Sobolev spaces

Abstract In this chapter we define Sobolev spaces which are the ‘natural’ spaces of functions in which to solve variational formulations of partial differential equations. Physically, Sobolev spaces can be interpreted as spaces of functions with finite energy. This chapter is the most ‘technical’ of this book and relies in part on a course of ‘pure’ mathematics Nevertheless, it is necessary to understand the results below well to follow the rest, of the course, including its more numerical aspects. In general, it is not necessary to know the proofs of these results (expect for the most simple and most useful).