Let $\Omega\subset\mathbb{R}^n$ be an open set. Let $H^{m,p}(\Omega)$ denote the Sobolev-space of at most $m$ times weakly differentiable functions whose weak derivatives are all $L^p$. Further let $C^{\infty}(\Omega)$ and $C_0^{\infty}(\Omega)$ denote respectively the smooth functions on $\Omega$ in the former case, and those who have compact support in the latter case.
There is a theorem that states that $C^{\infty}(\Omega)\cap H^{m,p}(\Omega)$ is dense in $H^{m,p}(\Omega)$. I'm wondering now if $C_0^{\infty}(\Omega)\cap H^{m,p}(\Omega)$ is dense in $H^{m,p}(\Omega)$.
Is this true or false? Could you please give me hints how to proove it or a link to literature on this?
Thank you in advance.