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Unstable Willmore surfaces of revolution subject to natural boundary conditions

by Dall'Acqua, A., Deckelnick, K., Wheeler, G..

Series: 2012-03, Preprints

35J40 Boundary value problems for higher-order elliptic equations
49J45 Methods involving semicontinuity and convergence; relaxation
49Q10 Optimization of shapes other than minimal surfaces

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.

Willmore surfaces of revolution