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

We establish the boundary pointwise Lipschitz regularity on exterior $C^{1,\mathrm{D}\mathrm{i}\mathrm{n}\mathrm{i}}$ domains and the Hopf lemma on interior $C^{1,\mathrm{D}\mathrm{i}\mathrm{n}\mathrm{i}}$ 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 $S^{\ast }(\lambda ,\mathrm{\Lambda },f)$. Furthermore, based on the boundary pointwise Lipschitz regularity, we obtain the global $W^{2,\delta }$ regularity on exterior $C^{1,\mathrm{D}\mathrm{i}\mathrm{n}\mathrm{i}}$ domains for any $0<\delta <1$, which is new even for the harmonic functions.