The boundary Harnack principle for nonlocal elliptic 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 $L{u}_1=L{u}_2=0$ in $\Omega \cap B_1$, $u_1=u_2=0$ in $B_1\setminus \Omega$, and $u_1,u_2\ge 0$ 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 Hölder continuous up to the boundary in $B_{1/2}$.