Schauder and Calder\'on-Zygmund type estimates for fully nonlinear parabolic equations under "small ellipticity aperture" and applications

Abstract

In this manuscript, we derive Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations \[ \partial_t u - F(x, t,D^2u) = f (x, t) \quad \text{in} \quad \mathrm{Q}_1 = B_1 \times (-1, 0], \] provided that the source $f$ and the coefficients of $F$ are H\"{o}lder continuous functions and $F$ enjoys a small ellipticity aperture. Furthermore, for problems with merely bounded data, we prove that such solutions are $C^{1, \text{Log-Lip}}$ smooth in the parabolic metric. We also address Calder\'{o}n-Zygmund estimates for such a class of non-convex operators. Finally, we connect our findings with recent estimates for fully nonlinear models in certain solution classes.