Asymptotic analysis of a semi-linear elliptic system in perforated domains: well-posedness and correctors for the homogenization limit
AAsymptotic analysis of a semi-linear elliptic system in perforated domains:well-posedness and correctors for the homogenization limit
Vo Anh Khoa a, ∗ , Adrian Muntean b a Mathematics and Computer Science Division, Gran Sasso Science Institute, L’Aquila, Italy b Department of Mathematics and Computer Science, Karlstad University, Sweden
Abstract
In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic systemposed in perforated media. The model presented here explores the interplay between stationary diffusion and bothsurface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling)for alike systems and particularly in justifying rigorously the obtained averaged descriptions. Essentially, we prove thewell-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrationsand then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergencerate of the asymptotic approximates to the macroscopic limit concentrations. Our techniques include Moser-like iterationtechniques, a variational formulation, two-scale asymptotic expansions as well as energy-like estimates.
Keywords:
Corrector estimates, Homogenization, Elliptic systems, Perforated domains
1. Introduction
We study the semi-linear elliptic boundary-value problem of the form( P ε ) : A ε u εi ≡ ∇ · ( − d εi ∇ u εi ) = R i ( u ε ) , in Ω ε ⊂ R d ,d εi ∇ u εi · n = ε ( a εi u εi − b εi F i ( u εi )) , on Γ ε ,u εi = 0 , on Γ ext , (1.1)for i ∈ { , ..., N } ( N ≥ , d ∈ { , } ). Following [1], this system models the diffusion in a porous medium as well asthe aggregation, dissociation and surface deposition of N interacting populations of colloidal particles indexed by u εi . Asshort-hand notation, u ε := ( u ε , ..., u εN ) points out the vector of these concentrations. Such scenarios arise in drug-deliverymechanisms in human bodies and often includes cross- and thermo-diffusion which are triggers of our motivation (compare [2] for the Sorret and Dufour effects and [3, 4] for related cross-diffusion and chemotaxis-like systems).The model (1.1) involves a number of parameters: d εi represents molecular diffusion coefficients, R i represents thevolume reaction rate, a εi , b εi are the so-called deposition coefficients, while F i indicates a surface chemical reaction for theimmobile species. We refer to (1.1) as problem ( P ε ).The main purpose of this paper is to obtain corrector estimates that delimitate the error made when homogenizing (averaging, upscaling, coarse graining...) the problem ( P ε ), i.e. we want to estimate the speed of convergence as ε → ∗ Corresponding author
Email addresses: [email protected], [email protected] (Vo Anh Khoa), [email protected] (Adrian Muntean)
October 2, 2018 a r X i v : . [ m a t h . A P ] D ec ustify rigorously the upscaled models derived in [1] and prepare the playground to obtain corrector estimates for thethermo-diffusion scenario discussed in [5]. From the corrector estimates perspective, the major mathematical difficultywe meet here is the presence of the nonlinear surface reaction term. To quantify its contribution to the corrector terms we use an energy-like approach very much inspired by [6]. The main result of the paper is Theorem 10 where we statethe corrector estimate. It is worth noting that this work goes along the line open by our works [7] (correctors via periodicunfolding) and [8] (correctors by special test functions adapted to the local periodicity of the microstructures). Analternative strategy to derive correctors for our scenario could in principle exclusively rely on periodic unfolding, refoldingand defect operators approach if the boundary conditions along the microstructure would be of homogeneous Neumann type; compare [9] and [10].The corrector estimates obtained with this framework can be further used to design convergent multiscale finite elementmethods for the studied PDE system (see e.g. [11] for the basic idea of the MsFEM approach and [12] for an applicationto perforated media).The paper is organized as follows: In Section 2 we start off with a set of technical preliminaries focusing especially on the working assumptions on the data and the description of the microstructure of the porous medium. The weaksolvability of the microscopic model is established in Section 3. The homogenization method is applied in Section 4 tothe problem ( P ε ). This is the place where we derive the corrector estimates and establish herewith the convergence rateof the homogenization process. A brief discussion (compare Section 5) closes the paper.
2. Preliminaries The geometry of our porous medium is sketched in Figure 2.1 (left), together with the choice of perforation (referredhere to also as ”microstructure”) cf. Figure 2.1 (right). We refer the reader to [13] for a concise mathematical represen-tation of the perforated geometry. In the same spirit, take Ω be a bounded open domain in R d with a piecewise smoothboundary Γ = ∂ Ω. Let Y be the unit representative cell, i.e. Y := (cid:40) d (cid:88) i =1 λ i (cid:126)e i : 0 < λ i < (cid:41) , where we denote by (cid:126)e i by i th unit vector in R d .Take Y the open subset of Y with a piecewise smooth boundary ∂Y in such a way that Y ⊂ Y . In the porous mediaterminology, Y is the unit cell made of two parts: the gas phase (pore space) Y \ Y and the solid phase Y .Let Z ⊂ R d be a hypercube. Then for X ⊂ Z we denote by X k the shifted subset X k := X + d (cid:88) i =1 k i (cid:126)e i , where k = ( k , ..., k d ) ∈ Z d is a vector of indices. Setting Y = Y \ Y , we now define the pore skeleton byΩ ε := (cid:91) k ∈ Z d (cid:8) εY k : Y k ⊂ Ω (cid:9) , where ε is observed as a given scale factor or homogenization parameter.It thus comes out that the total pore space is Ω ε := Ω \ Ω ε , εY k the ε -homotetic set of Y k , while the total pore surface of the skeleton is denoted byΓ ε := ∂ Ω ε = (cid:91) k ∈ Z d (cid:8) ε Γ k : Γ k ⊂ Ω (cid:9) . The exterior boundary of Ω ε is certainly a hypersurface in R d , denoted by Γ ext = ∂ Ω ε \ Γ ε , where it has a nonzero( d − ext ∩ Γ ε = ∅ and coincides with Γ. Moreover, n denotes the unit normal vector toΓ ε . Finally, our perforated domain Ω ε is assumed to be connected through the gas phase. Notice here that Γ ext is smooth. Ω Γ = ∂ ΩΩ ε Y YY Figure 2.1: Admissible two-dimensional perforated domain (left) and basic geometry of the microstructure (right).
