Preliminaries

Abstract

In this chapter we first recall several function spaces that will be relevant in order to tackle the questions of existence and regularity for (weak) solutions to elliptic problems. We further fix the notation and give some important properties, inequalities, and embedding theorems. Even though this theory would deserve to be developed in more detail and wider generality, we have decided to present the material in this chapter rather as a collection of the mathematical background. In particular, the proofs will be given only for some selected statements, which are of central interest in the course of these lecture notes or which are instructional and might serve as an illustration in order to understand the underlying concepts.