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The Kirchhoff-Plateau problem

Giulia Bevilacqua
Politecnico di Milano
Friday, December 6, 2019 - 14:30

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 [4] 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 into a system consisting by several rods linked in an arbitrary way and tied by a soap film and we perform some experiments to validate our result. Finally 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.

[1] G. Bevilacqua, L. Lussardi, A. Marzocchi, Soap film spanning electrically repulsive elastic protein links, Proceedings of School & Research Workshop Mathematical Modeling of Self-Organizations in Medicine, Biology and Ecology: from micro to macro, Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 96 (2018), suppl. 3, A1, 13pp;
[2] G. Bevilacqua, L. Lussardi, A. Marzocchi, Soap film spanning an elastic link, Quart. Appl. Math. 77 (3) (2019), 507–523;
[3] G. Bevilacqua, L. Lussardi, A. Marzocchi, Dimensional reduction of the Kirchhoff-Plateau problem, submitted;
[4] G.G. Giusteri, L. Lussardi, E. Fried, Solution of the Kirchhoff-Plateau problem, J. Nonlinear Sci. 27 (2017), 1043–1063.

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