.6) can be obtained by a Galerkin method as in Theorem 3.2. If we denote by {(wmε , wmε ∣∣ ∂Gjε )}, m = 1, 2, . . ., the orthonormal basis for the space Hδ ε = L2(T jε/4 \ G j ε) × L2(∂Gjε), then we find the Galerkin approximations (vδ,mε,φ , v δ,m ε,φ ∣∣ ∂Gjε ) ∈ C([0, T ];V δε ), where V δε = {(u, u ∣∣ ∂Gjε ) ∈ Hδ ε |u ∈ H1(T jε/4 \ G j ε, ∂T j ε/4)}, such that ∂t(v δ,m ε,φ , v δ,m ε,φ ∣∣ ∂Gjε ) ∈ L2(0, T ;Hδ ε ). Using similar considerations as in the proof of Theorem 3.2 we obtain the following estimates δmax [0,T ] ‖vδ,mε,φ ‖ 2 L2(T j ε/4 \Gjε) + ε−γ max [0,T ] ‖vδ,mε,φ ‖ 2 L2(∂Gjε) + ‖∇vδ,mε,φ ‖ 2 L2(0,T ;L2(T j ε/4 \Gjε)) ≤ Kεn ( ‖φ‖2 L2(0,T,C(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ) , (5.7) 14 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 δ‖∂tvδ,mε,φ ‖ 2 L2(0,T ;L2(T j ε/4 \Gjε)) + ε−γ‖∂tvδ,mε,φ ‖ 2 L2(0,T ;L2(∂Gjε)) + max [0,T ] ‖∇vδ,mε,φ ‖ 2 L2(T j ε/4 \Gjε) ≤ Kεn ( max Ω×[0,T ] |φ|2 + ‖∂tφ‖2L2(0,T ;C(Ω)) + ‖φ‖2α L2(0,T ;C(Ω)) + ‖g‖2α L2(0,T ;C(Ω)) + ‖g‖2 L2(0,T ;C(Ω)) ) , (5.8) where the constant K does not depend on ε, φ, g. From the estimates (5.7), (5.8) it follows that for some subsequence {ms}, vδ,msε,φ ⇀ vδ,jε,φ, weakly in L2(0, T ;H1(T jε/4 \G j ε)), ∂tv δ,ms ε,φ ⇀ ∂tv δ,j ε,φ, weakly in L2(0, T ;L2(T jε/4 \G j ε)), vδ,msε,φ → vδ,jε,φ in L2(0, T ;L2(T jε/4 \G j ε)), ∂tv δ,ms ε,φ ⇀ ∂tv δ,j ε,φ weakly in L2(0, T ;L2(∂Gjε)), as ms → ∞ and vδ,jε,φ is a strong solution of the problem (5.6) and the similar estimates hold for the limit function vδ,jε,φ. So for fixed ε > 0 by some subsequence {vδ ′,j ε,φ } as δ′ → 0, vδ ′,j ε,φ ⇀ vjε,φ, weakly in L2(0, T ;H1(T jε/4 \G j ε, ∂T j ε/4)), ∂tv δ′,j ε,φ ⇀ ∂tv j ε,φ, weakly in L2(0, T ;L2(∂Gjε)), vδ ′,j ε,φ ∣∣ ∂Gjε → vjε,φ ∣∣ ∂Gjε in L2(0, T ;L2(∂Gjε)). Taking into account that δ′ ∫ T 0 ∫ T j ε/4 \Gjε ∂tv δ′,j ε,φ ψ dx dt → 0, as δ′ → 0, we obtain that vjε,φ is a strong solution of the problem (5.5). From estimates (5.7), (5.8), and applying the Friederich inequality, we obtain the estimates in the Theorem 5.1. � In the following Section we will need some stronger regularity estimates on wjε,φ which we prove now. Theorem 5.2. As a matter of facts, wjε,φ ∈ L∞((T jε/4 \G j ε)× (0, T )) and we have ess sup (T j ε/4 \Gjε)×(0,T ) |wjε,φ| ≤ 2 max Ω×[0,T ] |φ(x, t)|+ 2 ∫ T 0 max Ω |g(x, t)|dt. (5.9) Proof. We introduce the function hjε,φ = φ(P jε , t)−w j ε,φ− ∫ t 0 g(P jε , s)ds. Then hjε,φ is a strong solution of the problem ∆hjε,φ = 0, x ∈ T jε/4 \G j ε, t ∈ (0, T ), ε−γ∂th j ε,φ + ∂νh j ε,φ + ε−γσ(hjε,φ + ∫ t 0 g(P jε , s)ds) = 0, x ∈ ∂Gjε, t ∈ (0, T ), hjε,φ = φ(P jε , t)− ∫ t 0 g(P jε , s)ds, x ∈ ∂T jε/4, t ∈ (0, T ), hjε,φ(x, 0) = 0, x ∈ ∂Gjε. (5.10) EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 15 We set K = maxΩ×[0,T ] |φ(x, t)|+ ∫ T 0 maxΩ |g(x, t)|dt. Then (hjε,φ −K)+ ∈ L2(0, T ;H1(T jε/4 \G j ε, ∂T j ε/4)) and taking it as test function in the corresponding integral identity associated with problem (5.10) we obtain ε−γ 2 ‖(hjε,φ −K)+(x, τ)‖2 L2(∂Gjε) + ∫ τ 0 ‖∇(hjε,φ −K)+‖2 L2(T j ε/4 \Gjε) dt + ε−γ ∫ τ 0 ∫ ∂Gjε σ(hjε,φ + ∫ t 0 g(x, s)ds)(hjε,φ −K)+dsdt = 0. (5.11) Using that hjε,φ + ∫ τ 0 g(x, s)ds ≥ 0 on the set where hjε,φ ≥ K and since σ is non- decreasing we obtain σ(hjε,φ + ∫ t 0 g(x, s)ds) ≥ 0. Thus on the left hand side of (5.11) we have a sum of nonnegative terms which is equal to zero. So we conclude that (hjε,φ−K)+ = 0 a.e. x ∈ T jε/4\G j ε and t ∈ (0, T ). Arguing in a similar way we obtain also that (hjε,φ + K)− = 0. Hence we obtain the statement of Theorem 5.2. � 6. G0-capacity for the exterior problem Now, in this section, we will prove the existence and uniqueness of w̃φ(x, y, t) solution of exterior problem (3.2), mentioned in the Introduction, by assuming g ∈ L2(0, T ;C(Ω)), and for a general test function φ ∈ L2(0, T ;L2(Ω)). To define the notion of strong solution of problem (3.2), we denote by M the set of functions w ∈ C∞(Rn \ G0) such that w(y) = 0 for y ∈ Rn \ T 0 R, for some ball T 0 R, such that G0 ⊂ T 0 R. We denote by M the closure of M with the norm ‖w‖M = ‖∇w‖L2(Rn\G0). By applying Hardy inequality on the rings T 0 R \ T 0 R0 and making R → ∞, we arrive to the following well-known result. Lemma 6.1. Let G0 ⊂ Rn, n ≥ 3, be a smooth bounded domain, star-shaped with respect to a ball T 0 R, T 0 R ⊂ G0. Then there exist a constant K0, only dependent of n, such that ∫ Rn\G0 |y|−2w2dy ≤ K0‖w‖2M, ∀w ∈M. (6.1) By a strong solution to (3.2) we mean a function w̃φ(x, y, t) such that for a.e. x ∈ Ω, w̃φ ∈ C([0, T ];L2(∂G0)), w̃φ ∈ L2(0, T ;M) and ∂tw̃φ ∈ L2(0, T ;L2(∂G0)), w̃φ(x, y, 0) = 0 for y ∈ ∂G0, and the following integral identity holds for any test function v ∈ L2(0, T ;M):∫ T 0 ∫ Rn\G0 ∇w̃φ∇vdydt+ C0 ∫ T 0 ∫ ∂G0 ∂tw̃φv ds dt+ C0 ∫ T 0 ∫ ∂G0 σ(w̃φ)v ds dt = ∫ T 0 φ(x, t) ∫ ∂G0 ∂νκvdsdt. (6.2) 16 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 Theorem 6.2. Assume σ is Hölder continuous satisfying (2.6) and let g belong to L2(0, T ;C(Ω)) and φ belong to L2(0, T ;L2(Ω)). Then problem (3.2) has a unique strong solution and for a.e. x ∈ Ω the following estimates hold ‖w̃φ‖C([0,T ];L2(∂G0)) + ‖w̃φ‖L2(0,T ;M) ≤ K(‖φ(·, x)‖L2(0,T ) + ‖g(·, x)‖L2(0,T )), ‖∂tw̃φ‖L2(0,T ;L2(∂G0)) ≤ K ( ‖φ(·, x)‖L2(0,T ) + ‖g(·, x)‖L2(0,T ) + ‖φ(·, x)‖αL2(0,T ) + ‖g(·, x)‖αL2(0,T ) ) . (6.3) In addition, if we define wφ(x, y, t) = κ(y)φ(x, t)− w̃φ(x, y, t), (6.4) with κ(y) solution of standard G0-capacity exterior problem (3.3) then, for a.e. x ∈ Ω, we have∫ T 0 (∫ ∂G0 ∂νwφ(x, y, t)dsy )2 dt ≤ K ( ‖φ(·, x)‖2L2(0,T ) + ‖g(·, x)‖2L2(0,T ) ) , (6.5) where the positive constant K does not depend on φ and g. Proof. It is a simple variation of the proof of Theorem 3.2 (with f ≡ 0 and Ω = Rn \G0). Notice that the application of the Hardy inequality (6.1) allows to extend the conclusion to this spatial domain even if it is unbounded. Then, for instance, the uniqueness results, once again, from the monotonicity of the operator mentioned in the proof of Theorem 3.2. Estimate (6.3) is consequence of the corresponding estimate (2.8). In addition, since w̃φ is a strong solution, we have∫ ∂G0 ∂νwφ(x, y, t)ds = ∫ Rn\G0 ∇wφ∇κdy = φ(x, t) ∫ Rn\G0 |∇κ(y)|2dy − ∫ Rn\G0 ∇w̃φ∇κdy = φ(x, t)λG0 − ∫ Rn\G0 ∇w̃φ∇κdy. From here we obtain(∫ ∂G0 ∂νwφds )2 ≤ 2λ2 G0 φ2(x, t) + 2‖w̃φ‖2Mλ2 G0 . Therefore,∫ T 0 (∫ ∂G0 ∂νwφ(x, y, t)ds )2 dt ≤ 2λ2 G0 (‖φ(·, x)‖2L2(0,T ) + ‖w̃φ‖2L2(0,T ;M)). Applying estimate of (6.3) we obtain (6.5). � Later, we will need some continuous dependence of wφ with respect to φ. Theorem 6.3. Assume σ is Hölder continuous satisfying (2.6), and let φ1, φ2 blong to L2(0, T ;L2(Ω)). Then, for a.e. x ∈ Ω, we have the following estimates ‖w̃φ1 − w̃φ2‖C([0,T ];L2(∂G0)) + ‖w̃φ1 − w̃φ2‖L2(0,T ;M) ≤ K‖[φ1 − φ2](·, x)‖L2(0,T ). (6.6) EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 17 ‖∂tw̃φ1 − ∂tw̃φ2‖L2(0,T ;L2(∂G0)) ≤ K ( ‖[φ1 − φ2](·, x)‖L2(0,T ) + ‖[φ1 − φ2](·, x)‖αL2(0,T ) ) ,∫ T 0 (∫ ∂G0 (∂νwφ1 − ∂νwφ2 )ds )2 dt ≤ K‖[φ1 − φ2](·, x)‖2L2(0,T ), (6.7) where the constant K is independent on function g. Proof. We set v = w̃φ1 − w̃φ2 . Then v is a solution of the problem ∆yv = 0, y ∈ Rn \G0, t ∈ (0, T ), C0∂tv + ∂νv + C0(σ(w̃φ1 )− σ(w̃φ2 )) = (φ1(x, t)− φ2(x, t))∂νκ(y), y ∈ ∂G0, t ∈ (0, T ), v(y, x, 0) = 0, y ∈ ∂G0, v → 0, |y| → ∞, (6.8) for a.e. x ∈ Ω. Now we can argue as in the proof of Theorem 3.2 and we obtain, for a.e. x ∈ Ω, ‖v‖2L2(0,T ;M) + max t∈[0,T ] ‖v(t, ·)‖2L2(∂G0) ≤ K‖[φ1 − φ2](·, x)‖2L2(0,T ), (6.9) ess supt∈[0,T ] ‖∇v(t, ·)‖2 L2(Rn\G0) + ‖∂tv‖2L2(0,T ;L2(∂G0)) ≤ K ( ‖[φ1 − φ2](·, x)‖2L2(0,T ) + ‖[φ1 − φ2](·, x)‖2αL2(0,T ) ) . (6.10) Taking into account that∫ ∂G0 ∂ν(wφ1 − wφ2 )ds = ∫ Rn\G0 ∇(wφ1 − wφ2 )∇κdy = λG0 (φ1(x, t)− φ2(x, t))− ∫ Rn\G0 ∇(w̃φ1 − w̃φ2 )∇κdy, and using estimate (6.9) we obtain (6.7). � As in [11, Lemma 4.6] we have a priori estimate for the solutions of problem (3.2) on the interior of the set Rn \G0. Theorem 6.4. Assume that φ(x, t) = ψ(x)η(t), ψ(x) ∈ C∞(Ω), η ∈ C1([0, T ]), QT = Ω× (0, T ). Then, a.e. t ∈ (0, T ), x ∈ Ω |w̃φ(x, y, t)| ≤ Rn−2 0 K(max QT |φ(x, t)|+ ∫ T 0 maxΩ |g(x, t)|dt) |y|n−2 , (6.11) for all y ∈ Rn \G0, and for y > 2R0, R0 = maxy∈∂G0 |y|, |∇ywφ(x, y, t)| ≤ K(max QT |φ(x, t)|+ ∫ T 0 maxΩ |g(x, t)|dt) |y|n−1 , (6.12) where wφ is defined by (6.4) and K is a positive constant only dependent on n. 18 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 Proof. Let wφ be defined by (6.4), i.e. w̃φ = φ(x, t)κ(y)−wφ. Then wφ is a strong solution of the problem ∆ywφ = 0, y ∈ Rn \G0, t ∈ (0, T ), C0∂twφ + ∂νwφ − C0σ(φ− wφ) = −C0g(x, t) + C0∂tφ(x, t), y ∈ ∂G0, t ∈ (0, T ), wφ(x, y, 0) = φ(x, 0), y ∈ ∂G0, wφ → 0, |y| → ∞. (6.13) We consider the sequence of solutions {wφ,R} to the problems ∆ywφ,R = 0, y ∈ T 0 R \G0, t ∈ (0, T ), C0∂twφ,R + ∂νwφ,R − C0σ(φ− wφ,R) = −C0g(x, t) + C0∂tφ(x, t), y ∈ ∂G0, t ∈ (0, T ), wφ,R(x, y, 0) = φ(x, 0), y ∈ ∂G0, wφ,R(x, y, t) = 0, y ∈ ∂T 0 R, t ∈ (0, T ). (6.14) Arguing as in the proof of the estimate (5.9) in Theorem 5.2, we find that |wφ,R| ≤ K0 ≡ 2 max Ω×[0,T ] |φ(x, t)|+ 2 ∫ T 0 max Ω |g(x, t)|dt. Using the maximum principle, we derive the estimate |wφ,R| ≤ K0R0 |y|n−2 , y ∈ T 0 R \G0, t ∈ (0, T ). (6.15) We define Wφ,R = { wφ,R, y ∈ T 0 R \G0, t ∈ (0, T ), 0, |y| ≥ R, t ∈ (0, T ). Noting that Wφ,R ∈ L2(0, T ;M) and taking into account that Wφ,R ⇀ wφ in L2(0, T ;M) as R→∞, we deduce the estimate |wφ| ≤ K0R0 |y|n−2 . From here we obtain (6.11). Using the inequality (6.11) and the mean-value theorem for the harmonic function ∂yiwφ(x, y, t), i = 1, . . . , n, on a ball we obtain estimate (6.12). � As the previous regularity result we have the following. Theorem 6.5. Assume (3.9). Let w̃φ(x1, y, t) be the weak solution to the problem (3.2). Then for any x1, x2 ∈ Ω, ‖w̃φ(x1, ·, ·)− w̃φ(x2, ·, ·)‖L2(0,T ;M) ≤ K{|x1 − x2|+ ‖φ(x1, ·)− φ(x2, ·)‖L2(0,T )}. Proof. The function w̃φ(x1, y, t)− w̃φ(x2, y, t) satisfies the following problem ∆y(w̃φ(x1, y, t)− w̃φ(x2, y, t)) = 0, y ∈ Rn \G0, C0∂t(w̃φ(x1, y, t)− w̃φ(x2, y, t)) + ∂ν(w̃φ(x1, y, t)− w̃φ(x2, y, t)) + C0(σ(w̃φ(x1, y, t))− σ(w̃φ(x2, y, t))) = C0(g(x1, t)− g(x2, t)) + (φ(x1, t)− φ(x2, t))∂νκ(y), y ∈ ∂G0, EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 19 w̃φ(x1, y, 0)− w̃φ(x2, y, 0) = 0, x ∈ ∂G0, w̃φ(x1, y, t)− w̃φ(x2, y, t)→ 0, |y| → ∞. From the definition of weak solution for this problem we deduce that∫ t 0 ∫ Rn\G0 |∇y(w̃φ(x1, y, t)− w̃φ(x2, y, t))|2dydτ + C0 2 ∫ ∂G0 |w̃φ(x1, y, t)− w̃φ(x2, y, t)|2ds + C0 ∫ t 0 ∫ ∂G0 (σ(w̃φ(x1, y, t))− σ(w̃φ(x2, y, t)))(w̃φ(x1, y, t)− w̃φ(x2, y, t)) ds dτ = C0 ∫ t 0 ∫ ∂G0 (g(x1, t)− g(x2, t))(w̃φ(x1, y, t)− w̃φ(x2, y, t)) ds dτ + ∫ t 0 ∫ ∂G0 (φ(x1, t)− φ(x2, t))∂νκ(y)(w̃φ(x1, t)− w̃φ(x2, y, t)) ds dτ. Using the monotonicity of σ, (3.9) and the Gronwall’s lemma we obtain the con- clusion of the Theorem. � The following result shows that the solution w̃φ(x, y, t) has a Lipschitz continuous dependence on function g without requiring the assumption (3.9). Theorem 6.6. Let w̃gφ(x, y, t), w̃ĝφ(x, y, t) be the weak solutions to the problem (3.2) corresponding to the boundary data g, ĝ ∈ L2(0, T ;C(Ω)). Then for a.e. x ∈ Ω, ‖w̃gφ(x, ·, ·)− w̃ĝφ(x, ·, ·)‖L2(0,T ;M) ≤ K‖g(x, ·)− ĝ(x, ·)‖L2(0,T ). Proof. The function w̃gφ(x, y, t)− w̃ĝφ(x, y, t) satisfies the problem ∆y(w̃gφ(x, y, t)− w̃ĝφ(x, y, t)) = 0, y ∈ Rn \G0, C0∂t(w̃ g φ(x, y, t)− w̃ĝφ(x, y, t)) + ∂ν(w̃gφ(x, y, t)− w̃ĝφ(x, y, t)) + C0(σ(w̃gφ(x, y, t))− σ(w̃ĝφ(x, y, t))) = C0(g(x, t)− ĝ(x, t)) y ∈ ∂G0, w̃φ(x1, y, 0)− w̃φ(x2, y, 0) = 0, x ∈ ∂G0, w̃φ(x1, y, t)− w̃φ(x2, y, t)→ 0, |y| → ∞. From the definition of weak solution for this problem (taking w̃gφ(x, y, t)−w̃ĝφ(x, y, t) as test function) we obtain∫ t 0 ∫ Rn\G0 |∇y(w̃gφ(x, y, τ)− w̃ĝφ(x, y, τ))|2dydτ + C0 2 ∫ ∂G0 |w̃gφ(x, y, t)− w̃ĝφ(x, y, t)|2ds + C0 ∫ t 0 ∫ ∂G0 (σ(w̃gφ(x, y, τ))− σ(w̃ĝφ(x, y, τ)))(w̃gφ(x, y, τ)− w̃ĝφ(x, y, τ)) ds dτ = C0 ∫ t 0 ∫ ∂G0 (g(x, τ)− ĝ(x, τ))(w̃gφ(x, y, τ)− w̃ĝφ(x, y, τ)) ds dτ. 20 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 Then, using the monotonicity of σ, Friedrich’s inequality and the Gronwall’s lemma we obtain the conclusion of the Theorem. � 7. Properties of the operator H[φ] By applying the results of previous sections we obtain different properties on the operator H[φ] defined by (3.1). Theorem 7.1. Assumeφ ∈ L2(0, T ;L2(Ω)). Then, for a.e. x ∈ Ω, ‖H[φ](·, x)‖L2(0,T ) ≤ K(‖φ(·, x)‖L2(0,T ) + ‖g(·, x)‖L2(0,T )). (7.1) Moreover, for φ1, φ2 ∈ L2(0, T ;L2(Ω)) and for a.e. x ∈ Ω we have ‖ (H[φ1]−H[φ2]) (·, x)‖L2(0,T ) ≤ K‖ (φ1 − φ2) (·, x)‖L2(0,T ). (7.2) Proof. By taking in the integral identity for w̃φ as a test function the solution of problem (3.3) we obtain∫ ∂G0 ∂νw̃φds = ∫ Rn\G0 ∇yw̃φ∇κdy. So, we have H[φ](x, t) = λG0φ(x, t)− ∫ Rn\G0 ∇yw̃φ∇κ dy. Applying estimate (6.3) we obtain ‖H[φ](·, x)‖L2(0,T ) ≤ K(‖φ(·, x)‖L2(0,T ) + ‖g(·, x)‖L2(0,T )), where K is independent from φ(·, x) and g(·, x). To obtain the inequality (7.2) we note that ∫ ∂G0 ∂ν(w̃φ1 − w̃φ2 )ds = ∫ Rn\G0 ∇(w̃φ1 − w̃φ2 )∇κdy. From here we conclude that H[φ1]−H[φ2] = (φ1 − φ2)λG0 − ∫ Rn\G0 ∇(w̃φ1 − w̃φ2)∇κdy. Applying (6.6) we obtain (7.2). � Remark 7.2. Note that if G0 is a ball, as in Remark 3.1, and H is a solution of the Cauchy problem (3.4), then we can prove, in a different way, that independently of the regularity of σ, the function H is Lipcshitz continuous. Indeed, for any φ1, φ2 ∈ L2(0, T ) we have ∂t(Hφ1 −Hφ2 ) + n− 2 C0 (Hφ1 −Hφ2 ) + (σ(Hφ1 )− σ(Hφ2 )) = n− 2 C0 (φ1 − φ2). Multiplying this equality by (Hφ1 −Hφ2 ), integrating on (0, t) and using the mono- tonicity property of σ, we obtain (Hφ1(x, t)−Hφ2(x, t))2 ≤ 2(n− 2) C0 ∫ t 0 (φ1 − φ2)(Hφ1 −Hφ2)dτ. Applying the Gronwall’s Lemma, we conclude that max [0,T ] |Hφ1 −Hφ2 | ≤ K‖φ1 − φ2‖L2(0,T ). EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 21 Remark 7.3. If we consider the similar problem for Robin type boundary condi- tions (see [15]), then, if G0 is a ball, the new nonlinear term in the homogenized problem must solve the functional equation H = λσ(u−H), λ = C0 n− 2 > 0. Setting V = u−H, we derive the equation λσ(V ) + V = u. Let us consider the difference of two equalities λσ(V1)+V1 = u1 and λσ(V2)+V2 = u2, i.e. λ(σ(V1)− σ(V2)) + V1 − V2 = u1 − u2. Multiplying this equality by V1 − V2, and using the monotonicity of σ, we obtain the Lipschitz continuity of V . Theorem 7.4. For any φ1(x, t), φ2(x, t), defined a.e. in Ω × (0, T ) such that for a.e. x ∈ Ω, φi(x, ·) ∈ L2(0, T ), for a.e. x ∈ Ω, i = 1, 2, we have∫ T 0 (H[φ1](x, t)−H[φ2](x, t))(φ1(x, t)− φ2(x, t))dt ≥ 0. (7.3) Proof. By denseness, without loss of generality we can assume that φi(x, ·) ∈ C1([0, T ]), i = 1, 2. We consider wφ1 , wφ2 . Then wφ1 − wφ2 is a weak solution to the problem ∆y(wφ1 − wφ2) = 0, y ∈ Rn \G0, C0∂t(wφ1 − wφ2 ) + ∂ν(wφ1 − wφ2 ) − C0(σ(φ1 − wφ1 )− σ(φ2 − wφ2 )) = C0∂t(φ1 − φ2), y ∈ ∂G0, t ∈ (0, T ), (wφ1 − wφ2)(x, y, 0) = φ1(x, 0)− φ2(x, 0), y ∈ ∂G0, wφ1 − wφ2 → 0, |y| → ∞. (7.4) Taking wφ1 −wφ2 as a test function in the integral identity corresponding to notion of weak solution of problem (7.4), we obtain ‖wφ1 − wφ2 ‖2L2(0,T ;M) + C0 ∫ T 0 ∫ ∂G0 ∂t(wφ1 − wφ2 )(wφ1 − wφ2 ) ds dt − C0 ∫ T 0 ∫ ∂G0 (σ(φ1 − wφ1 )− σ(φ2 − wφ2 ))(wφ1 − wφ2 ) ds dt = C0 ∫ T 0 ∫ ∂G0 ∂t(φ1 − φ2)(wφ1 − wφ2) ds dt. (7.5) From this equality we deduce that ‖wφ1 − wφ2 ‖2L2(0,T ;M) + C0 ∫ T 0 ∫ ∂G0 ∂t((wφ1 − φ1)− (wφ2 − φ2))((wφ1 − φ1)− (wφ2 − φ2)) ds dt + C0 ∫ T 0 ∫ ∂G0 (σ(φ1 − wφ1)− σ(φ2 − wφ2))((φ1 − wφ1)− (φ2 − wφ2)) ds dt 22 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 = C0 ∫ T 0 ∫ ∂G0 ∂t((φ1 − wφ1)− (φ2 − wφ2))(φ1 − φ2) ds dt + C0 ∫ T 0 ∫ ∂G0 (σ(φ1 − wφ1 )− σ(φ2 − wφ2 ))(φ1 − φ2) ds dt = ∫ T 0 (H[φ1]−H[φ2])(φ1 − φ2) dt. Using that σ is monotone we obtain that the left-hand side of last equality is nonnegative and completes the proof. � We obtain a stronger regularity on the operator H[φ] when the boundary datum g is more regular. Theorem 7.5. Assume (3.9) and let x1, x2 ∈ Ω. Then there exists K > 0 such that ‖H[φ](x1, t)−H[φ](x2, t)‖L2(0,T ) ≤ K{|x1 − x2|+ ‖φ(x1, t)− φ(x2, t)‖L2(0,T )}. Proof. Note that H[φ](x1, t)−H[φ](x2, t) = (φ(x1, t)− φ(x2, t))λG0 + ∫ Rn\G0 ∇(w̃φ(x1, y, t)− w̃φ(x2, y, t))∇κ(y) dy. Applying Theorem 6.5 to the right-hand side of this equality we obtain the state- ment of the theorem. � As in the previous section, we can prove that the operator H[φ] has a Lipschitz continuous dependence on function g without requiring assumption (3.9). Theorem 7.6. Let Hg φ(x, t),Hĝ φ(x, t) be the associate operators corresponding to the boundary data g, ĝ ∈ L2(0, T ;C(Ω)). Then there exists K > 0 such that for a.e. x ∈ Ω, ‖Hg φ(x, ·)−Hĝ φ(x, ·)‖L2(0,T ) ≤ K‖g(x, ·)− ĝ(x, ·)‖L2(0,T ) . Proof. Note that Hg φ(x, t)−Hĝ φ(x, t) = ∫ Rn\G0 ∇(w̃gφ(x, y, t)− w̃ĝφ(x, y, t))∇κ(y) dy. Then it suffices to apply Theorem 6.6 to the right-hand side of this equality to obtain the statement of the theorem. � 8. Asymptotic similarity between the capacity functions wjε,φ and wφ(P jε , x−P jε aε , t) The following results give some estimates on the differences of the two types of capacities functions introduced in previous sections. Theorem 8.1. Let vjε,φ = wjε,φ −wφ(P jε , x−P jε aε , t), where wφ(x, y, t) is the solution of (6.13). Then sup (T j ε/4 \Gjε)×(0,T ) |vjε,φ| ≤ sup ∂T j ε/4 ×(0,T ) ∣∣wφ(P jε , x− P jε aε , t) ∣∣. (8.1) Here, φ(x, t) = ψ(x)η(t), with ψ ∈ C∞(Ω), η ∈ C1([0, T ]). EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 23 The proof of the above theorem can be find in [34]. Theorem 8.2. ε−γ ∑ j∈Υε max t∈[0,T ] ‖vjε,φ‖ 2 L2(∂Gjε) + ∑ j∈Υε ‖∇vjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gjε)) ≤ Kε2(max QT φ2(x, t) + ∫ T 0 max Ω g2(x, t)dt), (8.2) ∑ j∈Υε ‖vjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gjε)) ≤ Kε4(max QT |φ(x, t)|2 + ∫ T 0 max Ω |g(x, t)|2dt), (8.3) where the positive constant K is independent of ε, φ and g. Proof. From the Green formulas we have∫ t 0 ∫ T j ε/4 \Gjε |∇vjε,φ| 2dxdτ + ε−γ ∫ t 0 ∫ ∂Gjε ∂tv j ε,φv j ε,φ ds dτ − ε−γ ∫ t 0 ∫ ∂Gjε ( σ(φ(P jε , τ)− wjε,φ)− σ(φ(P jε , τ) − wφ(P jε , x− P jε aε , τ)) ) vjε,φ ds dτ = − ∫ t 0 ∫ ∂T j ε/4 wφ(P jε , x− P jε aε , τ)∂νv j ε,φ ds dτ. (8.4) Taking into account that σ is monotone and applying the Green formula, we obtain∫ t 0 ∫ T j ε/4 \Gjε |∇vjε,φ| 2dxdτ + ε−γ 2 ‖vjε,φ‖ 2 L2(∂Gjε) ≤ − ∫ t 0 ∫ ∂T j ε/4 wφ(P jε , x− P jε aε , τ)∂νv j ε,φddτ = − ∫ t 0 ∫ T j ε/4 \T j ε/8 ∇vjε,φ∇wφdxdτ + ∫ t 0 ∫ ∂T j ε/8 ∂νv j ε,φwφ ds dτ. (8.5) Using the estimate (6.11) we obtain |vjε,φ(x, t)| ≤ sup ∂T j ε/4 ×(0,T ) ∣∣wφ(P jε , x− P jε aε , t) ∣∣ ≤ Kε2(max QT |φ(x, t)|+ ∫ T 0 max Ω |g(x, t)|dt), (8.6) where K is a constant independent on ε, φ and g. From (8.6) we deduce for x0 ∈ ∂T jε/8 |∂xiv j ε,φ(x0, t)| = |T x0 ε/16| −1 ∣∣ ∫ T x0 ε/16 ∂xiv j ε,φdx ∣∣ ≤ Kε(max QT |φ(x, t)|+ ∫ T 0 max Ω |g(x, t)|dt). (8.7) 24 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 Consequently,∣∣ ∫ T 0 ∫ ∂T j ε/8 ∂νv j ε,φwφ ds dt ∣∣ ≤ Kεn+2(max QT φ2(x, t) + ∫ T 0 max Ω g2(x, t)dt). (8.8) Applying (6.12) we obtain, that for x ∈ T jε/4 \ T j ε/8, ∣∣∇xwφ(P jε , x− P jε aε , t) ∣∣ ≤ Kε(max QT |φ(x, t)|+ ∫ T 0 max Ω |g(x, t)|dt). Therefore,∣∣ ∫ T 0 ∫ T j ε/4 \T j ε/8 ∇vjε,φ∇wφ dx dt ∣∣ ≤ 1 2 ∫ T 0 ∫ T j ε/4 \Gjε |∇vjε,φ| 2 dx dt+Kεn+2(max QT |φ(x, t)|2 + ∫ T 0 max Ω |g(x, t)|2dt). From here and (8.4) we obtain the estimate ε−γ max [0,T ] ‖vjε,φ‖ 2 L2(∂Gjε) + ‖∇vjε,φ‖ 2 L2(0,T ;L2(T j ε/4 \Gjε)) ≤ Kεn+2(max QT |φ(x, t)|2 + ∫ T 0 max Ω |g(x, t)|2dt). Hence, we obtain (8.2). The estimate (8.3) is then obtained from the Friedrichs inequality. � 9. Proof of Theorem 3.2 First step. We start by assuming the additional condition (3.9) on g. As mentioned in Section 5, the very weak formulation of problem (2.2) leads to the inequality ε−γ ∫ T 0 ∫ Sε ∂tφ(φ− uε) ds dt+ ∫ QTε ∇φ∇(φ− uε) dx dt + ε−γ ∫ T 0 ∫ Sε σ(φ)(φ− uε) ds dt ≥ ∫ QTε f(φ− uε) dx dt+ ε−γ ∫ T 0 ∫ Sε g(x, t)(φ(x, t)− uε(x, t)) ds dt − 1 2 ε−γ‖φ(x, 0)‖2L2(Sε) , (9.1) for any smooth test function φ(x, t). By density, we can assume, for instance, that φ(x, t) = ψ(x)η(t), for some ψ ∈ C∞0 (Ω), η ∈ C1([0, T ]). Second step. Again, under condition (3.9), we introduce the auxiliary function Wε,φ(x, t) = { wjε,φ(x, t), x ∈ T jε/4 \G j ε, t ∈ (0, T ), j ∈ Υε, 0, x ∈ Rn \ ∪j∈ΥεT j ε/4, t ∈ (0, T ), where wjε,φ is a solution of (5.3). Using Theorem 5.1 and the estimates from it we have that PεWε,φ ∈ L2(0, T ;H1 0 (Ω)) and PεWε,φ ⇀ 0 weakly in L2(0, T ;H1 0 (Ω)) as ε→ 0. EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 25 By taking (φ−Wε,φ) as a test function in (9.1), we obtain ε−γ ∫ T 0 ∫ Sε ∂t(φ−Wε,φ)(φ−Wε,φ − uε) ds dt + ∫ QTε ∇(φ−Wε,φ)∇(φ−Wε,φ − uε) dx dt + ε−γ ∫ T 0 ∫ Sε σ(φ−Wε,φ)(φ−Wε,φ − uε) ds dt ≥ ∫ QTε f(φ−Wε,φ − uε) dx dt+ ε−γ ∫ T 0 ∫ Sε g(φ−Wε,φ − uε) ds dt − 1 2 ε−γ‖φ(x, 0)−Wε,φ(x, 0)‖2L2(Sε) . (9.2) Taking into account properties of Wε,φ(x, t) we obtain lim ε→0 ∫ QTε f(φ−Wε,φ − uε) dx dt = ∫ QT f(φ− u0) dx dt, (9.3) lim ε→0 ∫ QTε ∇φ∇(φ−Wε,φ − uε)dxdt = ∫ QT ∇φ∇(φ− u0) dx dt. (9.4) Since wjε,φ is associated with the strong solution of problem (5.3), we have − ∫ QTε ∇Wε,φ∇(φ−Wε,φ − uε) dx dt = − ∑ j∈Υε ∫ T 0 ∫ T j ε/4 \Gjε ∇wjε,φ∇(φ− wjε,φ − uε) dx dt = −ε−γ ∑ j∈Υε ∫ T 0 ∫ ∂Gjε ( ∂t(φ− wjε,φ)− g + σ(φ(P jε , t)− w j ε,φ) ) × (φ− wjε,φ − uε) ds dt− ∑ j∈Υε ∫ T 0 ∫ ∂T j ε/4 ∂νw j ε,φ(φ− uε) ds dt. (9.5) Taking into account that as ε→ 0, we have Iε ≡ ε−γ ∑ j∈Υε ∫ T 0 ∫ ∂Gjε ( ∂t(φ(x, t)− φ(P jε , t)) + (σ(φ(x, t)− wjε,φ) − σ(φ(P jε , t)− w j ε,φ)− (g(x, t)− g(P jε , t)) ) (φ(x, t)− wjε,φ − uε) ds dt → 0, we conclude, that the sum of all integrals over Sε × (0, T ) tends to zero, as ε→ 0. Thus from here and from (9.2)-(9.5) we have that u0 satisfies the inequality∫ QT ∇φ∇(φ− u0) dx dt− lim ε→0 ∑ j∈Υε ∫ T 0 ∫ ∂T j ε/4 ∂νw j ε,φ(φ− uε) ds dt ≥ ∫ QT f(φ− u0) dx dt. (9.6) 26 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 Third step. We still assume condition (3.9).To estimate the second term we will apply the following Lemma. Lemma 9.1. Assume σ is Hölder continuous in (2.6). Let H[φ](x, t) be the value of H[φ] given by (3.1) for regular functions φ. Then, H[φ] ∈ L2(0, T ;L2(Ω)) and for any sequence {hε}, hε ∈ L2(0, T ;H1 0 (Ω)), such that hε ⇀ h weakly in L2(0, T ;H1 0 (Ω)) as ε→ 0, we have∣∣∣ ∑ j∈Υε ∫ T 0 ∫ ∂Gjε ∂νwφ(P jε , x− P jε aε , t)hε ds dt + Cn−2 0 ∫ T 0 ∫ Ω [∫ ∂G0 ∂νwφ(x, y, t)dsy ] h(x, t) dx dt ∣∣∣→ 0, (9.7) as ε→ 0, where ν is the unit outward normal vector to ∂T jε/4 (respectively to ∂G0) directed along the radius of the ball T jε/4 (respectively to the exterior of G0). The conclusion of this