# Pointwise Boundary Differentiability of Solutions of Elliptic Equations

## TL;DR

This paper establishes geometric boundary conditions that ensure the differentiability of solutions to elliptic equations at boundary points, providing necessary and sufficient criteria with counterexamples for optimality.

## Contribution

It introduces two geometric boundary conditions guaranteeing boundary differentiability of elliptic solutions and demonstrates their optimality through counterexamples.

## Key findings

- Boundary differentiability depends on proper blow up and exterior Dini hypersurface conditions.
- Conditions are proven to be optimal with counterexamples.
- Provides a geometric framework for boundary regularity of elliptic solutions.

## Abstract

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 $\Omega$ satisfies this two conditions at $x_0\in\partial \Omega$, the solution is differentiable at $x_0$. Furthermore, counterexamples show that the conditions are optimal (see Remark 3 and the counterexample in Section 2).