Convergence Rates in Homogenization of the Mixed Boundary Value Problems

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 in and , with no smoothness assumption on the coefficients.

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, the error estimate in was presented by Bensoussan, Lions, and Papanicolaou [1]. In 1987, Avellaneda and Lin [2] proved convergence by the method of maximum principle. At the same year, they [3] also obtained error estimate when 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 and 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 convergence results, which is based on Fourier analysis. In 2016, Shen [7] proved the 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 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 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 convergence rate of Steklov-type problems was studied in [11]. In 2017, Shen [12] also obtained the convergence rate with Dirichlet-Neumann mixed boundary value problem.

In homogenization problems for the Poisson equations in a domain with oscillating boundary, the 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 in and for oscillating operators with the mixed Dirichlet-Robin boundary condition. In particular, the 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, let be a bounded domain in . Suppose that , where and are two disjoint closed sets of . Let be a weak solution to the following problem:where is a number. Here denotes the conormal derivative with and is the outward unit normal to at the point .

Throughout this paper, the summation convention is used. We assume that the matrix with is real symmetric and satisfies the ellipticity condition, i.e.,where , and the periodicity conditionWe impose the smoothness conditionWithout loss of generality, we also assume the compatibility condition

Associated with (1) is the homogenized problemwhere the constant matrix is known as the homogenized matrix of and .

Recall that is called the weak solution of (1), if for any , function holds 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 converges to weakly in and strongly in , as .

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, 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 when , in the case of Robin problem when , and for the Neumann problem when and . 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 that and 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 where is the smoothing operator and is the solution of the cell problem.

Theorem 2. Under the conditions as Theorem 1, then there exists a constant C such that

The rest of the paper is organized as follows. Section 2 contains some basic formulas and useful propositions which play important roles to get convergence rates. In Section 3, we show that the solution of partial differential equation with mixed boundary value problems and 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.

Let denote an open ball with center and radius and . Since is Lipschitz, then there exists a bounded extension operator , such that is an extension of and . We set to be a smooth function and . We will also use to denote positive constant which may vary in different formulas.

Associated with operator in (1), the homogenized operator iswhere is a constant coefficient operator which is also called homogenized operator. The constant matrix is given by where . Function is a solution of the following cell problem:

Fix such that and . Define operator on as where . We also call it the smoothing operator.

Proposition 3. If , then and

Proof. These estimates have proved by Parseval’s Theorem and Hölder’s inequality, which may be found in [7].

Proposition 4. Let be a periodic function, . Suppose that and . Then there exists such that and .

Proof. This proposition had been proved by Kenig, Lin, and Shen [4].

Remark 5. Let Note that periodic function satisfies and . It follows from Proposition 4 that there exists a function , such that and

Remark 6. Under the assumption is bounded measurable in , it is known that . This implies that . In particular,

Proposition 7. If , then

Proof. By Fubini’s Theorem,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 establish and convergence rates of solutions.

LetIn order to prove Theorem 1, it suffices to show that .

By the represented formula of , then satisfies the following boundary value problem:where we have used (1) and (6) satisfied by and , respectively.

In view of the fact that we obtain It is easy to calculate that Then, it follows from the bilinear form that

To estimate , we note that, by Proposition 3,

Next, we shall estimate . Let . Note that is periodic and satisfies the conditions of Proposition 4. Then, in view of Remark 5, there exists a periodic function , such that , and .

Thus, by the divergence theorem, it gives Note that the second term vanishes in view of the antisymmetry of .

As a result, using Proposition 3 and Remark 6, we get that

It remains to estimate . It follows from Proposition 7 that

This, together with (25) and (27), gives that

By the coercive condition of bilinear form and duality argument, we get the desired result, which completes the proof of Theorem 1.

It follows from Theorem 1 and Proposition 3, by Minkowski’s inequality, that

This completes the proof of Theorem 2.

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 the and 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 the convergence rates, for any . 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 to -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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work has been supported by National Natural Science Foundation of China (No. 11626239), China Scholarship Council (No. 201708410483), and the Education Department of Henan Province (No. 18A110037). A part of this work was done while the first author was visiting school of mathematics and applied statistics, University of Wollongong, Australia.