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For a strictly elliptic differential operator $L=\sum_{i,j} D_i (a_{ij}D_ju) $ with strictly elliptic condition on open bounded smooth domain $\Omega$,and bounded coefficients, DiGiorgi-Nash-Moser states that for $\Omega_0\subset\subset\Omega$,

$||u||_{C^\alpha(\Omega_0)}<C||u||_{L^2(\Omega)}$.

How do we get from there to the global regularity result for the Dirichlet problem, i.e. = $u\in C^{\alpha}(\bar\Omega)$. Here $\|v\|_{C^{\alpha}}=\|v\|_{L^{\infty}}+[v]_{\alpha}$

Is there a reference which explains this step ?

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  • $\begingroup$ You need some boundary conditions for the global regularity. Did you have any specific in mind (Dirichlet, Neumann, etc.)? $\endgroup$ – Jose27 Mar 18 '15 at 22:06
  • $\begingroup$ Jose27: I meant it to be for Dirichlet problem. $\endgroup$ – user224319 Mar 18 '15 at 22:20
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    $\begingroup$ I'm pretty sure Gilbarg-Trudinger has what you want. After the section on the usual DeGiorgi-Nash-Moser interior estimates. $\endgroup$ – Jose27 Mar 18 '15 at 22:59
  • $\begingroup$ Ok, I'll take a look, thanks. $\endgroup$ – user224319 Mar 18 '15 at 23:36
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See Lemma 1.35 in "[Qing_Han,_Fanghua_Lin]_Elliptic_partial_differential_equations"

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