# American Institute of Mathematical Sciences

January  2022, 42(1): 403-423. doi: 10.3934/dcds.2021122

## The critical points of the elastic energy among curves pinned at endpoints

 Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

Received  January 2021 Revised  May 2021 Published  January 2022 Early access  September 2021

Fund Project: The author was supported by Grant-in-Aid for JSPS Fellows 19J2074

In this paper we find curves minimizing the elastic energy among curves whose length is fixed and whose ends are pinned. Applying the shooting method, we can identify all critical points explicitly and determine which curve is the global minimizer. As a result we show that the critical points consist of wavelike elasticae and the minimizers do not have any loops or interior inflection points.

Citation: Kensuke Yoshizawa. The critical points of the elastic energy among curves pinned at endpoints. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 403-423. doi: 10.3934/dcds.2021122
##### References:
 [1] S. S. Antman, Nonlinear Problems of Elasticity, Applied Mathematical Sciences, 107, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-4147-6.  Google Scholar [2] J. J. Arroyo, O. J. Garay and A. Pámpano, Boundary value problems for Euler-Bernoulli planar elastica. A solution construction procedure, J. Elasticity, 139 (2020), 359-388.  doi: 10.1007/s10659-019-09755-7.  Google Scholar [3] S. Avvakumov, O. Karpenkov and A. Sossinsky, Euler elasticae in the plane and the Whitney-Graustein theorem, Russ. J. Math. Phys., 20 (2013), 257-267.  doi: 10.1134/S1061920813030011.  Google Scholar [4] M. Bergner, A. Dall'Acqua and S. Fröhlich, Symmetric Willmore surfaces of revolution satisfying natural boundary conditions, Calc. Var. Partial Differential Equations, 39 (2010), 361-378.  doi: 10.1007/s00526-010-0313-7.  Google Scholar [5] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2$^{nd}$ edition, Die Grundlehren der mathematischen Wissenschaften, 67, Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar [6] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.  Google Scholar [7] T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592.  doi: 10.1137/S0036139901390088.  Google Scholar [8] A. Dall'Acqua, Uniqueness for the homogeneous Dirichlet Willmore boundary value problem, Ann. Global Anal. Geom., 42 (2012), 411-420.  doi: 10.1007/s10455-012-9320-6.  Google Scholar [9] A. Dall'Acqua, K. Deckelnick and H.-C. Grunau, Classical solutions to the Dirichlet problem for Willmore surfaces of revolution, Adv. Calc. Var., 1 (2008), 379-397.  doi: 10.1515/ACV.2008.016.  Google Scholar [10] A. Dall'Acqua, K. Deckelnick and G. Wheeler, Unstable Willmore surfaces of revolution subject to natural boundary conditions, Calc. Var. Partial Differential Equations, 48 (2013), 293-313.  doi: 10.1007/s00526-012-0551-y.  Google Scholar [11] A. Dall'Acqua, S. Fröhlich, H.-C. Grunau and F. Schieweck, Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data, Adv. Calc. Var., 4 (2011), 1-81.  doi: 10.1515/acv.2010.022.  Google Scholar [12] A. Dall'Acqua, M. Novaga and A. Pluda, Minimal elastic networks, Indiana Univ. Math. J., 69 (2020), 1909-1932.  doi: 10.1512/iumj.2020.69.8036.  Google Scholar [13] G. Dal Maso, I. Fonseca, G. Leoni and M. Morini, A higher order model for image restoration: The one-dimensional case, SIAM J. Math. Anal., 40 (2009), 2351-2391.  doi: 10.1137/070697823.  Google Scholar [14] F. Dayrens, S. Masnou and M. Novaga, Existence, regularity and structure of confined elasticae, ESAIM Control Optim. Calc. Var., 24, (2018), 25–43. doi: 10.1051/cocv/2016073.  Google Scholar [15] K. Deckelnick and H.-C. Grunau, Boundary value problems for the one-dimensional Willmore equation, Calc. Var. Partial Differential Equations, 30 (2007), 293-314.  doi: 10.1007/s00526-007-0089-6.  Google Scholar [16] K. Deckelnick and H.-C. Grunau, Stability and symmetry in the Navier problem for the one-dimensional Willmore equation, SIAM J. Math. Anal., 40 (2008/09), 2055-2076.  doi: 10.1137/07069033X.  Google Scholar [17] S. Eichmann and A. Koeller, Symmetry for Willmore surfaces of revolution, J. Geom. Anal., 27 (2017), 618-642.  doi: 10.1007/s12220-016-9692-0.  