Research article Special Issues

On the anisotropic Kirchhoff-Plateau problem

  • Received: 12 May 2021 Accepted: 27 May 2021 Published: 15 June 2021
  • We extend to the anisotropic setting the existence of solutions for the Kirchhoff-Plateau problem and its dimensional reduction.

    Citation: Antonio De Rosa, Luca Lussardi. On the anisotropic Kirchhoff-Plateau problem[J]. Mathematics in Engineering, 2022, 4(2): 1-13. doi: 10.3934/mine.2022011

    Related Papers:

  • We extend to the anisotropic setting the existence of solutions for the Kirchhoff-Plateau problem and its dimensional reduction.



    加载中


    [1] W. K. Allard, A characterization of the area integrand, Symp. Math., XIV (1974), 429–444.
    [2] W. K. Allard, An a priori estimate for the oscillation of the normal to a hypersurface whose first and second variation with respect to an elliptic integrand is controlled, Invent. Math., 73 (1983), 287–331. doi: 10.1007/BF01394028
    [3] F. J. Jr. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. Math., 87 (1968), 321–391. doi: 10.2307/1970587
    [4] F. J. Jr. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Am. Math. Soc., 4 (1976), 165.
    [5] S. S. Antman, Nonlinear problems of elasticity, In: Applied Mathematical Sciences, 2 Eds., New York: Springer, 2005.
    [6] F. Bernatzki, R. Ye, Minimal surfaces with an elastic boundary, Ann. Global Anal. Geom., 19 (2001), 1–9. doi: 10.1023/A:1006734619701
    [7] G. Bevilacqua, L. Lussardi, A. Marzocchi, Soap film spanning an elastic link, Quart. Appl. Math., 77 (2019), 507–523.
    [8] G. Bevilacqua, L. Lussardi, A. Marzocchi, Dimensional reduction of the Kirchhoff-Plateau problem, J. Elasticity, 140 (2020), 135–148. doi: 10.1007/s10659-020-09763-y
    [9] A. Biria, E. Fried, Buckling of a soap film spanning a flexible loop resistant to bending and twisting, Proc. R. Soc. Lond. A, 470 (2014), 20140368.
    [10] A. Biria, E. Fried, Theoretical and experimental study of the stability of a soap film spanning a flexible loop, Int. J. Eng. Sci., 94 (2015), 86–102. doi: 10.1016/j.ijengsci.2015.05.002
    [11] Y. C. Chen, E. Fried, Stability and bifurcation of a soap film spanning a flexible loop, J. Elasticity, 116 (2014), 75–100. doi: 10.1007/s10659-013-9458-x
    [12] P. G. Ciarlet, J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal., 97 (1987), 171–188. doi: 10.1007/BF00250807
    [13] R. Courant, The existence of minimal surfaces of given topological structure under prescribed boundary conditions, Acta Math., 72 (1940), 51–98. doi: 10.1007/BF02546328
    [14] C. De Lellis, A. De Rosa, F. Ghiraldin, A direct approach to the anisotropic Plateau's problem, Adv. Calc. Var., 12 (2017), 211–223.
    [15] C. De Lellis, F. Ghiraldin, F. Maggi, A direct approach to Plateau's problem, JEMS, 9 (2017), 2219–2240.
    [16] G. De Philippis, A. De Rosa, F. Ghiraldin, A direct approach to Plateau's problem in any codimension, Adv. Math., 288 (2016), 59–80. doi: 10.1016/j.aim.2015.10.007
    [17] G. De Philippis, A. De Rosa, F. Ghiraldin, Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Commun. Pure Appl. Math., 71 (2018), 1123–1148. doi: 10.1002/cpa.21713
    [18] G. De Philippis, A. De Rosa, F. Ghiraldin, Existence results for minimizers of parametric elliptic functionals, J. Geom. Anal., 30 (2020), 1450–1465. doi: 10.1007/s12220-019-00165-8
    [19] G. De Philippis, A. De Rosa, J. Hirsch, The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals, Discrete Cont. Dyn. A, 39 (2019), 7031–7056. doi: 10.3934/dcds.2019243
    [20] A. De Rosa, Minimization of anisotropic energies in classes of rectifiable varifolds, SIAM J. Math. Anal., 50 (2018), 162–181. doi: 10.1137/17M1112479
    [21] A. De Rosa, S. Gioffrè, Quantitative stability for anisotropic nearly umbilical hypersurfaces, J. Geom. Anal., 29 (2019), 2318–2346. doi: 10.1007/s12220-018-0079-2
    [22] A. De Rosa, S. Gioffrè, Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces, 2018, arXiv: 1803.09118.
    [23] A. De Rosa, S. Kolasinski, Equivalence of the ellipticity conditions for geometric variational problems, Commun. Pure Appl. Math., 73 (2020), 2473–2515. doi: 10.1002/cpa.21890
    [24] A. De Rosa, S. Kolasinski, M. Santilli, Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets, Arch. Rational Mech. Anal., 238 (2020), 1157–1198. doi: 10.1007/s00205-020-01562-y
    [25] H. Federer, Geometric measure theory, New York: Springer-Verlag New York Inc., 1969.
    [26] L. Giomi, L. Mahadevan, Minimal surfaces bounded by elastic lines, Proc. R. Soc. Lond. A, 468 (2012), 1851–1864.
    [27] G. G. Giusteri, P. Franceschini, E. Fried, Instability paths in the Kirchhoff–Plateau problem, J. Nonlinear Sci., 26 (2016), 1097–1132. doi: 10.1007/s00332-016-9299-4
    [28] G. G. Giusteri, L. Lussardi, E. Fried, Solution of the Kirchhoff-Plateau problem, J. Nonlinear Sci., 27 (2017), 1043–1063. doi: 10.1007/s00332-017-9359-4
    [29] O. Gonzalez, J. H. Maddocks, F. Schuricht, H. von der Mosel, Global curvature and self-contact of nonlinearly elastic curves and rods, Calc. Var., 14 (2002), 29–68. doi: 10.1007/s005260100089
    [30] J. Harrison, H. Pugh, Existence and soap film regularity of solutions to Plateau's problem, Adv. Calc. Var., 9 (2016), 357–394. doi: 10.1515/acv-2015-0023
    [31] J. Harrison, H. Pugh, General Methods of Elliptic Minimization, Calc. Var., 56 (2017), 123. doi: 10.1007/s00526-017-1217-6
    [32] T. M. Hoang, E. Fried, Influence of a spanning liquid film on the stability and buckling of a circular loop with intrinsic curvature or intrinsic twist density, Math. Mech. Solids, 23 (2016), 43–66.
    [33] H. Lewy, On mimimal surfaces with partially free boundary, Commun. Pure Appl. Math., 4 (1951), 1–13. doi: 10.1002/cpa.3160040102
    [34] F. Schuricht, Global injectivity and topological constraints for spatial nonlinearly elastic rods, J. Nonlinear Sci., 12 (2002), 423–444. doi: 10.1007/s00332-002-0462-8
    [35] L. Simon, Lectures on geometric measure theory, Canberra: Australian National University, Centre for Mathematical Analysis, 1983.
    [36] J. E. Taylor, Crystalline variational problems, Bull. Am. Math. Soc., 84 (1978), 568–588. doi: 10.1090/S0002-9904-1978-14499-1
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1792) PDF downloads(125) Cited by(4)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog