Characterizations of Sobolev Functions that vanish on a part of the   boundary
  Let $\Omega$ 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 ($\Omega$) vanishes on D in the sense of an interior trace if and only if it can be approximated within W 1,p ($\Omega$) 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. 
Moritz Egert (LM-Orsay), Patrick Tolksdorf
math.CA