# $C^2$ estimate for oblique derivative problem with mean Dini coefficients

## TL;DR

This paper proves that strong solutions to certain elliptic equations with oblique boundary conditions are twice differentiable up to the boundary under Dini continuity conditions on coefficients and boundary, extending previous results.

## Contribution

It establishes new regularity results for elliptic equations with oblique derivatives under mean Dini conditions, including an extension to fully nonlinear cases.

## Key findings

- Solutions are twice continuously differentiable up to the boundary.
- Dini continuity of coefficients and boundary derivatives is sufficient for regularity.
- Extension to fully nonlinear elliptic equations is achieved.

## Abstract

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.