08-24
Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data
by Dall\'Acqua, A.; Fröhlich, S.; Grunau, H.-Ch.; Schieweck, F.
Preprint series: 08-24, Preprints
- MSC:
- 49Q10 Optimization of the shape other than minimal surfaces, See also {73K40, 90C90}
- 53C42 Immersions (minimal, prescribed curvature, tight, etc.), See also {49Q05, 49Q10, 53A10, 57R40, 57R42}
- 35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
- 34L30 Nonlinear ordinary differential operators
Abstract: 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.
Keywords: Dirichlet boundary conditions, Willmore surfaces of revolution
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