Review of Contemporary Homogenization Methods

We give some examples of existing mathematical methods for homogenizing partial differential equations (PDEs). In particular we will explore homogenization in Fourier space. The Maxwell equations are homogenized as an example of the method. INTRODUCTION Homogenization of PDEs is a mathematical method in which one find homogeneous material approximations of heterogeneous materials. That is, for example, finding the effective macroscopic electromagnetic material properties for a fiber composite. Mathematically it is a question of introducing a parameter, ε, in the equation which describes the fine scale in the material. We get a PDE with rapidly oscillating coefficients with a corresponding family of solutions which converges to the solution of the homogenized PDE, when ε → 0. The homogenized equation has constant coefficients which corresponds to a model of a homogeneous material. In other words, the solution of the homogenized PDE is a good approximation of the heterogeneous material PDE solution if the fine scale is much smaller than all other scales in the problem. It is in that sense one should understand the homogenization results. This means for the Maxwell equations that the macroscopic (homogenized) properties are good approximations of the heterogeneous material if the electromagnetic wave length is much greater than the period in the heterogeneous material. In this paper we will give a short review of existing homogenization methods and demonstrate how homogenization can be done in Fourier space. BACKGROUND The theoretical foundation of homogenization has developed considerably since the first results in the late 60th, by Spagnolo [10], using the method of G-convergence. Later on Γ-convergence by De Girogio, Spagnolo and Franzoni [6] and [7] (see also [5]), and H-convergence, by Tartar [11] were introduced. Murat [8] and Tartar [12] introduced the compensated compactness theorem which is an important tool to prove convergence when the effective and local equations once where identified by multi-scale asymptotic expansions. In the late 80th the two-scale convergence method was introduced by Nguetseng [9] and further developed by Allaire [1]. The Floquet-Bloch expansion and the corresponding Bloch-wave homogenization method, (see [2], [4] and the references in [4]) is a high frequency method which recently has been further developed. It can be used to find the usual homogenization results, but more importantly, it provides dispersion relations for wave propagation in periodic structures (e.g. for photonic band gap structures). In [3] the periodic unfolding method was recently introduced which further simplifies the existing two-scale convergence proofs. The challenge in homogenization can shortly be described as follows: Let us consider a bounded sequence {uε} in L2(Ω), where Ω is a bounded (or unbounded) open set in Rn, n ≥ 1. By the weak sequential compactness in L2(Ω) there exists a subsequence, still denoted {uε}, such that, as ε → 0,