The Relative Capacity

Abstract

The purpose of this article is to introduce the relative $p$-capacity $\Cap_{p,Ω}$ with respect to an open set $Ω$ in $\IR^N$. It is a Choquet capacity on the closure of $Ω$ and extends the classical $p$-capacity $\Cap_p$ in the sense that $\Cap_{p,Ω}=\Cap_p$ 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 $\partialΩ$ which is then unique up to $\Cap_{p,Ω}$-polar set. As an application we prove a characterization of $W^{1,p}_0(Ω)$ for open sets $Ω\subset\IR^N$.