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Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data

by Dall'Acqua, A., Fröhlich, S., Grunau, H.-Ch., Schieweck, F..

Series: 2008-24, Preprints

49Q10 Optimization of shapes other than minimal surfaces
53C42 Immersions (minimal, prescribed curvature, tight, etc.)
35J65 Nonlinear boundary value problems for linear elliptic equations
34L30 Nonlinear ordinary differential operators

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.

Dirichlet boundary conditions, Willmore surfaces of revolution