We are in trouble with the following exercise, we are almost sure about it's related to Dirichlet's energy and Dirichlet's principle, but we don't know how to prove it. The exercise says:
Let $u\in C^2(\Omega)$, $u=0$ in $\partial \Omega$, being $\Omega$ a regular domain. Prove that:
$$ \int_\Omega |\nabla u|^2 dx\leq \epsilon \int_\Omega |\Delta u|^2 dx + \frac{1}{4\epsilon}\int_\Omega u^2 dx \text{ } \forall \epsilon>0$$
Using that $u=0$ on the boundary we get with integration by parts
$$\int_Ω \langle \nabla u, \nabla u \rangle \ dx=-\int_Ω u\ \Delta u \ dx \leq \int_Ω |u \ \Delta u| \ dx $$
Now using Young's inequality we get $$|u \cdot \Delta u|=|\frac{1}{\sqrt{2\epsilon}}u | \cdot|\sqrt{2\epsilon}\Delta u | \leq \frac{1}{2\epsilon} \frac{1}{2}|u|^2 +2\epsilon \frac{1}{2} |\Delta u|^2=\frac{1}{4\epsilon}u^2+\epsilon|\Delta u|^2$$
Just insert this into the integral above and the proof is completed.