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?