Google Scholar [18] L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti, (Latin) Edidit C. Carathédory. Societas Scientiarum Naturalium Helveticae, Bern, 1952.  Google Scholar [19] N. Koiso, Elasticae in a Riemannian submanifold, Osaka J. Math., 29 (1992), 539-543.   Google Scholar [20] J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differ. Geom., 20 (1984), 1-22.   Google Scholar [21] J. Langer and D. A. Singer, Knotted elastic curves in $\mathbb{R}^3$, J. Lond. Math. Soc. (2), 30 (1984), 512-520.  doi: 10.1112/jlms/s2-30.3.512.  Google Scholar [22] J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology, 24 (1985), 75-88.   Google Scholar [23] R. Levien, The Elastica: A mathematical History, Technical Report No. UCB/EECS-2008-10, University of California, Berkeley, 2008. Google Scholar [24] A. Linnér, Unified representations of nonlinear splines, J. Approx. Theory, 84 (1996), 315-350.  doi: 10.1006/jath.1996.0022.  Google Scholar [25] A. Linnér, Explicit elastic curves, Ann. Global Anal. Geom., 16 (1998), 445-475.  doi: 10.1023/A:1006526817291.  Google Scholar [26] E. A. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944.  Google Scholar [27] R. Mandel, Boundary value problems for Willmore curves in $\mathbb{R}^2$, Calc. Var. Partial Differential Equations, 54 (2015), 3905-3925.  doi: 10.1007/s00526-015-0925-z.  Google Scholar [28] R. Mandel, Explicit formulas and symmetry breaking for Willmore surfaces of revolution, Ann. Global Anal. Geom., 54 (2018), 187-236.  doi: 10.1007/s10455-018-9598-0.  Google Scholar [29] T. Miura, Elastic curves and phase transitions, Math. Ann., 376 (2020), 1629-1674.  doi: 10.1007/s00208-019-01821-8.  Google Scholar [30] T. Miura, Li-Yau type inequalities for curves in any codimension, preprint, arXiv: 2102.06597. Google Scholar [31] M. Müller and F. Rupp, A Li-Yau inequality for the 1-dimensional Willmore energy, to appear in Adv. Calc. Var., arXiv: 2101.08509. Google Scholar [32] D. Mumford, Elastica and computer vision, algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, (1994), 491–506.  Google Scholar [33] M. Murai, W. Matsumoto and S. Yotsutani, One can hear the shape of some non-convex drums, More Progress in Analysis, Proc. 5th ISAAC Congress, (2009), 863–872. Google Scholar [34] M. Murai, W. Matsumoto and S. Yotsutani, Representation formula for the plane closed elastic curves, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 2013, 9th AIMS Conference. Suppl., (2013), 565–585. doi: 10.3934/proc.2013.2013.565.  Google Scholar [35] J. C. C. Nitsche, Boundary value problems for variational integrals involving surface curvatures, Q. Appl. Math., 51 (1993), 363-387.  doi: 10.1090/qam/1218374.  Google Scholar [36] Y. L. Sachkov, Conjugate points in the Euler elastic problem, J. Dyn. Control Syst., 14 (2008), 409-439.  doi: 10.1007/s10883-008-9044-x.  Google Scholar [37] Y. L. Sachkov, Maxwell strata in the Euler elastic problem, J. Dyn. Control Syst., 14 (2008), 169-234.  doi: 10.1007/s10883-008-9039-7.  Google Scholar [38] Y. L. Sachkov, Closed Euler elasticae, Tr. Mat. Inst. Steklova, Steklov Inst. Math., 278 (2012), 218-232.  doi: 10.1134/s0081543812060211.  Google Scholar [39] Y. L. Sachkov and E. F. Sachkova, Exponential mapping in Euler's elastic problem, J. Dyn. Control Syst., 20 (2014), 443-464.  doi: 10.1007/s10883-014-9211-1.  Google Scholar [40] R. Schätzle, The Willmore boundary problem, Calc. Var. Partial Differential Equations, 37 (2010), 275-302.  doi: 10.1007/s00526-009-0244-3.  Google Scholar [41] D. A. Singer, Lectures on elastic curves and rods. Curvature and variational modeling in physics and biophysics, AIP Conf. Proc., Amer. Inst. Phys., Melville, NY, 1002 (2008), 3-32.  doi: 10.1063/1.2918095.  Google Scholar [42] C. Truesdell, The influence of elasticity on analysis: The classic heritage, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 293-310.  doi: 10.1090/S0273-0979-1983-15187-X.  Google Scholar [43] K. Watanabe, Planar $p$-elastic curves and related generalized complete elliptic integrals, Kodai Math. J., 37 (2014), 453-474.  doi: 10.2996/kmj/1404393898.  Google Scholar [44] H. Yanamoto, On the elastic closed plane curves, Kodai Math. J., 8 (1985), 224-235.  doi: 10.2996/kmj/1138037048.  Google Scholar

