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