The boundary Harnack principle for nonlocal elliptic operators in non-divergence form

TL;DR

This paper establishes a boundary Harnack inequality for nonlocal elliptic operators in non-divergence form, demonstrating comparability and boundary regularity of solutions in arbitrary open sets.

Contribution

It proves a boundary Harnack principle for nonlocal elliptic operators in non-divergence form with measurable coefficients, applicable to arbitrary open sets and showing boundary regularity in Lipschitz domains.

Findings

Solutions are comparable near the boundary in arbitrary open sets.

The quotient of solutions is Hölder continuous up to the boundary in Lipschitz domains.

The boundary Harnack inequality extends classical results to nonlocal operators in non-divergence form.

Abstract

We prove a boundary Harnack inequality for nonlocal elliptic operators $L$ in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if $Lu_1=Lu_2=0$ in $\Omega\cap B_1$, $u_1=u_2=0$ in $B_1\setminus\Omega$, and $u_1,u_2\geq0$ in $\mathbb R^n$, then $u_1$ and $u_2$ are comparable in $B_{1/2}$. The result applies to arbitrary open sets $\Omega$. When $\Omega$ is Lipschitz, we show that the quotient $u_1/u_2$ is H\"older continuous up to the boundary in $B_{1/2}$.