lemma can be obtained by different methods: here we point out that its proof is an easy variant of [11, Lemma 5.7] (which follows the main lines of a result of [25]). A detailed proof when σ is Lipschitz continuous can be obtained also thought [33, Lemma 9]. Nevertheless, the adaptation to the more general case of σ Hölder continuous is automatic thanks to the estimates given in Theorem 6.5 of the present paper. Using Theorem 7.5 and the previous Lemma we obtain 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(x, x− P jε aε , t)(φ− uε) ds dt = −Cn−2 0 ∫ QT H[φ](x, t)(φ− u0) dx dt. (9.8) From (9.6) and (9.8) we conclude that u0 satisfies∫ QT ∇φ∇(φ−u0) dx dt+Cn−2 0 ∫ QT H[φ](φ−u0) dx dt ≥ ∫ QT f(φ−u0) dx dt, (9.9) for any smooth test function φ(x, t) = ψ(x)η(t), ψ ∈ C∞0 (Ω), η ∈ C1[0, T ]. By denseness this inequality holds for any φ ∈ L2(0, T ;H1 0 (Ω)). Then, applying again the characterization of solutions given by monotone operators (see [6], or [21]) we deduce that u0 is a weak solution of problem (3.11). Fourth step. Assuming (3.9), to characterize the homogenized limit u0 it is impor- tant to prove that under the assumptions of Theorem 5.1 problem (3.11) has an unique solution. This is consequence of the following continuous dependence re- sult: if we suppose that u0,1 and u0,2 are two weak solutions of the problem (3.11) corresponding to f1, f2 satisfying (2.4) then, by multiplying by (u0,1 − u0,2) the corresponding equations, by the monotonicity of the operator H[φ](x, t), we obtain ‖∇(u0,1 − u0,2)‖L2(0,T ;L2(Ω)) ≤ ‖f1 − f2‖2L2(QT ). This proves the uniqueness of solutions and the proof of Theorem 5.1 under the additional condition (3.9) ends. EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 27 Fifth step. Given g ∈ L2(0, T ;C(Ω)), let gm ∈ L2(0, T ;W 1,∞(Ω)) (i.e., satisfying (3.9)) such that gm → g in L2(0, T ;C(Ω)) and such that ‖gm‖L2(0,T ;C(Ω)) ≤ ‖g‖L2(0,T ;C(Ω)) + 1. (9.10) By the monotonicity in of the abstract operator associated with problem (2.2) (see, e.g. [3]) we know that if uε,m is the solution of problem (2.2) corresponding to the boundary data gm, then ‖uε − uε,m‖2L2(0,T ;H1(Ωε,∂Ω)) + ε−γ‖uε − uε,m‖2L2(0,T ;L2(Sε)) ≤ K(‖g−gm‖2L2(0,T ;C(Ω)) ). (9.11) By applying the four previous steps to the boundary data gm we know the existence of a unique solution u0,m ∈ L2(0, T ;H1 0 (Ω)) of the family of problems (depending of the parameter t ∈ (0, T )) −∆xu0,m(x, t) + Cn−2 0 Hm[u0,m](x, t) = f(x, t), x ∈ Ω, t ∈ (0, T ), u0,m(x, t) = 0, x ∈ ∂Ω, t ∈ (0, T ), (9.12) with Hm[u0](x, t) defined by the nonlocal operator (3.1), corresponding to the boundary data gm, for a.e. x ∈ Ω. We define u0 ∈ L2(0, T ;H1 0 (Ω)) be the unique weak solution of the family of problems (depending of the parameter t ∈ (0, T )) associate to the boundary data g ∈ L2(0, T ;C(Ω)), −∆xu0(x, t) + Cn−2 0 H[u0](x, t) = f(x, t), x ∈ Ω, t ∈ (0, T ), u0(x, t) = 0, x ∈ ∂Ω, t ∈ (0, T ), (9.13) with H[u0](x, t) defined by the nonlocal operator (3.1) for a.e. x ∈ Ω. Note that u0 is well defined for g merely in L2(0, T ;C(Ω)) (i.e., without requiring condition (3.9)). Then we have ∆(u0 − u0,m) = Cn−2 0 [H[u0]−Hm[u0,m]] = Cn−2 0 [Hm[u0]−Hm[u0,m]] + Cn−2 0 [H[u0]−Hm[u0]] . As usual, by multiplying by u0 − u0,m and applying Poincaré inequality we obtain that u0,m → u0 in L2(0, T ;H1 0 (Ω)) (recall the estimates given in Theorems 7.1 and 7.6 and the fact that the corresponding constants are uniform in m). Then, given g ∈ L2(0, T ;C(Ω)) and a test function v ∈ L2(0, T ;H1 0 (Ω)) we have∫ T 0 ∫ Ω (Pεu g ε−u g 0)v dx dt = ∫ T 0 ∫ Ω (Pεu g ε − Pεugmε )v dx dt+ ∫ T 0 ∫ Ω (Pεu gm ε −u gm 0 )v dx dt + ∫ T 0 ∫ Ω (ugm0 −u g 0)v dx dt = I1 + I2 + I3. Thus, we obtain that for any δ > 0, and for any ε > 0 there exists a m0 ∈ N such that, for any m ≥ m0, we have |I1| ≤ δ (note that m0 is independent on ε since gm → g in L2(0, T ;C(Ω)), estimate (9.10) and the continuous dependence estimate (9.11)). On the other hand, thanks to the 28 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 fourth step, we know that for any δ > 0, there exists ε0 > 0 (independent on m) such that for any ε ≥ ε0, we have |I2| ≤ δ (the independence on m of ε0 comes from the fact that the a priori estimates for passing to the weak limit in Theorem 2.1 are only dependent on ‖g‖L2(0,T ;C(Ω)) and we know the uniform estimate (9.10)). Finally, we know that ugm0 → ug0 in L2(0, T ;H1 0 (Ω)) and thus, for any δ > 0, there exists a m̂0 ∈ N such that, for any m ≥ m̂0, we