Willmore obstacle problems under Dirichlet boundary conditions

2021/03/29 by Hans-Christoph Grunau, Grunau, Hans-Christoph, Shinya Okabe +1 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2103.15382

Abstract

We consider obstacle problems for the Willmore functional in the class of graphs of functions and surfaces of revolution with Dirichlet boundary conditions. We prove the existence of minimisers of the obstacle problems under the assumption that the Willmore energy with the unilateral constraint is below a universal bound. We address the question whether such bounds are necessary in order to ensure the solvability of the obstacle problems. Moreover, we give several instructive examples of obstacles such that minimisers exist.

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