Calculus of Variations and Geometric Measure Theory

Variational methods and applications

The Kirchhoff-Plateau problem and its generalizations

Giulia Bevilacqua (Università di Pisa)

created by paolini on 08 Sep 2021
modified on 14 May 2022

6 sep 2021 -- 15:00   [open in google calendar]


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. Giusteri, Lussardi and Fried in 5 established 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 points on the surface of the bounding loop to come into contact. In 1, 2, we use this result to generalize the situation studying a system composed by several rods linked in an arbitrary way and tied by a soap film and we perform some experiments to validate our result. We also study the Elastic Plateau problem, i.e. the above problem when the boundary is an elastic curve. In 3, we obtain the minimal energy solution of the Plateau problem with elastic boundary as a variational limit of the minima of the Kirchhoff-Plateau problems with a rod boundary when the cross-section of the rod vanishes. The limit boundary is a framed curve that can sustain bending and twisting. Finally, since computing the minimum of the energy functional is quite easy, we aim at characterizing the critical points, i.e. computing all the equilibrium configurations to complete understand the mechanical structure of the Plateau problem. Indeed, in 4, we propose a vari- ation of the Lagrange multiplier theorem in infinite dimension, obtaining a non-homogeneous first order differential system of equations for the only elastic curve. Joint work with Luca Lussardi (DISMA – Politecnico di Torino, and Alfredo Marzocchi (Universit`a Cattolica del Sacro Cuore,