have |I3| ≤ δ. In conclusion, for any δ > 0, and for any ε ≥ ε0 we know that by taking as intermediate step the approximation of g by gm, with m ≥ max{m0, m̂0}, we obtain that ∣∣ ∫ T 0 ∫ Ω (Pεu g ε−u g 0)v dx dt ∣∣ ≤ 3δ, which implies the weak convergence for a boundary datum g ∈ L2(0, T ;C(Ω)) and the proof of Theorem 5.1 is complete. Remark 9.2. Our treatment remains valid if G0 is the union of a finite family of sets Gm0 , m = 1, 2, . . . ,M , satisfying the same geometric properties (especially the critical size assumption 2.3). Indeed, the adaptation of the test function made in Step 1 of the proof of Theorem 3.2 is local and can be done, separately over disjoint neighborhoods of the associate Gmε , m = 1, 2, . . . ,M . In this way H[φ](x, t) = M∑ m=1 Hm[φ](x, t), with Hm[φ](x, t) = ∫ ∂Gm0 ∂νw m φ (x, y, t)dsy, where wmφ (y, t) is the corresponding solution of the associate problem (3.2). Remark 9.3. If n = 2 , aε = ε exp(−α2/ε2) and uε is a weak solution of the problem −∆uε = f(x, t), (x, t) ∈ QTε , β(ε)∂tuε + ∂νuε + β(ε)σ(x, uε) = β(ε)g(x, t), (x, t) ∈ Sε × (0, T ), uε = 0, (x, t) ∈ ∂Ω× (0, T ), uε(x, 0) = 0, x ∈ Sε, (9.14) where β(ε) = ε exp(α2/ε2), then, arguing as in [16] the pair (u0,H(u0)), defined in (2.9) and (3.1) respectively, is a weak solution of the problem −∆u0 + 2π α2 (u0(x, t)−H[u0](x, t)) = f(x, t), (x, t) ∈ QT , ∂tH[u0] + 2π α2|∂G0| H[u0] + σ(x,H[u0]) = g(x, t) + 2π α2|∂G0| u0, x ∈ Ω, t ∈ (0, T ), u0(x, t) = 0, (x, t) ∈ ∂Ω× (0, T ), H[u0](x, 0) = 0, x ∈ Ω. (9.15) Remark 9.4. We point out that although we do not need to justify the existence of solutions of problem (3.11), since u0 was built by passing to the limit in uε , nevertheless it is useful to analyze some properties of the associated operators since EJDE-2022/52 A STRANGE NON-LOCAL MONOTONE OPERATOR 29 they are useful , for instance, to the extension to the case in which f is merely in L1(0, T ;L1(Ω)) following, for instance, the abstract results of [7]. The existence of a weak solution to the problem (3.11) follows from the monotonicity of H[φ] understood in the sense of Theorem 7.1 and the method of monotone operators. If we introduce the operator A : L2(0, T ;H1 0 (Ω))→ L2(0, T ;H−1(Ω)) given by (A(u), v) = ∫ QT ∇u∇v dx dt+ Cn−2 0 ∫ QT H[u]v dx dt, then A is a monotone operator, since by Theorem 7.1, (A(u)−A(v), u−v) = ∫ QT |∇(u−v)|2 dx dt+Cn−2 0 ∫ QT (H[u]−H[v])(u−v) dx dt ≥ 0. Moreover, A is a coercive operator since (A(u), u) = ∫ QT |∇u|2 dx dt+ Cn−2 0 ∫ QT H[u]u dx dt ≥ ‖u‖2L2(0,T ;H1 0 (Ω)), and hence (A(u), u) ‖u‖L2(0,T ;H1 0 (Ω)) ≥ ‖u‖L2(0,T ;H1 0 (Ω)) →∞, as ‖u‖L2(0,T ;H1 0 (Ω)) →∞. From (7.1) we conclude that∫ T 0 ∫ Ω H2[u](x, t) dx dt ≤ K(‖u‖2L2(0,T ;L2(Ω)) + ‖g‖2L2(0,T ;L2(Ω))). (9.16) Using this inequality we obtain |(A(u), v)| ≤ ‖u‖L(0,T ;H1 0 (Ω))‖v‖L2(0,T ;H1 0 (Ω)) + (‖u‖L2(0,T ;L2(Ω)) + ‖g‖L2(0,T ;L2(Ω)))‖v‖L2(0,T ;L2(Ω)), which proves that A is a nonlinear bounded operator. Notice that this also implies the existence of a weak solution of (3.11) even if f ∈ L2(0, T ;H−1(Ω)) (see [21]). Remark 9.5. We conjecture that the main result of this paper remains valid when σ : D(σ) ⊂ R → P(R) is a maximal monotone graph of R2 such that σ(0) 3 0 for x ∈ Ω, as, for instance, σ corresponds to the zero-order chemical reactions (α = 0) or when σ represents the case of Signorini type boundary conditions. By adapting to this framework some abstract results on the Cauchy problem associated with subdifferential operators (see, e.g., [17, 18, 3] and their references) it is possible to prove that, for any ε > 0, problem (2.2) has a unique strong solution and the following estimate holds ε−γ‖∂tuε‖L2(0,T ;L2(Sε)) +‖uε‖L2(0,T ;H1(Ωε,∂Ω)) +ε−γ‖uε‖L2(0,T ;L2(Sε)) ≤ K, (9.17) where K is a constant independent on ε. Indeed, such as shown in [17] and [3], we know that the operator uε → ∂νuε + σ(uε) is not only a maximal monotone operator on L2(Sε) (such as it can be deduced from the results of [21]) but, in fact, it is the subdifferential of the convex function is the subdifferential of the lower semicontinuous convex and proper function Φt : L2(Sε)→ R, Φt(vε) =  εγ 2 ∫ Ωε ‖∇uε‖2dx− εγ ∫ Ωε f(x, t)uε(x) dx+ ∫ Sε j(vε(σ)) dσ if uε ∈ H1(Ωε, ∂Ω), uε = vε, j(vε) ∈ L1(Sε), +∞ otherwise, 30 J. I. DÍAZ, T. A. SHAPOSHNIKOVA, M. N. ZUBOVA EJDE-2022/52 for each t ∈ [0, T ], where ∂j(s) = σ(s) for any s in the domain of σ. Note that the t-dependence in Φt(vε) because of the presence of the term f(x, t) satisfying (2.4), which obviously