## L^p estimate of lapacian operator

Refering to GT’s book corollary 9.10

$\textbf{Thm:}$ $\Omega$ is a bounded domain in $\mathbb{R}^n$. If $u\in W^{2,p}(\Omega)$ and vanish on the boundary, then

(1) $||D^2u||_p\leq ||\Delta u||_p$ , $1.

where $C=C(n,p)$. If $p=2$

(2) $||D^2u||_2= ||\Delta u||_2$

The proof of this theorem is using the following fact, if $u\in C^2(\mathbb{R}^n)$ has compact support, then

$\displaystyle u(x)=\int_{\mathbb{R}^n}\Gamma(x-y)\Delta u(y)dy$

$\Gamma(x)$ is the fundamental solution of $\Delta$(Give a second thought why the RHS has compact support). This is exactly the Newtonian potential. By the result of Thm 9.9

$||D^2u||_p\leq C||\Delta u||_p$

is true for $u\in C^2_0(\Omega)$. Since $C^2_0(\Omega)$ is dense in $W^{2,p}(\Omega)$ with zero boundary, then 9.10 is true.