Partial H\"older regularity for fully nonlinear nonlocal parabolic equations with integrable kernels

TL;DR

This paper establishes partial regularity results for solutions to fully nonlinear nonlocal parabolic equations with integrable kernels, extending previous linear and elliptic findings to a nonlinear, time-dependent context.

Contribution

It introduces a robust partial regularity estimate for solutions with truncated kernels, generalizing earlier linear and elliptic results to nonlinear parabolic equations.

Findings

Solutions exhibit partial regularity up to the truncation scale.

Regularity estimates are stable under kernel truncation.

Extends linear and elliptic results to nonlinear parabolic equations.

Abstract

In this work, we consider solutions to (fully nonlinear) parabolic integro-differential equations with integrable interaction kernels. A typical equation would be that obtained by starting with, for $s\in(0,1)$, the $s$-fractional heat equation, but replacing the interaction kernel in the integro-differential term with one which has been truncated, for $\rho>0$, at the value $\rho^{-d-2s}$, hence integrable. We show that solutions to these equations have a partial regularity estimate which captures differences of the solution up to the scale at which the kernel has a truncation in its singularity. The estimates we provide are robust with respect to the truncation parameter, and they include the existing results for the original operators without truncation. There are some earlier results for linear and elliptic cases of this situation of integrable interaction kernels, and so our work is a generalization of those to the nonlinear and parabolic setting.