help to prove $||u||_{W^{2,2}(\Omega) }\le C ||\Delta u ||_{L^2(\Omega )} $

Question

Can some one give a reference or hint for proving

$$||u||_{W^{2,2}(\Omega)} \le C ||\Delta u ||_{L^2(\Omega )} $$

@Jose27: you are clearly right, but I think here we are assuming $u_{|\partial\Omega}=0$ or something similar, so this is just the Poincaré-Wirtinger inequality. — Jack D'Aurizio, Jan 17 '15 at 19:59
http://math.stackexchange.com/questions/246790/sobolov-space-w2-2-cap-w1-2-0-norm-equivalence — Fin8ish, Jul 10 '15 at 11:20

score 1 · Accepted Answer · answered Jan 18 '15 at 00:32

Suppose $u\in W_0^{1,2}(\Omega)\cap W^{2,2}(\Omega)$ and $\partial\Omega\in C^2$, then the inequality will be hold. Indeed, we have the PDE \begin{cases} -\Delta u=-\Delta u &x\in\Omega\\ u=0&x\in\partial \Omega \end{cases} has solution $u\in W_0^{1,2}(\Omega)$ and hence by outer regularity we have $$ \|u\|_{W^{2,2}(\Omega)}\leq C\|\Delta u\|_{L^2(\Omega)} $$ which is the desired result.

help to prove $||u||_{W^{2,2}(\Omega) }\le C ||\Delta u ||_{L^2(\Omega )} $

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