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

by Grunau, H.-Ch..

Series: 2010-29, Preprints

49Q10 Optimization of shapes other than minimal surfaces
53C42 Immersions (minimal, prescribed curvature, tight, etc.)
35J65 Nonlinear boundary value problems for linear elliptic equations
34L30 Nonlinear ordinary differential operators

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

Dirichlet boundary conditions, Willmore surfaces of revolution, asymptotic shape, boundary layer