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The Boundary Harnack Principle for Nonlocal Elliptic Operators in Non-divergence Form
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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 $B_1 \backslash \Omega$, and $u_1, u_2 \geq 0$ 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 $B_{1 / 2}$. These results will be used in forthcoming works on obstacle-type problems for nonlocal operators.
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ROS, Xavier and SERRA MONTOLÍ, Joaquim. The Boundary Harnack Principle for Nonlocal Elliptic Operators in Non-divergence Form. Potential Analysis. 2019. Vol. 51, num. 315-331. ISSN 0926-2601. [consulted: 9 of August of 2026]. Available at: https://hdl.handle.net/2445/194027