The Asymptotic Shape of a Boundary Layer of Symmetric Willmore Surfaces of Revolution

Abstract

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 arctanβ may be prescribed. Energy minimising solutions u _α,β have been previously constructed and for fixed β∈ℝ, the limit $\lim_{\alpha \searrow0 }u_{\alpha,\beta}(x) =\sqrt{1 - x^{2}}$ 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↘0 in a boundary layer of width kα, k>0 fixed, close to ±1. After rescaling $x\mapsto\frac{1}{\alpha}u_{\alpha,\beta }(\alpha(x-1)+1)$ one has convergence to a suitably chosen cosh on [1−k,1].

Financial support of “Deutsche Forschungsgemeinschaft” for the project “Randwertprobleme für Willmoreflächen – Analysis, Numerik und Numerische Analysis”, (DE 611/5-2) is gratefully acknowledged.