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

**Series:** 2005-31, Preprints

- MSC:
- 34B15 Nonlinear boundary value problems
- 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions
- 35J65 Nonlinear boundary value problems for linear elliptic equations

**Abstract:**

The one-dimensional Willmore equation is studied under Navier

as well as under Dirichlet boundary conditions. We are interested in

smooth graph solutions, since for suitable boundary data, we expect

the stable solutions to be among these.

In the first part, classical symmetric solutions for symmetric

boundary data are studied and closed expressions are deduced.

In the Navier case, one has existence of precisely two solutions

for boundary data below a suitable threshold, precisely one solution

on the threshold and no solution beyond the threshold. This effect reflects

that we have a bending point in the corresponding bifurcation diagram and

is {\it not} due to that we restrict ourselves to graphs. Under Dirichlet

boundary conditions we always have existence of precisely one symmetric

solution.

In the second part, we consider boundary value problems with

nonsymmetric data. Solutions are constructed by rotating and rescaling suitable parts

of the graph of an explicit symmetric solution.

One basic observation for the symmetric case

can already be found in Euler's work. It is one goal of the

present paper to make Euler's observation more accessible and to develop

it under the point of view of boundary value problems.

Moreover,

general existence results are proved.

**Keywords:**

Willmore equation, one dimensional, boundary value problem

**This paper was published in:**

Calc. Var. Partial Differ. Equ. 30, 293-314 (2007).