CEA - EDF - INRIA school on homogenization , 13 - 16 December 2010 LECTURE 1 INTRODUCTION TO HOMOGENIZATION THEORY

This lecture is devoted to a brief introduction to the mathematical theory of homogenization. For a more advanced presentation of homogenization, the reader is referred to the books [2], [5], [6], [11], [12], [21], [27], [32] and [33]. Roughly speaking, homogenization is a rigorous version of what is known as averaging. In other words, homogenization extracts homogeneous effective parameters from disordered or heterogeneous media. Homogenization has first been developed for periodic structures. Indeed, in many fields of science and technology one has to solve boundary value problems in periodic media. Quite often the size of the period is small compared to the size of a sample of the medium, and, denoting by ǫ their ratio, an asymptotic analysis, as ǫ goes to zero, is called for. Starting from a microscopic description of a problem, we seek a macroscopic, or effective, description. This process of making an asymptotic analysis and seeking an averaged formulation is called homogenization. The first chapter will focus on the homogenization of periodic structures. The method of two-scale asymptotic expansions is presented, and its mathematical justification will be briefly discussed. However we emphasize that homogenization is not restricted to the periodic case and can be applied to any kind of disordered media. This is the focus of the second chapter where the notion of Gor H-convergence is introduced. It allows to consider any possible geometrical situation without any specific assumptions like periodicity or randomness.