Consider a uniformly elliptic equation $$ \sum_{i,j=1}^n a_{ij}(x)\partial_{ij}u+\sum_{i=1}^n b_{i}(x)\partial_{i}u+c(x)u=0 $$ say, in an open ball $B\subset \mathbb R^n$, where coefficients are Hölder continuous in $\bar B$ with some exponent $\alpha\in(0,1).$ There are interior Schauder estimates, of course. But they are conditional, supposing that a solution is from $C^{2+\alpha}$ locally then giving an estimate for it.

Is somewhere stated that all solutions of this equation has to be classical in $B$, i.e. from $C^2(B)$? Or can it be derived from some other considerations? They should be even from $C^{2+\alpha}(B)$, but I can't find a reference.

The same question goes for solutions of a uniformly parabolic equation $$ \partial_tu-\sum_{i,j=1}^n a_{ij}(x,t)\partial_{ij}u-\sum_{i}^n b_{i}(x,t)\partial_{i}u-c(x,t)u=0 $$ with Hölder continuous coefficients.