The Euler–Helfrich functional

Palmer, Bennett; Pámpano, Álvaro

doi:10.1007/s00526-022-02188-6

Cited by 7 publications

(3 citation statements)

References 27 publications

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“…(63) § § The interplay between F H and edge strain energies has been explored in a series of papers, some of which are recent. [41][42][43][44][45][46][47][48][49] ¶ ¶ As appropriately remarked, 23 the same result had also been proved, albeit with different methods, in the differential geometry literature, [51][52][53] but it was unknown to plates and shells practitioners.…”

Section: Bending-neutrality Compatibilitymentioning

confidence: 83%

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Pure measures of bending for soft plates

Virga

2024

Soft Matter

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Section: Bending-neutrality Compatibilitymentioning

confidence: 83%

“…(63) § § The interplay between F H and edge strain energies has been explored in a series of papers, some of which are recent. [41][42][43][44][45][46][47][48][49] 1…”

Section: Bending-neutrality Compatibilitymentioning

confidence: 99%

Pure measures of bending for soft plates

Virga

2024

Soft Matter

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“…However, in the particular case p = 2 (respectively, α = −2) and n = 2, the same relation holds for surfaces under less restrictive assumptions. In [39], it was proven that any surface satisfying (7) for α = −2 also satisfies the Euler-Lagrange equation associated with W 2,µ,0 , while the converse holds for disc type surfaces of revolution.…”

Section: Generalized Singular Minimal Hypersurfacesmentioning

confidence: 99%

A relation between critical points of Willmore-type energies, weighted areas and vertical potential energies

López¹,

Pámpano²

2022

Preprint

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This paper considers the energies of three different physical scenarios and obtains relations between them in a particular case. The first family of energies consists of the Willmore-type energies involving the integral of powers of the mean curvature which extends the Willmore and Helfrich energies. A second family of energies is the area functionals arising in weighted manifolds, following the theory developed by Gromov, when the density is a power of the height function. The third one is the free energies of a fluid deposited in a horizontal hyperplane when the potentials depend on the height with respect to this hyperplane. In this paper we find relations between each of them when the critical point is a hypersurface of cylindrical type. Cylindrical hypersurfaces are determined by their generating planar curves and for each of the families of energies, these curves satisfy suitable ordinary differential equations. For the Willmore-type energies, the equation is of fourth order, whereas it is of order two in the other two cases. We prove that the generating curves coincide for the Willmore-type energies without area constraint and for weighted areas, and the similar result holds for the generating curves of Willmore-type energies and of the vertical potential energies, after suitable choices of the physical parameters. In all the cases, generating curves are critical points for a family of energies extending the classical bending energy. In the final section of the paper, we analyze the stability of a liquid drop deposited on a horizontal hyperplane with vertical potential energies. It is proven that if the free interface of the fluid is a graph on this hyperplane, then the hypersurface is stable in the sense that it is a local minimizer of the energy. In fact, we prove that the hypersurface is a global minimizer in the class of all graphs with the same boundary.

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Stability of Membranes

Palmer,

Pámpano

2024

J Geom Anal

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The Euler–Helfrich functional

Cited by 7 publications

References 27 publications

Pure measures of bending for soft plates

Pure measures of bending for soft plates

A relation between critical points of Willmore-type energies, weighted areas and vertical potential energies

Stability of Membranes

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