If $\Omega \subset \mathbb{R^n}$ be a bounded set then $(\int (\Delta u)^2)$ gives a norm on $H^2 \cap H_0^1$. I need to show that this norm is equivalent to the usual $H^2$ norm in $H^2 \cap H_0^1$.
The equivalence is easy to prove in the $H_0^2$ space (Since we have the poincare inequality). But the poincare inequality is not valid for $u \in H^2$.