Recently, I am learning Schauder estimates for elliptic equations and I come across a proposition as follows
Let $ \alpha\in (0,1) $ and $ \Omega $ be a bounded $ C^2(\Omega) $ domain on $ \mathbb{R}^n $, let $ u\in C^{2,\alpha}(\Omega) $ be a weak solution to \begin{eqnarray} \Delta u&=&f \text{ in }\Omega\\ u&=&g\text{ on }\partial\Omega, \end{eqnarray} for $ f\in C^{0,\alpha}(\Omega) $ and $ g\in C^{0,\alpha}(\partial\Omega) $. Then
\begin{eqnarray} \left\|u\right\|_{C^{2,\alpha}(\Omega)}\leq C\left(\left\|f\right\|_{C^{0,\alpha}(\Omega)}+\left\|g\right\|_{C^{2,\alpha}(\partial\Omega)}\right), \end{eqnarray} for some constant $ C $ depending only on $ \alpha, n, $ and $ \Omega $.
This proposition is a boundary estimates for the solution. I want to prove it and here is my try. First I notice that the estimate can be decomposed to two parts, the interior part and the boundary parts. That is, if we can prove that for all $ x_0\in \partial \Omega $, there exists a ball $ B(x_0,r) $ with radius $ r>0 $, center $ x_0 $ and a constant $ C $ depending only on $ \alpha, n, $ and $ \Omega $ such that \begin{eqnarray} \left\|u\right\|_{C^{2,\alpha}(B(x_0,r/2)\cap\Omega)}\leq C\left(\left\|f\right\|_{C^{0,\alpha}(B(x_0,r/2)\cap\Omega)}+\left\|g\right\|_{C^{2,\alpha}(B(x_0,r/2)\cap\partial\Omega)}\right),(*) \end{eqnarray} I can use the property that $ \partial\Omega $ is compact and construct a finite covering $ \left\{B(x_i,r_i)\right\}_{i=1}^{N} $ for $ \partial\Omega $. Then I can choose $ K $ which is a compact subset of $ \Omega $ and $ \operatorname{dist}(K,\partial\Omega)>0 $ such that $ \Omega\subset\cup_{i}^{N}B(x_i,r_i)\cup K $. By using the classical Schauder estimates on $ K $, I can obtain that \begin{eqnarray} \left\|u\right\|_{C^{2,\alpha}(K)}\leq C\left(\left\|f\right\|_{C^{0,\alpha}(\Omega)}+\left\|u\right\|_{L^{\infty}(\Omega)}\right), \end{eqnarray} where $ C $ is a constant depending only on $ n, K, \Omega $ and $ \alpha $. We can directly obtain \begin{eqnarray} \left\|u\right\|_{C^{2,\alpha}(K)}\leq C\left(\left\|f\right\|_{C^{0,\alpha}(\Omega)}+\left\|g\right\|_{C^{0,\alpha}(\partial\Omega)}\right), \end{eqnarray} by using the maximum principle for laplacian. However I have some trouble in getting the boundary estimate (*). In the book "Regularity Theory for Elliptic PDE" written by Fernandez-Real, the author give a instruction we should use the blow up method and use the Liouville's theorem in the half-space, I tried but failed at some details. Can you give me some hints on dealing with the boundary terms or give me some references on it?
