Let $\Omega \subseteq R^{n}$ bounded domain. I need to prove that for $u\in H^{2}(\Omega)\cap H^{1}\_{0}(\Omega)$: 
\begin{equation}
\|\nabla u\|\_{L^{2}(\Omega)}^{2}\leq \|u\|\_{L^{2}(\Omega)}\|\Delta u\|\_{L^{2}(\Omega)}
\end{equation}
I know that I should prove it for a $u\in C^{2}\_{0}(\Omega)$ and then use the Global Approximation Theorem with smooth functions to extend $u\in C^{2}\_{0}(\Omega)$ to $u\in H^{2}(\Omega)\cap H^{1}\_{0}(\Omega)$. Also, I know the following expression could be useful: 
\begin{equation}
\Delta(\frac{u^{2}}{2})=|\nabla u|^{2}+u\Delta u 
\end{equation}



Interpolation Inequality's Proof
Use integration by parts in each coordinate direction to write the Left hand side as the integral of u times the Laplacian of u. Then apply Cauchy- Schwarz. 


