Solvability of divergence equation in Lipschitz spaces

Abstract

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