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 $S^*(\lambda, \Lambda, f)$. 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.