# Convergence Rates in Homogenization of the Mixed Boundary Value Problems

Authors: Jie Zhao; Juan Wang
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

We will study the convergence rates of solutions for homogenization of the mixed boundary value problems. By utilizing the smoothing operator as well as duality argument, we deal with the mixed boundary conditions in a uniform fashion. As a consequence, we establish the sharp rate of convergence inH1 and L2, with no smoothness assumption on the coefficients.


---

## Body

## 1. Introduction

Convergence rates estimates of solutions are one of the main questions in homogenization theory. There are many papers about convergence of solutions for elliptic homogenization problems. Assume that all of functions are smooth enough, theO(ε) error estimate in L∞ was presented by Bensoussan, Lions, and Papanicolaou [1]. In 1987, Avellaneda and Lin [2] proved Lp convergence by the method of maximum principle. At the same year, they [3] also obtained L∞ error estimate when f is less regular than Bensoussan, Lions, and Papanicolaou’s. Recently, there were many activities in the theory of homogenization with error estimates. In 2012, Kenig, Lin, and Shen [4] obtained convergence of solutions in L2 and H1/2 in Lipschitz domains with Dirichlet or Neumann boundary conditions. In 2014, they [5] have also studied the asymptotic behavior of the Green and Neumann functions obtaining some error estimates of solutions. In 2015, the first author [6] obtained the pointwise as well as W1,p convergence results, which is based on Fourier analysis. In 2016, Shen [7] proved the H1 convergence rates with Dirichlet or Neumann conditions.The problems of changing type of boundary conditions in homogenization have been studied extensively in various settings in the past years. In [8] the H1 weak convergence of solutions was obtained in homogenization problems of multi-level-junction type. In 2011, Cardone [9] considered the homogenization with mixed boundary value problems in a thin periodically perforated plate and obtained the logarithmic rate of convergence of solutions. In the monograph [10], the H1 convergence rate was shown by the method of potentials for the solutions to the Dirichler-Fourier mixed boundary value problem in the perforated domain. The L2 convergence rate of Steklov-type problems was studied in [11]. In 2017, Shen [12] also obtained the L2 convergence rate with Dirichlet-Neumann mixed boundary value problem.In homogenization problems for the Poisson equations in a domain with oscillating boundary, theH1 convergence rate have been studied in [13–15], while work [16] deals with the multilevel oscillation of the boundary with different conflicting boundary conditions. See also [17–19] for more results on the asymptotic behavior of eigenvalues for the boundary value problems in domains with oscillating boundaries or interfaces.In this paper, we shall establish the sharp rates of convergence inH1 and L2 for oscillating operators with the mixed Dirichlet-Robin boundary condition. In particular, the L2 estimate was proved by Griso in [20, 21] for Dirichlet or Neumann boundary conditions, using the periodic unfolding method. Our results, on one hand, extend the classical Laplace operator to oscillating operator; on the other hand they extend the classical boundary value problems to a broader mixed boundary conditions settings in homogenization. Meanwhile, our approach is utilizing the smoothing operator which is much more simple and direct than periodic unfolding method.More precisely, letΩ be a bounded C1,1 domain in Rn. Suppose that ∂Ω=Γ1∪Γ2, where Γ1 and Γ2 are two disjoint closed sets of ∂Ω. Let uε∈H1(Ω) be a weak solution to the following problem:(1)Lεuε=-div⁡Axε∇uε=-∂∂xiaijxε∂uε∂xj=finΩ,uε=gonΓ1,∂uε∂νε+kuε=honΓ2,where k≥0 is a number. Here ∂uε/∂νε=niaij∂uε/∂xj denotes the conormal derivative with Lε and n(x) is the outward unit normal to ∂Ω at the point x.Throughout this paper, the summation convention is used. We assume that the matrixA(y)=(aij(y)) with 1≤i,j≤n is real symmetric and satisfies the ellipticity condition, i.e.,(2)aijy=ajiy,λξ2≤aijyξiξj≤1λξ2,fory∈Rnandξ=ξi∈Rn,where λ>0, and the periodicity condition(3)Ay+z=Ayfory∈Rnandh∈Zn.We impose the smoothness condition(4)Ayisboundedmeasurable,f∈L2Ω,g∈H1∂Ω,h∈H-1/2∂Ω.Without loss of generality, we also assume the compatibility condition(5)∫∂Ωuεxdσx=∫Ωfxdx+∫Γ2hxdσx=0.Associated with (1) is the homogenized problem(6)L0u0=-qij∂2u0∂xi∂xj=finΩ,u0=gonΓ1,∂u0∂ν0+ku0=honΓ2,where the constant matrix qij is known as the homogenized matrix of aij(y) and ∂u0/∂ν0=niqij∂u0/∂xj.Recall thatuε is called the weak solution of (1), if for any ϕ∈H1(Ω), function uε-g∈H1(Ω) holds (7)∫ΩAε∇uε·∇ϕdx+k∫∂Ωuεϕdσx=∫Ωfϕdx+h,ϕH-1/2∂Ω×H1/2∂Ω.The existence and uniqueness of the weak solution to the mixed boundary value problem (1) follow from Lax-Milgram theorem. It is well known that the solution uε converges to u0 weakly in H1(Ω) and strongly in L2(Ω), as ε→0.For the Dirichlet or Neumann boundary value problems, the regularity estimates of solutions in quantitative homogenization have been studied extensively. By the compactness method, interior and boundary Hölder’s estimates,W1,p estimates, and Lipschitz estimates, the regularity of solutions for second-order elliptic systems or equations was established by Avellaneda, Kenig, Lin, Shen, and Suslina in a serious of papers [2, 3, 22–26]. For the case of homogenization with mixed boundary value problems, the uniform interior estimates and boundary Hölder’s estimates have already been established in [27], and the sharp boundary regularity estimates have obtained in [28]. See also [29–32] for more related results on uniform regularity estimates.The novelty of this paper lies in the fact that it deals with the mixed Dirichlet-Robin boundary condition which is a more general settings, for instance, in the case of Dirichlet problem whenΓ2=∅, in the case of Robin problem when Γ1=∅, and for the Neumann problem when Γ1=∅ and k=0. As far as the author knows, very few convergence rates results are known for (1) of such mixed boundary value problems.The following are the main results of this paper.Theorem 1.
Suppose thatuε∈H1(Ω) and u0∈H2(Ω) are the weak solutions of the mixed boundary value problems (1) and (6), respectively. Then, under the assumptions (2)–(5), there exists a constant C such that (8)uε-u0-εχTε∇u0H1Ω≤Cεu0H2Ω,where Tε is the smoothing operator and χ is the solution of the cell problem.Theorem 2.
Under the conditions as Theorem1, then there exists a constant C such that (9)uε-u0L2Ω≤Cεu0H2Ω.The rest of the paper is organized as follows. Section2 contains some basic formulas and useful propositions which play important roles to get convergence rates. In Section 3, we show that the solution uε of partial differential equation with mixed boundary value problems H1(Ω) and L2(Ω) convergence to the solutions of the corresponding homogenized problems is based on using of smoothing operator. Finally, we summarize our results and discuss possible further development in Section 4.


