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<p<\infty.

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.

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