Homogenization of a variational problem in three-dimension space

In this paper, we investigate the variational problem for a sequence of 3-dimensional domains with highly oscillating boundaries. Using the unfolding method and the averaging method, we obtain the result of the homogenization problem, that is, a sequence of solutions of Eq. (3.1) converges to the solution of Eq. (3.4) as the periodic length approaches zero. It is noteworthy that the convergence is in the strong sense. The periodic unfolding method was introduced in [6] by Cioranescu et al. for the study of classical periodic homogeniza-tion in the case of fixed domains and further described in [1–4,8,9,11]. This method was also applied to problems with holes and truss-like structures or in linearized elasticity. The homogenization of periodic structures was carried out in the last 30 years for various kinds of problems involving differential equations[12–15] and systems, as well as integral energies. But most of these works all got the weak convergence. Recently, there was a break in [5,7], where the achievement of strong convergence was obtained. In [10], the unfolding method was applied to a linear elliptic equation in the oscillating boundary cases in two-dimension space, and the new result of strong convergence was obtained. The purpose of this paper is to generalize the work in [10], i.e. we apply the periodic unfolding method to a variational problem in the oscillating boundary cases in three-dimension space, and obtain the strong convergence result. The symbols used in this paper are the same as the ones to those in [10]. We will work on domains which are constructed as follows. Let b 2 R such that b 2 ð0; 1Þ; 1=e ¼ N, where N is a positive integer. Define