## 2. Preliminaries

We begin by specifying our notations.LetBr(x)={y∈Rn:y-x<r} denote an open ball with center x and radius r and Ω~ε={x∈Rn:dist(x,∂Ω)≤ε}. Since Ω is Lipschitz, then there exists a bounded extension operator E:H2(Ω)→H2(Rn), such that u~0=E(u0) is an extension of u0 and u~0H2(Rn)≤Cu0H2(Ω). We set φ∈C0∞(Rn) to be a smooth function and φH1(Rn)≤CφH1(Ω). We will also use C to denote positive constant which may vary in different formulas.Associated with operatorLε in (1), the homogenized operator is(10)L0=-qij∂2∂xi∂xjinΩ,where L0 is a constant coefficient operator which is also called homogenized operator. The constant matrix Q=(qij) is given by (11)qij=∫Yaijy+aiky∂χjy∂ykdy, where Y=[0,1)n≃Rn/Zn. Function χ(y)=(χj(y)) is a solution of the following cell problem: (12)-∂∂yiaiky∂χjy∂yk+aijy=0,inY,χjy+h=χjy,fory∈Rn,h∈Yn,∫Yχjydy=0,Fixψ∈C0∞(B1(0)) such that ψ≥0 and ∫Rnψdx=1. Define operator Tε on L2 as(13)Tεux=u∗ψε=∫Rnux-yψεydy, where ψε(x)=ε-nψ(x/ε). We also call it the smoothing operator.Proposition 3.
Ifu0∈H2(Rn), then (14)∇u0-Tε∇u0L2Rn≤Cε∇2u0L2Rnand (15)Tε∇2u0L2Rn≤C∇2u0L2RnProof.
These estimates have proved by Parseval’s Theorem and Hölder’s inequality, which may be found in [7].Proposition 4.
LetFij(y)∈L2(Y) be a periodic function, Y=[0,1)n. Suppose that ∫YFij(y)dy=0 and ∂/∂yi(Fij(y))=0. Then there exists Φkij∈H1(Y) such that Fij=∂Φkij/∂yk and Φkij=-Φikj.Proof.
This proposition had been proved by Kenig, Lin, and Shen [4].Remark 5.
Let(16)Fijy=qij-aijy-aiky∂χjy∂yk. Note that periodic function Fij(y) satisfies ∫YFij(y)dy=0 and ∂/∂yiFij=0. It follows from Proposition 4 that there exists a function Φkij(y), such that Φkij=-Φikj and Fij=∂Φkij/∂yk.Remark 6.
Under the assumptionA(y) is bounded measurable in Rn, it is known that χ(y)∈C0,α(Rn). This implies that Φ(y)∈C0,α(Rn). In particular, χL∞(Rn)+ΦL∞(Rn)+≤C.Proposition 7.
Ifu0∈H2(Rn), then(17)Tε∇u0L2Ω~ε≤Cε1/2u0H2Rn.Proof.
By Fubini’s Theorem,(18)∫Ω~εTε∇u02dx≤C∫Ω~ε∫B10∇u0x-εy2dydx≤C∫B10dy∫Ω~2ε∇u0x2dx≤Cεu0H2Rn2,where we have used the well-known estimate for the last inequality. See [33] or [12] for the proof.


