Higher-order boundary regularity estimates for nonlocal parabolic equations

TL;DR

This paper proves new higher-order boundary regularity estimates for solutions to nonlocal parabolic equations involving operators like the fractional Laplacian, extending the understanding of solution smoothness up to the boundary.

Contribution

It provides the first higher-order boundary regularity estimates for nonlocal parabolic equations, applicable even in smooth domains for fractional Laplacians.

Findings

Solutions are $C^{s+eta}$ up to the boundary when $f$ is $C^eta$ in space.

Solutions are $C^{1+eta/2s}$ in time up to the boundary.

Estimates hold in $C^{2,eta}$ domains for fractional Laplacian operators.

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.