Boundary value problems for the Willmore and the Helfrich functional for surfaces of revolution

Abstract: This talk concerns joint works with A. Dall'Acqua, K. Deckelnick, M. Doemeland, S. Eichmann, and S. Okabe.

A special form of the Helfrich energy for a sufficiently smooth (two dimensional) surface $ S \subset \mathbb{R} ^3 $ (with or without boundary) is defined by $$ {\mathscr H}_\varepsilon(S) := \int_{S} H^2 \, d S + \varepsilon \int_{S} \, d S , $$ where $H$ denotes the mean curvature of $S$. The first integral may be considered as a bending energy and the second as surface (stretching) energy. ${\mathscr W} (S):={\mathscr H}_0 (S)$ is called the Willmore functional. We consider surfaces of revolution $ S $ $$ (x,\varphi)\mapsto \big(x,u(x)\cos \varphi, u(x)\sin \varphi \big) \, , \quad x\in[-1,1],~\varphi\in[0,2\pi], $$ with smooth strictly positive profile curve $u$ subject to Dirichlet boundary conditions $$ u(-1)=\alpha,\quad u(1)=\beta,\quad u'(\pm1)=0 $$ and aim at minimising ${\mathscr H}_\varepsilon$. Thanks to these boundary conditions the Gauss curvature integral $\int_{S} K\, d S $ becomes a constant and needs not to be considered.

In the first part of the lecture I shall consider the Willmore case, i.e. $\varepsilon=0$. After briefly recalling minimisation in the symmetric case $\alpha=\beta$ (see [1,4]) I shall show how much more complicated the problem becomes for $\alpha\not=\beta$. Only when $\alpha$ and $\beta$ do not differ too much, the profile curve will remain a graph while in general it will become a nonprojectable curve, see [3].

In the second part, ${\mathscr H}_\varepsilon$ is considered for $\varepsilon\in[0,\infty)$, but again in the symmetric setting $\alpha=\beta$. For $\alpha \ge \alpha_m = c_m \cosh(\frac{1}{c_m})\approx 1.895$ with $c_m\approx 1.564$ the unique solution of the equation $ \frac{2}{c} = 1 + e ^ {-2/ c} $, when one has a catenoid $v_\alpha$ which globally minimises the surface energy, we find minimisers $u_\varepsilon$ for any $\varepsilon\ge 0$ and show uniform and locally smooth convergence $u_\varepsilon \to v_\alpha$ under the singular limit $\varepsilon \to \infty$. These results are collected in [2].

At the end I shall briefly mention recent work on obstacle problems [5].

[1] A. Dall'Acqua, K. Deckelnick, and H.-Ch. Grunau, Classical solutions to the Dirichlet problem for Willmore surfaces of revolution, Adv. Calc. Var. 1 (2008), 379-397.

[2] K. Deckelnick, H.-Ch. Grunau, and M. Doemeland, Boundary value problems for the Helfrich functional for surfaces of revolution - Existence and asymptotic behaviour, Calc. Var. Partial Differ. Equ. 60 (2021), Article number 32.

[3] S. Eichmann and H.-Ch. Grunau, Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions, Adv. Calc. Var. 12 (2019), 333–361.

[4] H.-Ch. Grunau, The asymptotic shape of a boundary layer of symmetric Willmore surfaces of revolution. In: C. Bandle et al. (eds.), Inequalities and Applications 2010. International Series of Numerical Mathematics 161 (2012), 19-29.

[5] H.-Ch. Grunau and S. Okabe, Willmore obstacle problems under Dirichlet boundary conditions, submitted.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysismetric geometrynumerical analysisoptimization and control

Audience: researchers in the topic

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