Regularity for fully nonlinear nonlocal parabolic equations with rough kernels

TL;DR

This paper establishes space and time regularity for solutions of fully nonlinear nonlocal parabolic equations with rough kernels, extending regularity results to equations with less smooth kernels.

Contribution

It proves new regularity results for viscosity solutions of fully nonlinear nonlocal parabolic equations with rough kernels, using Liouville theorems and blow-up techniques.

Findings

Solutions are $C^eta$ in space and $C^{eta/\sigma}$ in time.

Regularity holds for solutions bounded in the entire space.

Results apply to equations with kernels of minimal smoothness.

Abstract

We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $u_t = \I u$, where $\I$ is translation invariant and elliptic with respect to the class $\mathcal L_0(\sigma)$ of Caffarelli and Silvestre, $\sigma\in(0,2)$ being the order of $\I$. We prove that if $u$ is a viscosity solution in $B_1 \times (-1,0]$ which is merely bounded in $\R^n \times (-1,0]$, then $u$ is $C^\beta$ in space and $C^{\beta/\sigma}$ in time in $\overline{B_{1/2}} \times [-1/2,0]$, for all $\beta< \min\{\sigma, 1+\alpha\}$, where $\alpha>0$. Our proof combines a Liouville type theorem ---relaying on the nonlocal parabolic $C^\alpha$ estimate of Chang and D\'avila--- and a blow up and compactness argument.