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