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