# Homogenization Problem in a Domain with Double Oscillating Boundary

Authors: Jie Zhao; Juan Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, we study the convergence of solutions for homogenization problems about the Poisson equation in a domain with double oscillating locally periodic boundary. Such a problem arises in the processing of devices with very small features. We utilize second-order Taylor expansion of boundary data in combination with boundary correctors to obtain the convergence rate inH1-norm. This work explores the domain with double oscillating boundary and also shows the influence of the amplitudes and periods of the oscillations to convergence rates of solutions.


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## Body

## 1. Introduction

Several important problems arising in physics and engineering lead to considering boundary value problems in domains with oscillating boundaries. Such problems arise in the context of fluid flows over a rough surface [1, 2], of reinforcement by a thin layer [3], or of electromagnetic scattering by an obstacle with a periodic coating [4, 5]. Indeed, the oscillating boundary results can be applied to the homogenization of neutronic diffusion or transport equation [6–8]. Studying the oscillating boundary is also the key for determining interface transmission conditions in various mechanical problems [9–11].In this paper, we are interested in the following boundary value problem for the Poisson equation in a domain with double oscillating locally periodic boundary:(1)-∆uε=fxinΩε,∂uε∂νε+px′,x′εαuε=gx′,x′εαonΓ1ε,∂uε∂νε+qx′,x′εβuε=hx′,x′εβonΓ2ε,uε=0onΓ3.where x′=(x1,x2⋯,xn-1)∈(0,1)n-1 and νε is the outward unit normal. The domain Ωε is a bounded domain in Rn and ∂Ωε=Γ1ε⋃Γ2ε⋃Γ3, where hypersurfaces(2)Γ1ε:xn=εαSx′,x′εα,α>0,and(3)Γ2ε:xn=1-εβTx′,x′εβ,β>0.Without loss of generality, we assume thatΩε is a domain bounded by hyperplanes {xi=0} and {xi=1}(i=1,2⋯n-1) and hypersurfaces Γ1ε and Γ2ε.We suppose that(4)px′,ξ′≥0,qx′,η′≥0,Sx′,ξ′≥0,Tx′,η′≥0,whereξ′=x′εαandη′=x′εβ.The conditions p(x′,ξ′)≥0 and q(x′,η′)≥0 ensure the existence and uniqueness of solution for problem (1). This can be guaranteed by Lax-Milgram Theorem. The assumptions S(x′,ξ′)≥0 and T(x′,η′)≥0 ensure that 0≤xn≤1. This makes the proof a bit more simpler. In fact, all the results of this paper remain valid even without the nonnegativity of S(x′,ξ′) and T(x′,η′).We also impose the smoothness condition and periodicity condition(5)px′,ξ′,gx′,ξ′andSx′,ξ′aresmoothfunctionsinx′,ξ′and1-periodicinξ′,qx′,η′,hx′,η′andTx′,η′aresmoothfunctionsinx′,η′and1-periodicinη′.Here we do not aim to obtain the optimal smoothness but rather focus on the method itself.In order to avoid technical difficulties, we assume that(6)gx′,ξ′≡0,hx′,η′≡0,Sx′,ξ′≡0,Tx′,η′≡0,onxi=0andxi=1,where i=1,2⋯n-1. The conditions on g and h are compatibility conditions. The assumptions about S and T are needed for dealing with the situation when the oscillating boundary intersects the flat boundary.As the authors are aware, there are many papers about results of convergence rates for elliptic homogenization problems with oscillating boundary data. In 1997, A. Friedman, B. Hu, and Y. Liu [12] studied two-dimensional domain, whose boundary is oscillating according to three scales. They extended the results of Belyaev [13, 14] to the three-scale oscillating boundary. In 1999, G. A. Chechkin, A. Friedman, and A. L. Piatnitski [15] considered such problem including some parameters and obtained some error estimates in H1-norm. Their [15] method is following a general procedure in homogenization but without using correctors.Recently, there has been a surge of activity in the theory of homogenization in domain with oscillating boundary data. In 2012, D. Gérard and N. Masmoudi [16] studied the homogenization of elliptic system with Dirichlet boundary conditions, when the coefficients of the system and the boundary data are ε-periodic. They obtained the solutions convergence in L2 with a power rate in ε. In 2013, H. Aleksanyan, H. Shahgholian, and P. Sjölin [17] studied the the boundary value homogenization for Dirichlet problem. In particular, they proved pointwise and Lp convergence results. Their method is based on analysis of oscillatory integrals. The papers [18, 19] were devoted to the investigations of the homogenization problem for the Poisson equation in a thin domain with an oscillating boundary.For such domain with rapidly oscillating boundary, there are many directions enabling us to consider. Boundary-value problems involving rapidly oscillating boundaries or interfaces appear in many fields of physics and engineering sciences, and it will be interesting to investigate the asymptotic behavior of the solutions of spectral problems. Y. Amirat, G. A. Chechkin, and R. R. Gadyl’shin were devoted to prove theH1 convergence of the eigenvalues and eigenfunctions to the eigenvalues and eigenfunctions of the homogenization problem, via the traditional method of asymptotic expansions. Moreover, it is worth noticing that many mathematical works have been contributed to the asymptotic analysis of problems in domains with random microstructure. For instance, in 2011, Y. Amirat and other scholars, after adding some extra assumptions on the random variables, were able to obtain the convergence results of solutions in H1. This is also an interesting problem.One may consult several outstanding sources [20–26] for background and overview of the homogenization theory.The main difference of the present work in relation to previous existing work is that this work explores the domain with double oscillating boundary and also shows the influence of the amplitudes and periods of the oscillations to convergence rates of solutions. Meanwhile, with different traditional asymptotic expansions method, this paper improves the convergence rate results inH1-norm by virtue of boundary correctors that can be used to obtain effective approximation.We now describe the outline of this paper. Section2 contains some basic formulas and estimates which are important to obtain error estimates. In Section 3, we show that the solution uε of problem (1) converges to the solution u0 of the corresponding homogenized problem in the H1-norm with error estimate up to order of O(εα/2+εβ/2). In Section 4, we construct correctors that play important role in improving the power in ε. Next, we improve the error estimate up to order of O(εα+εβ) in Section 5. This can be obtained via the correctors. Following the same line of research, in Section 6, we obtain an approximation to uε with error estimate up to order of (ε3α/2+ε3β/2) in H1-norm by using correctors and second-order Taylor’s expansion.


