Research Article | Open Access

Jie Zhao, Juan Wang, "Convergence Rates in Homogenization of the Mixed Boundary Value Problems", *Mathematical Problems in Engineering*, vol. 2019, Article ID 2680657, 6 pages, 2019. https://doi.org/10.1155/2019/2680657

# Convergence Rates in Homogenization of the Mixed Boundary Value Problems

**Academic Editor:**Weimin Han

#### 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.

#### References

- A. Bensoussan, J.-L. Lions, and G. Papanicolaou,
*Asymptotic Analysis for Periodic Structures*, North-Holland, 1978. View at: MathSciNet - M. Avellaneda and F.-H. Lin, “Homogenization of elliptic problems with boundary data,”
*Applied Mathematics & Optimization*, vol. 15, no. 2, pp. 93–107, 1987. View at: Publisher Site | Google Scholar | MathSciNet - M. Avellaneda and F.-H. Lin, “Compactness methods in the theory of homogenization,”
*Communications on Pure and Applied Mathematics*, vol. 40, no. 6, pp. 803–847, 1987. View at: Publisher Site | Google Scholar | MathSciNet - C. E. Kenig, F. Lin, and Z. Shen, “Convergence rates in for elliptic homogenization problems,”
*Archive for Rational Mechanics and Analysis*, vol. 203, no. 3, pp. 1009–1036, 2012. View at: Publisher Site | Google Scholar | MathSciNet - C. E. Kenig, F. Lin, and Z. Shen, “Periodic homogenization of Green and Neumann functions,”
*Communications on Pure and Applied Mathematics*, vol. 67, no. 8, pp. 1219–1262, 2014. View at: Publisher Site | Google Scholar | MathSciNet - J. Zhao, “Homogenization of the boundary value for the Neumann problem,”
*Journal of Mathematical Physics*, vol. 56, no. 2, Article ID 021508, 9 pages, 2015. View at: Publisher Site | Google Scholar | MathSciNet - Z. Shen, “Boundary estimates in elliptic homogenization,”
*Analysis & PDE*, vol. 10, no. 3, pp. 653–694, 2017. View at: Publisher Site | Google Scholar | MathSciNet - T. A. Melnik and G. A. Chechkin, “Asymptotic analysis of boundary value problems in thick three-dimensional multilevel junctions,”
*Sbornik: Mathematics*, vol. 200, no. 3, pp. 49–74, 2009. View at: Publisher Site | Google Scholar | MathSciNet - G. Cardone, S. A. Nazarov, and A. L. Piatnitski, “On the rate of convergence for perforated plates with a small interior Dirichlet zone,”
*Zeitschrift für angewandte Mathematik und Physik ZAMP*, vol. 62, no. 3, pp. 439–468, 2011. View at: Publisher Site | Google Scholar | MathSciNet - G. A. Chechkin,
*Topics on Concentration Phenomena and Problems with Multiple Scales*, vol. 2, Springer, Berlin, Germany, 2006. View at: Publisher Site - Y. Amirat, O. Bodart, G. A. Chechkin, and A. L. Piatnitski, “Asymptotics of a spectral-sieve problem,”
*Journal of Mathematical Analysis and Applications*, vol. 435, no. 2, pp. 1652–1671, 2016. View at: Publisher Site | Google Scholar | MathSciNet - Z. Shen and J. Zhuge, “Convergence rates in periodic homogenization of systems of elasticity,”
*Proceedings of the American Mathematical Society*, vol. 145, no. 3, pp. 1187–1202, 2017. View at: Publisher Site | Google Scholar - Y. Amirat, O. Bodart, G. A. Chechkin, and A. L. Piatnitski, “Boundary homogenization in domains with randomly oscillating boundary,”
*Stochastic Processes and Their Applications*, vol. 121, no. 1, pp. 1–23, 2011. View at: Publisher Site | Google Scholar | MathSciNet - G. A. Chechkin and T. P. Chechkina, “On homogenization of problems in domains of the ”infusorium” type,”
*Journal of Mathematical Sciences*, vol. 120, pp. 1470–1482, 2003. View at: Google Scholar | MathSciNet - Y. Amirat, O. Bodart, U. De Maio, and A. Gaudiello, “Asymptotic approximation of the solution of the Laplace equation in a domain with highly oscillating boundary,”
*SIAM Journal on Mathematical Analysis*, vol. 35, no. 6, pp. 1598–1616, 2004. View at: Publisher Site | Google Scholar | MathSciNet - G. A. Chechkin, C. D'Apice, and U. De Maio, “On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary,”
