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,