# Interpolation Inequality's Proof

Let $$\Omega \subseteq R^{n}$$ bounded domain. I need to prove that for $$u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$$: $$$$\|\nabla u\|_{L^{2}(\Omega)}^{2}\leq \|u\|_{L^{2}(\Omega)}\|\Delta u\|_{L^{2}(\Omega)}$$$$ I know that I should prove it for a $$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)$$. Also, I know the following expression could be useful: $$$$\Delta(\frac{u^{2}}{2})=|\nabla u|^{2}+u\Delta u$$$$