show all references

##### References:
 [1] S. S. Antman, Nonlinear Problems of Elasticity, Applied Mathematical Sciences, 107, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-4147-6.  Google Scholar [2] J. J. Arroyo, O. J. Garay and A. Pámpano, Boundary value problems for Euler-Bernoulli planar elastica. A solution construction procedure, J. Elasticity, 139 (2020), 359-388.  doi: 10.1007/s10659-019-09755-7.  Google Scholar [3] S. Avvakumov, O. Karpenkov and A. Sossinsky, Euler elasticae in the plane and the Whitney-Graustein theorem, Russ. J. Math. Phys., 20 (2013), 257-267.  doi: 10.1134/S1061920813030011.  Google Scholar [4] M. Bergner, A. Dall'Acqua and S. Fröhlich, Symmetric Willmore surfaces of revolution satisfying natural boundary conditions, Calc. Var. Partial Differential Equations, 39 (2010), 361-378.  doi: 10.1007/s00526-010-0313-7.  Google Scholar [5] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2$^{nd}$ edition, Die Grundlehren der mathematischen Wissenschaften, 67, Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar [6] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.  Google Scholar [7] T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592.  doi: 10.1137/S0036139901390088.  Google Scholar [8] A. Dall'Acqua, Uniqueness for the homogeneous Dirichlet Willmore boundary value problem, Ann. Global Anal. Geom., 42 (2012), 411-420.  doi: 10.1007/s10455-012-9320-6.  Google Scholar [9] A. Dall'Acqua, K. Deckelnick and H.-C. Grunau, Classical solutions to the Dirichlet problem for Willmore surfaces of revolution, Adv. Calc. Var., 1 (2008), 379-397.  doi: 10.1515/ACV.2008.016.  Google Scholar [10] A. Dall'Acqua, K. Deckelnick and G. Wheeler, Unstable Willmore surfaces of revolution subject to natural boundary conditions, Calc. Var. Partial Differential Equations, 48 (2013), 293-313.  doi: 10.1007/s00526-012-0551-y.  Google Scholar [11] A. Dall'Acqua, S. Fröhlich, H.-C. Grunau and F. Schieweck, Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data, Adv. Calc. Var., 4 (2011), 1-81.  doi: 10.1515/acv.2010.022.  Google Scholar [12] A. Dall'Acqua, M. Novaga and A. Pluda, Minimal elastic networks, Indiana Univ. Math. J., 69 (2020), 1909-1932.  doi: 10.1512/iumj.2020.69.8036.  Google Scholar [13] G. Dal Maso, I. Fonseca, G. Leoni and M. Morini, A higher order model for image restoration: The one-dimensional case, SIAM J. Math. Anal., 40 (2009), 2351-2391.  doi: 10.1137/070697823.  Google Scholar [14] F. Dayrens, S. Masnou and M. Novaga, Existence, regularity and structure of confined elasticae, ESAIM Control Optim. Calc. Var., 24, (2018), 25–43. doi: 10.1051/cocv/2016073.  Google Scholar [15] K. Deckelnick and H.-C. Grunau, Boundary value problems for the one-dimensional Willmore equation, Calc. Var. Partial Differential Equations, 30 (2007), 293-314.  doi: 10.1007/s00526-007-0089-6.  Google Scholar [16] K. Deckelnick and H.-C. Grunau, Stability and symmetry in the Navier problem for the one-dimensional Willmore equation, SIAM J. Math. Anal., 40 (2008/09), 2055-2076.  doi: 10.1137/07069033X.  Google Scholar [17] S. Eichmann and A. Koeller, Symmetry for Willmore surfaces of revolution, J. Geom. Anal., 27 (2017), 618-642.  doi: 10.1007/s12220-016-9692-0.  Google Scholar [18] L. Euler, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti, (Latin) Edidit C. Carathédory. Societas Scientiarum Naturalium Helveticae, Bern, 1952.  Google Scholar [19] N. Koiso, Elasticae in a Riemannian submanifold, Osaka J. Math., 29 (1992), 539-543.   Google Scholar [20] J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differ. Geom., 20 (1984), 1-22.   Google Scholar [21] J. Langer and D. A. Singer, Knotted elastic curves in $\mathbb{R}^3$, J. Lond. Math. Soc. (2), 30 (1984), 512-520.  doi: 10.1112/jlms/s2-30.3.512.  Google Scholar [22] J. Langer and D. A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology, 24 (1985), 75-88.   Google Scholar [23] R. Levien, The Elastica: A mathematical History, Technical Report No. UCB/EECS-2008-10, University of California, Berkeley, 2008. Google Scholar [24] A. Linnér, Unified representations of nonlinear splines, J. Approx. Theory, 84 (1996), 315-350.  