Homogenization Problem in a Domain with Double Oscillating Boundary

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 in -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.

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:where and is the outward unit normal. The domain is a bounded domain in and , where hypersurfacesand

Without loss of generality, we assume that is a domain bounded by hyperplanes and and hypersurfaces and .

We suppose thatThe conditions and ensure the existence and uniqueness of solution for problem (1). This can be guaranteed by Lax-Milgram Theorem. The assumptions and ensure that . This makes the proof a bit more simpler. In fact, all the results of this paper remain valid even without the nonnegativity of and .

We also impose the smoothness condition and periodicity conditionHere we do not aim to obtain the optimal smoothness but rather focus on the method itself.

In order to avoid technical difficulties, we assume thatwhere . The conditions on and are compatibility conditions. The assumptions about and 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 -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 -periodic. They obtained the solutions convergence in with a power rate in . 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 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 the 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 . 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 in -norm by virtue of boundary correctors that can be used to obtain effective approximation.

We now describe the outline of this paper. Section 2 contains some basic formulas and estimates which are important to obtain error estimates. In Section 3, we show that the solution of problem (1) converges to the solution of the corresponding homogenized problem in the -norm with error estimate up to order of . In Section 4, we construct correctors that play important role in improving the power in . Next, we improve the error estimate up to order of in Section 5. This can be obtained via the correctors. Following the same line of research, in Section 6, we obtain an approximation to with error estimate up to order of in -norm by using correctors and second-order Taylor’s expansion.

2. Preliminaries

We consider the homogenized problem associated with problem (1) in the formwhere and is the outward unit normal. Here and . Functions , and are defined as follows:

As a preliminary step, we shall prove some propositions.

Proposition 1. There exists a constant independent of such that, for any , the following inequalitiesandhold.

Proof. Without loss of generality, we may assume . A simple calculation then gives It follows from Hölder’s inequality that Integrating over , we obtain (9). The proof of (10) is similar to (9). This completes the proof.

Proposition 2. There exists a constant independent of such that, for any , the following estimateis valid.

Proof. Note thatwhere .
It is easy to see thatLet , and, integrating over , we obtainA similar computation yields the following result:where .
Combining these two terms, we obtain (13). This completes the proof.

Proposition 3. Let , be 1-periodic in , and satisfyrespectively. Then, these inequalities are satisfied.

Proof. This proposition has been proved by G. A. Chechkin, A. Friedman, and A. L. Piatnitski in [15].

Proposition 4. There exists a constant independent of such that, for all , the following estimatesandtake place.

Proof. A direct computation shows thatThis, together with Propositions 1 and 3, yields estimate (20). Similarly, one can prove (21).

Proposition 5. There exists a constant independent of such that, for all , the following inequalitiesandhold true.

Proof. This proposition can be proved in the same way as Proposition 4.

3. Error Estimate up to

In this section, we will prove the error estimate up to the order of . Our main result is the following theorem.

Theorem 6 (let ). Assume that is a solution of problem (1)-(3). Suppose that , , , , , and satisfy (4)-(6). Then there exists a constant C which does not depend on , such that

Remark 7. Following general procedure in homogenization, we introduce the associated bilinear form whereWe shall establishThen, we extend into without increasing the -norm by more than a multiplicative constant. What remains is just to take and use Poincaré’s inequality. Then this theorem will be proved.

Proof. Since , every function can be easily extended by 0 in to become a function of . For any and on , it is easy to see thatIt follows from Proposition 2 that andwhere we have used the uniform boundedness of .
Then, according to Propositions 4 and 5, we conclude thatChoosing and using condition (4) () and Poincaré’s inequality, we obtain the desired result. This completes the proof.

4. Construction of Correctors and

In order to improve the power of , in this section, we shall construct the correctors and .

Firstly, we introduce the harmonic functions , and is treated as a parameter, as solutions ofwhereandThis system was first introduced by A. G. Belyaev [13, 14].

To ensure these solutions exist, we need to verify the compatibility condition holds true. A simple calculation then giveswhere we have used (5) andTherefore, these harmonic functions exist.

Then we define as follows:

Using the same technique, we construct corrector as follows.

Assume harmonic functions , is treated as a parameter, as solutions ofwhereand the hypersurface

It is easy to verify the compatibility condition holds true.

We define as follows:

5. Error Estimate up to

In this section, we will prove the error estimate up to the order of . The main technique is using the correctors and . Our main result is the following theorem.

Theorem 8 (let ). Assume that is a solution of problem (1)-(3). Suppose that , and satisfy (4)-(6). Then there exists a constant C which does not depend on , such thatwhere correctors and are defined in (38) and (42), respectively.

Proof. Analogously to Section 3, for any and on , we considerFirstly, let us estimate the term . Clearly,It follows from (7) thatHenceThis givesIn view of (44), we obtainWe shall next deal with the term . Using the change of variables, we find thatwhereIn view of integration by parts and the fact that , we get whereIt follows from (38) thatwhereAlso, note thatwhere we have used the change of variables . HenceNext, we shall evaluate the term .
Also, note that since and ,By the definition of and in (54), this implies thatIn a similar way, we obtain that whereUsing the same technique, we also haveCombining all these terms and choosing , we conclude thatwhere we have used Poincaré’s inequality and the fact where .
This completes the proof.

6. Error Estimate up to

In this section, we will improve the error estimate up to the order of . The main technique is using the correctors and second-order Taylor’s expansion. Our main result is the following theorem.

Theorem 9 (let ). Assume that is a solution of problem (1)-(3). Suppose that , , , , , and satisfy (4)-(6). Then there exists a constant that does not depend on , such thatwhere correctors and are given by (88) and (93).

Proof. Following the same line of research, we considerThis, together with (7) and the second-order Taylor’s expansion, givesIt follows that