12-03
Unstable Willmore surfaces of revolution subject to natural boundary conditions
A. Dall'Acqua, K. Deckelnick, G. Wheeler
Preprint series: 12-03, Preprints
- MSC:
- 35J40 Boundary value problems for higher-order, elliptic equations
- 49J45 Problems involving semicontinuity and convergence
- 49Q10 Optimization of the shape other than minimal surfaces, See also {73K40, 90C90}
Abstract:In the class of surfaces with fixed boundary, critical points of the Willmore functional are naturally found to be those solutions of the Euler-Lagrange equation where the mean curvature on the boundary vanishes. We consider the case of symmetric surfaces of revolution in the setting where there are two families of stable solutions given by the catenoids. In this paper we demonstrate the existence of a third family of solutions which are unstable critical points of the Willmore functional, and which spatially lie between the upper and lower families of catenoids. Our method does not require any kind of smallness assumption, and allows us to derive some additional interesting qualitative properties of the solutions.
Keywords:Willmore surfaces of revolution
Upload:2012-02-14-02-14