Stability and Symmetry in the Navier Problem for the One-Dimensional Willmore Equation

We consider the one-dimensional Willmore equation subject to Navier boundary conditions, i.e. the position and the curvature are prescribed on the boundary. In a previous work, explicit symmetric solutions to symmetric data have been constructed. Within a certain range of boundary curvatures one has precisely two symmetric solutions while for boundary curvatures outside the closure of this range there are none. The solutions are ordered; one is “small”, the other “large”. In the first part of this paper we address the stability problem and show that the small solution is (linearized) stable in the whole open range of admissible boundary curvatures, while the large one is unstable and has Morse index 1. A second goal is to investigate whether the small solution is minimal for the corresponding Willmore functional. It turns out that for a certain subrange of admissible boundary curvatures the small solution is the unique minimum, while for curvatures outside that range the minimum is not attained. As a by–product of our argument we show that for any admissible function there exists a symmetric function with smaller Willmore energy.