Global Holder regularity from DiGiorgi-nash-moser

Question

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 ?

Answer 1

See Lemma 1.35 in "[Qing_Han,_Fanghua_Lin]_Elliptic_partial_differential_equations"