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

We give closed expressions for classical symmetric 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, 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 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’s the goal of the present report to make Euler’s observations more accessible and to develop them under the point of view of boundary value problems.