The Relative Capacity

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The purpose of this article is to introduce the relative p-capacity ₚ,Ω with respect to an open set Ω in ⁿ. It is a Choquet capacity on the closure of Ω and extends the classical p-capacity ₚ in the sense that ₚ,Ω=ₚ if Ω=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 ∂Ω which is then unique up to ₚ,Ω-polar set. As an application we prove a characterization of W¹,ᵖ₀(Ω) for open sets Ω⊂ⁿ.