*Mathematical Methods in the Applied Sciences*, vol. 33, no. 17, pp. 2019–2036, 2010. View at: Publisher Site | Google Scholar | MathSciNet - G. Griso, “On the spectrum of deformations of compact double-sided flat hypersurfaces,”
*Analysis & PDE*, vol. 6, no. 5, pp. 1051–1088, 2013. View at: Publisher Site | Google Scholar | MathSciNet - Y. Amirat, G. A. Chechkin, and R. Gadyl'shin, “Spectral boundary homogenization in domains with oscillating boundaries,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 6, pp. 4492–4499, 2010. View at: Publisher Site | Google Scholar | MathSciNet - M. Lobo, S. A. Nazarov, and E. Perez, “Eigen-oscillations of contrasting non-homogeneous elastic bodies: asymptotic and uniform estimates for eigenvalues,”
*IMA Journal of Applied Mathematics*, vol. 70, no. 3, pp. 419–458, 2005. View at: Publisher Site | Google Scholar | MathSciNet - G. Griso, “Error estimate and unfolding for periodic homogenization,”
*Asymptotic Analysis*, vol. 40, no. 3-4, pp. 269–286, 2004. View at: Google Scholar | MathSciNet - G. Griso, “Interior error estimate for periodic homogenization,”
*Analysis and Applications*, vol. 4, no. 1, pp. 61–79, 2006. View at: Publisher Site | Google Scholar | MathSciNet - M. Avellaneda and F.-H. Lin, “Compactness methods in the theory of homogenization. II. Equations in nondivergence form,”
*Communications on Pure and Applied Mathematics*, vol. 42, no. 2, pp. 139–172, 1989. View at: Publisher Site | Google Scholar | MathSciNet - M. Avellaneda and F.-H. Lin, “ bounds on singular integrals in homogenization,”
*Communications on Pure and Applied Mathematics*, vol. 44, no. 8-9, pp. 897–910, 1991. View at: Publisher Site | Google Scholar | MathSciNet - C. E. Kenig, F. Lin, and Z. Shen, “Homogenization of elliptic systems with Neumann boundary conditions,”
*Journal of the American Mathematical Society*, vol. 26, no. 4, pp. 901–937, 2013. View at: Publisher Site | Google Scholar | MathSciNet - T. A. Suslina, “Homogenization of the Dirichlet problem for elliptic systems: -operator error estimates,”
*Mathematika. A Journal of Pure and Applied Mathematics*, vol. 59, no. 2, pp. 463–476, 2013. View at: Publisher Site | Google Scholar | MathSciNet - T. Suslina, “Homogenization of the Neumann problem for elliptic systems with periodic coefficients,”
*SIAM Journal on Mathematical Analysis*, vol. 45, no. 6, pp. 3453–3493, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. Gu and Z. Shen, “Homogenization of stokes systems and uniform regularity estimates,”
*SIAM Journal on Mathematical Analysis*, vol. 47, no. 5, pp. 4025–4057, 2015. View at: Publisher Site | Google Scholar | MathSciNet - S. Gu and Q. Xu, “Optimal boundary estimates for Stokes systems in homogenization theory,”
*SIAM Journal on Mathematical Analysis*, vol. 49, no. 5, pp. 3831–3853, 2017. View at: Publisher Site | Google Scholar | MathSciNet - J. Geng, “ estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains,”
*Advances in Mathematics*, vol. 229, no. 4, pp. 2427–2448, 2012. View at: Publisher Site | Google Scholar | MathSciNet - J. Geng, Z. Shen, and L. Song, “Uniform estimates for systems of linear elasticity in a periodic medium,”
*Journal of Functional Analysis*, vol. 262, no. 4, pp. 1742–1758, 2012. View at: Publisher Site | Google Scholar | MathSciNet - S. N. Armstrong and Z. Shen, “Lipschitz estimates in almost-periodic homogenization,”
*Communications on Pure and Applied Mathematics*, vol. 69, no. 10, pp. 1882–1923, 2016. View at: Publisher Site | Google Scholar | MathSciNet - S. N. Armstrong and C. K. Smart, “Quantitative stochastic homogenization of convex integral functionals,”
*Annales Scientifiques de l'Ecole Normale Superieure*, vol. 49, no. 2, pp. 423–481, 2016. View at: Publisher Site | Google Scholar | MathSciNet - M. A. Pakhnin and T. A. Suslina, “Operator error estimates for the homogenization of the elliptic Dirichlet problem in a bounded domain,”
*Algebra i Analiz*, vol. 24, no. 6, pp. 139–177, 2012. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2019 Jie Zhao and Juan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.