Characterizations of Sobolev Functions that vanish on a part of the boundary

Let $Ω$ be a bounded domain in R n with a Sobolev extension property around the complement of a closed part D of its boundary. We prove that a function u $\in$ W 1,p ($Ω$) vanishes on D in the sense of an interior trace if and only if it can be approximated within W 1,p ($Ω$) by smooth functions with support away from D. We also review several other equivalent characterizations, so to draw a rather complete picture of these Sobolev functions vanishing on a part of the boundary.

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