Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data

We consider the Willmore boundary value problem for surfaces of revolution where, as Dirichlet boundary conditions, any symmetric set of position and angle may be prescribed. Using direct methods of the calculus of variations, we prove existence and regularity of minimising solutions. Moreover, we estimate the optimal Willmore energy and prove a number of qualitative properties of these solutions. Besides convexity-related properties we study in particular the limit when the radii of the boundary circles converge to 0, while the “length” of the surfaces of revolution is kept fixed. This singular limit is shown to be the sphere, irrespective of the prescribed boundary angles. These analytical investigations are complemented by presenting a numerical algorithm, based on C 1-elements, and numerical studies. They intensively interact with geometric constructions in finding suitable minimising sequences for the Willmore functional.