## 2. Preliminaries

We consider the homogenized problem associated with problem (1) in the form(7)-∆u0=fxinΩ0,∂u0∂ν0+Px′u0=Gx′onΓ1,∂u0∂ν0+Qx′u0=Hx′onΓ2,u0=0onΓ3,where Ω0⊂(0,1)n and ν0 is the outward unit normal. Here Γ1={x′∈[0,1]n-1,xn=0} and Γ2={x′∈[0,1]n-1,xn=1}. Functions P(x′),Q(x′),G(x′), and H(x′) are defined as follows:(8)Px′=∫01⋯∫011+▽ξ′Sx′,ξ′21/2px′,ξ′dξ′,Gx′=∫01⋯∫011+▽ξ′Sx′,ξ′21/2gx′,ξ′dξ′,Qx′=∫01⋯∫011+▽η′Tx′,η′21/2qx′,η′dη′,Hx′=∫01⋯∫011+▽η′Tx′,η′21/2hx′,η′dη′.As a preliminary step, we shall prove some propositions.Proposition 1.
There exists a constantC independent of ε such that, for any v∈H1(Ω0), the following inequalities(9)vx′,εαSx′,x′εα-vx′,0L2Γ1≤Cεα/2vH1Ω0and(10)vx′,1-εβTx′,x′εβ-vx′,1L2Γ2≤Cεβ/2vH1Ω0hold.Proof.
Without loss of generality, we may assumev∈C∞(Ω0). A simple calculation then gives (11)vx′,εαSx′,x′εα-vx′,0=∫0εαSx′,x′/εα∂vx′,t∂tdt.It follows from Hölder’s inequality that (12)vx′,εαSx′,x′εα-vx′,02≤Cεα∫0εαSx′,x′/εα∂vx′,t∂t2dt.Integrating over Γ1, we obtain (9). The proof of (10) is similar to (9). This completes the proof.Proposition 2.
There exists a constantC independent of ε such that, for any v∈H1(Ω0), the following estimate(13)vL2Ω0∖Ωε≤Cεα/2+εβ/2vH1Ω0is valid.Proof.
Note that(14)vx′,1-vx′,τ=∫τ1∂vx′,t∂tdt,where τ∈[1-εβT(x′,x′/εβ),1].
It is easy to see that(15)v2x′,τ≤2v2x′,1+21-τ∫τ1∂vx′,t∂t2dt.Let τ=xn,2=1-εβT(x′,x′/εβ), and, integrating over (0,1)n-1, we obtain(16)∫01⋯∫01∫xn,21v2x′,xn,2dx≤2εβmax⁡TvL20,1n-1+2ε2βmax⁡T2∫01⋯∫01∫xn,21∂vx′,t∂t2dt.
A similar computation yields the following result:(17)∫01⋯∫01∫0xn,1v2x′,xn,1dx≤2εαmax⁡SvL20,1n-1+2ε2αmax⁡S2∫01⋯∫01∫0xn,1∂vx′,t∂t2dt,where xn,1=εαS(x′,x′/εα).
Combining these two terms, we obtain (13). This completes the proof.Proposition 3.
Letm(x′,ξ′), n(x′,η′) be 1-periodic in ξ′, η′ and satisfy(18)∫01⋯∫01mx′,ξ′dξ′=0and∫01⋯∫01nx′,η′dη′=0respectively. Then, these inequalities (19)∫Γ1mx′,x′εαvx′dx′≤Cεα/2vH1/2Γ1and∫Γ2nx′,x′εβvx′dx′≤Cεβ/2vH1/2Γ2are satisfied.Proof.
This proposition has been proved by G. A. Chechkin, A. Friedman, and A. L. Piatnitski in [15].Proposition 4.
There exists a constantC independent of ε such that, for all v∈H1(Ω0), the following estimates(20)∫Γ1εgx′,x′εαvx′,εαSx′,x′εαdσ1-∫Γ1Gx′vx′,0dx′≤Cεα/2vH1Ω0and(21)∫Γ2εhx′,x′εβvx′,1-εβTx′,x′εβdσ2-∫Γ2Hx′vx′,1dx′≤Cεβ/2vH1Ω0.take place.Proof.
A direct computation shows that(22)∫Γ1εgx′,x′εαvx′,εαSx′,x′εαdσ1-∫Γ1Gx′vx′,0dx′=∫Γ1gx′,x′εαvx′,εαSx′,x′εα1+▽ξ′S21/2dx′-∫Γ1Gx′vx′,0dx′+OεαvH1Ω0≤∫Γ1gx′,x′εαvx′,εαSx′,x′εα-vx′,01+▽ξ′S21/2dx′+∫Γ1gx′,x′εαvx′,01+▽ξ′S21/2dx′-∫Γ1Gx′vx′,0dx′+CεαvH1Ω0.This, together with Propositions 1 and 3, yields estimate (20). Similarly, one can prove (21).Proposition 5.
There exists a constantC independent of ε such that, for all v∈H1(Ω0), the following inequalities(23)∫Γ1εpx′,x′εαu0x′,x′εαvx′,εαSx′,x′εαdσ1-∫Γ1Px′u0x′,0vx′,0dx′≤Cεα/2u0H1Ω0vH1Ω0and(24)∫Γ2εqx′,x′εβu0x′,x′εβvx′,1-εβTx′,x′εβdσ2-∫Γ2Qx′u0x′,1vx′,1dx′≤Cεβ/2u0H1Ω0vH1Ω0.hold true.Proof.
This proposition can be proved in the same way as Proposition4.


