  We consider second-order elliptic equations in non-divergence form with
oblique derivative boundary conditions. We show that any strong solutions to
such problems are twice continuously differentiable up to the boundary provided
that the mean oscillations of coefficients satisfy the Dini condition and the
boundary is locally represented by a $C^1$ function whose first derivatives are
Dini continuous. This improves a recent result in [6]. An extension to fully
nonlinear elliptic equations is also presented.
