  We obtain conditions for the differentiability of weak solutions for a
second-order uniformly elliptic equation in divergence form with a homogeneous
co-normal boundary condition. The modulus of continuity for the coefficients is
assumed to satisfy the square-Dini condition and the boundary is assumed to be
differentiable with derivatives also having this modulus of continuity.
Additional conditions for the solution to be Lipschitz continuous or
differentiable at a point on the boundary depend upon the stability of a
dynamical system that is derived from the coefficients of the elliptic
equation.
