Let $U$ be an open set in $R^d$. I am confused about the differences between $$W^{k,p}(U):=\{u\in L^p(U): D^{\alpha}u\in L^p(U) \text{ for all } |\alpha|\le k\}$$ and $$W^{k,p}_0(U):=\overline{C_c^{\infty}(U)}$$ with the closure taken in the usual $W^{k,p}(U)$-norm.
$C_c^{\infty}(U)$ is dense in $W^{k,p}(U)$ using the $L^p$-norm but it seems that it is not dense with the $W^{k,p}(U)$-norm.
My question is: What are the main differences between the two spaces and what functions are in $W^{k,p}(U)$ that are not in $W^{k,p}_0(U)$?
Thanks.