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

by H.-Ch. Grunau


Preprint series: 10-29 , Preprints

49Q10 Optimization of the shape other than minimal surfaces, See also {73K40, 90C90}
53C42 Immersions (minimal, prescribed curvature, tight, etc.), See also {49Q05, 49Q10, 53A10, 57R40, 57R42}
35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
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

Upload: 2010-11-16


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Letzte Änderung: 01.03.2018 - Ansprechpartner:

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