Boundary Regularity Estimates for Nonlocal Elliptic Equations in C and C Domains

We establish sharp boundary regularity estimates in C and C domains for nonlocal problems of the form Lu = f in Ω, u = 0 in Ω. Here, L is a nonlocal elliptic operator of order 2s, with s ∈ (0, 1). First, in C domains we show that all solutions u are C up to the boundary and that u/d ∈ C(Ω), where d is the distance to ∂Ω. In C domains, solutions are in general not comparable to d, and we prove a boundary Harnack principle in such domains. Namely, we show that if u1 and u2 are positive solutions, then u1/u2 is bounded and Hölder 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 [CRS15].