Skip to main content

Advertisement

SpringerLink
Log in

The elastica problem under area constraint

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We show that the elastic energy \(E(\gamma )\) of a closed curve \(\gamma \) has a minimizer among all plane simple regular closed curves of given enclosed area \(A(\gamma )\), and that the minimum is attained for a circle. The proof is of a geometric nature and deforms parts of \(\gamma \) in a finite number of steps to construct some related convex sets with smaller energy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Arreaga, G., Capovilla, R., Chryssomalakos, C., Guven, J.: Area-constrained planar elastica. Phys. Rev. E 65(3), 031801–14 (2002)

    Article  Google Scholar 

  2. Avvakumov, S., Karpenkov, O., Sossinsky, A.: Euler elasticae in the plane and the Whitney-Graustein theorem. Russ. J. Math. Phys. 20(3), 257–267 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bianchini, C., Henrot, A., Takahashi, T.: Elastic energy of a convex body. Math. Nachr. arXiv:1406.7305

  4. Bucur, D., Henrot, A.: A new isoperimetric inequality for the elasticae. arXiv:1412.4536 (2014)

  5. Dondl, P.W., Mugnai, L., Röger, M.: Confined elastic curves. SIAM J. Appl. Math. 71(6), 2205–2226 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dziuk, G., Kuwert, E., Schätzle, R.: Evolution of elastic curves in \(\mathbb{R}^n\): existence and computation. SIAM J. Math. Anal. 33(5), 1228–1245 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Enomoto, K.: Squared curvatures are almost minimal. Geom. Dedic. 65, 193–201 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Enomoto, K., Okura, M.: The total squared curvature of curves and approximation by piecewise circular curves. Results Math. 64(1–2), 215–228 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gage, M.E.: An isoperimetric inequality with applications to curve shortening. Duke Math. J. 40(4), 1225–1229 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gage, M.E.: Positive centers and the Bonnesen inequality. Proc. Am. Math. Soc. 110(4), 1041–1048 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gage, M.E., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23(1), 69–96 (1986)

    MathSciNet  MATH  Google Scholar 

  12. Giomi, L., Mahadevan, L.: Minimal surfaces bounded by elastic lines. Proc. R. Soc. A Math. Phys. Eng. Sci. 468(2143), 1851–1864 (2012)

    Article  MathSciNet  Google Scholar 

  13. Hopf, H.: Über die drehung der tangenten und sehnen ebener Kurven. Compos. Math. 2, 50–62 (1935)

    MathSciNet  MATH  Google Scholar 

  14. Koiso, N.: Elasticae in a Riemannian submanifold. Osaka J. Math. 29(3), 539–543 (1992)

    MathSciNet  MATH  Google Scholar 

  15. Langer, J., Singer, D.A.: The total squared curvature of closed curves. J. Differ. Geom. 20(1), 1–22 (1984)

    MathSciNet  MATH  Google Scholar 

  16. Langer, J., Singer, D.A.: Knotted elastic curves in \(\mathbb{R}^3\). J. Lond. Math. Soc. 30(2), 512–520 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Langer, J., Singer, D.A.: Curves in the hyperbolic plane and mean curvature of tori in 3-space. Bull. Lond. Math. Soc. 16, 531–534 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Langer, J., Singer, D.A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24(1), 75–88 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Levien, R.: The elastica: a mathematical history. In: Tech. Rep. UCB/EECS-2008-103. EECS Department, University of California, Berkeley (2008)

  20. Linnér, A.: Curve-straightening and the Palais-Smale condition. Trans. Am. Math. Soc. 350(9), 3743–3765 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lin, Y.-C., Tsai, D.-H.: Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves. J. Evol. Equ. 12(4), 833–854 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Okabe, S.: The motion of elastic planar closed curves under the area-preserving condition. Indiana Univ. Math. J. 56(4), 1871–1913 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Osserman, R.: Bonnesen-style isoperimetric inequalities. Am. Math. Mon. 86(4), 1–29 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Radon, J.: Zum Problem von Lagrange. Abh. Math. Sem. Univ. Hambg. 6(1), 273–299 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sachkov, YuL: Closed Euler elasticae. Proc. Stekl. Inst. Math. 278(1), 218–232 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Singer, D.A.: Lectures on elastic curves and rods. Curvature and variational modeling in physics and biophysics. In: AIP Conf. Proc., 1002, Am. Inst. Phys., Melville, NY, pp. 2–32 (2008)

  27. Smith, L.: The asymptotics of the heat equation for a boundary value problem. Invent. Math. 63(3), 467–493 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  28. Truesdell, C.: The influence of elasticity on analysis: the classic heritage. Bull. Am. Math. Soc. (N.S.) 9(3), 293–310 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  29. Veerapaneni, Shravan K., Raj, Ritwik, Biros, George, Purohit, Prashant K.: Analytical and numerical solutions for shapes of quiescent two-dimensional vesicles Int. J. Non Linear. Mech. 44(3), 257–262 (2009)

    Article  Google Scholar 

  30. Watanabe, K.: Plane domains which are spectrally determined. Ann. Glob. Anal. Geom. 18(1), 447–475 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Watanabe, K.: Plane domains which are spectrally determined II. J. Inequal. Appl. 7(1), 25–47 (2002)

    MathSciNet  MATH  Google Scholar 

  32. Wen, Y.Z.: Curve straightening flow deforms closed plane curves with nonzero rotation number to circles. J. Differ. Equ. 120(1), 89–107 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Willmore, T.: Curves. In: Handbook of Differential Geometry, vol. 1, pp. 997–1023. North-Holland, Amsterdam (2000)

Download references

Acknowledgments

We thank A. Henrot for helpful discussions during a visit to Napoli in 2013. After we had completed this manuscript in November 2014 (arXiv:1411.6100) and sent it to colleagues for comments, we were kindly informed by A. Henrot that he and D. Bucur had answered the open problem addressed in the present paper with a different proof. Their written proof has appeared in December 2014 on CVGMT (http://cvgmt.sns.it/paper/2582/) see also [4]. C. Nitsch was financially supported by an Alexander-von-Humboldt Grant and by Progetto Star “SInECoSINE”. First and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Author information

Authors and Affiliations

  1. Università degli Studi di Napoli Federico II, Naples, Italy

    Vincenzo Ferone & Carlo Nitsch

  2. Universität zu Köln, Cologne, Germany

    Bernd Kawohl & Carlo Nitsch

Authors
  1. Vincenzo Ferone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernd Kawohl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carlo Nitsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernd Kawohl.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferone, V., Kawohl, B. & Nitsch, C. The elastica problem under area constraint. Math. Ann. 365, 987–1015 (2016). https://doi.org/10.1007/s00208-015-1284-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-015-1284-y

Mathematics Subject Classification

Search

Navigation