Sobolev spaces on different domains

Question

Let $U\subset\subset V\subset\subset \mathbb R^n$. Let $f:\mathbb R^n\to \mathbb R$ such that $f=0$ ouside $U$. If $f\in W^{k,p}(U)$, is it true that $f\in W^{k,p}(V)$ ? or any counterexample ? It might be not true, or otherwise, the extension theorem of section 5.4 in Evans' PDE book is meaningless (may take $E=$identity).

Answer 1

As long as you do not pose any 'global' conditions on $f$, this is not true. An easy counterexample is $$ f(x) = \begin{cases} 1 & \text{for } x \in U,\\ 0 & \text{else}.\end{cases}$$ This satisfies your conditions, but does not belong to $W^{k,p}(V)$ if $k > 0$.