THE KIRCHHOFF-PLATEAU PROBLEM AND ITS GENERALIZATIONS - GIULIA BEVILACQUA – MOX - POLITECNICO DI MILANO
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 [6] 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 [2, 3], 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 Plateau problem, i.e. the boundary is an elastic curve. In [4], 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, in [5] and [1], we study the critical points of the Plateau problem to characterize its entire mechanical structure and to compute all the equilibrium configurations. First, we propose a slight variation of the Lagrange multiplier theorem in infinite dimension, obtaining a non-homogeneous first order differential system of equations [5]. Then, in [1], using an induction argument, we construct infinitely many critical points for a functional depending only on the curvature and both on the curvature and the torsion.
[1] G. Bevilacqua, A. De Rosa, L. Lussardi, Critical points of the Plateau problem, in preparation.
[2] 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.
[3] G. Bevilacqua, L. Lussardi, A. Marzocchi, Soap film spanning an elastic link, Quart. Appl. Math. 77 (3) (2019), 507 - 523.
[4] G. Bevilacqua, L. Lussardi, A. Marzocchi, Dimensional reduction of the Kirchhoff-Plateau problem, to appear on J. of Elasticity.
[5] G. Bevilacqua, L. Lussardi, A. Marzocchi, Variational analysis of inextensible elastic curves, in preparation.
[6] G.G. Giusteri, L. Lussardi, E. Fried, Solution of the Kirchhoff-Plateau problem, J. Nonlinear Sci. 27 (2017), 1043 - 1063.
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