Solvability of divergence equation in Lipschitz spaces

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