The Second-Order \(L^{2}\)-Flow of Inextensible Elastic Curves with Hinged Ends in the Plane

  • Chun-Chi LinEmail author
  • Yang-Kai Lue
  • Hartmut R. Schwetlick


In this article, we study the evolution of open inextensible planar curves with hinged ends. We obtain long time existence of \(C^{\infty}\)-smooth solutions during the evolution, given the initial curves that are only \(C^{2}\)-smooth with vanishing curvature at the boundary. Moreover, the asymptotic limits of this flow are inextensible elasticae. Our method and result extend the work by Wen (Duke Math. J. 70(3):683–698, 1993).


Geometric flow Second-order parabolic equation Hinged boundary conditions Elastic energy Willmore functional 

Mathematics Subject Classification (2010)

35B65 35K51 53A04 53C44 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angenent, S.: On the formation of singularities in the curve shortening flow. J. Differ. Geom. 33(3), 601–633 (1991) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Applied Mathematical Sciences, vol. 107. Springer, New York (2005) zbMATHGoogle Scholar
  3. 3.
    Bryant Robert, R., Griffiths, P.: Reduction for constrained variational problems and \(\int\frac {1}{2}\vec{\kappa}^{2}ds\). Am. J. Math. 108(3), 525–570 (1986) CrossRefzbMATHGoogle Scholar
  4. 4.
    Brunnett, G., Wendt, J.: Elastic splines with tension control. In: Mathematical Methods for Curves and Surfaces, II, Lillehammer, 1997. Innov. Appl. Math., pp. 33–40. Vanderbilt Univ. Press, Nashville (1998) Google Scholar
  5. 5.
    Cannon, J.R.: The One-Dimensional Heat Equation. Encyclopedia of Mathematics and Its Applications, vol. 23. Addison-Wesley, Reading (1984). Advanced Book Program. With a foreword by Felix E. Browder CrossRefzbMATHGoogle Scholar
  6. 6.
    Dall’Acqua, A., Pozzi, P.: A Willmore–Helfrich \(L^{2}\)-flow of curves with natural boundary conditions. Commun. Anal. Geom. 22(4), 617–669 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dall’Acqua, A., Lin, C.-C., Pozzi, P.: Evolution of open elastic curves in \(\mathbb{R}^{n}\) subject to fixed length and natural boundary conditions. Analysis (Berlin) 34(2), 209–222 (2014) MathSciNetGoogle Scholar
  8. 8.
    Dall’Acqua, A., Lin, C.-C., Pozzi, P.: A gradient flow for open elastic curves with fixed length and clamped ends. Preprint (2014) Google Scholar
  9. 9.
    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) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23(1), 69–96 (1986) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Golomb, M., Jerome, J.: Equilibria of the curvature functional and manifolds of nonlinear interpolating spline curves. SIAM J. Math. Anal. 13, 421–458 (1982) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hearst, J.E., Shi, Y.: The elastic rod provides a model for DNA and its functions. In: Mathematical Approaches to Biomolecular Structure and Dynamics, Minneapolis, MN, 1994. IMA Vol. Math. Appl., vol. 82, pp. 59–70. Springer, New York (1996) CrossRefGoogle Scholar
  13. 13.
    Koiso, N.: On the motion of a curve towards elastica. In: Actes de la Table Ronde de Géométrie Différentielle, Luminy, 1992. Sémin. Congr., vol. 1, pp. 403–436. Soc. Math. France, Paris (1996) Google Scholar
  14. 14.
    Langer, J., Singer, D.A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24(1), 75–88 (1985) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Langer, J., Singer, D.A.: Lagrangian aspects of the Kirchhoff elastic rod. SIAM Rev. 38(4), 605–618 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lin, C.-C.: \(L^{2}\)-flow of elastic curves with clamped boundary conditions. J. Differ. Equ. 252(12), 6414–6428 (2012) CrossRefzbMATHGoogle Scholar
  17. 17.
    Linnér, A.: Some properties of the curve straightening flow in the plane. Trans. Am. Math. Soc. 314(2), 605–618 (1989) CrossRefzbMATHGoogle Scholar
  18. 18.
    Mumford, D.: Elastica and Computer Vision, West Lafayette, IN, 1990. Algebraic Geometry and Its Applications, pp. 491–506. Springer, New York (1994) Google Scholar
  19. 19.
    Novaga, M., Okabe, S.: Curve shortening–straightening flow for non-closed planar curves with infinite length. J. Differ. Equ. 256(3), 1093–1132 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Oelz, D., Schmeiser, C.: Derivation of a model for symmetric lamellipodia with instantaneous cross-link turnover. Arch. Ration. Mech. Anal. 198(3), 963–980 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Oelz, D.: On the curve straightening flow of inextensible, open, planar curves. SeMA Journal 54, 5–24 (2011) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Polden, A.: Curves and surfaces of least total curvature and fourth-order flows. Ph.D. dissertation, Universität Tübingen, Tübingen, Germany (1996) Google Scholar
  23. 23.
    Starostin, E.L., van der Heijden, G.H.M.: The shape of a Möbius strip. Nat. Mater. 6, 563–567 (2007) CrossRefGoogle Scholar
  24. 24.
    Wen, Y.: \(L^{2}\) flow of curve straightening in the plan. Duke Math. J. 70(3), 683–698 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wen, Y.: Curve straightening flow deforms closed plane curves with nonzero rotation number to circles. J. Differ. Equ. 120(1), 89–107 (1995) CrossRefzbMATHGoogle Scholar
  26. 26.
    Willmore, T.: Curves. In: Handbook of Differential Geometry, vol. I, pp. 997–1023. North-Holland, Amsterdam (2000) (English summary) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Chun-Chi Lin
    • 1
    • 2
    Email author
  • Yang-Kai Lue
    • 1
  • Hartmut R. Schwetlick
    • 3
  1. 1.Department of MathematicsNational Taiwan Normal UniversityTaipeiTaiwan
  2. 2.2F of Astronomy-Mathematics BuildingNational Taiwan UniversityTaipeiTaiwan
  3. 3.Department of Mathematical SciencesUniversity of BathBathUK

Personalised recommendations