2 - Elements of Sobolev Space Theory

Summary

We provide a brief, but self-contained, introduction to the theory of Sobolev spaces. We prove some facts from the calculus with weak derivatives, including product and chain rules. We discuss various kinds of Sobolev inequalities, including those by Gagliardo–Nirenberg, Poincaré, Friedrichs, and Hardy, both on the whole space and on domains, and include some information on their sharp constants. Furthermore, we discuss Rellich’s compactness theorem, the Sobolev extension property of a domain, as well as homogeneous Sobolev spaces.