Does anyone know if elliptic regularity still holds in the d-dimensional cube $Q:=[0,1]^d$ with Neumann boundary conditions ? More precisely, let $\rho\in C^{0,\alpha}(Q)$ (let say scalar) with $\alpha\in (0,1)$ and consider the $H^1(Q)$-solution $f$ of $$ \left\{ \begin{array}{ll} -\nabla\cdot \rho\nabla f=g & \text{in $Q$} \\ \partial_n f\equiv 0 & \mbox{on $\partial Q$}, \end{array} \right. $$ where $\partial_n$ denotes the normal derivative and $g\in L^{\infty}(Q)$. Do I have for $\alpha'<\alpha$ $$\|\nabla f\|_{C^{0,\alpha'}(Q)}\leq C \|g\|_{L^{\infty}(Q)},$$ for a constant $C$ depending of $\|\rho\|_{C^{0,\alpha}(Q)}$ ? If yes, does anyone have a reference for that ?
Thank you !