Bloch-Fourier Approach to Classical Homogenization Problems
Carlos Conca · 2004
In this mini-course we will go into the so-called Bloch-Fourier approach to understand homogenization of periodic structures. The aims are to state the main results with motivations, highlight various phenomena in the Fourier space and indicate possible gains over other methods. Main ideas will be given trying to avoid technicalities. It is fair to say that ever since the publication of the book [4], there is a renewed and vigorous activity in homogenization problems which form an important area of Applied Mathematics. In order to tackle the key questions in homogenization, several methods have been devised and there is an enormous literature on the subject. Our goal in these series of lectures is to address certain basic questions which we consider to be fundamental and to present a way to answer them which can be classified inside category (B) below. Homogenization methods can be broadly categorized as follows: (A) Physical space methods, (B) Fourier space methods, and (C) Phase space methods. There are many methods falling in the class (A) and let us mention some successful ones: Multiscale expansion [4], Multiscale convergence [1], [21], Method of oscillating test functions [19],