Elastic catenoids

Scholtes, Sebastian

doi:10.1524/anly.2011.1088

Cited by 6 publications

(12 citation statements)

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“…2. First existence results for (1) have been obtained in [43] and [18]. The goal of this paper is to extend these results in several directions including a description of the qualitative behaviour of solutions when possible.…”

Section: Introductionmentioning

confidence: 95%

“…4 on the local picture around a Helfrich cylinder, i.e. u(x) ≡ α, which is an explicit solution and the unique Helfrich minimiser in the case ε = ε α := 1 4α 2 (see [43,Lemma 4.1]; red curve in Fig. 1).…”

Section: Introductionmentioning

confidence: 98%

“…. The red curve where ε = ε α = 1 4α 2 displays the Helfrich cylinders (see [43,Lemma 4.1]). The orange domain sketches existence of solutions obtained by perturbing the Helfrich cylinders (see Theorem 3).…”

Section: Introductionmentioning

confidence: 99%

“…The orange domain sketches existence of solutions obtained by perturbing the Helfrich cylinders (see Theorem 3). The purple domain is obtained with the help of gluing techniques (see Theorem 7) and may be compared to the green domain implicitly found in [43,Theorem 4.14] (colour figure online) because they are heavily based on its conformal invariance and on explicitly known Willmore minimisers. Nevertheless, by applying the direct method and utilising an energy gap of the Willmore energy to 4π we find in Theorem 2 solutions of (1) for any α > 0 and ε ≥ 0 not too large, see the blue regions in Fig.…”

Section: Introductionmentioning

confidence: 99%

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

2021

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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.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2021

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“…In the class of surfaces of revolution existence results for Willmore surfaces were obtained in the series of the papers [10], [11] and [15]. These results even have some extensions to the Helfrich energy in [31] or [14]. In the class of graphs existence of a very weak minimiser in the class of functions of bounded variation for the Helfrich and Willmore energy was obtained in [12].…”

Section: Introductionmentioning

confidence: 98%

The Helfrich boundary value problem

Eichmann

2019

Calc. Var.

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We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy for oriented varifolds. The main contribution of this paper is a lower semicontinuity result with respect to oriented varifold convergence for the Helfrich energy and a minimising sequence. For arbitrary sequences this is false by a counterexample of Große-Brauckmann.Keywords. Dirichlet boundary conditions, Helfrich immersion, Existence MSC. 35J35, 35J40, 58J32, 49Q20, 49Q10 * The author thanks Prof. Reiner Schätzle for discussing the Helfrich energy and providing insight into geometric measure theory.

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Elastic graphs with clamped boundary and length constraints

Dall’Acqua,

Deckelnick

2023

Annali di Matematica

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We study two minimization problems concerning the elastic energy on curves given by graphs subject to symmetric clamped boundary conditions. In the first, the inextensible problem, we fix the length of the curves while in the second, the extensible problem, we add a term penalizing the length. This can be considered as a one-dimensional version of the Helfrich energy. In both cases, we prove existence, uniqueness and qualitative properties of the minimizers. A key ingredient in our analysis is the use of Noether identities valid for critical points of the energy and derived from the invariance of the energy functional with respect to translations. These identities allow us also to prove curvature bounds and ordering of the minimizers even though the problem is of fourth order and hence in general does not allow for comparison principles.

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Elastic catenoids

Cited by 6 publications

References 0 publications

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

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

The Helfrich boundary value problem

Elastic graphs with clamped boundary and length constraints

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