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We examine a steepest energy descent flow with obstacle constraint in higherorder energy frameworks where the maximum principle is not available. Weconstruct the flow under general assumptions using De Giorgi's minimizingmovement scheme. Our main application will be the elastic flow of graph curveswith Navier boundary conditions for which we study long-time existence andasymptotic behavior.

On Gradient Flows with Obstacles and Euler's Elastica