Schauder Estimates – Schauder Estimates with Boundary Conditions

Question

For the elliptic equation with non-divergence form
$$
\sum_{i,j=1}^na_{ij}(x)\partial_{ij}^2u=f\text{ in }B(0,1)\quad\text{and}\quad u=g\text{ on }\partial B(0,1),
$$
where $ \{a_{ij}(x)\} $ is a matrix-valued function such that for any $ \xi\in\mathbb{R}^n $ and $ x\in B(0,1) $
$$
\mu|\xi|^2\leq\sum_{i,j=1}^na_{ij}(x)\xi_i\xi_j\leq\mu^{-1}|\xi|^2,\text{ with }\mu>0,
$$
and $ a_{ij}(x)\in C^{0,\alpha}(\overline{B(0,1)}) $ with $ \|a\|_{C^{0,\alpha}(B(0,1))}\leq M $, $ M>0 $. $ g\in C^{2,\alpha}(\overline{B(0,1)}) $ and $ f\in C^{0,\alpha}(\overline{B(0,1)}) $. Then Schauder estimates imply that
$$
\|u\|_{C^{2,\alpha}(B(0,1))}\leq C\left\{\|f\|_{C^{0,\alpha}(B(0,1))}+\|g\|_{C^{2,\alpha}(B(0,1))}\right\}.
$$
I know that to show it, we can consider $ u-g $ and assume that $ g=0 $. I want to ask that if $ g\in C^{0,\alpha}(\overline{B(0,1)}) $ and $ f=0 $, what regularity of $ u $ can I get? I wonder if $ u\in C^{0,\alpha}(\overline{B(0,1)}) $ and satisfies
$$
\|u\|_{C^{0,\alpha}(B(0,1))}\leq C\|g\|_{C^{0,\alpha}(B(0,1))}.
$$
Can you give me some hints or references?

Answer 1

The result is true. Let $L=\sum_{ij}a_{ij}D_{ij}$ and consider $$L^{-1}: C^{2+\alpha}(\partial \Omega) \mapsto C^{2+\alpha}(\bar \Omega)$$ with $L^{-1}f=u$ is $Lu=0$ and $u=f$ at the boundary.
$L^{-1}$ is bounded from $C^{2+\alpha}(\partial \Omega)$ to $ C^{2+\alpha}(\bar \Omega)$ by the Schauder theory and from $C(\partial \Omega)$ to $C(\bar \Omega)$, by the maximum principle. By interpolation it is bounded from $C^\alpha (\partial \Omega)$ to $C^\alpha (\bar \Omega)$ (see for example Chapter 1 in the book "Analytic semigroups and optimal regularity in parabolic problems", by A. Lunardi).