Soap film spanning an elastic link
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
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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
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- 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
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- 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
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- Amiya Mukherjee, Differential topology, 2nd ed., Hindustan Book Agency, New Delhi; Birkhäuser/Springer, Cham, 2015. MR 3379695
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- 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
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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 0225243, 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 $\textbf {R}^n$: 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 0114145, 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
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Additional Information
Giulia Bevilacqua
Affiliation:
MOX - Dipartimento di Matematica, Politecnico di Milano, Italy
Email:
giulia.bevilacqua@polimi.it
Luca Lussardi
Affiliation:
Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Italy
MR Author ID:
805041
Email:
luca.lussardi@polito.it
Alfredo Marzocchi
Affiliation:
Dipartimento di Matematica e Fisica “Niccolò Tartaglia”, Università Cattolica del Sacro Cuore, Italy
MR Author ID:
120815
Email:
alfredo.marzocchi@unicatt.it
Received by editor(s):
November 22, 2017
Received by editor(s) in revised form:
April 28, 2018
Published electronically:
June 25, 2018
Article copyright:
© Copyright 2018
Brown University