Higher-order boundary regularity estimates for nonlocal parabolic equations

Abstract

We establish sharp higher-order H\"older 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 H\"older 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.