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2010-29

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

by Grunau, H.-Ch..


Series: 2010-29, Preprints

MSC:
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

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 $\alpha$ 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 $[1-k,1]$.

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