Boundary value problems for a special Helfrich functional for surfaces of revolution: existence and asymptotic behaviour

Abstract

The central object of this article is (a special version of) the Helfrich functional which is the sum of the Willmore functional and the area functional times a weight factor $$\varepsilon \ge 0$$ ε ≥ 0 . We collect several results concerning the existence of solutions to a Dirichlet boundary value problem for Helfrich surfaces of revolution and cover some specific regimes of boundary conditions and weight factors $$\varepsilon \ge 0$$ ε ≥ 0 . These results are obtained with the help of different techniques like an energy method, gluing techniques and the use of the implicit function theorem close to Helfrich cylinders. In particular, concerning the regime of boundary values, where a catenoid exists as a global minimiser of the area functional, existence of minimisers of the Helfrich functional is established for all weight factors $$\varepsilon \ge 0$$ ε ≥ 0 . For the singular limit of weight factors $$ \varepsilon \nearrow \infty $$ ε ↗ ∞ they converge uniformly to the catenoid which minimises the surface area in the class of surfaces of revolution.

Axisymmetric membranes with edges under external force: buckling, minimal surfaces, and tethers

“…Our work is complementary to recent work in the mathematics community on the shapes that are critical points of the bending energy, such as the study of axisymmetric shapes with zero mean curvature at the edges and with no constraint on the area 27 , or the study of axisymmetric shapes with fixed tension 28 . The paper of Deckelenick and Grunau 29 is an important precursor for our present article since their numerical experiments suggest a rich collection of possible shapes in the case of no area constraint and Fig.…”

The Euler-Helfrich Functional

“…The second derivative of f with respect to evaluated at = 0, after using once again (6) and the definitions K := −r /r and κ g := −r /r, is given by…”

“…This proves case (i). If b = 0, we use (6) to rewrite K in terms of a, b and (H + c o ) 2 , concluding with cases (ii) and (iii). q.e.d.…”

“…Axially symmetric surfaces which are critical for the Helfrich energy have already been studied by many authors, for example [6], [18] and [25], but with boundary conditions distinct from those considered here. The reference [25] shows that axially symmetric critical surfaces for the Helfrich energy satisfy a third order differential equation which is equivalent to equation (15) below.…”

Variational models for the interaction of surfactants with curvature -- existence and regularity of minimizers in the case of flexible curves

“…There are several contributions on the minimization problem [22,35], even in the case of more than one surface [12,14,15]. Moreover there is no lack of stability results [8,23] and also the associated Dirichlet boundary value problem has been considered [17,21]. The Helfrich functional can be interpreted as the singular limit of a suitable approximating functional defined on diffuse interfaces [7,27,28].…”