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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2021

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Volume 2021 |Article ID 8036814 | https://doi.org/10.1155/2021/8036814

Xiong Liu, Wenming He, "A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients", Journal of Function Spaces, vol. 2021, Article ID 8036814, 6 pages, 2021. https://doi.org/10.1155/2021/8036814

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

Academic Editor: Chuanjun Chen
Received29 Apr 2021
Accepted14 Jun 2021
Published19 Jun 2021

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 [13]). 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, 49]). There also are many works (see [3, 1016]) 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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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.

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