Fractional Sobolev Spaces

Abstract

Chapter 4 is not essential for solving the elliptic problems of Chapters 5 and 6, but it does generalize the notion of trace we introduced earlier. We define all fractional Sobolev spaces, expanding on those of Chapter 3. We note that when the open set is $\mathbb{R}^{N}$ and p=2, we can use the Fourier transform to define the spaces W ^s,2 with noninteger s. This leads to simple proofs of density theorems for regular functions and of embedding theorems into more regular spaces. We establish these theorems for all W ^s,p, p>1. In contrast to the previous chapter, in which we could use properties carried over from the Sobolev spaces with integer exponents, the proofs in this chapter often use the fundamental solution of the Laplacian and properties of convolutions. As in the previous chapters, we give compactness results for the bounded subsets of W ^s,p(Ω) when Ω is bounded.

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