Solvability of divergence equation in Lipschitz spaces

Abstract

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