Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $0<\sigma<1$, $L$ be a 2nd order strictly elliptic linear differential operator of divergence form: \begin{eqnarray} Lf=\left(a^{ij}\dfrac{\partial f}{\partial x^i}\right)_{x_j}\ \ \ (f\in H^1(\Omega)) \end{eqnarray} on $\Omega$ with $a^{ij}\in C^{0,\sigma}(\Omega)$.
I'm not able to find any proofs of Theorem A.2.3 (ii) below in [113] or [166]. Which textbook shows it? The figure below is p.579 of Jost, Riemannian Geometry and Geometric Analysis 6th edition. Thank you.
A reference I have found very useful is the book "Second Order Elliptic Equations and Elliptic Systems'' by Ya-Zhe Chen and Lan-Cheng Wu (English translation). The proofs are complete and well organized. In chapter 9, theorem 2.6 the theorem is proven when $k=\nabla \cdot F$ and $F$ is $C^{0,\alpha}$. The result also holds when $k\in L^p$ and $p>n$ and you can think about it as there being some $F\in W^{1,p}$ that is Holder continuous by embedding.
Very similar material is in the lecture notes (2012 in English and available online) by Mariano Giaquinta and Luca Martinazzi, "An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs." I think the results in chapter 5 encompass the result you ask about, and they also carry the results out to the boundary.
This is all in the context of systems of equations, which is natural as the result does not require an application of the maximum principle. I am interested in how general such results hold, for example for the Stokes equations similar results are shown in a 1982 paper by M. Giaquinta and G. Modica, "Non linear systems of the type of the stationary Navier-Stokes system."
I believe the the methods for this version of the Schauder estimates are very much due to S. Campanato in the 1960's, however I have not seen any translations of his works. The iteration arguments and embedding inequalities developed by C. Morrey also play a crucial role.