## 3. Error Estimate up toO(εα/2+εβ/2)

In this section, we will prove the error estimate up to the order ofO(εα/2+εβ/2). Our main result is the following theorem.Theorem 6 (letf∈L2(Ωε)).
Assume thatuε is a solution of problem (1)-(3). Suppose that p(x′,ξ′), g(x′,ξ′), q(x′,η′), h(x′,η′), S(x′,ξ′), and T(x′,η′) satisfy (4)-(6). Then there exists a constant C which does not depend on ε, such that(25)uε-u0H1Ωε≤Cεα/2+εβ/2.Remark 7.
Following general procedure in homogenization, we introduce the associated bilinear form(26)Bu,v≐∫Ωε▽u·▽vdx+∫Γ1εpuvdσ1+∫Γ2εquvdσ2,where(27)dσ1=1+∑i=1n-1εα∂S∂xi+∂S∂ξi21/2dx′=1+▽ξ′S21/2dx′+Oεαdx′,dσ2=1+∑i=1n-1εβ∂T∂xi+∂T∂ξi21/2dx′=1+▽η′T21/2dx′+Oεβdx′.We shall establish(28)Buε-u0,v≤Cεα/2+εβ/2vH1Ω0.Then, we extend uε-u0 into Ω0 without increasing the H1-norm by more than a multiplicative constant. What remains is just to take v=uε-u0 and use Poincaré’s inequality. Then this theorem will be proved.Proof.
SinceΩε⊂Ω0, every function f∈L2(Ωε) can be easily extended by 0 in Ω0∖Ωε to become a function of L2(Ω0). For any v∈H1(Ω0) and v=0 on Γ3, it is easy to see that(29)Buε-u0,v=∫Ωε▽uε-u0·▽vdx+∫Γ1εpuε-u0vdσ1+∫Γ2εquε-u0vdσ2=∫Ωεfvdx+∫Γ1εgvdσ1-∫Γ1εpu0vdσ1-∫Ωε▽u0·▽vdx+∫Γ2εhvdσ2-∫Γ2εqu0vdσ2=-∫Ω0▽u0·▽vdx-∫Γ1Pu0vdx′-∫Γ2Qu0vdx′+∫Γ1Pu0vdx′-∫Γ1εpu0vdσ1+∫Γ2Qu0vdx′-∫Γ2εqu0vdσ2+∫Ωεfvdx+∫Γ1εgvdσ1+∫Γ2εhvdσ2+∫Ω0∖Ωε▽u0·▽vdx=-∫Ω0∖Ωεfvdx+∫Ω0∖Ωε▽u0·▽vdx+∫Γ1εgvdσ1-∫Γ1Gvdx′+∫Γ2εhvdσ2-∫Γ2Hvdx′+∫Γ1Pu0vdx′-∫Γ1εpu0vdσ1+∫Γ2Qu0vdx′-∫Γ2εqu0vdσ2.It follows from Proposition 2 that (30)∫Ω0∖Ωεfvdx≤Cεα/2+εβ/2fL2Ω0vH1Ω0and(31)∫Ω0∖Ωε▽u0·▽vdxdx≤Cεα/2+εβ/2vH1Ω0,where we have used the uniform boundedness of u0H2(Ω0).
Then, according to Propositions4 and 5, we conclude that(32)∫Ωε▽uε-u0·▽vdx+∫Γ1εpuε-u0vdσ1+∫Γ2εquε-u0vdσ2≤Cεα/2+εβ/2vH1Ω0.Choosing v=uε-u0 and using condition (4) (p≥0,q≥0) and Poincaré’s inequality, we obtain the desired result. This completes the proof.


