Higher-order boundary regularity estimates for nonlocal parabolic equations

Abstract

We establish sharp higher-order Hölder regularity estimates up to the boundary for solutions to equations of the form ${\partial }_{t}u-Lu=f(t,x)$ in $I\times \Omega$ where $I\subset \mathbb{R}$, $\Omega \subset {\mathbb{R}}^n$ and $f$ is Hölder continuous. The nonlocal operators $L$ considered are those arising in stochastic processes with jumps such as the fractional Laplacian $(-\Delta {)}^s$, $s\in (0,1)$. Our main result establishes that, if $f$ is $C^{\gamma }$ is space and $C^{\gamma /2s}$ in time, and $\Omega$ is a $C^{2,\gamma }$ domain, then $u/d^s$ is $C^{s+\gamma }$ up to the boundary in space and $u$ is $C^{1+\gamma /2s}$ up the boundary in time, where $d$ is the distance to $\partial \Omega$. This is the first higher order boundary regularity estimate for nonlocal parabolic equations, and is new even for the fractional Laplacian in $C^{\infty }$ domains.