Boundary regularity estimates for nonlocal elliptic equations in $C^1$ and $C^{1,\alpha}$ domains

TL;DR

This paper derives sharp boundary regularity estimates for nonlocal elliptic equations in $C^1$ and $C^{1,eta}$ domains, including boundary behavior and Harnack principles, crucial for free boundary problems.

Contribution

It provides new boundary regularity results for nonlocal elliptic equations in less smooth domains, extending previous understanding and establishing tools for free boundary problem analysis.

Findings

Solutions are $C^s$ up to the boundary in $C^{1,eta}$ domains.

Boundary Harnack principle holds in $C^1$ domains.

Analogous regularity results for equations with measurable coefficients.

Abstract

We establish sharp boundary regularity estimates in $C^1$ and $C^{1,\alpha}$ domains for nonlocal problems of the form $Lu=f$ in $\Omega$, $u=0$ in $\Omega^c$. Here, $L$ is a nonlocal elliptic operator of order $2s$, with $s\in(0,1)$. First, in $C^{1,\alpha}$ domains we show that all solutions $u$ are $C^s$ up to the boundary and that $u/d^s\in C^\alpha(\bar\Omega)$, where $d$ is the distance to $\partial\Omega$. In $C^1$ domains, solutions are in general not comparable to $d^s$, and we prove a boundary Harnack principle in such domains. Namely, we show that if $u_1$ and $u_2$ are positive solutions, then $u_1/u_2$ is bounded and H\"older continuous up to the boundary. Finally, we establish analogous results for nonlocal equations with bounded measurable coefficients in non-divergence form. All these regularity results will be essential tools in a forthcoming work on free boundary problems for nonlocal elliptic operators \cite{CRS-obstacle}.