On compact embeddings of Sobolev spaces and extension operators which preserve some smoothness

Abstract: It is well-known that there are bounded domains $\Omega \subset {\Bbb R^{n}}$ whose boundaries $\partial \Omega $ are not smooth enough for there to exist a bounded linear extension of the Sobolev space $ W_{p}^{1}(\Omega)$ into $W_{p}^{1}({\Bbb R^{n}})$ but the embedding $W_{p}^{1}(\Omega) \subset L_{p}(\Omega)$ is nevertheless compact. For a Lip $\gamma$ $(0<\gamma<1)$ boundary there ]exists an extension of $W_{p}^{1}(\Omega)$ into $W_{p}^{\gamma}({\Bbb R^{n}})$, but not into $W_{p}^{1}({\Bbb R^{n}})$ in general, and the smoothness retained by this extension is enough to ensure that the embedding $W_{p}^{1}(\Omega) \subset L_{p}(\Omega)$ is compact. It is natural to ask if this is typical for bounded domains which are such that $W_{p}^{1}(\Omega) \subset L_{p}(\Omega)$ is compact, that is, there exists a bounded extension into a space of functions in ${\Bbb R^{n}}$ which enjoy adequate smoothness. This is the question which will be discussed in the lecture. A central feature on the analysis is a Hardy-type inequality for differences.

Full text of the article:

• Compressed DVI file (13926 Bytes)
• Compressed Postscript file (62069 Bytes)