N.B. This paper aims at understanding the problem in two or three space dimensions. However, all our results holdalso for d ≥
3. Throughout this paper, C denotes a generic constant which can change from line to line. If not otherwisestated, the constant C is independent of the choice of ε . We denote by x ∈ Ω ε the macroscopic variable and by y = x/ε the microscopic variable representing fast variationsat the microscopic geometry. With this convention in view, we write d εi ( x ) = d i (cid:16) xε (cid:17) = d i ( y ) . A similar meaning is given to all involved ”oscillating” data, e.g. to a εi ( x ), b εi ( x ). We now make the following set of assumptions:(A ) the diffusion coefficient d εi ∈ L ∞ (cid:0) R d (cid:1) is Y -periodic, and it exists a positive constant α i such that d i ( y ) ξ i ξ j ≥ α i | ξ | for any ξ ∈ R d . ) the deposition coefficients a εi , b εi ∈ L ∞ (Γ ε ) are positive and Y -periodic.(A ) the reaction rates R i : Ω ε × [0 , ∞ ) N → R and F i : Γ ε × [0 , ∞ ) → R are Carath´eodory functions, i.e. they are,respectively, continuous in [0 , ∞ ) N and [0 , ∞ ) with respect to x variable (in the “almost all” sense), and measurable inΩ ε and Γ ε with essential boundedness with respect to concentrations u εi ≥ (A ) The chemical rate R i and F i are sublinear in the sense that for any p = ( p , ..., p N ) R i ( p ) ≤ C N (cid:88) j =1 ,j (cid:54) = i p i p j for p ≥ ,F i ( p i ) ≤ C (1 + p i ) for p i ≥ , for any p = ( p , ..., p N ).Furthermore, assume that R i ( p ) /p i is decreasing and F i ( p i ) /p i is increasing in p i for any p > ) For every ε >
0, there exist vectors ( x -dependent) r ε , r ε ∞ , f ε , f ε ∞ whose elements are r ε ,i = lim u εi → + R i ( u ε ) u εi , r ε ∞ ,i = lim u εi →∞ R i ( u ε ) u εi ,f ε ,i = lim u εi → + ε (cid:18) a εi − b εi F i ( u εi ) u εi (cid:19) , f ε ∞ ,i = lim u εi →∞ ε (cid:18) a εi − b εi F i ( u εi ) u εi (cid:19) . (A ) R i and F i satisfy the growth conditions: | R i ( x, p ) | ≤ C N (cid:88) i =1 (1 + p i ) for p ≥ , (2.1) | a εi p i − b εi F i ( p i ) | ≤ C (1 + p i ) for p i ≥ . (2.2)Let us define the function space V ε := (cid:8) v ∈ H (Ω ε ) | v = 0 on Γ ext (cid:9) , which is a closed subspace of the Hilbert space H (Ω ε ), and thus endowed with the semi-norm (cid:107) v (cid:107) V ε = (cid:32) d (cid:88) i =1 (cid:90) Ω ε (cid:12)(cid:12)(cid:12)(cid:12) ∂v∂x i (cid:12)(cid:12)(cid:12)(cid:12) dx (cid:33) / for all v ∈ V ε . Obviously, this norm is equivalent to the usual H -norm by the Poincar´e inequality. Moreover, this equivalence isuniform in ε (cf. [6, Lemma 2.1]).We introduce the Hilbert spaces H (Ω ε ) = L (Ω ε ) × ... × L (Ω ε ) , V ε = V ε × ... × V ε , with the inner products defined respectively by (cid:104) u, v (cid:105) H (Ω ε ) := N (cid:88) i =1 (cid:90) Ω ε u i v i dx, u = ( u , ..., u N ) , v = ( v , ..., v N ) ∈ H (Ω ε ) , (cid:104) u, v (cid:105) V ε := N (cid:88) i =1 n (cid:88) j =1 (cid:90) Ω ε ∂u i ∂x j ∂v i ∂x j dx, u = ( u , ..., u N ) , v = ( v , ..., v N ) ∈ V ε . Furthermore, the notation H (Γ ε ) indicates the corresponding product of L (Γ ε ) spaces. For q ∈ (2 , ∞ ], the followingspaces are also used W q (Ω ε ) = L q (Ω ε ) × ... × L q (Ω ε ) , W q (Γ ε ) = L q (Γ ε ) × ... × L q (Γ ε ) . . Well-posedness of the microscopic model Before studying the asymptotics behaviour as ε → introduced in [15] an energy minimization approach to guarantee the existence, uniqueness and positivity results for thesemi-linear elliptic problem with zero Dirichlet boundary conditions. Very recently, Garc´ıa-Meli´an et al. [16] extended theresult in [15] (and also of other previous works including [17, 18]) to problems involving nonlinear boundary conditions ofmixed type. For what we are concerned here, we will use Moser-like iterations technique (see the original works by Moser[19, 20]) to prove L ∞ -bounds for all concentrations and then follow the strategy provided by Brezis and Oswald [15] to study the well-posedness of ( P ε ). Definition 1.
A function u ε ∈ V ε is a weak solution to ( P ε ) provided that N (cid:88) i =1 (cid:90) Ω ε ( d εi ∇ u εi ∇ ϕ i − R i ( u ε ) ϕ i ) dx − N (cid:88) i =1 ε (cid:90) Γ ε ( a εi u εi − b εi F i ( u εi )) ϕ i dS ε = 0 for all ϕ ∈ V ε . (3.1) Definition 2.
By means of the usual variational characterization, the principal eigenvalue of ( P ε ) is defined by λ ( p ε , q ε ) := inf u ε ∈V ε , N (cid:88) i =1 | u εi | (cid:54) =0 N (cid:88) i =1 (cid:18) α (cid:90) Ω ε |∇ x u εi | dx − N (cid:90) Ω ε p εi | u εi | dx − N (cid:90) Γ ε q εi | u εi | dS ε (cid:19) N (cid:88) i =1 (cid:90) Ω ε | u εi | dx , (3.2)where p εi and q εi are measurable such that either they are simultaneously bounded from above or from below (this leadsto λ ∈ ( −∞ , ∞ ] or λ ∈ [ −∞ , ∞ ), correspondingly). Here, we denote α := min { α , ..., α N } . Lemma 3.
Assume ( A ) - ( A ) and replace ( A ) by ( A ) . Let u ε ∈ V ε ∩ H (Γ ε ) be a weak solution to ( P ε ) , then u ε ∈W ∞ (Ω ε ) and it exists an ε -independent constant C > such that (cid:107) u ε (cid:107) W ∞ (Ω ε ) ≤ C (cid:16) (cid:107) u ε (cid:107) H (Ω ε ) + (cid:107) u ε (cid:107) H (Γ ε ) (cid:17) . Proof.
Let β ≥ k i > i = 1 , N . We begin by introducing a vector ϕ ε of test functions ϕ εi = min (cid:110) v β + i , k β + i (cid:111) − v i = u εi + 1 with u εi as in (3.1). Thus, it is straightforward to show that ϕ ε ∈ V ε ∩ H (Γ ε ). We have α (cid:18) β + 12 (cid:19) N (cid:88) i =1 (cid:90) { v i 2) if d ≥ 3, and 2 ∗ ∂ Ω ε = ∞ if d = 2 (cf. [21]). Therefore, given q ∈ (2 , ∗ ] we apply this embeddingto (3.3) with the aid of (3.4) and then obtain4 α (cid:0) β + (cid:1)(cid:0) β + (cid:1) N (cid:88) i =1 (cid:34)(cid:18)(cid:90) Γ ε | ψ i | q dS ε (cid:19) q − (cid:90) Ω ε | ψ i | dx (cid:35) ≤ C N (cid:88) i =1 (cid:18)(cid:90) Ω ε v β + i dx + (cid:90) Γ ε v β + i dS ε (cid:19) . (3.5)We see that ψ i ≤ v β + and also 1 β + ≤ (cid:0) β + (cid:1)(cid:0) β + (cid:1) ≤ β ≥ 1. As a result, (3.5) yields N (cid:88) i =1 (cid:18)(cid:90) Γ ε | ψ i | q dS ε (cid:19) q ≤ Cα − (cid:18) β + 32 (cid:19) N (cid:88) i =1 (cid:18)(cid:90) Ω ε v β + i dx + (cid:90) Γ ε v β + i dS ε (cid:19) . (3.6)Our next aim is to show that if for some s ≥ u ε ∈ W s (Ω ε ) ∩ W s (Γ ε ), then u ε ∈ W ks (Ω ε ) ∩ W ks (Γ ε ) for k > ε -level. In fact, assume that u ε ∈ W β + (Ω ε ) ∩ W β + (Γ ε ) then letting k → ∞ in (3.6) gives N (cid:88) i =1 (cid:18)(cid:90) Γ ε | v i | q ( β + ) dS ε (cid:19) q ≤ C (cid:18) β + 32 (cid:19) N (cid:88) i =1 (cid:18)(cid:90) Ω ε v β + i dx + (cid:90) Γ ε v β + i dS ε (cid:19) . (3.7)One obtains in the same manner that by the embedding H (Ω ε ) ⊂ L q (Ω ε ) (this is valid for 1 ≤ q ≤ ∗ Ω ε where2 ∗ Ω ε = 2 d/ ( d − 2) if d ≥ 3, and 2 ∗ Ω ε = ∞ if d = 2; thus q given before is definitely valid), we are led to the followingestimate N (cid:88) i =1 (cid:18)(cid:90) Ω ε | v i | q ( β + ) dx (cid:19) q ≤ C (cid:18) β + 32 (cid:19) N (cid:88) i =1 (cid:18)(cid:90) Ω ε v β + i dx + (cid:90) Γ ε v β + i dS ε (cid:19) . (3.8)Combining (3.7), (3.8) and the Minkowski inequality enables us to get (cid:18)(cid:90) Ω ε | v i | q ( β + ) dx + (cid:90) Γ ε | v i | q ( β + ) dS ε (cid:19) q ≤ C (cid:18) β + 32 (cid:19) N (cid:88) i =1 (cid:18)(cid:90) Ω ε v β + i dx + (cid:90) Γ ε v β + i dS ε (cid:19) , for all i ∈ { , ..., N } , which easily leads to, by raising to the power 1 / (cid:0) β + (cid:1) , the fact that u εi ∈ L q ( β + ) (Ω ε ) ∩ L q ( β + ) (Γ ε ) for all i ∈ { , ..., N } ; and hence u ε ∈ W q ( β + ) (Ω ε ) ∩ W q ( β + ) (Γ ε ).The constant k is indicated by q/ > 1. Thus, if we choose q and β such that β + 32 = 2 (cid:16) q (cid:17) n for n = 0 , , , ..., and iterating the above estimate, we obtain, by induction, that (cid:107) v (cid:107) ( q ) n ≤ n (cid:89) j =0 (cid:18) (cid:16) q (cid:17) j C (cid:19) ( q ) j (cid:107) v (cid:107) , (3.9)where we have denoted by (cid:107) v (cid:107) r := N (cid:88) i =1 (cid:18)(cid:90) Ω ε | v i | r dx + (cid:90) Γ ε | v i | r dS ε (cid:19) r . 6t is interesting to point out that since the series (cid:80) ∞ n =0 (cid:16) q (cid:17) n and (cid:80) ∞ n =0 n (cid:16) q (cid:17) n are convergent for q > 2, we have n (cid:89) j =0 (cid:18) (cid:16) q (cid:17) j C (cid:19) ( q ) j < (cid:113) (2 C ) (cid:80) ∞ n =0 ( q ) n q (cid:80) ∞ n =0 n ( q ) n = C. Therefore, the constant in the right-hand side of (3.9) is indeed independent of n , and by passing n → ∞ in (3.9), i.e.in the inequality, (cid:107) v (cid:107) W ( q ) n (Ω ε ) ≤ C (cid:16) (cid:107) v (cid:107) H (Ω ε ) + (cid:107) v (cid:107) H (Γ ε ) (cid:17) , we finally obtain (cid:107) v (cid:107) W ∞ (Ω ε ) ≤ C (cid:16) (cid:107) v (cid:107) H (Ω ε ) + (cid:107) v (cid:107) H (Γ ε ) (cid:17) . Consequently, recalling v i = u εi + 1, we have: (cid:107) u ε (cid:107) W ∞ (Ω ε ) ≤ C (cid:16) (cid:107) u ε (cid:107) H (Ω ε ) + (cid:107) u ε (cid:107) H (Γ ε ) (cid:17) . This step completes the proof of the lemma. Remark . Using the trace inequality (cf. [6, Lemma 2.31]) and the norm equivalence between V ε and H (Ω ε ), if u ε ∈ V ε then the result in Lemma 3 reads (cid:107) u ε (cid:107) W ∞ (Ω ε ) ≤ C (cid:16) ε − / (cid:107) u ε (cid:107) H (Ω ε ) + (cid:107) u ε (cid:107) V ε (cid:17) ≤ C (cid:16) ε − / (cid:107) u ε (cid:107) V ε (cid:17) . Lemma 5. Assume ( A ) - ( A ) and that λ ( r ε ∞ , f ε ∞ ) > and λ ( r ε , f ε ) < hold. We define the following functional J [ u ε ] := 12 N (cid:88) i =1 (cid:90) Ω ε d εi |∇ u εi | dx − N (cid:88) i =1 (cid:90) Ω ε R i ( x, u ε ) dx − N (cid:88) i =1 (cid:90) Γ ε F i ( x, u εi ) dS ε , where R i ( x, u ε ) := (cid:90) u εi R i ( x, u ε , ...s i , ..., u εN ) ds i , F i ( x, u εi ) := (cid:90) u εi ( a εi s − b εi F i ( s )) ds, and the nonlinear terms are extended to be R i ( x, and F i ( x, for u εi ≤ . Then J is coercive on V ε and lowersemi-continuous for V ε . Moreover, there exists φ ∈ V ε such that J [ φ ] < .Proof. Step 1: (Coerciveness)Suppose, by contradiction, that it exists a sequence { u ε,m } ⊂ V ε such that (cid:107) u ε,m (cid:107) V ε → ∞ while J [ u ε,m ] ≤ C . Setting s i,m = (cid:18)(cid:90) Γ ε | u ε,mi | dS ε (cid:19) / , t i,m = (cid:18)(cid:90) Ω ε | u ε,mi | dx (cid:19) / , (3.10)we say that (cid:80) Ni =1 t i,m → ∞ up to a subsequence as m → ∞ . Indeed, the assumption J [ u ε,m ] ≤ C yields that12 N (cid:88) i =1 (cid:90) Ω ε d εi |∇ u ε,mi | dx ≤ N (cid:88) i =1 (cid:90) Ω ε R i ( x, u ε,m ) dx + N (cid:88) i =1 (cid:90) Γ ε F i ( x, u ε,mi ) dS ε + C, (3.11)which, in combination with (3.10) and (A ), leads to12 N (cid:88) i =1 (cid:90) Ω ε d εi |∇ u ε,mi | dx ≤ C ( N ) (cid:32) N (cid:88) i =1 t i,m + N (cid:88) i =1 s i,m (cid:33) . (3.12)7ere, if (cid:80) Ni =1 t i,m is convergent, then (cid:80) Ni =1 s i,m cannot be bounded. While putting v i,m = u ε,mi / N (cid:88) i =1 s i,m , it enables us to derive that N (cid:88) i =1 (cid:90) Ω ε |∇ v i,m | dx = N (cid:88) i =1 (cid:90) Ω ε |∇ u ε,mi | dx (cid:32) N (cid:88) i =1 s i,m (cid:33) ≤ N (cid:88) i =1 (cid:90) Ω ε |∇ u ε,mi | dx N (cid:88) i =1 s i,m . (3.13)If we assign α := min { α , ..., α N } > 0, then it follows from (3.12) and (3.13) that α N (cid:88) i =1 (cid:90) Ω ε |∇ v i,m | dx ≤ N (cid:88) i =1 (cid:90) Ω ε d εi |∇ u ε,mi | dx N (cid:88) i =1 s i,m ≤ C ( N ) N (cid:88) i =1 t i,mN (cid:88) i =1 s i,m + 1 N (cid:88) i =1 s i,m ≤ C ( N ) . Now, we claim that there exists v i ∈ V ε such that v i,m (cid:42) v i weakly in V ε , and then strongly in L (Ω ε ) and in L (Γ ε ).However, it implies here a contradiction. It is because we have v i ≡ ε for all i = 1 , N while N (cid:88) i =1 (cid:90) Γ ε | v i | dS ε = (cid:32) N (cid:88) i =1 s i (cid:33) − N (cid:88) i =1 (cid:90) Γ ε | u εi | dS ε ≥ N − > . Let us now assume that (cid:80) Ni =1 t i,m is divergent. By putting w i,m = u ε,mi / N (cid:88) i =1 t i,m , we have, in the same manner, that α N (cid:88) i =1 (cid:90) Ω ε |∇ w i,m | dx ≤ C ( N ) N (cid:88) i =1 t i,m + N (cid:88) i =1 s i,mN (cid:88) i =1 t i,m . From (3.10), we know that N (cid:88) i =1 (cid:90) Ω ε | w i,m | dx = (cid:32) N (cid:88) i =1 t i,m (cid:33) − N (cid:88) i =1 (cid:90) Ω ε | u ε,mi | dx ≤ , (3.14)and N (cid:88) i =1 (cid:90) Γ ε | w i,m | dS ε ≥ N − (cid:32) N (cid:88) i =1 t i,m (cid:33) − N (cid:88) i =1 (cid:90) Γ ε | u ε,mi | dS ε ≥ N (cid:88) i =1 s i,m N N (cid:88) i =1 t i,m . (3.15)8ombining the trace inequality (cf. [6, Lemma 2.31]) with (3.14) and (3.15), we obtain N (cid:88) i =1 s i,mN (cid:88) i =1 t i,m ≤ N N (cid:88) i =1 (cid:90) Γ ε | w i,m | dS ε ≤ CN (cid:32) N (cid:88) i =1 (cid:18)(cid:90) Ω ε | w i,m | dx (cid:19) / (cid:18)(cid:90) Ω ε |∇ w i,m | dx (cid:19) / + ε − N (cid:88) i =1 (cid:90) Ω ε | w i,m | dx (cid:33) ≤ CN (cid:32) N (cid:88) i =1 (cid:18)(cid:90) Ω ε |∇ w i,m | dx (cid:19) / + ε − (cid:33) . It yields that N (cid:88) i =1 (cid:90) Ω ε |∇ w i,m | dx ≤ C ( N ) α N (cid:88) i =1 (cid:18)(cid:90) Ω ε |∇ w i,m | dx (cid:19) / + C ( ε ) + (cid:32) N (cid:88) i =1 t i,m (cid:33) − , which finally leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18)(cid:90) Ω ε |∇ w i,m | dx (cid:19) / − C ( N ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( N, ε ) (cid:32) N (cid:88) i =1 t i,m (cid:33) − / for all i = 1 , N . (3.16)Therefore, (cid:82) Ω ε |∇ w i,m | dx is bounded by the inequality (3.16). So, up to a subsequence, w i,m (cid:42) w i weakly in V ε ,and then strongly in L (Ω ε ) and L (Γ ε ). In addition, it can be proved that (cid:80) Ni =1 (cid:82) Ω ε | w i | dx ≥ N − > 0, and from(3.11), it gives us that α N (cid:88) i =1 (cid:90) Ω ε |∇ w i,m | dx ≤ C N (cid:88) i =1 t i,m + N (cid:88) i =1 (cid:90) Ω ε R i ( x, u ε,m ) N (cid:88) i =1 t i,m dx + N (cid:88) i =1 (cid:90) Γ ε F i ( x, u ε,mi ) N (cid:88) i =1 t i,m dS ε . (3.17)We now consider the second integral on the right-hand side of the above inequality, then the third one is totallysimilar. Using the fact that w i,m → w i strongly in L (Ω ε ) and the assumptions (A )-(A ) in combination with the Fatoulemma, we get lim sup m →∞ N (cid:88) i =1 (cid:90) Ω ε R i ( x, u ε,m ) N (cid:88) i =1 t i,m dx ≤ N N (cid:88) i =1 (cid:90) Ω ε ∩{ w> } r ε ∞ ,i | w i,m | dx, where we have also applied the following inequalities N − (cid:32) N (cid:88) i =1 t i,m (cid:33) − ≤ (cid:32) N (cid:88) i =1 t i,m (cid:33) − ≤ | w i,m | | u ε,mi | − , lim sup u εi →∞ R i ( x, u ε ) | u εi | ≤ r ε ∞ ,i ( x ) for a.e. x ∈ Ω ε . Thus, passing to the limit in (3.17) we are led to α N (cid:88) i =1 (cid:90) Ω ε |∇ w i | dx ≤ N (cid:32) N (cid:88) i =1 (cid:90) Ω ε ∩{ w> } r ε ∞ ,i | w i | dx + N (cid:88) i =1 (cid:90) Γ ε ∩{ w> } f ε ∞ ,i | w i | dS ε (cid:33) . Recall that λ ( r ε ∞ , f ε ∞ ) > 0, it then gives us that w + i ≡ i = 1 , N . As a consequence, w i ≡ (cid:80) Ni =1 (cid:82) Ω ε | w i | dx ≥ N − .Hence, J is coercive. 9 tep 2: (Lower semi-continuity) It can be proved as in [15, 16] that: if u ε,m (cid:42) u ε in V ε , then we obtainlim sup m →∞ (cid:90) Ω ε R i ( x, u ε,m ) dx ≤ (cid:90) Ω ε R i ( x, u ε ) dx, lim sup m →∞ (cid:90) Γ ε F i ( x, u ε,mi ) dS ε ≤ (cid:90) Γ ε F i ( x, u εi ) dS ε , by using the growth assumptions (A ) in combination with the Fatou lemma. Thus, J is lower semi-continuous.This result tells us that J achieves the global minimum at a function u ε ∈ V ε . If we replace u ε by ( u ε ) + , u ε can besupposed to be non-negative. Moreover, the last step shows that u ε is non-trivial. Step 3: (Non-triviality of the minimisers)What we need to prove now is that there exists φ ∈ V ε such that J [ φ ] < 0. In fact, given ψ ∈ V ε ∩ W ε satisfying (cid:107) ψ (cid:107) W ε = 1 and α N (cid:88) i =1 (cid:90) Ω ε |∇ ψ i | dx < N N (cid:88) i =1 (cid:18)(cid:90) Ω ε r ε ,i | ψ i | dx + (cid:90) Γ ε f ε ,i | ψ i | dS ε (cid:19) . In fact, here we assume that ψ is non-negative. By the assumptions (A )-(A ), we havelim inf δ → + R i ( x, δψ ) δ ≥ r ε ,i ( x ) | ψ | ≥ r ε ,i ( x ) | ψ i | for a.e. x ∈ Ω ε , and lim inf δ → + F i ( x, δψ i ) δ ≥ f ε ,i ( x ) | ψ i | for a.e. x ∈ Γ ε . This coupling with the Fatou lemma enable us to obtain the following N (cid:88) i =1 (cid:18) lim inf δ → + (cid:90) Ω ε R i ( x, δψ ) δ dx + lim inf δ → + F i ( x, δψ i ) δ (cid:19) ≥ N (cid:88) i =1 (cid:18)(cid:90) Ω ε r ε ,i | ψ i | dx + (cid:90) Γ ε f ε ,i | ψ i | dS ε (cid:19) , which leads to lim sup δ → + J [ δψ ] δ < . Hence, to complete the proof, we need to choose φ = δψ . Theorem 6. Assume ( A ) - ( A ) and λ ( r ε ∞ , f ε ∞ ) > , λ ( r ε , f ε ) < hold. Then ( P ε ) admits at least a non-negativeweak solution u ε ∈ V ε ∩ W ∞ (Ω ε ) .Proof. We begin the proof by introducing the approximate system (cid:0) P k,ε (cid:1) : ∇ · ( − d εi ∇ u εi ) = R ki ( u ε ) , in Ω ε ⊂ R d ,d εi ∇ u εi · n = G ki ( u εi ) , on Γ ε ,u εi = 0 , on Γ ext , in which we have defined that for each integer k > R ki ( u ε ) := max {− ku εi , R i ( u ε ) } , if u εi ≥ ,R i (0) , if u εi < , G ki ( u εi ) := ε max {− ku εi , a εi u εi − b εi F i ( u εi ) } , if u εi ≥ , − εb εi F i (0) , if u εi < . It is easy to check that our truncated functions R ki and G ki fulfill both (A ) and (A ). In addition, if we set elements R k ,i , R k ∞ ,i , G k ,i , G k ∞ ,i as functions in (A ) by R ki and G ki , one may prove that r ε ,i ≤ R k ,i , r ε ∞ ,i ≤ R k ∞ ,i , f ε ,i ≤ G k ,i , f ε ∞ ,i ≤ G k ∞ ,i for all i ∈ { , ..., N } , and λ (cid:0) R k , G k (cid:1) < λ (cid:0) R k ∞ , G k ∞ (cid:1) > k large (see, e.g. [16]).Thanks to Lemma 5, the problem (cid:0) P k,ε (cid:1) admits a global non-trivial and non-negative minimizer, denoted by u k,ε ,which belongs to V ε and it is associated with the following functional J k [ u ε ] := 12 N (cid:88) i =1 (cid:90) Ω ε d εi |∇ u εi | dx − N (cid:88) i =1 (cid:90) Ω ε R ki ( x, u ε ) dx − N (cid:88) i =1 (cid:90) Γ ε F ki ( x, u εi ) dS ε . Furthermore, u k,ε defines a weak solution to the problem (cid:0) P k,ε (cid:1) for every k and thus, u k,ε ∈ W ∞ (Ω ε ) by Lemma 3.Now, we assign a vector v ε whose elements are defined by v εi := min (cid:110) u εi , u k,εi (cid:111) where u ∈ V ε is the global minimizer constructed from the functional J . We shall prove that J [ v ε ] ≤ J [ u ε ]. Note that when doing so, v ε ∈ W ∞ (Ω ε ) and thendefine a weak solution u ∈ V ε ∩ W ∞ (Ω ε ) to ( P ε ).In fact, one has J k (cid:2) u k,ε (cid:3) ≤ J [ φ ] for all φ ∈ V ε . Then by choosing φ such that φ i := max (cid:110) u εi , u k,εi (cid:111) we have N (cid:88) i =1 (cid:90) { u k,εi i ∈ { , ..., N } , then ( P ε ) has at least a positive, non-trivial and bounded weak solution u ε by the Hopfstrong maximum principle. Furthermore, one may prove in the same vein in [16, Lemma 13] that the solution is uniqueby using vectors of test functions ϕ εδ and ψ εδ whose elements are given by ϕ εδ,i = ( u εi + δ ) − ( v εi + δ ) u εi + δ , ψ εδ,i = ( u εi + δ ) − ( v εi + δ ) v εi + δ , where u εi and v εi are two solutions of ( P ε ) at each layer i ∈ { , ..., N } , which are expected to equal to each other. Remark . In the case of zero Neumann boundary condition on Γ ε , if the nonlinearity R i is globally Lipschitzi with theLipschitz constant, denoted by L i , independent of the scale ε for any i ∈ { , ..., N } , then we