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

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 $S^{\ast }(\hat{I}»,\hat{I}›,f)$. Furthermore, based on the boundary pointwise Lipschitz regularity, we obtain the global $W^{2,\hat{I}´}$ regularity on exterior $C^{1,\mathrm{Dini}}$ domains for any $0<\hat{I}´<1$, which is new even for the harmonic functions.