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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
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  • 687 Downloads

Abstract

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).

Keywords

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

Mathematics Subject Classification (2010)

35B65 35K51 53A04 53C44 

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References

  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

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