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2009-02

A Navier boundary value problem for Willmore surfaces of revolution

by Deckelnick, K., Grunau, H.-Ch..


Series: 2009-02, Preprints

MSC:
53C42 Immersions (minimal, prescribed curvature, tight, etc.)
34B15 Nonlinear boundary value problems
35J65 Nonlinear boundary value problems for linear elliptic equations
35B32 Bifurcation

Abstract:
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.

Keywords:
Willmore surfaces, natural boundary value problem, surfaces of revolution, bifurcation, Clifford torus, Newton's method