Sobolev Spaces

Abstract

In this chapter we develop the elements of the theory of Sobolev spaces, a tool that, together with methods of functional analysis, provides for numerous successful attacks on the questions of existence and smoothness of solutions to many of the basic partial differential equations. For a positive integer k, the Sobolev space ${H}^{k}({\mathbb{R}}^{n})$ is the space of functions in ${L}^{2}({\mathbb{R}}^{n})$ such that, for |α| ≤ k, D^α u, regarded a priori as a distribution, belongs to ${L}^{2}({\mathbb{R}}^{n})$. This space can be characterized in terms of the Fourier transform, and such a characterization leads to a notion of ${H}^{s}({\mathbb{R}}^{n})$ for all $s\in \mathbb{R}$. For s < 0, ${H}^{s}({\mathbb{R}}^{n})$ is a space of distributions. There is an invariance under coordinate transformations, permitting an invariant notion of H^s(M) whenever M is a compact manifold. We also define and study H^s(Ω) when Ω is a compact manifold with boundary.

The tools from Sobolev space theory discussed in this chapter are of great use in the study of linear PDE; this will be illustrated in the following chapter. Chapter 13 will develop further results in Sobolev space theory, which will be seen to be of use in the study of nonlinear PDE.