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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 $u \in H^2$.

tori
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  • I think this can be proved using elliptic regularity. – tori Nov 26 '16 at 12:33
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    I asked a more generalized version of this problem couple years before. Read it, it might be helpful. http://math.stackexchange.com/questions/1095972/a-generalization-of-the-problem-delta-u-l2-is-an-equivalent-norm-for – spatially Nov 27 '16 at 17:59

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