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 and 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.

[2]

J. J. ArroyoO. 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.

[3]

S. AvvakumovO. 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.

[4]

M. BergnerA. 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.

[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.

[6]

T. ChanA. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.

[7]

T. F. ChanS. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592.  doi: 10.1137/S0036139901390088.

[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.

[9]

A. Dall'AcquaK. 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.

[10]

A. Dall'AcquaK. 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.

[11]

A. Dall'AcquaS. FröhlichH.-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.

[12]

A. Dall'AcquaM. Novaga and A. Pluda, Minimal elastic networks, Indiana Univ. Math. J., 69 (2020), 1909-1932.  doi: 10.1512/iumj.2020.69.8036.

[13]

G. Dal MasoI. FonsecaG. 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.

[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.

[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.

[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.

[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.

[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.

[19]

N. Koiso, Elasticae in a Riemannian submanifold, Osaka J. Math., 29 (1992), 539-543. 

[20]

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

[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.

[22]

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

[23]

R. Levien, The Elastica: A mathematical History, Technical Report No. UCB/EECS-2008-10, University of California, Berkeley, 2008.

[24]

A. Linnér, Unified representations of nonlinear splines, J. Approx. Theory, 84 (1996), 315-350.  doi: 10.1006/jath.1996.0022.

[25]

A. Linnér, Explicit elastic curves, Ann. Global Anal. Geom., 16 (1998), 445-475.  doi: 10.1023/A:1006526817291.

[26]

E. A. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944.

[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.

[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.

[29]

T. Miura, Elastic curves and phase transitions, Math. Ann., 376 (2020), 1629-1674.  doi: 10.1007/s00208-019-01821-8.

[30]

T. Miura, Li-Yau type inequalities for curves in any codimension, preprint, arXiv: 2102.06597.

[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.

[32]

D. Mumford, Elastica and computer vision, algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, (1994), 491–506.

[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.

[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.

[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.

[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.

[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.

[38]

Y. L. Sachkov, Closed Euler elasticae, Tr. Mat. Inst. Steklova, Steklov Inst. Math., 278 (2012), 218-232.  doi: 10.1134/s0081543812060211.

[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.

[40]

R. Schätzle, The Willmore boundary problem, Calc. Var. Partial Differential Equations, 37 (2010), 275-302.  doi: 10.1007/s00526-009-0244-3.

[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.

[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.

[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.

[44]

H. Yanamoto, On the elastic closed plane curves, Kodai Math. J., 8 (1985), 224-235.  doi: 10.2996/kmj/1138037048.

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.

[2]

J. J. ArroyoO. 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.

[3]

S. AvvakumovO. 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.

[4]

M. BergnerA. 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.

[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.

[6]

T. ChanA. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.

[7]

T. F. ChanS. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592.  doi: 10.1137/S0036139901390088.

[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.

[9]

A. Dall'AcquaK. 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.

[10]

A. Dall'AcquaK. 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.

[11]

A. Dall'AcquaS. FröhlichH.-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.

[12]

A. Dall'AcquaM. Novaga and A. Pluda, Minimal elastic networks, Indiana Univ. Math. J., 69 (2020), 1909-1932.  doi: 10.1512/iumj.2020.69.8036.

[13]

G. Dal MasoI. FonsecaG. 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.

[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.

[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.

[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.

[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.

[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.

[19]

N. Koiso, Elasticae in a Riemannian submanifold, Osaka J. Math., 29 (1992), 539-543. 

[20]

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

[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.

[22]

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

[23]

R. Levien, The Elastica: A mathematical History, Technical Report No. UCB/EECS-2008-10, University of California, Berkeley, 2008.

[24]

A. Linnér, Unified representations of nonlinear splines, J. Approx. Theory, 84 (1996), 315-350.  doi: 10.1006/jath.1996.0022.

[25]

A. Linnér, Explicit elastic curves, Ann. Global Anal. Geom., 16 (1998), 445-475.  doi: 10.1023/A:1006526817291.

[26]

E. A. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944.

[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.

[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.

[29]

T. Miura, Elastic curves and phase transitions, Math. Ann., 376 (2020), 1629-1674.  doi: 10.1007/s00208-019-01821-8.

[30]

T. Miura, Li-Yau type inequalities for curves in any codimension, preprint, arXiv: 2102.06597.

[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.

[32]

D. Mumford, Elastica and computer vision, algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, (1994), 491–506.

[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.

[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.

[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.

[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.

[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.

[38]

Y. L. Sachkov, Closed Euler elasticae, Tr. Mat. Inst. Steklova, Steklov Inst. Math., 278 (2012), 218-232.  doi: 10.1134/s0081543812060211.

[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.

[40]

R. Schätzle, The Willmore boundary problem, Calc. Var. Partial Differential Equations, 37 (2010), 275-302.  doi: 10.1007/s00526-009-0244-3.

[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.

[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.

[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.

[44]

H. Yanamoto, On the elastic closed plane curves, Kodai Math. J., 8 (1985), 224-235.  doi: 10.2996/kmj/1138037048.

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
Figure 2.  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 $.
Figure 3.  For any $ n\in \mathbb{N}\cup\{0\} $ the curves $ \check{ \gamma}^+_n $ and $ \check{ \gamma}^-_n $ has a loop
Figure 4.  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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