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TR 2005-02

Boundary value problems for the one-dimensional Willmore equation - Almost explicit solutions

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


Series: TR 2005-02, Technical Reports

MSC:
34B15 Nonlinear boundary value problems
34A05 Explicit solutions and reductions

Abstract:
We give closed expressions for classical solutions of boundary value problems for the one-dimensional Willmore equation.
Navier as well as Dirichlet boundary conditions are considered. In the first case, one has existence of precisely
two solutions for boundary data below a suitable threshold.
This effect reflects that we have a bending point in the
corresponding bifurcation diagram and is not due to that we
restrict ourselves to graphs. Under Dirichlet boundary
conditions we always have existence of precisely one
symmetric solution. Parts of the material can already be
found in Euler's work. It is the goal of the present report
to make Euler's observations more accesible and to develop
them under the point of view of boundary value problems.

Keywords:
Willmore functional, Willmore equation, elastic curves, boundary value problems, explicit solutions