Willmore obstacle problems under Dirichlet boundary conditions

TL;DR

This paper investigates obstacle problems for the Willmore functional with Dirichlet boundary conditions, proving existence of minimisers under energy bounds and providing examples of obstacles where solutions exist.

Contribution

It establishes conditions for the existence of minimisers in Willmore obstacle problems and explores the necessity of energy bounds for solvability.

Findings

Existence of minimisers under a universal energy bound.

Examples of obstacles with guaranteed minimiser existence.

Discussion on the necessity of energy bounds for solvability.

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.