by Deckelnick, K., Grunau, H.-Ch..
Series: 2009-02, Preprints
We study a boundary value problem for Willmore surfaces of revolution, where the position and the mean curvature H=0 are prescribed as boundary data. The latter is a natural datum when considering critical points of the Willmore
functional in classes of functions where only the position at the boundary is fixed.
For specific boundary positions, catenoids and a suitable part of the Clifford torus are explicit solutions. Numerical experiments, however, suggest a much richer bifurcation diagram. In the present paper we verify analytically some properties of the expected bifurcation diagram. Furthermore, we present a finite element method which allows the calculation of critical points of the Willmore functional irrespective of their stability properties.
Willmore surfaces, natural boundary value problem, surfaces of revolution, bifurcation, Clifford torus, Newton's method