Floquet Theory for Hypoelliptic Equations and Systems in the Whole Space

Abstract

This chapter is the place where the analytic tools developed in chapters 1 and 2 start to work for periodic partial differential equations. The main notions of Floquet theory (Floquet-Bloch solutions, quasimomentums, etc.) are introduced in section 3.1. In the same section we study analytic properties of the sets of quasimomentums and Floquet exponents. The main theorem that provides a Floquet type expansion for solutions of scalar periodic elliptic equations with smooth coefficients is proved in section 3.2. This theorem deals only with solutions of exponential growth. Solutions of some faster growth are considered in section 3.3. We describe in section 3.4 how to reduce the restrictions on equations. The last section contains some comments and references.