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

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

Answer 1

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.