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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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    Do you have any assumptions a$b$out the regularity of the boundary? – 2012-10-29

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