is independent on vε. Then, problem (2.2) can be rewritten as a particular case of an abstract Cauchy problem of the form dvε dt + ∂Φt(vε) 3 g(., t) in L2(Sε), for t ∈ (0, T ), vε(0) = 0. (9.18) Then, since vε(0) = 0 ∈ D(Φt), a.e. t ∈ (0, T ), and g ∈ L2(0, T ;L2(Sε)), by [5, Théorème 3.6] (if Φt is time independent) or [30, Theorem 1] we know that the mild solution vε ∈ C([0, T ];L2(Sε)) of the Cauchy problem (9.18) is, in fact, a strong solution such that dvε dt ∈ L2(0, T ;L2(Sε)) and that the application t → Φt(vε) is absolutely continuous on [0, T ]. It seems possible to generalize all the arguments used in this paper to get the conclusion of Theorem 5.1. That was carried out in [13] for the stationary case associated with (2.2) by means of an argument of regularization of σ (as to be Lipschitz continuous) and then by passing to the limit. Notice that all the arguments of the present paper were obtained directly for the case of σ non-Lipschitz continuous but satisfying merely the Hölder regularity condition (2.6). The extension to the case σ a maximal monotone graph of R2, including in particular the case of Signorini type boundary conditions, will be the object of a separated study. Remark 9.6. Another possible generalization of the results of this paper concerns the consideration of nonzero initial conditions in the formulation of problem (2.2): −∆xuε = f(x, t), (x, t) ∈ QTε , ε−γ∂tuε + ∂νuε + ε−γσ(uε) = ε−γg(x, t), (x, t) ∈ STε , uε(x, t) = 0, (x, t) ∈ ΓT , uε(x, 0) = U0(x), x ∈ Sε, (9.19) for some U0 ∈ H1/2(Sε), U0 6= 0. Such as it was pointed out in [14, Remark B.2] a new curious “strange phenomenon” arise then: Pεuε ⇀ U , weakly in L2(0, T ;H1 0 (Ω)), in fact U ∈ C([0, T ];H1 0 (Ω)), the homogenized equation is ex- actly of the same type than problem (3.11) if t > 0 but, in general, U(x, 0) 6= U0(x) since U(x, 0) solves the modified problem −∆xU0(x, 0) +A0(U0(x, 0)− U0(x)) = f(x, 0), x ∈ Ω, U0(., 0) = 0, x ∈ ∂Ω, . (9.20) for a suitable constant A0 > 0. The main steps of the proof of this result (getting a “linear strange term”) were indicated in [14] (see, for instance, page 12 and notice that in our case p = 2). To avoid additional technical details we are not developing this property here. Notice that, in fact, if f(x, 0) 6= 0, and U0 ≡ 0 this “strange initial datum” arises since u0(x, 0) 6= 0 on Ω, even if uε(x, 0) = 0 on Sε. Remark 9.7. 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Yotsutani; Evolution equations associated with subdifferentials, J. Math. Soc. Japan 31 (1978), 623-646. [31] M. N. Zubova, T. A. Shaposhnikova; Homogenization of boundary value problems in perfo- rated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem, Diff. Eq. 2011. V. 47. N.1. P. 1-13. [32] M. N. Zubova, T. A. Shaposhnikova; Homogenization limit for the diffusion equation in a domain perforated along (n − 1) dimensional manifold with dynamic condition on the boundary of the perforations: critical case. Doklady Mathematics, Doklady Mathematics, 99,3(2019), p. 245-251. [33] M. N. Zubova, T. A. Shaposhnikova; Homogenization of a boundary value problem in a domain perforated by cavities of arbitrary shape with a general nonlinear boundary condition on their boundaries: the case of critical values of the parameters. Journal of Math. Science, 244 (2), (2020), 235-253. [34] M. N. Zubova, T. A. Shaposhnikova; Appearance of a nonlocal monotone operator in trans- mission conditions when homogenizing the Poisson equations in a domain perforated along an (n-1) - dimensional manifold by sets of arbitrary shape and critical size with nonlin- ear dynamic boundary conditions on the boundary of perforations. Journal of Mathematics Sciences, V. 255, No. 4, 2021. DOI: 10.1007/s10958-021-05382-7. Jesús Ildefonso D́ıaz Instituto de Matematica Interdisciplinar, Universidad Complutense, Madrid, 28040 Spain Email address: ji diaz@ucm.es Tatiana A. Shaposnikova Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Rus- sia Email address: shaposh.tan@mail.ru Maria N. Zubova Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Rus- sia Email address: zubovnv@mail.ru 1. Introduction 2. Problem statement 3. Statement of main results 4. Proof of Theorem 3.2 5. Auxiliary Gj-capacity problems to adapt the global test function 6. G0-capacity for the exterior problem 7. Properties of the operator H[] 8. Asymptotic similarity between the capacity functions w ,j and w(P j,x-Pja,t) 9. Proof of Theorem 3.2 Acknowledgements References