Boundary estimates for non-divergence equations in $C^1$ domains

TL;DR

This paper establishes boundary regularity and nondegeneracy estimates for solutions to non-divergence equations in $C^1$ domains, extending classical results and unifying theories for different boundary regularities.

Contribution

It provides explicit boundary estimates for non-divergence equations in $C^1$ domains, extending classical lemmas and unifying boundary regularity theories.

Findings

Extended Hopf-Oleinik lemma to $C^1$ domains

Derived explicit boundary regularity estimates

Unified boundary regularity results for Lipschitz and $C^{1, ext{Dini}}$ domains

Abstract

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence equations in $C^1$ domains, providing an explicit modulus of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for flat Lipschitz domains, unifying both theories with a single proof.