## 4. Construction of Correctorsu1 and u2

In order to improve the power ofε, in this section, we shall construct the correctors u1 and u2.Firstly, we introduce the harmonic functionsMi(x′,ξ), and x′ is treated as a parameter, as solutions of(33)∆Mi=0inD~,∂M0∂νξ=g-pPGonΓ~,∂Mj∂νξ=-11+▽ξ′Sx′,ξ′21/2∂Sx′,ξ′∂ξjj=1,2⋯n-1onΓ~,∂Mn∂νξ=11+▽ξ′Sx′,ξ′21/2-pPonΓ~,Mi≤Cexp⁡-λξii=0,1,⋯,n,λ>0,Mix′,ξis1-periodic  inξ′,where(34)D~=ξ′,ξn:ξ′∈0,1n-1,Sx′,ξ′<ξn<+∞,and(35)Γ~:ξn=Sx′,ξ′.This system was first introduced by A. G. Belyaev [13, 14].To ensure these solutionsMi exist, we need to verify the compatibility condition holds true. A simple calculation then gives(36)∫∂D~∂M0∂νξdσξ=∫01⋯∫01g-pPG1+▽ξ′Sx′,ξ′21/2dξ′=0,∫∂D~∂Mj∂νξdσξ=∫01⋯∫01-∂Sx′,ξ′∂ξjdξ′=0,where we have used (5) and(37)∫∂D~∂Mn∂νξdσξ=∫01⋯∫011-pP1+▽ξ′Sx′,ξ′21/2dξ′=0.Therefore, these harmonic functions Mi exist.Then we defineu1 as follows:(38)u1x,ξ=M0x′,ξ+∑k=1n∂u0∂xkxMkx′,ξ.Using the same technique, we construct correctoru2 as follows.Assume harmonic functionsNi(x′,η), x′ is treated as a parameter, as solutions of(39)∆Ni=0inD^,∂N0∂νη=h-qQHonΓ^,∂Nj∂νη=-11+▽η′Tx′,η′21/2∂Tx′,η′∂ηjj=1,2,…,n-1onΓ^,∂Nn∂νη=-11+▽η′Tx′,η′21/2+qQonΓ^,Ni≤Cexp⁡-θηii=0,1,…,n,θ>0,Nix′,η′is1-periodicinη′,where(40)D^=η′,ηn:η′∈0,1n-1,-∞<ηn<1-Tx′,η′,νη=νη1,νη2⋯,νηnis  the  outward  normal,and the hypersurface(41)Γ^:ηn=Tx′,η′.It is easy to verify the compatibility condition holds true.We defineu2 as follows:(42)u2x,η=N0x′,η+∑k=1n∂u0∂xkxNkx′,η.


