BOUNDARY REGULARITY FOR FULLY NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS
Xavier Ros‐Oton, Joaquim Serra · 2014
Abstract. We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s ∈ (0, 1). We consider the class of nonlocal operators L ∗ ⊂ L0, which consists of infini-tesimal 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 Iu = f in Ω, u = 0 in Rn \\ Ω, satisfy u/ds ∈ Cs+γ(Ω), where d is the distance to ∂Ω and f ∈ Cγ. 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 like ds. The constants in all the estimates in this paper 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. 1. Introduction and