When $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, $C^k_0(\bar{\Omega})$ is dense in $W^{k,p}(\Omega)$(Sobolev space, k-differentiable and each term estimated by p-norm).
I am wondering if it holds for general open set $\Omega\subset \mathbb{R}^d$.
Since for general $\Omega$, $C^k(\bar{\Omega})\cap W^{k,p}(\Omega)$ dense in $W^{k,p}(\Omega)$.
So can it be proved by showing $C^k_0(\bar{\Omega})$ dense $C^k(\bar{\Omega})$ (in $L^p$ norm) holds?