Solvability of divergence equation in Lipschitz spaces

We study the solvability of the divergence equation
$$
\operatorname{div} \u = f
$$
in bounded $C^2$ domains under homogeneous Dirichlet boundary conditions for data
$f\in C^{0,\alpha}(\Omega)$
satisfying the compatibility condition
$
\int_\Omega f =0.
$
We construct a solution $\u$ such that for every $0<\beta<\alpha$
$$
\u\in C^{1,\beta}(\Omega)^n
$$
satisfies
$$
\|\u\|_{C^{1,\beta}(\Omega)}
\le
C\|f\|_{C^{0,\alpha}(\Omega)}.
$$
The proof combines localization techniques with a boundary flattening procedure reducing the problem to a model half-cube.