may use an iterative schemeto deal with the existence and uniqueness of solutions to our problem. In fact, for n ∈ N such an iterative scheme is givenby ( P εn ) : ∇ · (cid:16) − d εi ∇ u ε,n +1 i (cid:17) = R i ( u ε,n ) , in Ω ε ,d εi ∇ u ε,n +1 i · n = 0 , on Γ ε ,u ε,n +1 i = 0 , on Γ ext , (3.22)where the starting point is u ε, = 0.This global Lipschitz assumption is an alternative to (A ) for R i and it is termed as (cid:0) A (cid:48) (cid:1) . Theorem 9. Assume ( A ) and ( A ) hold (without F i ) and suppose that the nonlinearity R i satisfy (cid:0) A (cid:48) (cid:1) replaced by ( A ) . Then, the problem ( P ε ) with zero Neumann boundary condition on Γ ε has a unique solution in V ε if the constant α − max ≤ i ≤ N { L i } N is small enough.Proof. It is worth noting that the problem (3.22) admits a unique solution in V ε for any n . Then, the functional w ε,ni = u ε,n +1 i − u ε,ni ∈ V ε satisfies the following problem: ∇ · ( − d εi ∇ w ε,ni ) = R i ( u ε,n ) − R i (cid:0) u ε,n − (cid:1) , in Ω ε ,d εi ∇ w ε,ni · n = 0 , on Γ ε ,w ε,ni = 0 . on Γ ext , Using the test function ψ i ∈ V ε we arrive at (cid:104) d εi w ε,ni , ψ i (cid:105) V ε = (cid:10) R i ( u ε,n ) − R i (cid:0) u ε,n − (cid:1) , ψ i (cid:11) L (Ω ε ) We may consider an estimate for the above expression: α N (cid:88) i =1 (cid:12)(cid:12) (cid:104) w ε,ni , ψ i (cid:105) V ε (cid:12)(cid:12) ≤ N (cid:88) i =1 L i N (cid:12)(cid:12)(cid:12)(cid:12)(cid:68) w ε,n − i , ψ i (cid:69) L (Ω ε ) (cid:12)(cid:12)(cid:12)(cid:12) (3.23)Thanks to H¨older’s and Poincar´e inequalities, we have N (cid:88) i =1 (cid:12)(cid:12) (cid:104) w ε,ni , ψ i (cid:105) V ε (cid:12)(cid:12) ≤ C p α − max ≤ i ≤ N { L i } N (cid:13)(cid:13) w ε,n − (cid:13)(cid:13) V ε (cid:107) ψ (cid:107) V ε , where C p > ε , but the dimension d of the media (see, e.g. [6,Lemma 2.1] and [22]). 12t this point, if the constant α − max ≤ i ≤ N { L i } N is small enough such that κ p := C p α − max ≤ i ≤ N { L i } N < ψ i = w ε,ni for i ∈ { , ..., N } we obtain that (cid:107) w ε,n (cid:107) V ε ≤ κ p (cid:13)(cid:13) w ε,n − (cid:13)(cid:13) V ε . Consequently, for some k ∈ N we get (cid:13)(cid:13) u ε,n + k − u ε,n (cid:13)(cid:13) V ε ≤ (cid:13)(cid:13) u ε,n + k − u ε,n + k − (cid:13)(cid:13) V ε + ... + (cid:13)(cid:13) u ε,n +1 − u ε,n (cid:13)(cid:13) V ε ≤ κ n + k − p (cid:13)(cid:13) u ε, − u ε, (cid:13)(cid:13) V ε + ... + κ np (cid:13)(cid:13) u ε, − u ε, (cid:13)(cid:13) V ε ≤ κ np (cid:0) κ k − p + κ k − p + ... + 1 (cid:1) (cid:13)(cid:13) u ε, (cid:13)(cid:13) V ε ≤ κ np (cid:0) − κ kp (cid:1) − κ p (cid:13)(cid:13) u ε, (cid:13)(cid:13) V ε . (3.24)Therefore, { u ε,n } is a Cauchy sequence in V ε , and then there exists uniquely u ε ∈ V ε such that u ε,n → u ε strongly in V ε as n → ∞ . Remarkably, this convergence combining with the Lipschitz property of R i leads to the fact that R i ( u ε,n ) → R i ( u ε ) strongly in V ε as n → ∞ . As a result, the function u ε is the solution of the problem ( P ε ) whenpassing to the limit in n .In addition, when k → ∞ , it follows from (3.24) that (cid:107) u ε,n − u ε (cid:107) V ε ≤ κ np − κ p (cid:13)(cid:13) u ε, (cid:13)(cid:13) V ε , which implies the convergence rate of the linearization and guarantees the stability of the problem ( P ε ). 4. Homogenization asymptotics. Corrector estimates For every i ∈ { , ..., N } , we introduce the following M th-order expansion ( M ≥ u εi ( x ) = M (cid:88) m =0 ε m u i,m ( x, y ) + O (cid:0) ε M +1 (cid:1) , x ∈ Ω ε , (4.1)where u i,m ( x, · ) is Y -periodic for 0 ≤ m ≤ M .It follows from (4.1) that ∇ u εi = (cid:0) ∇ x + ε − ∇ y (cid:1) (cid:32) M (cid:88) m =0 ε m u i,m + O (cid:0) ε M +1 (cid:1)(cid:33) = ε − ∇ y u i, + M − (cid:88) m =0 ε m ( ∇ x u i,m + ∇ y u i,m +1 ) + O (cid:0) ε M (cid:1) . (4.2)Using the relation of the operator A ε and (4.2), we compute that A ε u εi = (cid:0) ∇ x + ε − ∇ y (cid:1) · (cid:32) − d i ( y ) (cid:34) ε − ∇ y u i, + M − (cid:88) m =0 ε m ( ∇ x u i,m + ∇ y u i,m +1 ) (cid:35)(cid:33) + O (cid:0) ε M − (cid:1) , then after collecting those having the same powers of ε , we obtain A ε u εi = ε − ∇ y · ( − d i ( y ) ∇ y u i, )+ ε − [ ∇ x · ( − d i ( y ) ∇ y u i, ) + ∇ y · ( − d i ( y ) ( ∇ x u i, + ∇ y u i, ))]+ M − (cid:88) m =0 ε m [ ∇ x · ( − d i ( y ) ( ∇ x u i,m + ∇ y u i,m +1 ))+ ∇ y · ( − d i ( y ) ( ∇ x u i,m +1 + ∇ y u i,m +2 ))] + O (cid:0) ε M − (cid:1) . (4.3)13n the same vein, we take into consideration the boundary condition at Γ ε as follows: − d εi ∇ u εi · n := − d i ( y ) (cid:32) ε − ∇ y u i, + M − (cid:88) m =0 ε m ( ∇ x u i,m + ∇ y u i,m +1 ) (cid:33) · n= εb i ( y ) F i (cid:32) M − (cid:88) m =0 ε m u i,m (cid:33) − a i ( y ) M − (cid:88) m =0 ε m +1 u i,m + O (cid:0) ε M (cid:1) . (4.4)It is worth noting that in order to investigate the convergence analysis, we give assumptions that allow to pull the ε -dependent quantities out of the nonlinearities R i and F i : R i (cid:32) M (cid:88) m =0 ε m u ,m , ..., M (cid:88) m =0 ε m u N,m (cid:33) = M (cid:88) m =0 ε m ¯ R i ( u ,m , ..., u N,m ) + O (cid:0) ε M +1 (cid:1) , (4.5) F i (cid:32) M (cid:88) m =0 ε m u i,m (cid:33) = M (cid:88) m =0 ε m ¯ F i ( u i,m ) + O (cid:0) ε M +1 (cid:1) , (4.6)in which ¯ R i and ¯ F i are global Lipschitz functions corresponding to the Lipschitz constant L i and K i , respectively, for i ∈ { , ..., N } .From now on, collecting the coefficients of the same powers of ε in (4.3) and (4.4) in combination with using (4.5) and(4.6), we are led to the following systems of elliptic problems, which we refer to the auxiliary problems: A u i, = 0 , in Y , − d i ( y ) ∇ y u i, · n = 0 , on ∂Y ,u i, is Y − periodic in y, (4.7) A u i, = −A u i, , in Y , − d i ( y ) ( ∇ x u i, + ∇ y u i, ) · n = 0 , on ∂Y ,u i, is Y − periodic in y, (4.8) A u i,m +2 = ¯ R i ( u m ) − A u i,m +1 − A u i,m , in Y , − d i ( y ) ( ∇ x u i,m +1 + ∇ y u i,m +2 ) · n = b i ( y ) ¯ F i ( u i,m ) − a i ( y ) u i,m , on ∂Y ,u i,m +2 is Y − periodic in y, (4.9)for 0 ≤ m ≤ M − u m is ascribed to the vector containing elements u i,m for all i ∈ { , ..., N } , and we have denoted by A := ∇ y · ( − d i ( y ) ∇ y ) , A := ∇ x · ( − d i ( y ) ∇ y ) + ∇ y · ( − d i ( y ) ∇ x ) , A := ∇ x · ( − d i ( y ) ∇ x ) . For the first auxiliary problem (4.7), it is trivial to prove that the solution to (4.7) is independent of y , and hence weobtain u