When is $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?

Question

Let $\Omega\subset\mathbb{R}$, $m\in\mathbb{N}$ and $1\leq p<\infty$.

What are the (most common) sufficient conditions (if such conditions exists) that we can impose on $\Omega$, $m$ and $p$ to get
(1) $C_0^\infty(\Omega)$ dense in $W^{m,p}(\Omega)$?
(2) $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?
(3) $W^{m,p}_0(\Omega)$ dense in $L^p(\Omega)$ ?

What are the most important cases? What is the answer if we replace $\Omega$ by $\overline{\Omega}$?

Thanks.

Answer 1

(1): Only if $m=0$. Otherwise, the closure of $C_0^\infty(\Omega)$ in $W^{m,p}(\Omega)$ is $W^{m,p}_0(\Omega)$, a genuine subset of $W^{m,p}(\Omega)$.

(2): Always true. Even $C^\infty_0(\Omega)$, with bound $L^p$-norm is dense in $L^p(\Omega)$.

(3): Always true - see the answer for (2).