Boundary regularity estimates for nonlocal elliptic equations in $$C^1$$ C 1 and $$C^{1,\alpha }$$ C 1 , α domains

Abstract

We establish sharp boundary regularity estimates in \(C^1\) and \(C^{1,\alpha }\) domains for nonlocal problems of the form \(Lu=f\) in \(\Omega \), \(u=0\) in \(\Omega ^c\). Here, L is a nonlocal elliptic operator of order 2s, with \(s\in (0,1)\). First, in \(C^{1,\alpha }\) domains we show that all solutions u are \(C^s\) up to the boundary and that \(u/d^s\in C^\alpha (\overline{\Omega })\), where d is the distance to \(\partial \Omega \). In \(C^1\) domains, solutions are in general not comparable to \(d^s\), and we prove a boundary Harnack principle in such domains. Namely, we show that if \(u_1\) and \(u_2\) are positive solutions, then \(u_1/u_2\) is bounded and Holder continuous up to the boundary. Finally, we establish analogous results for nonlocal equations with bounded measurable coefficients in nondivergence form. All these regularity results will be essential tools in a forthcoming work on free boundary problems for nonlocal elliptic operators (Caffarelli et al., in Invent Math, to appear).