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02 November 2024 

   

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A relation between critical points of Willmoretype energies, weighted areas and vertical potential energies  Rafael López
; Álvaro Pámpano
;  Date: 
2 Jun 2022  Abstract:  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 Willmoretype 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 Willmoretype 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 Willmoretype energies without area
constraint and for weighted areas, and the similar result holds for the
generating curves of Willmoretype 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.  Source:  arXiv, 2206.01070  Services:  Forum  Review  PDF  Favorites 


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