i, ( x, y ) = ˜ u i, ( x ) . (4.10)For the second auxiliary problem (4.8), we recall the result in [23, Lemma 2.1] to ensure the existence and uniquenessof periodic solutions to the elliptic problem, which is called the solvability condition. In this case, this condition satisfies14tself because we easily get from the PDE in (4.8) that − (cid:90) ∂Y d i ( y ) ∇ y u i, · n dS y = (cid:90) ∂Y d i ( y ) ∇ x ˜ u i, · n dS y , by Gauß’s theorem. Thus, it claims the existence of a unique weak solution to (4.8).Moreover, this solution is sought by using separation of variables: u i, ( x, y ) = − χ i ( y ) · ∇ x ˜ u i, ( x ) + C i ( x ) . (4.11)Substituting (4.11) into (4.8), we obtain the i th cell problem: A χ i = ∇ y d i ( y ) , in Y , − d i ( y ) ∇ y χ i · n = d i ( y ) · n , on ∂Y ,χ i is Y − periodic in y, (4.12)in which the field χ i ( y ) is called cell function. Additionally, by the definition of the mean value, we have M Y ( χ i ) := 1 | Y | (cid:90) Y χ i dy = 0 . (4.13)As a consequence, it can be proved that χ i belongs to the space H ( Y ) / R and satisfies (4.13).Now, it only remains to consider the third auxiliary problem (4.9). Assume that we have in mind the functions u m and u m +1 , then to find u m +2 let us remark that the right-hand side of the PDE in (4.9) can be rewritten as¯ R i ( u m ) − A u i,m +1 − A u i,m = ¯ R i ( u m ) + ∇ y ( d i ( y ) ∇ x u i,m +1 )+ ∇ x ( d i ( y ) ( ∇ x u i,m + ∇ y u i,m +1 )) . (4.14)We define the operator L i ( ψ ) for i ∈ { , ..., N } by multiplying (4.14) by a test function ψ ∈ C ∞ ( Y ), as follows: L i ( ψ ) = (cid:90) Y ¯ R i ( u m ) ψdy + (cid:90) Y ∇ y ( d i ( y ) ∇ x u i,m +1 ) ψdy + (cid:90) Y ∇ x ( d i ( y ) ( ∇ x u i,m + ∇ y u i,m +1 )) ψdy = (cid:90) Y ¯ R i ( u m ) ψdy − (cid:90) Y d i ( y ) ∇ x u i,m +1 ∇ y ψdy + (cid:90) Y ∇ x ( d i ( y ) ( ∇ x u i,m + ∇ y u i,m +1 )) ψdy. To apply the Lax-Milgram type lemma provided by [6, Lemma 2.2], we need L i ( ψ ) = L i ( ψ ) for ψ , ψ ∈ H ( Y ) / R with ψ (cid:39) ψ , or it is equivalent to (cid:90) Y ¯ R i ( u m ) ( ψ − ψ ) dy + (cid:90) Y ∇ x ( d i ( y ) ( ∇ x u i,m + ∇ y u i,m +1 )) ( ψ − ψ ) dy = 0 . (4.15)Note that ψ − ψ is independent of y . Hence, (4.15) becomes (cid:90) Y ∇ x ( − d i ( y ) ( ∇ x u i,m + ∇ y u i,m +1 )) dy = (cid:90) Y ¯ R i ( u m ) dy. (4.16)For simplicity, we first take m = 0. Remind from (4.10) and (4.11) that u i, and u i, are known, while the term R i ( u )depends on x only, then one has (cid:90) Y ∇ x ( − d i ( y ) ( −∇ y χ i ∇ x ˜ u i, + ∇ x ˜ u i, )) dy = | Y | ¯ R i ( u ) , 15r equivalently, (cid:90) Y ∇ x ( − d i ( y ) ( −∇ y χ i + I ) ∇ x ˜ u i, ) dy = | Y | ¯ R i ( u ) . Thus, if we set the homogenized (or effective) coefficient q i = 1 | Y | (cid:90) Y d i ( y ) ( −∇ y χ i + I ) dy, the ˜ u i, must satisfy (in the “almost all” sense) −∇ x ( q i ∇ x ˜ u i, ) = | Y | − | Y | ¯ R i ( u ) , in Ω . (4.17)On the other hand, it is associated with ˜ u i, = 0 at Γ ext and still satisfies the ellipticity condition.Let us now determine u i, . At first, the PDE in (4.9) (for m = 0) is given by A u i, = ¯ R i ( u ) − d i ( y ) ∇ y χ i ∇ x ˜ u i, − ∇ y ( d i ( y ) χ i ) ∇ x ˜ u i, + d i ( y ) ∇ x ˜ u i, , in Y . (4.18)Next, the boundary condition reads − d i ( y ) ∇ y u i, · n = b i ( y ) ¯ F i ( u i, ) − a i ( y ) u i, − d i ( y ) χ i ∇ x ˜ u i, · n , on ∂Y . Note that (4.18) can be rewritten as A u i, − ∇ y (cid:0) d i ( y ) χ i ∇ x ˜ u i, (cid:1) = ¯ R i ( u ) − d i ( y ) ( ∇ y χ i − I ) ∇ x ˜ u i, . Using the relation (4.17), we have A u i, + A (cid:0) χ i ∇ x ˜ u i, (cid:1) = − | Y | − | Y | ∇ x ( q i ∇ x ˜ u i, ) − d i ( y ) ( ∇ y χ i − I ) ∇ x ˜ u i, . (4.19)Therefore, (4.19) allows us to look for u i, of the form u i, ( x, y ) = θ i ( y ) ∇ x ˜ u i, , (4.20)in which such a function θ i is the solution of the following problem A ( ∇ y θ i − χ i ) = − | Y | − | Y | q i − d i ( y ) ( ∇ y χ i − I ) , in Y , − d i ( y ) ( ∇ y θ i − χ i ) · n = b i ( y ) ¯ F i ( u i, ) − a i ( y ) u i, , on ∂Y ,θ i is Y − periodic in y. (4.21)In conclusion, we have derived an expansion of u εi ( x ) up to the second-order corrector. In particular, we deduced that u εi ( x ) = ˜ u i, ( x ) − εχ i (cid:16) xε (cid:17) · ∇ x ˜ u i, ( x ) + ε θ i (cid:16) xε (cid:17) ∇ x ˜ u i, ( x ) + O (cid:0) ε (cid:1) , x ∈ Ω ε , (4.22)where ˜ u i, can be solved by the microscopic problem (4.7), χ i satisfies the cell problem (4.12), and θ i satisfies the cellproblem (4.21). Moreover, the homogenized equation is defined in (4.17).For the time being, it remains to derive the macroscopic equation from the PDE for u i, in (4.9) for m = 0. Whendoing so, the following solvability condition has to be fulfilled: (cid:90) Y (cid:0) ¯ R i ( u ) − A u i, − A ˜ u i, (cid:1) dy = (cid:90) ∂Y (cid:0) b i ( y ) ¯ F i (˜ u i, ) − a i ( y ) ˜ u i, + d i ( y ) ∇ x u i, · n (cid:1) dS y . (4.23)16he left-hand side of (4.23) can be rewritten as (cid:90) Y ¯ R i ( u ) dy + (cid:90) Y ∇ y ( d i ( y ) ∇ x u i, ) dy + (cid:90) Y ∇ x ( d i ( y ) ( ∇ x ˜ u i, + ∇ y u i, )) dy. (4.24)Let us consider the last two integrals in (4.24). In fact, we have (cid:90) Y ∇ x ( d i ( y ) ∇ x ˜ u i, ) dy = ∇ x · (cid:20)(cid:18)(cid:90) Y d i ( y ) dy (cid:19) ∇ x ˜ u i, (cid:21) = (cid:18)(cid:90) Y d i ( y ) dy (cid:19) : ∇ x ∇ x ˜ u i, , (4.25)where we have used the inner product (or exactly, double dot product) between two matrices A : B := tr (cid:0) A T B (cid:1) = (cid:88) ij a ij b ij . In addition, by periodicity and Gauß’s theorem we obtain (cid:90) Y ∇ y ( d i ( y ) ∇ x u i, ) dy = (cid:90) ∂Y d i ( y ) ∇ x u i, · n dS y . (4.26)Next, employing the double dot product again, we get (cid:90) Y ∇ x ( d i ( y ) ∇ y u i, ) dy = − (cid:90) Y ( d i ( y ) ∇ y χ i ) dy : ∇ x ∇ x ˜ u i, . (4.27)Combining (4.23), (4.25)-(4.27) yields the macroscopic equation: (cid:18)(cid:90) Y ( d i ( y ) − d i ( y ) ∇ y χ i ) dy (cid:19) : ∇ x ∇ x ˜ u i, = (cid:104) b i (cid:105) ¯ F i (˜ u i, ) − (cid:104) a i (cid:105) ˜ u i, − | Y | ¯ R i ( u ) , where we have denoted by (cid:104) a i (cid:105) := (cid:90) ∂Y a i ( y ) dy, (cid:104) b i (cid:105) := (cid:90) ∂Y b i ( y ) dy. Furthermore, this equation is associated with the boundary condition ˜ u i, = 0 at Γ ext . From the point of view of applications, upper bound estimates on convergence rates over the space V ε in terms ofquantitative analysis tells how fast one can approximate