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 is an element of (0, 1). We consider the class of nonlocal operators L-* subset of L-0, which consists of infinitesimal generators of stable Levy processes belonging to the class L-0 of Caffarelli-Silvestre. For fully nonlinear operators I elliptic with respect to L-*, we prove that solutions to Iu = f in Omega, u = 0 in R-n \ Omega, satisfy u / d(s) is an element of c(s+gamma) ((Omega) over bar), where d is the distance to partial derivative Omega and f is an element of C-gamma. We expect the class L-* to be the largest scale-invariant subclass of L-0 for which this result is true. In this direction, we show that the class L-0 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 up arrow 1, we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.