1. A Navier boundary value problem for Willmore surfaces of revolution
- Author
-
Hans-Christoph Grunau and Klaus Deckelnick
- Subjects
Numerical Analysis ,Mean curvature ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Bifurcation diagram ,Boundary knot method ,Willmore energy ,Minimal surface of revolution ,Mathematics::Differential Geometry ,Boundary value problem ,Surface of revolution ,Analysis ,Mathematics - 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.
- Published
- 2009