Let $\Omega\subset\mathbb{R}^n$ be an limited open set of class $C^1$ and $1\leq p<\infty$. Show that $$\bigcap_{m=0}^{\infty}W^{m,p}(\Omega)=C^{\infty}(\overline{\Omega}).$$
An exercise about Sobolev Spaces
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sobolev-spaces
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0Have you shown that $C^{\infty}(\overline{\Omega})$ is contained in the intersection, using the fact that the open set is bounded? – 2012-07-07
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0For the other inclusion you can think at the [Morrey's inequality](http://en.wikipedia.org/wiki/Sobolev_inequality#Morrey.27s_inequality) – 2012-07-07
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0The inclusion $\supset$ I used the fact that the support is compact in $\overline{\Omega}$. But the other inclusion is not clear, since we have $\Omega$ of $C^1$ class. I dont´t know how to use Morrey's Inequality. :( – 2012-07-08