The asymptotic shape of a boundary layer of symmetric Willmore surfaces of revolution*

We consider the Willmore boundary value problem for surfaces of revolution over the interval [−1, 1] where, as Dirichlet boundary conditions, any symmetric set of position α and angle tanβ may be prescribed. Energy minimising solutions uα,β have been previously constructed and for fixed β ∈ R, the limit limαց0 uα,β(x) = √ 1 − x has been proved locally uniformly in (−1, 1), irrespective of the boundary angle. Subject of the present note is to study the asymptotic behaviour for fixed β ∈ R and α ց 0 in a boundary layer of width kα, k > 0 fixed, close to ±1. After rescaling x 7→ 1 α uα,β(α(x− 1) + 1) one has convergence to a suitably chosen cosh on [1 − k, 1].