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.
Minimization problems involving nonlocal functionals: nonlocal minimal surfaces and a free boundary problem.https://arxiv.org/abs/1811.09746
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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.