J. Rákosník (ed.),
Function spaces, differential operators and nonlinear analysis.
Proceedings of the conference held in Paseky na Jizerou, September 3-9, 1995.
Mathematical Institute, Czech Academy of Sciences, and Prometheus Publishing House, Praha 1996
p. 17 - 26

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

Viktor I. Burenkov, W. Desmond Evans

Department of Mathematics College of Cardiff, University of Wales, 23 Senghennydd Rd, Cardiff CF2 4YH, United Kingdom burenkov@cardiff.ac.uk

Department of Mathematics College of Cardiff, University of Wales, 23 Senghennydd Rd, Cardiff CF2 4YH, United Kingdom w.d.evans@cardiff.ac.uk


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.

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