## 5. Error Estimate up toO(εα+εβ)

In this section, we will prove the error estimate up to the order ofO(εα+εβ). The main technique is using the correctors u1 and u2. Our main result is the following theorem.Theorem 8 (letf∈L2(Ωε)).
Assume thatuε is a solution of problem (1)-(3). Suppose that p(x′,ξ′),g(x′,ξ′),q(x′,η′),h(x′,η′),S(x′,ξ′), and T(x′,η′) satisfy (4)-(6). Then there exists a constant C which does not depend on ε, such that(43)uε-u0-εαu1-εβu2H1Ωε≤Cεα+εβ,where correctors u1 and u2 are defined in (38) and (42), respectively.Proof.
Analogously to Section3, for any v∈H1(Ω0) and v=0 on Γ3, we consider(44)Buε-u0-εαu1-εβu2,v=Buε-u0,v-∫Ωε▽ξu1·▽vdx-∫Ωε▽ηu2·▽vdx+Oεα+εβvH1Ω0.
Firstly, let us estimate the termB(uε-u0,v). Clearly,(45)Buε-u0,v=∫Ωε▽uε-u0·▽vdx+∫Γ1εpuε-u0vdσ1+∫Γ2εquε-u0vdσ2=∫Γ1ε-pu0-∂u0∂νε+gvdσ1+∫Γ2ε-qu0-∂u0∂νε+hvdσ2It follows from (7) that(46)∂u0∂ν0+Px′u0=Gx′onΓ1,∂u0∂ν0+Qx′u0=Hx′onΓ2.Hence(47)pu0+pP∂u0∂ν0-GΓ1ε≤Cεα,qu0+qQ∂u0∂ν0-HΓ2ε≤Cεβ.This gives(48)Buε-u0,v=∫Γ1εg-pPGvdσ1+∫Γ1εpP∂u0∂ν0-∂u0∂νεvdσ1+OεαvH1Ω0+∫Γ2εh-qQHvdσ2+∫Γ2εqQ∂u0∂ν0-∂u0∂νεvdσ2+OεβvH1Ω0.In view of (44), we obtain(49)Buε-u0-εαu1-εβu2,v=∫Γ1εg-pPGvdσ1+∫Γ1εpP∂u0∂ν0-∂u0∂νεvdσ1-∫Ωε▽ξu1·▽vdx+OεαvH1Ω0+∫Γ2εh-qQHvdσ2+∫Γ2εqQ∂u0∂ν0-∂u0∂νεvdσ2-∫Ωε▽ηu2·▽vdx+OεβvH1Ω0.
We shall next deal with the term∫Ωε‍▽ξu1·▽vdx. Using the change of variables, we find that(50)∫Ωε▽ξu1·▽vdx=εn-1α∑i∫Ωi~▽ξu1·▽ξvdξ,where(51)Ωi~=ξ′,ξn∣ξ′∈0,1εαn-1,S<ξn<1εα-εβ-αT∩ξ′∈Aiεα,Ai+1εαn-1,whereAi+1-Ai=εα,i=0,1,2⋯,and  we  assume  thatA0=0.
In view of integration by parts and the fact thatΔξu1=0, we get (52)∫Ωε▽ξu1·▽vdx=εn-1α∑i∫Γi~∂u1∂νξvdσξ,where(53)Γi~:ξn=Sx′,ξ′,ξi∈Aiεα,Ai+1εαn-1.
