I was studying De Giorgi's regularity theorem for elliptic systems, and there is something I don't quite understand. The main part of the theorem is dedicated to the proof that certain functions (that belong to the so called "De Giorgi class" and satisfy a particular integral inequality, but whatever) are holder continuous, and from holder continuity there is a bootstrap-like argument that gives you $C^\infty$ of solution provided suitable regularity of the lagrangian, which I don't understand. I'm not being specific here because it really doesn't matter.
What I can't understand is, as I said, the bootstrap argument. It uses a theorem by Schauder that basically says that if I have an elliptic equation like:
$$ \sum_{\alpha, \beta} \partial_{\alpha}(A^{\alpha\beta}(x) \partial_{\beta}u)= \sum_{\alpha} \partial_\alpha F^{\alpha}$$
with $A$ $\alpha$-holder continuous which also satisfies a certain elliptic hypothesis and $u$ is a solution, then $F \in C^{0,\alpha}_{loc} \implies \nabla u \in C^{0,\alpha}_{loc}$.
I have seen the proof of this fact and I understand it. What I don't underrstand is how to pass from $C^{0,\alpha}$ to $C^{n, \alpha}$ and how to use this result to obtain $C^{\infty}$ regularity.