# An exercise about Sobolev Spaces

Let $\Omega\subset\mathbb{R}^n$ be an limited open set of class $C^1$ and $1\leq p<\infty$. Show that $$\bigcap_{m=0}^{\infty}W^{m,p}(\Omega)=C^{\infty}(\overline{\Omega}).$$

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Have you shown that $C^{\infty}(\overline{\Omega})$ is contained in the intersection, using the fact that the open set is bounded? –  Davide Giraudo Jul 7 '12 at 20:36
For the other inclusion you can think at the Morrey's inequality –  Giuseppe Negro Jul 7 '12 at 20:42
The 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. :( –  user23069 Jul 8 '12 at 20:40