Schauder and Calderón–Zygmund type estimates for fully nonlinear parabolic equations under “small ellipticity aperture” and applications

Abstract

In this manuscript, we derive some Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations of the form: ${\partial }_{t}u-F(x,t,D^{2}u)=f(x,t)\text{in}{Q}_1=B_1\times (-1,0],$provided that the source $f$ and the coefficients of $F$ are Hólder continuous functions, and $F$ enjoys a small ellipticity aperture. Furthermore, for problems with merely bounded data, we prove that such solutions are $C^{1,\text{Log-Lip}}$-regular. We also obtain Calderón-Zygmund estimates for such a class of non-convex operators. Finally, we connect our results and recent estimates for fully nonlinear models in certain solution classes.