The Mechanics of Ribbons and Möbius Bands pp 263-291 | Cite as
The Second-Order \(L^{2}\)-Flow of Inextensible Elastic Curves with Hinged Ends in the Plane
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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 functionalMathematics Subject Classification (2010)
35B65 35K51 53A04 53C44Preview
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References
- 1.Angenent, S.: On the formation of singularities in the curve shortening flow. J. Differ. Geom. 33(3), 601–633 (1991) MathSciNetzbMATHGoogle Scholar
- 2.Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Applied Mathematical Sciences, vol. 107. Springer, New York (2005) zbMATHGoogle Scholar
- 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.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.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.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.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.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.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.Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23(1), 69–96 (1986) MathSciNetzbMATHGoogle Scholar
- 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.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.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.Langer, J., Singer, D.A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24(1), 75–88 (1985) MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Langer, J., Singer, D.A.: Lagrangian aspects of the Kirchhoff elastic rod. SIAM Rev. 38(4), 605–618 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
- 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.Linnér, A.: Some properties of the curve straightening flow in the plane. Trans. Am. Math. Soc. 314(2), 605–618 (1989) CrossRefzbMATHGoogle Scholar
- 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.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.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.Oelz, D.: On the curve straightening flow of inextensible, open, planar curves. SeMA Journal 54, 5–24 (2011) MathSciNetCrossRefGoogle Scholar
- 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.Starostin, E.L., van der Heijden, G.H.M.: The shape of a Möbius strip. Nat. Mater. 6, 563–567 (2007) CrossRefGoogle Scholar
- 24.Wen, Y.: \(L^{2}\) flow of curve straightening in the plan. Duke Math. J. 70(3), 683–698 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
- 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.Willmore, T.: Curves. In: Handbook of Differential Geometry, vol. I, pp. 997–1023. North-Holland, Amsterdam (2000) (English summary) Google Scholar
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