both u ε , the solution of systems ( P ε ), and ∇ u ε by the asymptoticexpansion (4.22). On the other hand, it also gives rise to the question that: how much information on data will we needvia such averaging techniques? We introduce the well-known cut-off function m ε ∈ C ∞ c (Ω) such that ε |∇ m ε | ≤ C and m ε ( x ) := , if dist ( x, Γ) ≤ ε, , if dist ( x, Γ) ≥ ε. For i ∈ { , ..., N } , we define the function Ψ εi byΨ εi := ϕ εi + (1 − m ε ) (cid:0) εu i, + ε u i, (cid:1) , where we have denoted by ϕ εi := u εi − (cid:0) u i, + εu i, + ε u i, (cid:1) . A ε ϕ εi = R i ( u ε ) − ¯ R i ( u ) − ε ( A u i, + A u i, ) − ε A u i, , x ∈ Ω ε , (4.28)while on the boundary Γ ε , the function ϕ εi satisfies − d εi ∇ x ϕ εi · n = ε d εi ∇ x u i, · n + ε (cid:2) a εi ( u i, − u εi ) + b εi (cid:0) F i ( u εi ) − ¯ F i ( u i, ) (cid:1)(cid:3) . (4.29)Rewriting the above information, the function ϕ εi satisfies the following system: A ε ϕ εi = ¯ R i ( u ε ) − ¯ R i ( u ) + εg εi , in Ω ε , − d εi ∇ x ϕ εi · n = ε h εi · n + εl εi , on Γ ε , (4.30)where we have denoted by g εi := − d i (cid:16) xε (cid:17) χ i (cid:16) xε (cid:17) ∇ x ˜ u i, + d i (cid:16) xε (cid:17) θ i (cid:16) xε (cid:17) ∇ x ˜ u i, + ∇ y (cid:16) d i (cid:16) xε (cid:17) θ i (cid:16) xε (cid:17)(cid:17) ∇ x ˜ u i, + εd i (cid:16) xε (cid:17) θ i (cid:16) xε (cid:17) ∇ x ˜ u i, ,h εi := d i (cid:16) xε (cid:17) θ i (cid:16) xε (cid:17) ∇ x ˜ u i, ,l εi := a i (cid:16) xε (cid:17) (˜ u i, − u εi ) + b i (cid:16) xε (cid:17) (cid:0) F i ( u εi ) − ¯ F i (˜ u i, ) (cid:1) . Now, multiplying the PDE in (4.30) by ϕ i ∈ V ε for i ∈ { , ..., N } and integrating by parts, we get that (cid:104) d εi ϕ εi , ϕ i (cid:105) V ε = (cid:10) ¯ R i ( u ε ) − ¯ R i ( u ) , ϕ i (cid:11) L (Ω ε ) + ε (cid:104) g εi , ϕ i (cid:105) L (Ω ε ) − ε (cid:104) l εi , ϕ i (cid:105) L (Γ ε ) − ε (cid:90) Γ ε h εi · n ϕ i dS ε . (4.31)To guarantee all the derivatives appearing in g εi (up to higher order correctors), ˜ u i, , which is the solution to (4.17), needsto be smooth enough, says L ∞ (Ω ε ) (cf. [24]), and the cell functions χ i and θ i to (4.12) and (4.21), respectively, belongat least to H ( Y ) as derived above. Consequently, it allows us to estimate g εi by an ε -independent constant, i.e. (cid:107) g εi (cid:107) L (Ω ε ) ≤ C for all i ∈ { , ..., N } . (4.32)Furthermore, it is easily to estimate the integral including h εi in (4.31) by the following (see, e.g. [6]): (cid:90) Γ ε h εi · n dS ε ≈ Cε − , which leads to (cid:107) h εi · n (cid:107) L (Γ ε ) ≤ Cε − / . (4.33)Now, it remains to estimate the third integral in (4.31). Thanks to (A ) and (4.6), we may have (cid:12)(cid:12)(cid:12) (cid:104) l εi , ϕ i (cid:105) L (Γ ε ) (cid:12)(cid:12)(cid:12) ≤ C (cid:0) K i (cid:1) (cid:107) u εi − ˜ u i, (cid:107) L (Γ ε ) (cid:107) ϕ i (cid:107) L (Γ ε ) . (4.34)In the same vein, we get: (cid:12)(cid:12)(cid:12)(cid:10) R i ( u ε ) − ¯ R i ( u ) , ϕ i (cid:11) L (Ω ε ) (cid:12)(cid:12)(cid:12) ≤ C ¯ L i (cid:107) u ε − ˜ u (cid:107) V ε (cid:107) ϕ i (cid:107) L (Ω ε ) . (4.35)Combining (4.31)-(4.35) with (A ) and putting ¯ L := max (cid:8) ¯ L , ..., ¯ L N (cid:9) and ¯ K := 1 + max (cid:8) ¯ K , ..., ¯ K N (cid:9) , we are led tothe estimate: α N (cid:88) i =1 |(cid:104) ϕ εi , ϕ i (cid:105) V ε | ≤ C (cid:0) ¯ L (cid:107) u ε − ˜ u (cid:107) V ε + ε (cid:1) (cid:107) ϕ (cid:107) V ε + C (cid:16) ¯ K (cid:107) u ε − ˜ u (cid:107) H (Γ ε ) + ε / (cid:17) (cid:107) ϕ (cid:107) H (Γ ε ) ≤ C (cid:16) ε + ε / (cid:17) (cid:107) ϕ (cid:107) V ε ≤ Cε / (cid:107) ϕ (cid:107) V ε , (4.36)18here we have made use of the trace inequality (cid:107) ϕ (cid:107) H (Γ ε ) ≤ Cε − / (cid:107) ϕ (cid:107) V ε and the Poincar´e inequality (cid:107) ϕ (cid:107) H (Ω ε ) ≤ C (cid:107) ϕ (cid:107) V ε . Recall that our aim is to estimate (cid:107) Ψ ε (cid:107) V ε , it remains to control the term (cid:10) (1 − m ε ) (cid:0) εu i, + ε u i, (cid:1) , ϕ i (cid:11) V ε for ϕ i ∈ V ε .In fact, one easily has that N (cid:88) i =1 (cid:12)(cid:12)(cid:10) (1 − m ε ) (cid:0) εu i, + ε u i, (cid:1) , ϕ i (cid:11) V ε (cid:12)(cid:12) ≤ Cε (cid:107)∇ (1 − m ε ) (cid:107) H (Ω ε ) (cid:107) ϕ (cid:107) V ε + C (cid:13)(cid:13) (1 − m ε ) ∇ (cid:0) εu + ε u (cid:1)(cid:13)(cid:13) H (Ω ε ) (cid:107) ϕ (cid:107) V ε ≤ C (cid:16) ε / + ε / (cid:17) (cid:107) ϕ (cid:107) V ε ≤ Cε / (cid:107) ϕ (cid:107) V ε , (4.37)where we have used (cid:107)∇ (1 − m ε ) (cid:107) H (Ω ε ) ≤ N (cid:32)(cid:90) Ω ε ∩ { x | dist ( x, Γ) ≤ ε } |∇ m ε | dx (cid:33) ≤ Cε − , (cid:13)(cid:13) (1 − m ε ) ∇ (cid:0) εu + ε u (cid:1)(cid:13)(cid:13) H (Ω ε ) ≤ N ε | Ω ε | (cid:90) Ω ε ∩ { x | dist ( x, Γ) ≤ ε } |∇ m ε | dx ≤ Cε . Hence, by using the triangle inequality in (4.36) and (4.37) yields that N (cid:88) i =1 |(cid:104) Ψ εi , ϕ i (cid:105) V ε | ≤ Cε / (cid:107) ϕ (cid:107) V ε , which finally leads to (cid:107) Ψ ε (cid:107) V ε ≤ Cε / , by choosing ϕ = Ψ ε .Summarizing, we can now state of the following theorem. Theorem 10. Let u ε be the solution of the elliptic system ( P ε ) with assumptions ( A ) − ( A ) and (4.5) - (4.6) up to M = 2 .Suppose that the unique pair ( u , u m ) ∈ W ∞ (Ω ε ) × W ∞ (cid:16) Ω ε ; H ( Y ) / R (cid:17) for m ∈ { , } . The following corrector withsecond order for the homogenization limit holds: (cid:13)(cid:13) u ε − u − m ε (cid:0) εu + ε u (cid:1)(cid:13)(cid:13) V ε ≤ Cε / , where u , u and u are vectors whose elements are defined by (4.10) , (4.11) and (4.20) , respectively. 