## L^p estimate of lapacian operator

Refering to GT’s book corollary 9.10

is a bounded domain in . If and vanish on the boundary, then

(1) , .

where . If

(2)

The proof of this theorem is using the following fact, if has compact support, then

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

is true for . Since is dense in with zero boundary, then 9.10 is true.

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