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For a classical solution of Dirichlet problem, by Qing Han & Fanghua Lin's Elliptic PDEs, 2nd Edition, we have Global $W^{2,p}$ Estimate:

Let $\Omega$ be a bounded $C^{1,1}$-domain in ${\mathbb R}^n$, $a_{ij}$ be continuous functions in $\Omega$, and $b_i$ and $c$ be bounded functions in $\Omega$. For some constant $p>1$, suppose $u\in W^{2,p}(\Omega)$ is a solution of classical Dirichlet problem for some $f\in L^p(\Omega)$ and $\varphi \in W^{2,p}(\Omega)$. Then $${\|u\|}_{W^{2,p}(\Omega)}\le C\{ {\|u\|}_{L^p(\Omega)}+ {\|f\|}_{L^p(\Omega)}+{\|\varphi\|}_{W^{2,p}(\Omega)}\},$$where $C$ is a positive constant depending only on $n,p,\lambda,\Omega$, the moduli of continuity of $a_{ij}$, and the $L^{\infty}(\Omega)$-norms of $a_{ij}$, $b_i$, and $c$.

NB: The Dirichlet Problem is $$Lu=f \ {\mathrm {in} } \ \Omega,$$ $$u=\varphi \ {\mathrm {on} } \ \partial \Omega.$$ $a^{ij},b^{i}, c$ are coefficients of $L$.

The theorem is stated without any proof for the use of next chapters. I have only read about the proof of Interior $W^{2,p}$ Estimate from Gilbarg & Trudinger's Elliptic PDEs of 2nd Order. Can anyone please give an outline of the proof of this global version?

Thank you very much!

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  • $\begingroup$ That's a tall order, and probably more than can fit in a stackexchange answer. The global version should be in Gilbarg&Trudinger. $\endgroup$ – Jeff Jan 9 '18 at 19:09
  • $\begingroup$ @Jeff oh, I will look through G&T again then. $\endgroup$ – Ivon Jan 9 '18 at 19:27
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    $\begingroup$ It will involve flattening the boundary and proving regularity up to the boundary for a halfspace problem. Should be similar to the proof in Evans for $p=2$. $\endgroup$ – Jeff Jan 9 '18 at 20:58

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