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 $C^{k,\gamma }$ domains which satisfy local Neumann conditions on the boundary. Such solutions typically blow up at the boundary like $v\asymp d^{s-1}$ and are sometimes called large solutions. In this setup we prove optimal regularity results for the quotients $v\mathrm{/}{d}^{s-1}$, 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.