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.
For each open bounded set $U$, we have that $\chi_U\in W^{k,p}(U)$ but it's not in $W^{k,p}_0(U)$ (it would contradict Poincaré inequality).
However, if the boundary of $U$ is smooth enough, we have that $W^{k,p}(U)$ is the closure of $C^{\infty}(\overline U)$, that is, the smooth functions such that the derivatives of each order admit a continuous extension to the boundary.