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.
We give a sharp lower bound for the self-intersection of a nef line bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dimension of X, which we call the generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X) >= 2n!chi(omega(X)) and it is sharp.