As s approaches 0, nonlocal minimal graphs in the obstacle problem adhere completely to the obstacle and leave the domain asymptotically empty, providing counterexamples to continuity.
Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and globals-minimal surfaces.Interfaces Free Bound., 20(2):261–296
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Asymptotics as $s\searrow 0$ of the nonlocal nonparametric Plateau problem with obstacles
As s approaches 0, nonlocal minimal graphs in the obstacle problem adhere completely to the obstacle and leave the domain asymptotically empty, providing counterexamples to continuity.