Homogenization of partial differential equations : from multiple scale expansions to Tartar's H-measures

Abstract

Homogenization theory has emerged over the last decades as a fundamental tool in the study of mathematical problems arising in processes taking place in highly heterogeneous media, such as composite materials, ow through porous medium, living tissues, just to cite a few. The main feature of these problems is the presence of multiple scales, notably microscopic and macroscopic scales. A prominent and simpli ed theory of homogenization is period homogenization based on assumptions of periodic structure in the problems investigated. Since its inception, several challenges had to be overcome in the evolution of the theory. My dissertation was aimed at covering these challenges and the corresponding deep methods that were invented subsequently. First, we study elliptic partial di erential equations with periodic coe cients using the multiscale expansion and Tartar's method of oscillating test functions. Then we discuss nonlinear homogenization using the div-curl lemma, compensated compactness, Young measures and H-measures. We shall endeavour to motivate the emergence of these methods along their historical flow.