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We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in . More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of admissible curves in a way that an existence result can be obtained by a penalization argument.
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Marius Müller, An obstacle problem for elastic curves: Existence results. Interfaces Free Bound. 21 (2019), no. 1 pp. 87–129