Asymptotics as $s\searrow 0$ of the nonlocal nonparametric Plateau problem with obstacles

In this paper, we introduce a functional and a geometric setting for an obstacle problem for nonlocal minimal graphs. In particular we study existence of solutions, a priori estimates, and we prove the equivalence of the two settings. We then observe a striking stickiness phenomena when the fractional parameter is small and the data at infinity is not too large: the nonlocal minimal graphs adhere entirely to the obstacle and leave the remainder of the domain asymptotically empty. We thus provide a class of examples where continuity of nonlocal minimal graphs across the boundary and across the obstacle may fail.