Suppose $U,V$ are open sets in $\mathbb{R}^n$ and $\overline{V} \subseteq U$. Suppose $f \in W^{n,p} (U)$ is $0$ on $U-\overline{V}$. Can we say that the restriction of $f$ to $V$ (denoted again by $f$) is in $W^{n,p}_0 (V)$?
I suspect that this is true (at least if $p \neq \infty$) since $W^{n,p}_0$ intuitively corresponds to functions which are $0$ on $\partial V$. But I cannot formalize my idea.
Any help will be fully appreciated. Thank you.