Sobolev Spaces and Embedding Theorems

Abstract

In Chapter 2, we define the spaces L ^p and the Sobolev spaces, using differentiation in the sense of distributions. We prove density theorems for the regular functions, allowing us to work with regular functions in most of the later proofs instead of working with functions that are only almost everywhere differentiable. In particular, we use this to prove the embedding theorems into “more regular” spaces. For example, for p sufficiently large, the space of functions W ^1,p, that is, the space of functions in L ^p whose first derivatives are in L ^p, is contained in L ^∞. Following this, we give compactness theorems that allow us to extract from a sequence of bounded functions in W ^1,p(Ω), where Ω is bounded and p>1, a subsequence that converges strongly in L ^p. This plays an important role in the convergence of minimizing sequences in Chapter 5.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4471-2807-6_8