Equality of Sobolev spaces

Question

It is well known that if $\Omega\subset \mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N-1}(F)=0$,where $\mathcal{H}^{N-1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{1,p}(\Omega)=W^{1,p}(\Omega\backslash F)$ for $p\in (1,+\infty)$.

Question: Assume $F$ and $\Omega$ satisfy the conditions prescribed above. In this case, shall we have the following equality $W^{1,p}_{0}(\Omega)=W^{1,p}_{0}(\Omega\backslash F)$ ?

Thanks for any suggestion.

Answer 1

This is not true for $p>N$ due to Sobolev embeddings: They imply that functions from $W^{1,p}_0(\Omega\setminus F)$ have zero boundary values on $F$.