doi: 10.1006/jath.1996.0022.  Google Scholar [25] A. Linnér, Explicit elastic curves, Ann. Global Anal. Geom., 16 (1998), 445-475.  doi: 10.1023/A:1006526817291.  Google Scholar [26] E. A. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944.  Google Scholar [27] R. Mandel, Boundary value problems for Willmore curves in $\mathbb{R}^2$, Calc. Var. Partial Differential Equations, 54 (2015), 3905-3925.  doi: 10.1007/s00526-015-0925-z.  Google Scholar [28] R. Mandel, Explicit formulas and symmetry breaking for Willmore surfaces of revolution, Ann. Global Anal. Geom., 54 (2018), 187-236.  doi: 10.1007/s10455-018-9598-0.  Google Scholar [29] T. Miura, Elastic curves and phase transitions, Math. Ann., 376 (2020), 1629-1674.  doi: 10.1007/s00208-019-01821-8.  Google Scholar [30] T. Miura, Li-Yau type inequalities for curves in any codimension, preprint, arXiv: 2102.06597. Google Scholar [31] M. Müller and F. Rupp, A Li-Yau inequality for the 1-dimensional Willmore energy, to appear in Adv. Calc. Var., arXiv: 2101.08509. Google Scholar [32] D. Mumford, Elastica and computer vision, algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, (1994), 491–506.  Google Scholar [33] M. Murai, W. Matsumoto and S. Yotsutani, One can hear the shape of some non-convex drums, More Progress in Analysis, Proc. 5th ISAAC Congress, (2009), 863–872. Google Scholar [34] M. Murai, W. Matsumoto and S. Yotsutani, Representation formula for the plane closed elastic curves, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 2013, 9th AIMS Conference. Suppl., (2013), 565–585. doi: 10.3934/proc.2013.2013.565.  Google Scholar [35] J. C. C. Nitsche, Boundary value problems for variational integrals involving surface curvatures, Q. Appl. Math., 51 (1993), 363-387.  doi: 10.1090/qam/1218374.  Google Scholar [36] Y. L. Sachkov, Conjugate points in the Euler elastic problem, J. Dyn. Control Syst., 14 (2008), 409-439.  doi: 10.1007/s10883-008-9044-x.  Google Scholar [37] Y. L. Sachkov, Maxwell strata in the Euler elastic problem, J. Dyn. Control Syst., 14 (2008), 169-234.  doi: 10.1007/s10883-008-9039-7.  Google Scholar [38] Y. L. Sachkov, Closed Euler elasticae, Tr. Mat. Inst. Steklova, Steklov Inst. Math., 278 (2012), 218-232.  doi: 10.1134/s0081543812060211.  Google Scholar [39] Y. L. Sachkov and E. F. Sachkova, Exponential mapping in Euler's elastic problem, J. Dyn. Control Syst., 20 (2014), 443-464.  doi: 10.1007/s10883-014-9211-1.  Google Scholar [40] R. Schätzle, The Willmore boundary problem, Calc. Var. Partial Differential Equations, 37 (2010), 275-302.  doi: 10.1007/s00526-009-0244-3.  Google Scholar [41] D. A. Singer, Lectures on elastic curves and rods. Curvature and variational modeling in physics and biophysics, AIP Conf. Proc., Amer. Inst. Phys., Melville, NY, 1002 (2008), 3-32.  doi: 10.1063/1.2918095.  Google Scholar [42] C. Truesdell, The influence of elasticity on analysis: The classic heritage, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 293-310.  doi: 10.1090/S0273-0979-1983-15187-X.  Google Scholar [43] K. Watanabe, Planar $p$-elastic curves and related generalized complete elliptic integrals, Kodai Math. J., 37 (2014), 453-474.  doi: 10.2996/kmj/1404393898.  Google Scholar [44] H. Yanamoto, On the elastic closed plane curves, Kodai Math. J., 8 (1985), 224-235.  doi: 10.2996/kmj/1138037048.  Google Scholar
], these curves are called wavelike elasticae">Figure 1.  Critical points of $\mathcal{W}$ in $\mathcal{A}_{l, L}$ are given by Theorem 1.1. According to [41], these curves are called wavelike elasticae
The relation between critical curves and the ratio $l/L$. The number of inflection points (where the sign of the curvature changes) in $(0, l)$ is given by $n\in \mathbb{N}\cup\{0\}$. The curve $\hat{ \gamma}^{\pm}_n$ ($n\in \mathbb{N}$) can be constructed from $\hat{ \gamma}^{\pm}_0$.
For any $n\in \mathbb{N}\cup\{0\}$ the curves $\check{ \gamma}^+_n$ and $\check{ \gamma}^-_n$ has a loop
The red curve is the graph of $2\frac{ \mathrm{E}(p)}{ \mathrm{K}(p)}-1$; $0\leq p \leq p_0$ and $-2\frac{ \mathrm{E}(p)}{ \mathrm{K}(p)}+1$; $p_0< p <1$
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