Boundary Regularity for Fully Nonlinear Integro-differential Equations

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s 2 .0; 1/. We consider the class of nonlocal operators L L0, which consists of infinitesimal generators of stable Lévy processes belonging to the class L0 of Caffarelli–Silvestre. For fully nonlinear operators I elliptic with respect to L , we prove that solutions to IuD f in , uD 0 in R n , satisfy u=d s 2 C sC . N /, where d is the distance to @ and f 2 C . We expect the class L to be the largest scale-invariant subclass of L0 for which this result is true. In this direction, we show that the class L0 is too large for all solutions to behave as d s . The constants in all the estimates in this article remain bounded as the order of the equation approaches 2. Thus, in the limit s " 1, we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.