= Solution
The <Interior Schauder estimate> is
$$
\boxed{\|u\|_{C^{2,\alpha}(B_{1/2})}
\leq C\left(\|u\|_{C^0(B_1)}+
\|f\|_{C^{0,\alpha}(B_1)}\right),}
$$
where $C$ depends only on $n$, $\alpha$, the ellipticity constant, and the stated coefficient bounds.
It is enough first to prove the estimate for the Hölder seminorm of $D^2u$. The <Interpolation inequality in Holder spaces> controls $Du$ and $D^2u$ by a small multiple of $[D^2u]_{\alpha}$ plus $\|u\|_0$; the small term is later absorbed. Freezing $a^{ij}$ at the centre of each ball and moving coefficient differences and lower-order terms to the right reduces the local estimate to
$$
r^{2+\alpha}[D^2u]_{\alpha;B_{r/2}}
\leq \varepsilon r^{2+\alpha}[D^2u]_{\alpha;B_r}
+C\left(\|u\|_{0;B_r}+r^2\|f\|_{0,\alpha;B_r}\right).
$$
Once this is known, the <Simon absorption lemma> gives the desired estimate on $B_{1/2}$.
For completeness, prove the frozen-coefficient estimate by contradiction. If it failed, choose solutions $u_k$, points $x_k$, and scales $\rho_k\downarrow0$ at which the scale-invariant Hölder quotient is almost maximal. Let $q_k$ be the quadratic Taylor polynomial of $u_k$ at $x_k$ and define
$$
v_k(y)=
\frac{u_k(x_k+\rho_ky)-q_k(x_k+\rho_ky)}
{\rho_k^{2+\alpha}[D^2u_k]_{\alpha}}.
$$
Then $v_k(0)=Dv_k(0)=D^2v_k(0)=0$, the Hessians have uniformly bounded local $C^{0,\alpha}$ seminorms, and the normalization makes their oscillation nonzero on a fixed ball. The localized equation is
$$
\widetilde a_k^{ij}D_{ij}v_k=g_k,
$$
where the coefficient matrices converge locally uniformly to one constant positive-definite matrix and the given normalized error $g_k$ tends locally uniformly to zero.
The <Arzela-Ascoli theorem> produces a locally $C^2$ convergent subsequence with limit $v$ satisfying a constant-coefficient elliptic equation on $\mathbb R^n$. A linear rotation and scaling turn it into a <harmonic function>. The normalization gives $D^2v(0)=0$ but a nonconstant Hessian, while maximality of the scaled quotient gives growth at most $C(1+|y|^{2+\alpha})$. Applying the <Liouville theorem> to derivatives shows that every second derivative is constant because $0<\alpha<1$. This contradicts the normalized Hessian oscillation. The frozen estimate follows, and interpolation plus absorption completes the proof.
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