Regularity for nonlocal equations with local Neumann boundary conditions

In this article we establish fine results on the boundary behavior of solutions to nonlocal equations in C^k, domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like v d^s-1 and are sometimes called large solutions. In this setup we prove optimal regularity results for the quotients v/d^s-1, depending on the regularity of the domain and on the data of the problem. The results of this article will be important in a forthcoming work on nonlocal free boundary problems.