# Dense property of $C^k_0(\Omega)$

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?

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It's true when $\Omega$ has a $C^1$ boundary (you have to use charts to see this). –  Davide Giraudo Jun 5 '12 at 17:19
@DavideGiraudo true for all statements above? –  newbie Jun 5 '12 at 17:26