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We consider obstacle problems for the Willmore functional in the class ofgraphs of functions and surfaces of revolution with Dirichlet boundaryconditions. We prove the existence of minimisers of the obstacle problems underthe assumption that the Willmore energy with the unilateral constraint is belowa universal bound. We address the question whether such bounds are necessary inorder to ensure the solvability of the obstacle problems. Moreover, we giveseveral instructive examples of obstacles such that minimisers exist.

Willmore obstacle problems under Dirichlet boundary conditions