Journal of Function Spaces
Volume 2021 (2021), Article ID 8036814, 6 pages
https://doi.org/10.1155/2021/8036814

Research Article

A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients

Xiong Liu and Wenming He

School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China

Correspondence should be addressed to Xiong Liu; liuxiong980211@163.com

Received 29 April 2021; Accepted 14 June 2021; Published 19 June 2021

Academic Editor: Chuanjun Chen

Copyright © 2021 Xiong Liu and He Wenming. 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.

Abstract.

In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form . Noticing the fact that the classic homogenization theory presented by Oleinik has a high demand for the smoothness of the homogenization solution , we present a new estimate for the homogenization method under the weaker smoothness that homogenization solution satisfies than the classical homogenization theory needs.

1. Introduction

Many people investigated the second-order elliptic problem with a fixed boundary. As far as we know, there is not any work related to the elliptic problem with periodic boundary (see [1–3]). In this article, we will consider the following multiscale elliptic model problem:

Here, is a bounded domain, and the matrix of coefficients is symmetric and satisfies the following conditions:

Assume that . By the homogenization method, Oleinik et al. (see [4, 5]) obtained the 1-order approximation of as follows:

where is a 1-periodic function and satisfies the following equations:

and the homogenization solution satisfies the problem as follows:

Oleinik et al. (see [5], p. 28) proved the following result.

We end this section with the details of some notations. Throughout this paper, the Einstein summation convention is used: summation is taken over repeated indices, and denotes the distance between and .

2. Some Useful Lemmas

Lemma 1. Under the assumption that , there holds

There are numerous literatures discussing the homogenization method (see [1, 2, 4–9]). There also are many works (see [3, 10–16]) discussing the numerical methods of the multiscale homogenization problem. We observe that most of them are based on the assumption , which is unrealistic for some problems. For example, when . Let . As far as we know, it is the first time for us to estimate under the assumption that the homogenization solution belongs to the Sobolev space for the case that .

Lemma 2. Assume that . Then,

Proof. One observes that can be split into

We first estimate . Assume that and . Note that the definition of implies . By the definitions of and , we have, for any ,

Using (10), we obtain

Furthermore, from the definition of and (11), it follows that

Note that

This, together with (13), gives

Next we estimate . Assume that . Set . Let or . We have

Let . Note that whenever . By (16), one observes that can be decomposed into

We need estimates and . Assume that . Note that . One observes that . Then, we have

By the definition of and (18), we have

Note that

Inserting (19) and (20) into (17), we have

We turn now to the estimation of . We split into

We need to estimate the two items of the right-hand side of (22). Note that

By (22) and (23), we have

To estimate , we have

Plugging the above two estimates into (22), we obtain

This, together with (17), gives

Inserting (21) and (27) into (17), we have

Furthermore, let , we have

where we have used (28). Then, (7) follows by combining (15) and (29). We turn now to the estimation of . We decompose into

We first estimate . Assume that . By (10), we have, for any ,

Note that . By the definition of , we have . Then, by (28) and (31), we have

Finally, similarly to (15), by (32), we have

We turn now to the estimation of . Similarly to (17), we have

Note that the definition of implies . Therefore, let , from (34), it follows that

The desired result (8) follows by combining (33) and (35).☐

3. A New Estimate for Multiscale Homogenization Method

In this section, we give the main results as follows.

Theorem 3. Assume that and . Assume also that and for some . Then,

Assume that is the cutoff function satisfying , and if , and if . Let . One observes that and for all . Set . In the process of proving Theorem 3, we need the above Lemma 2.

Based on Lemma 2, we can prove Theorem 3 as follows:

Proof. Assume that is defined as in Lemma 2. Set

We introduce by the following problem:

One observes that and are the homogenization solution of (38) and the 1-order approximation of , respectively. We decompose into

We first estimate . Let . Note that satisfies the following problem:

One observes that can be split into

where and satisfies the following problem:

From the combination of the definition of and (7), it follows that

To estimate , by (8) and the definitions of and , one observes that

Combining (39), (41), (43), and (44), we have

Next, we estimate . Set . By the method of asymptotic expansion (see [7], p. 27), one finds that can be split into , where and are defined by

respectively, where

We first estimate . Note that (8) implies

By the method of asymptotic expansion (see ([7], p. 27), from (48), it follows that

Assume that is a cutoff function satisfying if , and if , and . We split into

where and satisfies the following problem:

To estimate , one has

We now estimate . Assume that and denotes the volume of if , or the area of if . One observes that

The combination of (8), (52), and (53) gives

Using (51) and (54), we derive

The above two estimates, together with (50), imply

Furthermore, by (49) and (56), we have

We next estimate . Assume that . Note that the definitions of and imply . By (7) and (47), we have

Assume that . This, together with (39), (45), and (57), gives the desired result (36).☐

Data Availability

The paper’s data available through the email liuxiong980211@163.com or from the author’s ORCID: 0000-0002-5452-653X, and other data is given to the journal of functional spaces https://orcid.org/0000-0002-5452-653X.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China under grants (No. 11671304) and the PhD Start-up Fund of Lingnan Normal University (No. ZL1810).

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