Soap film spanning an elastic link

Bevilacqua, Giulia; Lussardi, Luca; Marzocchi, Alfredo

doi:10.1090/qam/1510

Authors: Giulia Bevilacqua, Luca Lussardi and Alfredo Marzocchi
Journal: Quart. Appl. Math. 77 (2019), 507-523
MSC (2010): Primary 49Q05, 49Q20, 74K10; Secondary 74B20
DOI: https://doi.org/10.1090/qam/1510
Published electronically: June 25, 2018
MathSciNet review: 3962579
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the equilibrium problem of a system consisting of several Kirchhoff rods linked in an arbitrary way and tied by a soap film, using techniques of the Calculus of Variations. We prove the existence of a solution with minimum energy, which may be quite irregular, and perform experiments confirming the kind of surface predicted by the model.

References

F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2) 87 (1968), 321–391. MR 225243, DOI https://doi.org/10.2307/1970587
Stuart S. Antman, Nonlinear problems of elasticity, 2nd ed., Applied Mathematical Sciences, vol. 107, Springer, New York, 2005. MR 2132247
Philippe G. Ciarlet and Jindřich Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 97 (1987), no. 3, 171–188. MR 862546, DOI https://doi.org/10.1007/BF00250807
Bernard Dacorogna, Direct methods in the calculus of variations, 2nd ed., Applied Mathematical Sciences, vol. 78, Springer, New York, 2008. MR 2361288
Guy David, Should we solve Plateau’s problem again?, Advances in analysis: the legacy of Elias M. Stein, Princeton Math. Ser., vol. 50, Princeton Univ. Press, Princeton, NJ, 2014, pp. 108–145. MR 3329849

C. De Lellis, A. De Rosa, and F. Ghiraldin, A direct approach to the anisotropic Plateau’s problem, Adv. Calc. Var. arXiv:1602.08757, DOI:https://doi.org/10.1515/acv-2016-0057.

C. De Lellis, F. Ghiraldin, and F. Maggi, A direct approach to Plateau’s problem, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 8, 2219–2240. MR 3668059, DOI https://doi.org/10.4171/JEMS/716

Thierry De Pauw, Size minimizing surfaces, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 1, 37–101 (English, with English and French summaries). MR 2518893, DOI https://doi.org/10.24033/asens.2090

G. De Philippis, A. De Rosa, and F. Ghiraldin, A direct approach to Plateau’s problem in any codimension, Adv. Math. 288 (2016), 59–80. MR 3436382, DOI https://doi.org/10.1016/j.aim.2015.10.007

G. De Philippis, A. De Rosa, and F. Ghiraldin, Existence results for minimizers of parametric elliptic functionals, arXiv:1704.07801.

Antonio De Rosa, Minimization of anisotropic energies in classes of rectifiable varifolds, SIAM J. Math. Anal. 50 (2018), no. 1, 162–181. MR 3742687, DOI https://doi.org/10.1137/17M1112479

Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660

Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325

Giulio G. Giusteri, Luca Lussardi, and Eliot Fried, Solution of the Kirchhoff-Plateau problem, J. Nonlinear Sci. 27 (2017), no. 3, 1043–1063. MR 3638328, DOI https://doi.org/10.1007/s00332-017-9359-4

J. Harrison and H. Pugh, Existence and soap film regularity of solutions to Plateau’s problem, arXiv:1310.0508 (2013).

J. Harrison, Soap film solutions to Plateau’s problem, J. Geom. Anal. 24 (2014), no. 1, 271–297. MR 3145925, DOI https://doi.org/10.1007/s12220-012-9337-x

Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR 658490

Amiya Mukherjee, Differential topology, 2nd ed., Hindustan Book Agency, New Delhi; Birkhäuser/Springer, Cham, 2015. MR 3379695

James R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. Second edition of [ MR0464128]. MR 3728284

J. Plateau, Experimental and theoretical statics of liquids subject to molecular forces only, Gauthier- Villars (1873).

David Preiss, Geometry of measures in ${\bf R}^n$: distribution, rectifiability, and densities, Ann. of Math. (2) 125 (1987), no. 3, 537–643. MR 890162, DOI https://doi.org/10.2307/1971410

E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. MR 114145, DOI https://doi.org/10.1007/BF02547186

Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. MR 0515288

F. Schuricht, Global injectivity and topological constraints for spatial nonlinearly elastic rods, J. Nonlinear Sci. 12 (2002), no. 5, 423–444. MR 1923387, DOI https://doi.org/10.1007/s00332-002-0462-8