It follows from (38) that(54)∫Ωε▽ξu1·▽vdx=εn-1α∑i∫Γi~∂M0∂νξvdσξ+∑j=1n-1∫Γi~∂Mj∂νξ∂u0∂xjvdσξ+∫Γi~∂Mn∂νξ∂u0∂xnvdσξ≐A1+A2+A3,where(55)A1=εn-1α∑i∫Γi~∂M0∂νξvdσξ=εn-1α∑i∫Γi~g-pPGvdσξ=εn-1α∑i∫Ai/εαAi+1/εα⋯∫Ai/εαAi+1/εαg-pPGv1+▽ξ′Sx′,ξ′21/2dξ′,A2=εn-1α∑j=1n-1∑i∫Γi~∂Mj∂νξ∂u0∂xjvdσξ=εn-1α∑j=1n-1∑i∫Γi~-11+▽ξ′Sx′,ξ′21/2∂Sx′,ξ′∂ξj∂u0∂xjvdσξ=εn-1α∑j=1n-1∑i∫Ai/εαAi+1/εα⋯∫Ai/εαAi+1/εα-∂Sx′,ξ′∂ξj∂u0∂xjvdξ′,andA3=εn-1α∑i∫Γi~∂Mn∂νξ∂u0∂xnvdσξ=εn-1α∑i∫Γi~11+▽ξ′Sx′,ξ′21/2-pP∂u0∂xnvdσξ=εn-1α∑i∫Ai/εαAi+1/εα⋯∫Ai/εαAi+1/εα1-pP1+▽ξ′Sx′,ξ′21/2∂u0∂xnvdξ′.Also, note that(56)∫Γ1εg-pPGvdσ1=∫Γ1g-pPGv1+▽ξ′Sx′,ξ′21/2dx′+OεαvH1Ω0=εn-1α∑i∫Ai/εαAi+1/εα⋯∫Ai/εαAi+1/εαg-pPGv1+▽ξ′Sx′,ξ′21/2dξ′+OεαvH1Ω0,where we have used the change of variables x′=εαξ′. Hence(57)∫Γ1εg-pPGvdσ1-A1=OεαvH1Ω0.
Next, we shall evaluate the term∫Γ1ε(p/P)(∂u0/∂ν0)-∂u0/∂νεvdσ1.
Also, note that sinceν0=(0,…,0,-1) and νε=(▽ξ′S,-1)(1/[1+▽ξ′Sx′,ξ′2]1/2)+O(εα),(58)∫Γ1εpP∂u0∂ν0-∂u0∂νεvdσ1=∫Γ1εpP▽u0·ν0-▽u0·νεvdσ1=∫Γ1ε∂u0∂xn11+▽ξ′Sx′,ξ′21/2-pPvdσ1-∑j=1n-1∫Γ1ε∂u0∂xj11+▽ξ′Sx′,ξ′21/2∂S∂ξjvdσ1+OεαvH1Ω0=∫Γ1∂u0∂xn1-pP1+▽ξ′Sx′,ξ′21/2vdx′-∑j=1n-1∫Γ1∂u0∂xj∂S∂ξjvdx′+OεαvH1Ω0=εn-1α∑i∫Ai/εαAi+1/εα⋯∫Ai/εαAi+1/εα∂u0∂xn1-pP1+▽ξ′Sx′,ξ′21/2vdξ′-εn-1α∑j=1n-1∑i∫Ai/εαAi+1/εα⋯∫Ai/εαAi+1/εα∂u0∂xj∂S∂ξjvdξ′+OεαvH1Ω0.By the definition of A2 and A3 in (54), this implies that(59)∫Γ1εpP∂u0∂ν0-∂u0∂νεvdσ1-A2-A3=OεαvH1Ω0.
In a similar way, we obtain that(60)∫Ωε▽ηu2·▽vdx≐B1+B2+B3,where(61)B1=εn-1α∑i∫Bi/εβBi+1/εβ⋯∫Bi/εβBi+1/εβh-qQHv1+▽η′Tx′,η′21/2dη′,B2=εn-1α∑j=1n-1∑i∫Bi/εβBi+1/εβ⋯∫Bi/εβBi+1/εβ-∂Tx′,η′∂ηj∂u0∂xjvdη′,B3=εn-1α∑i∫Bi/εβBi+1/εβ⋯∫Bi/εβBi+1/εβ-1+qQ1+▽η′Tx′,η′21/2∂u0∂xnvdη′.
Using the same technique, we also have(62)∫Γ2εh-qQHvdσ2-B1=OεβvH1Ω0and∫Γ2εqQ∂u0∂ν0-∂u0∂νεvdσ2-B2-B3=OεβvH1Ω0.
Combining all these terms and choosingv=uε-u0-εα(u1-u1∣x1=1)-εβ(u2-u2∣x1=1), we conclude that(63)uε-u0-εαu1-εβu2H1Ωε≤Cεα+εβ,where we have used Poincaré’s inequality and the fact (64)εαu1∣xi=1+εβu2∣xi=1≤Cεα+εβexp⁡-λεα+exp⁡-θεβ,where i=1,2…,n-1.
This completes the proof.


## 6. Error Estimate up toOε3α/2+ε3β/2

In this section, we will improve the error estimate up to the order ofO(ε3α/2+ε3β/2). The main technique is using the correctors and second-order Taylor’s expansion. Our main result is the following theorem.Theorem 9 (letf∈L2(Ωε)).
Assume thatuε is a solution of problem (1)-(3). Suppose that p(x′,ξ′), g(x′,ξ′), q(x′,η′), h(x′,η′), S(x′,ξ′), and T(x′,η′) satisfy (4)-(6). Then there exists a constant C that does not depend on ε, such that(65)uε-u0-εαu1-εβu2+εαϕ1+εβϕ2H1Ωε≤Cε3α/2+ε3β/2,where correctors ϕ1 and ϕ2 are given by (88) and (93).Proof.
Following the same line of research, we consider(66)Buε-u0,v=∫Γ1ε-pu0-∂u0∂νε+gvdσ1+∫Γ2ε-qu0-∂u0∂νε+hvdσ2.This, together with (7) and the second-order Taylor’s expansion, gives(67)∂u0∂ν0+Px′u0-Gx′Γ1ε=εαSx′,ξ′∂∂xn∂u0∂ν0+Px′u0-Gx′Γ1+Oε2α,and∂u0∂ν0+Qx′u0-Hx′Γ2ε=-εβTx′,η′∂∂xn∂u0∂ν0+Qx′u0-Hx′Γ2+Oε2β.It follows that(68)Buε-u0,v=∫Γ1εg-pPGvdσ1+∫Γ1εpP∂u0∂ν0-∂u0∂νεvdσ1-∫Γ1εεαpPSx′,ξ′∂∂xn∂u0∂ν0+Px′u0vdσ1+Oε2αvH1Ω0+∫Γ2εh-qQHvdσ2+∫Γ2εqQ∂u0∂ν0-∂u0∂νεvdσ2+∫Γ2εεβqQTx′,η′∂∂xn∂u0∂ν0+Qx′u0vdσ2+Oε2βvH1Ω0.Note that(69)B-εαu1-εβu2,v=∫Ωεεα∆u1vdx-∫Γ1εεα∂u1∂νεvdσ1-∫Γ1εεαpu1vdσ1+∫Ωεεβ∆u2vdx-∫Γ2εεβ∂u2∂νεvdσ2-∫Γ2εεβqu2vdσ2=∫Ωεεα∆xu1vdx+∫Ωε2▽x·▽ξu1vdx-∫Γ1εεα∂u1∂νεvdσ1+∫Ωεεβ∆xu2vdx+∫Ωε2▽x·▽ηu2vdx-∫Γ1εεαpu1vdσ1-∫Γ2εεβ∂u2∂νεvdσ2-∫Γ2εεβqu2vdσ2,where we have used ∆ξu1=0 and ∆ηu2=0.
Thus we obtain(70)
B
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+
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.Note that on Γ1ε(71)ν0=0,⋯,0,-1,νε=▽ξ′S,-111+▽ξ′S21/2+εα▽x′S,▽x′S·▽ξ′S11+▽ξ′S23/2+Oε2αand on Γ2ε(72)ν0=0,⋯,0,1,(73)νε=▽η′T,111+▽η′T21/2+εβ▽x′T,-▽x′T·▽η′T11+▽η′T23/2+Oε2β.
In view of (33)-(42), we obtain(74)∫Ωεεα∆xu1+εβ∆xu2vdx≤Cεα∫Ωεexp⁡-λx1εαvdx+Cεβ∫Ωεexp⁡-θx1εβvdx≤Cε3α/2+ε3β/2vH1Ω0,(75)2∫Ωε▽x·▽ξu1vdx=2εα∫Γ1ε▽xu1·νεvdσ1-2εα∫Ωε▽xu1·▽xvdx≤2εα∫Γ1ε▽xu1·νεvdσ1+Cεα∫Ωεexp⁡-λξ1▽xvdx≤2εα∫Γ1ε▽xu1·▽ξ′S,-111+▽ξ′S21/2vdσ1+Cε3α/2vH1Ω0+2ε2α∫Γ1ε▽xu1·▽x′S,▽x′S·▽ξ′S11+▽ξ′S23/2vdσ1and(76)2∫Ωε▽x·▽ηu2vdx≤2εβ∫Γ2ε▽xu2·▽η′T,111+▽η′T21/2vdσ2+Cε3β/2vH1Ω0+2ε2β∫Γ2ε▽xu2·▽x′T,-▽x′T·▽η′T11+▽η′T23/2vdσ2.
It follows from (38) that(77)εα∂u1∂νε∣Γ1ε=▽ξu1·νε+εα▽xu1·νε∣Γ1ε=A~1+A~2+Oε2α,where(78)A~1=g-pPG-▽x′u0·▽ξ′S11+▽ξ′S21/2+∂u0∂xn11+▽ξ′S21/2-pPandA~2=εα▽xu1·▽ξ′S,-111+▽ξ′S21/2.
Combining all these terms in (77), we have proved that(79)∫Γ1εεα∂u1∂νεvdσ1=∫Γ1εA~1vdσ1+∫Γ1εA~2vdσ1+Oε2αvH1Ω0.
A similar calculation yields the following result:(80)∫Γ1εpP∂u0∂ν0-∂u0∂νεvdσ1=∫Γ1ε-pP∂u0∂xnvdσ1-A~+Oε2αvH1Ω0,where(81)A~=∫Γ1ε11+▽ξ′S21/2▽x′u0·▽ξ′S-∂u0∂xn+εα▽u0·▽x′S,▽x′S·▽ξ′S11+▽ξ′S23/2vdσ1.
