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

TL;DR

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

Contribution

It provides explicit boundary estimates for parabolic equations in $C^1$ domains, extending classical lemmas and unifying boundary regularity results with a single proof.

Findings

Extended Hopf-Oleinik lemma to $C^{1, ext{Dini}}$ domains.

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

Provided explicit moduli of continuity for solutions.

Abstract

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence form parabolic equations in parabolic $C^1$ domains, providing explicit moduli 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 parabolic Lipschitz domains, unifying both regimes with a single proof.