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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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    @DavideGiraudo true for all statements above?2012-06-05

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