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The Boundary Harnack Principle for Nonlocal Elliptic Operators in Non-divergence Form

2018; Springer Science+Business Media; Volume: 51; Issue: 3 Linguagem: Inglês

10.1007/s11118-018-9713-7

1572-929X

Xavier Ros‐Oton, Joaquim Serra,

Differential Equations and Boundary Problems

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 Lu1 = Lu2 = 0 in Ω ∩ B1, u1 = u2 = 0 in B1 ∖Ω, and u1,u2 ≥ 0 in ℝn, then u1 and u2 are comparable in B1/2. The result applies to arbitrary open sets Ω. When Ω is Lipschitz, we show that the quotient u1/u2 is Hölder continuous up to the boundary in B1/2. These results will be used in forthcoming works on obstacle-type problems for nonlocal operators.