For a compact bounded set $\Omega$, for the expression $\int_\Omega \Delta u (\nabla u \cdot \nabla f)$ where $u \in H^2$ and $f \in C^\infty$, is it possible to show that the expression is $\geq$ (greater than or equal to) some expression NOT involving $-\lVert \nabla u\rVert_{L^2}$? Notice the minus sign; with a plus sign it's fine. I can't see any way. So basically I want the expression $\geq \pm\lVert u \rVert_{L^2}$ or $\geq \lVert u \rVert_{H^2}$, with some multiplicative positive constants in the inequalities.
Any ideas appreciated.