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.
Frontiere orientate di misura minima.Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61 Editrice Tecnico Scientifica, Pisa, page 57 pp
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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.