Calculus of Variations and Geometric Measure Theory

G. G. Giusteri - L. Lussardi - E. Fried

Solution of the Kirchhoff-Plateau problem

created by lussardi on 16 Jun 2016
modified on 24 Apr 2017


Published Paper

Inserted: 16 jun 2016
Last Updated: 24 apr 2017

Journal: J. Nonlinear Sci.
Volume: 27
Pages: 1043-1063
Year: 2017
Doi: 10.1007/s00332-017-9359-4


The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for self-contact of the filament. In our treatment, the bounding filament retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the filament is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust itself into a configuration that complies with the physical constraints encountered in experiments.

Keywords: Plateau problem, minimal surface, Kirchhoff rod, Stable equilibria, Liquid film, Topological constraints