Pointwise Boundary Differentiability of Solutions of Elliptic Equations

In this paper, we give pointwise geometric conditions on the boundary which guarantee the differentiability of the solution at the boundary. Precisely, the geometric conditions are two parts: the proper blow up condition (see Definition 1) and the exterior Dini hypersurface condition (see Definition 2). If $\mathrm{\Omega }$ satisfies this two conditions at $x_0\in \mathrm{\partial }\mathrm{\Omega }$, the solution is differentiable at $x_0$. Furthermore, counterexamples show that the conditions are optimal (see Remark 3 and the counterexample in Section 2).