Smooth Solutions to a Class of Nonlocal Fully Nonlinear Elliptic Equations

Abstract

We show that a certain class of fully nonlinear nonlocal equations have smooth solutions as long as the right-hand side is nice and the boundary datum is bounded. To this end we follow the classical strategy. We first show that solutions are $C^{σ+α}$ continuous, then develop a bootstrap argument robust enough for operators that may depend on the solution and its σ-derivatives.

20 pages. In v.2 Definition 2.10 is included at the suggestion of a very careful referee. To appear in Indiana Univ. Math. Journal