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I am struggling to prove the following.

Let $\Omega$ be bounded set in $\bf{R}^n$ and $u$ a $C^2$ function on $\Omega$, such that $u=0$ in $\partial \Omega$. Prove that there is a constant $C$ depending on $\Omega$ such that $$\int_{\Omega}u^2\leq C\cdot \int_{\Omega}|\nabla u|^2.$$

Can anyone help me with this?

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Well, as you wrote it this statement is actually false, unless you can assume something about $u$. Usually the extra assumption is that that $u$ vanishes on $\partial \Omega$. Another possible assumption is that $u$ has zero average on $\Omega$. – bartgol Oct 29 '12 at 5:30
Yes, I forgot to add the condition. I will edit it now. Thanks! – dmm Oct 29 '12 at 5:59
Do you have any assumptions about the regularity of the boundary? – Lukas Geyer Oct 29 '12 at 13:06

I'll give you just a hint: try to use the divergence theorem on the function $\varphi=\underline{x}u^2$

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