Interpolation Inequality's Proof

Question

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}

Answer 1

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.