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.