## 3. Proofs Theorems

The goal of this section is to establishH1 and L2 convergence rates of solutions.Let(19)Mε=uε-u0-εχTε∇u~0.In order to prove Theorem 1, it suffices to show that MεH1(Ω)≤Cεu0H2(Ω).By the represented formula ofMε, then Mε satisfies the following boundary value problem:(20)LεMε=L0u0-Lεu0-LεεχTε∇u~0inΩ,Mε=-εχTε∇u~0onΓ1,∂Mε∂νε+kMε=∂u0∂ν0-∂u0∂νε-∂εχTε∇u~0∂νε-kεχTε∇u~0onΓ2,where we have used (1) and (6) satisfied by uε and u0, respectively.In view of the fact that(21)∫ΩAε∇uε·∇φdx+k∫∂Ωuεφdσx=∫ΩQ∇u0·∇φdx+k∫∂Ωu0φdσx, we obtain (22)∫ΩAε∇Mε·∇φdx+k∫∂ΩMεφdσx=∫ΩQ∇u0-Aε∇u0-εAε∇χTε∇u~0·∇φdx-k∫∂ΩεAεχTε∇u~0φdσx. It is easy to calculate that (23)Q∇u0-Aε∇u0-εAε∇χTε∇u~0=Q∇u0-Tε∇u~0-Aε∇u0-Tε∇u~0-εAεχTε∇2u~0+Q-Aε-Aε∇χTε∇u~0.Then, it follows from the bilinear form that(24)∫ΩAε∇Mε·∇φdx+k∫∂ΩMεφdσx=∫ΩQ∇u0-Tε∇u~0-Aε∇u0-Tε∇u~0-εAεχTε∇2u~0·∇φdx+∫ΩQ-Aε-Aε∇χTε∇u~0·∇φdx-k∫∂ΩεAεχTε∇u~0φdσx≐I1+I2+I3.To estimateI1, we note that, by Proposition 3,(25)I1≤Cεu0H2Ω∇φL2Ω.Next, we shall estimateI2. Let Fij=qij-aij(y)-aik(y)∂χj(y)/∂yk. Note that Fij is periodic and satisfies the conditions of Proposition 4. Then, in view of Remark 5, there exists a periodic function Φkij∈H1(Y), such that Φkij=-Φikj, and qij-aij(y)-aik(y)∂χj(y)/∂yk=∂Φkij/∂yk.Thus, by the divergence theorem, it gives(26)I2=∫Ω∂∂xkεΦkijTε∂u~0∂xj·∂φ∂xidx=-∫ΩεΦkijTε∂2u~0∂xk∂xj∂φ∂xidx-∫ΩεΦkijTε∂u~0∂xj∂2φ∂xk∂xidx=-∫ΩεΦkijTε∂2u~0∂xk∂xj∂φ∂xidx. Note that the second term vanishes in view of the antisymmetry of Φkij.As a result, using Proposition3 and Remark 6, we get that(27)I2≤Cεu0H2Ω∇φL2Ω.It remains to estimateI3. It follows from Proposition 7 that(28)I3≤k∫∂ΩεAεχTε∇u~0φdσx≤Cε∫Ω~εTε∇u~0·φdx≤CεTε∇u~0L2Ω~εφL2Ωε≤Cε3/2u0H2ΩφL2Ωε.This, together with (25) and (27), gives that(29)∫ΩAε∇Mε·∇φdx+k∫∂ΩMεφdσx≤Cεu0H2Ω∇φL2Ω.By the coercive condition of bilinear form and duality argument, we get the desired result, which completes the proof of Theorem1.It follows from Theorem1 and Proposition 3, by Minkowski’s inequality, that(30)uε-u0L2Ω≤Cεu0H2Ω+εχTε∇u0L2Ω≤Cεu0H2Ω.This completes the proof of Theorem2.


## 4. Conclusions and Perspectives

In this paper, we research the convergence rates of solutions for homogenization of the mixed Dirichlet-Robin boundary value problems. Our approach is utilizing the smoothing operator, which is much more simple and direct to deal with boundary discrepancies. As a consequence, we obtain theH1 and L2 convergence rates results, which extend the classical boundary value problems to a broader mixed boundary condition settings.Indeed, it is expected that one could obtain theW1,p convergence rates, for any 1≤p<∞. To the best of our knowledge, such estimates for the mixed boundary value problems in homogenization have not been reported so far in the literature. Hence, how to utilize the smoothing operator and avoid difficult of the terms from boundary discrepancies for such problems are an interesting problem. This is one further possible direction to be developed.Generally, many other types of equations with the mixed boundary conditions settings could be considered by this method. One may naturally try to extend the classical second-order equations to2m-order higher-order equations or nonlinear elliptic equations. It is expected that the method of this work could contribute to a better solving of the mixed boundary value problem in homogenization.



---

*Source: 2680657-2019-03-05.xml*