I don't understand why $W_0^{k,p}(U)$ is not the same that $W^{k,p}(U)$ on a bounded domain $U$.
$W_0^{k,p}(U)$ is the closure of $C_0^{\infty}(U)$ in $W^{k,p}(U)$.
Can someone give an example of a function that is in $W_0^{k,p}(U)$ but not in $W^{k,p}(U)$ on a bounded domain? Thanks.