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.
Regularity properties of nonlocal minimal surfaces via limiting arguments
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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.