# Global Holder regularity from DiGiorgi-nash-moser

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 ?

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