The Relative Capacity

Abstract

The purpose of this article is to introduce the relative $p$-capacity ${⋒}_{p,\Omega }$ with respect to an open set $\Omega$ in $\IR^N$. It is a Choquet capacity on the closure of $\Omega$ and extends the classical $p$-capacity ${⋒}_p$ in the sense that ${⋒}_{p,\Omega }={⋒}_p$ if $Ω=\IR^N$. The importance of the relative $p$-capacity stems from the fact that a large class of Sobolev functions defined on a 'bad domain' admit a trace on the boundary $\partial \Omega$ which is then unique up to ${⋒}_{p,\Omega }$-polar set. As an application we prove a characterization of $W_0^{1,p}(\Omega )$ for open sets $Ω\subset\IR^N$.

20 pages

Cited by in corpus (1)

• Lattice Homomorphisms between Sobolev Spaces