Boundary Regularity for Fully Nonlinear Parabolic equations on $C^{1,\mathrm{Dini}}$ Domains

TL;DR

This paper proves boundary regularity and the Hopf lemma for fully nonlinear parabolic equations on $C^{1, ext{Dini}}$ domains, extending to Pucci's class solutions and establishing new global regularity results even for harmonic functions.

Contribution

It introduces a unified perturbation method to establish boundary regularity and the Hopf lemma for fully nonlinear parabolic equations on $C^{1, ext{Dini}}$ domains, including Pucci's class solutions.

Findings

Boundary pointwise Lipschitz regularity on exterior $C^{1, ext{Dini}}$ domains.

Hopf lemma validity on interior $C^{1, ext{Dini}}$ domains.

Global $W^{2, ext{delta}}$ regularity for solutions, even for harmonic functions.

Abstract

We establish the boundary pointwise Lipschitz regularity on exterior $C^{1,\mathrm{Dini}}$ domains and the Hopf lemma on interior $C^{1,\mathrm{Dini}}$ domains for fully nonlinear parabolic equations by a unified perturbation method. In fact, above two regularity hold for more general solution sets, i.e., the Pucci's class $S^*(\lambda, \Lambda, f)$. Furthermore, based on the boundary pointwise Lipschitz regularity, we obtain the global ${W}^{2,\delta}$ regularity on exterior $C^{1,\mathrm{Dini}}$ domains for any $0<\delta<1$, which is new even for the harmonic functions.