5. Discussion In real-world applications, the nonlinear reaction term R i is often locally Lipschitz. However, relying on Lemma 3 the L ∞ -type estimate of the positive solution makes the nonlinearity globally Lipschitz. For example, we choose N = 2 andonly consider the R ( u , u ) = u u − u . We have | R ( u , u ) − R ( v , v ) | ≤ max {(cid:107) u (cid:107) L ∞ , (cid:107) u (cid:107) L ∞ + (cid:107) v (cid:107) L ∞ } ( | u − v | + | u − v | ) . In addition, for M = 1 we compute that R ( u , + εu , , u , + εu , ) = u , u , + ε ( u , u , + u , u , − u , u , ) + O (cid:0) ε (cid:1) . (5.1)19onsequently, it follows from (5.1) that R (cid:88) m ∈{ , } ε m u ,m , (cid:88) m ∈{ , } ε m u ,m = (cid:88) m ∈{ , } ε m [(1 − m ) u , u , + , + m ( u , u , + u , u , − u , u , )] + O (cid:0) ε (cid:1) . which implies ¯ R := (1 − m ) u , u , + m ( u , u , + u , u , − u , u , ).If u i,m , v i,m ∈ L ∞ (Ω ε ) for all i, m we thus arrive at (cid:12)(cid:12) ¯ R ( u , , u , , u , , u , ) − ¯ R ( v , , v , , v , , v , ) (cid:12)(cid:12) ≤ L (cid:88) m ∈{ , } ,i ∈{ , } | u i,m − v i,m | , where L = 4 max (cid:110) (cid:107) u , (cid:107) L ∞ (Ω ε ) , (cid:107) v , (cid:107) L ∞ (Ω ε ) , (cid:107) v , (cid:107) L ∞ (Ω ε ) , (cid:107) v , (cid:107) L ∞ (Ω ε ) , (cid:107) u , (cid:107) L ∞ (Ω ε ) , (cid:111) .A similar discussion for the nonlinear surface rates F i . In particular, note that that if L ∞ bounds are available (up to the boundary) then also the exponential function F ( u ) = e u can be treated conveniently.We may repeat the homogenization procedure by the auxiliary problems (4.7)-(4.9) to obtain not only the generalexpansion of the concentrations and corresponding problems, but also the higher order of corrector estimate due to the˜ u -based construction of u m . Taking the M -level expansion (4.1) into consideration, the general corrector can be foundeasily. Indeed, by induction we have from (4.28) that for x ∈ Ω ε A ε ϕ εi = A ε u εi − ε − A u i, − ε − ( A u i, + A u i, ) − M − (cid:88) m =0 ε m ( A u i,m +2 + A u i,m +1 + A u i,m ) − ε M − ( A u i,M + A u i,M − ) − ε M A u i,M = R i ( u ε ) − M − (cid:88) m =0 ε m ¯ R i ( u m ) − ε M − ( A u i,M + A u i,M − ) − ε M A u i,M , while (4.29) becomes − d εi ∇ x ϕ εi · n = ε M d εi ∇ x u i,M + ε (cid:34) a εi (cid:32) M − (cid:88) m =0 ε m u i,m − u εi (cid:33) + b εi (cid:32) F ( u εi ) − M − (cid:88) m =0 ε m ¯ F ( u i,m ) (cid:33)(cid:35) . Thanks to the assumptions (4.5) and (4.6), we are totally in a position to prove the generalization of Theorem 10.Since we just need to follow the above procedure, we shall give the following theorem while skipping the proof. Theorem 11. Let u ε be the solution of the elliptic system ( P ε ) with assumptions ( A ) − ( A ) and (4.5) - (4.6) up to M -level of expansion. Suppose that the unique pair ( u , u m ) ∈ W ∞ (Ω ε ) × W ∞ (cid:16) Ω ε ; H ( Y ) / R (cid:17) for all ≤ m ≤ M .The following correctors for the homogenization limit hold: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u ε − M (cid:88) m =0 ε m u m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ε ≤ C (cid:16) ε M − + ε M − / (cid:17) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u ε − u − m ε M (cid:88) m =1 ε m u m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ε ≤ C M (cid:88) m =1 ε m − / . Acknowledgment AM thanks NWO MPE ”Theoretical estimates of heat losses in geothermal wells” (grant nr. 657.014.004) for funding.VAK gratefully acknowledges the hospitality of Department of Mathematics and Computer Science, Eindhoven University of Technology in The Netherlands. 20 eferencesReferences [1] O. Krehel, A. Muntean, P. Knabner, Multiscale modeling of colloidal dynamics in porous media including aggregationand deposition, Advances in Water Resources 86 (2015) 209–216. [2] S. de Groot, P. Mazur, Non-equilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam, 1962.[3] T. Funaki, H. Izuhara, M. Mimura, C. Urabe, A link between microscopic and macroscopic models of self-organizedaggregation, Networks and Heterogeneous Media 7 (4) (2012) 705–740.[4] V. Vanag, I. Epstein, Cross-diffusion and pattern formation in reaction-diffusion systems, Physical Chemistry Chem-ical Physics 11 (6) (2009) 897–912. [5] O. Krehel, T. Aiki, A. Muntean, Homogenization of a thermo-diffusion system with Smoluchowski interactions,Networks and Heterogeneous Media 9 (4) (2014) 739–762.[6] D. Cioranescu, J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, 1999.[7] T. Fatima, A. Muntean, M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predictingconcrete corrosion, Applicable Analysis 91 (6) (2012) 1129–1154. [8] A. Muntean, T. van Noorden, Corrector estimates for the homogenization of a locally-periodic medium with areasof low and high diffusivity, European Journal of Applied Mathematics 24 (5) (2012) 657–677.[9] A. Mielke, S. Reichelt, M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slowdiffusion, Networks Heterogeneous Media 9 (2014) 353–382.[10] S. Reichelt, Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales, Tech. rep., WIAS Berlin (2015).[11] T. Hou, X. H. Wu, Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidlyoscillating coefficients, Mathematics of Computation 68 (227) (1999) 913–943.[12] C. L. Bris, F. Legoll, A. Lozinski, An MsFEM type approach for perforated domains, SIAM Multiscale Modelingand Simulation 12 (3) (2014) 1046–1077. [13] U. Hornung, W. J¨ager, Diffusion, convection, adsorption, and reaction of chemicals in porous media, Journal ofDifferential Equations 92 (1991) 199–225.[14] H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, Journal of FunctionalAnalysis 11 (3) (1972) 346–384.[15] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis: Theory, Methods & Applications 10 (1) (1986) 55–64.[16] J. Garc´ıa-Meli´an, J. Rossi, J. S. D. Lis, Existence and uniqueness of positive solutions to elliptic problems withsublinear mixed boundary conditions, Communications in Contemporary Mathematics 11 (4) (2009) 585–613.2117] S. Cano-Casanova, J. L´opez-G´omez, Properties of the principal eigenvalues of a general class of non-classical mixedboundary value problems, Journal of Differential Equations 178 (2002) 123–211. [18] E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet–Neumann boundary conditions, Journal ofFunctional Analysis 199 (2) (2003) 468–507.[19] J. Moser, A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations,Communications on Pure and Applied Mathematics 13 (1960) 457–468.[20] J. Moser, A Harnack inequality for parabolic differential equations, Communications on Pure and Applied Mathe- matics 17 (1964) 101–134.[21] X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, Journal of Mathematical Analysisand Applications 339 (2008) 1395–1412.[22] A. Damlamian, P. Donato, Which sequences of holes are admissible for periodic homogenization with Neumannboundary condition?, ESAIM: Control, Optimisation and Calculus of Variations 8 (2002) 555–585. [23] L. Persson, L. Persson, N. Svanstedt, J. Wyller, The Homogenization Method: An Introduction, Chartwell Bratt,Sweden, 1993.[24] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equationssatisfying general boundary value conditions I, Communications on Pure and Applied Mathematics 12 (1959) 623–727.215