In the same way, it follows from (42) that(82)εβ∂u2∂νε∣Γ2ε=▽ηu2·νε+εβ▽xu2·νε∣Γ2ε=B~1+B~2+Oε2β,where(83)B~1=h-qQH-▽x′u0·▽η′T11+▽ξ′T21/2+∂u0∂xn-11+▽η′T21/2+qQandB~2=εβ▽xu2·▽η′T,111+▽η′T21/2.
Combining all these terms in (82), we have proved that(84)∫Γ2εεβ∂u2∂νεvdσ2=∫Γ2εB~1vdσ2+∫Γ2εB~2vdσ2+Oε2βvH1Ω0.
Similarly, we obtain the following result:(85)∫Γ2εqQ∂u0∂ν0-∂u0∂νεvdσ2=∫Γ2εqQ∂u0∂xnvdσ2-B~+Oε2βvH1Ω0,where(86)B~=∫Γ2ε11+▽η′T21/2▽x′u0·▽η′T+∂u0∂xn+εβ▽u0·▽x′T,-▽x′T·▽η′T11+▽η′T23/2vdσ2.
Next, we shall introduce two new correctors so as to cancel lower power ofεα and εβ.
For any given function of the formg~, suppose that u~ε is the solution of the following problem:(87)-∆u~ε=0inΩε,∂u~ε∂νε+px′,x′εαu~ε=g~x′,x′εαonΓ1ε,u~ε=0on∂Ωε∖Γ1ε.
Meanwhile, we assume thatϕ1 is the solution of the homogenized problem:(88)-∆ϕ1=0inΩ0,∂ϕ1∂ν0+Px′ϕ1=G~x′onΓ1,ϕ1=0on∂Ω0\Γ1,where Px′ and Γ1ε are defined in (8) and (2), (89)G~x′=∫01⋯∫011+▽ξ′Sx′,ξ′21/2g~x′,ξ′dξ′.Clearly, it is a special case of problem (1).
We may invoke Theorem8 to conclude that(90)u~ε-ϕ1-εαu~1H1Ωε≤Cεα,where u~1 follows essentially the same steps as Section 4, with the obvious modifications.
It follows that(91)Bεαϕ1,v=Bεαϕ1-u~ε+εαu~1,v+Bεαu~ε-ε2αu~1,v=Bεαu~ε,v+Oε2αvH1Ω0.
In a similar way, for any given function of the formh~, suppose that u^ε is the solution of the following problem:(92)-∆u^ε=0inΩε,∂u^ε∂νε+qx′,x′εβu^ε=h^x′,x′εβonΓ2ε,u^ε=0on∂Ωε∖Γ2ε.
Analogously toϕ1, let ϕ2 be the solution of the homogenized problem(93)-∆ϕ2=0inΩ0,∂ϕ2∂ν0+Qx′ϕ2=H^x′onΓ2,ϕ2=0on∂Ω0\Γ2,where Qx′ and Γ2ε are defined in (8) and (3): (94)H^x′=∫01⋯∫011+▽η′Tx′,η′21/2h^x′,η′dη′.
In view of Theorem8,(95)u~ε-ϕ2-εβu^2H1Ωε≤Cεβ,where u^2 follows essentially the same steps as Section 4, with the obvious modifications.
It follows that(96)Bεβϕ2,v=Bεβu^ε,v+Oε2βvH1Ω0.
We now write(97)Buε-u0-εαu1-εβu2+εαϕ1+εβϕ2,v=εα∫Γ1g~vdx′+εβ∫Γ2h^vdx′-εα∑i=13Ii-εβ∑i=13Li+Oε2α+ε2βvH1Ω0,where(98)I1=∫Γ1▽u0·▽x′S,▽x′S·▽ξ′Svdx′,I2=∫Γ1εpPS∂∂xn∂u0∂ν0+Pu0∣Γ1vdσ1,I3=-∫Γ1▽xu1·▽x′S,-1vdx′,andL1=∫Γ2▽u0·▽x′T,-▽x′T·▽η′Tvdx′,L2=-∫Γ2εqQT∂∂xn∂u0∂ν0+Qu0∣Γ2vdσ2,L3=-∫Γ2▽xu2·▽x′T,1vdx′.
Then we just need to take that(99)g~=∑i=13I~iandh^=∑i=13L^i,where(100)I~1=▽u0·▽x′S,▽x′S·▽ξ′S,I~2=pPS∂∂xn∂u0∂ν0+Pu0∣Γ11+▽ξ′Sx′,ξ′21/2,I~3=-▽xu1·▽x′S,-1.L^1=▽u0·▽x′T,-▽x′T·▽η′T,L^2=-qQT∂∂xn∂u0∂ν0+Qu0∣Γ21+▽η′Tx′,η′21/2,L^3=-▽xu2·▽x′T,1.
Combining all these terms (70)-(97) and choosing v=uε-u0-εα(u1-u1∣xi=1)-εβ(u2-u2∣xi=1)+εαϕ1+εβϕ2, the proof is completed.



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*Source: 3746562-2018-10-21.xml*