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PDE - How is "the space of Wn,p Sobolev functions that are 0 at the boundary of the domain" usually constructed?

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The space you’re talking about is usually constructed by taking the closure of smooth compactly supported functions with respect to the Sobolev norm. In 1 dimension, every W^m,p function admits a continuous representative*, which you can require to vanish at the boundary.

*This is because W^1,1 in 1 dimension consists of precisely the absolutely continuous functions, and W^1,p consists of Holder continuous functions by Morrey’s embedding. For higher Sobolev spaces the situation only improves.
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There's a theorem called sobolev trace theorem which states if the function satisfy certain properties then there exist an equivalent  sufficiently smooth function function thats equal to 0 on the boundary
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Some equivalence relation.

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