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Boundary value problems for the one-dimensional Willmore equation

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

Series: 2005-31, Preprints

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

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

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.
general existence results are proved.

Willmore equation, one dimensional, boundary value problem

This paper was published in:
Calc. Var. Partial Differ. Equ. 30, 293-314 (2007).