The Bloch transform and applications

Abstract

The Fourier Transform is one of the basic concepts in the mathematical analysis of partial differential equations in Physics. As is well-known, one of its most curious properties is the fact that it converts derivatives into multiplications. This makes it a very useful tool, as it transforms differential equations into algebraic ones. Naturally, it also has its limits. Particularly, it is not evident that it can be applied to differential equations with variable coefficients, that is, in the treatment of heterogeneous media. In this Lecture, we show one possible way to generalize the ideas of Fourier Analysis, so as to make them accessible to the study of non-homogeneous periodic media. We would like to generate a new technique, oriented to understanding certain mathematical phenomena in Homogenization Theory. To do so, we use a special class of functions, known as Bloch waves, which are commonly used in Solid State Physics. The resulting methodology is illustrated through two applications. We see how it applies in the classical problem of homogenization of elliptic operators in arbitrary domains of R with periodically oscillating coefficients. We also use it to study the asymptotic behaviour of the spectrum of some periodic structures, and more precisely, we consider it in the context of the wave equation in a bounded periodic heterogeneous medium. This Lecture reviews and unifies recent joint works of the authors.