Homogenization Problem in a Domain with Double Oscillating Boundary

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 inH1norm.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.


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][7][8].Studying the oscillating boundary is also the key for determining interface transmission conditions in various mechanical problems [9][10][11].
We also impose the smoothness condition and periodicity condition  (  ,   ) ,  (  ,   ) and  (  ,   ) are smooth functions in (  ,   ) and 1 − periodic in   ,  (  ,   ) , ℎ (  ,   ) and  (  ,   ) are smooth functions in (  ,   ) and 1 − periodic in   . (5) Here we do not aim to obtain the optimal smoothness but rather focus on the method itself.
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  1norm.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  2 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  1 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  1 .This is also an interesting problem.
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  1 -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  0 of the corresponding homogenized problem in the  1 -norm with error estimate up to order of ( /2 +  /2 ).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 ( 3/2 +  3/2 ) in  1 -norm by using correctors and second-order Taylor's expansion.

Preliminaries
We consider the homogenized problem associated with problem (1) in the form where Ω 0 ⊂ (0, 1)  and ] 0 is the outward unit normal. Here 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 V ∈ hold.
Proposition 2. There exists a constant  independent of  such that, for any V ∈  1 (Ω 0 ), the following estimate is valid.
Combining these two terms, we obtain (13).This completes the proof.
respectively.Then, these inequalities          ∫ and 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 V ∈  1 (Ω 0 ), the following estimates and take place.
Proof.A direct computation shows that This, 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 V ∈  1 (Ω 0 ), the following inequalities and hold true.
Proof.This proposition can be proved in the same way as Proposition 4.

Error Estimate up to 𝑂(𝜀
In this section, we will prove the error estimate up to the order of ( /2 +  /2 ).Our main result is the following theorem.

Construction of Correctors 𝑢 1 and 𝑢 2
In order to improve the power of , in this section, we shall construct the correctors  1 and  2 .Firstly, we introduce the harmonic functions   (  , ), and   is treated as a parameter, as solutions of Mathematical Problems in Engineering where and This 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 gives where we have used (5) and Therefore, these harmonic functions   exist.
Then we define  1 as follows: Using the same technique, we construct corrector  2 as follows.
Assume harmonic functions   (  , ),   is treated as a parameter, as solutions of where and the hypersurface It is easy to verify the compatibility condition holds true.We define  2 as follows:

Error Estimate up to 𝑂(𝜀 𝛼 +𝜀 𝛽 )
In this section, we will prove the error estimate up to the order of (  +   ).The main technique is using the correctors  1 and  2 .Our main result is the following theorem.
Proof.Analogously to Section 3, for any V ∈  1 (Ω 0 ) and V = 0 on Γ 3 , we consider Firstly, let us estimate the term (  −  0 , V).Clearly, Mathematical Problems in Engineering 7 It follows from ( 7) that This gives In view of (44), we obtain We shall next deal with the term ∫ Ω     1 ⋅ V.Using the change of variables, we find that where where  +1 −   =   ,  = 0, 1, 2 ⋅ ⋅ ⋅ , and we assume that  0 = 0. (51) In view of integration by parts and the fact that Δ   1 = 0, we get where It follows from (38) that where Also, note that where we have used the change of variables   =     .Hence Next, we shall evaluate the term Also, note that since ] 0 = (0, . . ., 0, −1) and ]  = (   , By the definition of  2 and  3 in (54), this implies that In a similar way, we obtain that where Using the same technique, we also have Combining all these terms and choosing where we have used Poincaré's inequality and the fact where  = 1, 2 . . .,  − 1.This completes the proof.

Error Estimate up to 𝑂(𝜀
In this section, we will improve the error estimate up to the order of ( 3/2 +  3/2 ).The main technique is using the correctors and second-order Taylor's expansion.Our main result is the following theorem.
Proof.Following the same line of research, we consider (66) This, together with (7) and the second-order Taylor's expansion, gives +  ( 2 ) , and It follows that Mathematical Problems in Engineering Note that where we have used    1 = 0 and    2 = 0.
Thus we obtain Note that on Γ  1 ] 0 = (0, ⋅ ⋅ ⋅ , 0, −1) , In view of (33)-(42), we obtain and It follows from (38) that where Combining all these terms in (77), we have proved that A similar calculation yields the following result: where In the same way, it follows from (42) that (86) Next, we shall introduce two new correctors so as to cancel lower power of   and   .