Schauder estimate

The Schauder estimates refer to an interior regularity result assuming that the coefficients of a uniformly elliptic equation are Holder continuous. More precisely, for the equation $$a_{ij}(x) u_{ij} = f \qquad \text{in a domain } D$$ with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$ in the interior of $D$.