Abstract
This paper is devoted to a classical variational problem for planar elastic curves of clamped endpoints, so-called Euler’s elastica problem. We investigate a straightening limit that means enlarging the distance of the endpoints, and obtain several new results concerning properties of least energy solutions. In particular we reach a first uniqueness result that assumes no symmetry. As a key ingredient we develop a foundational singular perturbation theory for the modified total squared curvature energy. It turns out that our energy has almost the same variational structure as a phase transition energy of Modica–Mortola type at the level of a first order singular limit.
Similar content being viewed by others
References
Antman, S.S.: The influence of elasticity on analysis: modern developments. Bull. Amer. Math. Soc. (N.S.) 9(3), 267–291 (1983)
Antman, S.S.: Nonlinear problems of elasticity. Springer-Verlag, New York (1995)
Ardentov, A.A., Sachkov, YuL: Solution of Euler’s elastica problem. Autom. Remote Control 70(4), 633–643 (2009)
Audoly, B., Pomeau, Y.: Elasticity and Geometry: From Hair Curls to the Non-Linear Response of Shells. Oxford University Press, Oxford (2010)
Avvakumov, S., Karpenkov, O., Sossinsky, A.: Euler elasticae in the plane and the Whitney–Graustein theorem. Russ. J. Math. Phys. 20(3), 257–267 (2013)
Bergner, M., Dall’Acqua, A., Fröhlich, S.: Symmetric Willmore surfaces of revolution satisfying natural boundary conditions. Calc. Var. Part. Differ. Equ. 39(3–4), 361–378 (2010)
Bernard, Y.: Analysis of constrained Willmore surfaces. Commun. Part. Differ. Equ. 41(10), 1513–1552 (2016)
Bernard, Y., Rivière, T.: Energy quantization for Willmore surfaces and applications. Ann. Math. (2) 180(1), 87–136 (2014)
Born, M.: Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen, PhD thesis, University of Göttingen, (1906)
Braides, A.: Local minimization, variational evolution and \(\Gamma \)-convergence. Springer, Cham (2014)
Brunnett, G.: A new characterization of plane elastica. In: Mathematical methods in computer aided geometric design, II. Academic Press, Boston, pp. 43–56 (1992)
Bucur, D., Henrot, A.: A new isoperimetric inequality for the elasticae. J. Eur. Math. Soc. 19(11), 3355–3376 (2017)
Caffarelli, L.A., Córdoba, A.: Uniform convergence of a singular perturbation problem. Commun. Pure Appl. Math. 48(1), 1–12 (1995)
Carr, J., Gurtin, M.E., Slemrod, M.: Structured phase transitions on a finite interval. Arch. Ration. Mech. Anal. 86(4), 317–351 (1984)
Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys 28, 258–267 (1958)
Dall’Acqua, A.: Uniqueness for the homogeneous Dirichlet Willmore boundary value problem. Ann. Global Anal. Geom. 42(3), 411–420 (2012)
Dall’Acqua, A., Deckelnick, K.: An obstacle problem for elastic graphs, Preprint No. 2 Universität Magdeburg (2017), 20 pp
Dall’Acqua, A., Deckelnick, K., Grunau, H.-C.: Classical solutions to the Dirichlet problem for Willmore surfaces of revolution. Adv. Calc. Var. 1(4), 379–397 (2008)
Dall’Acqua, A., Deckelnick, K., Wheeler, G.: Unstable Willmore surfaces of revolution subject to natural boundary conditions. Calc. Var. Part. Differ. Equ. 48(3–4), 293–313 (2013)
Dall’Acqua, A., Fröhlich, S., Grunau, H.-C., Schieweck, F.: Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data. Adv. Calc. Var. 4(1), 1–81 (2011)
Dal Maso, G.: An introduction to \(\Gamma \)-convergence. Birkhäuser, Boston (1993)
Dayrens, F., Masnou, S., Novaga, M.: Existence, regularity and structure of confined elasticae. ESAIM Control Optim. Calc. Var. 24(1), 25–43 (2018)
Dall’Acqua, A., Pluda, A.: Some minimization problems for planar networks of elastic curves. Geom. Flows 2, 105–124 (2017)
Dall’Acqua, A., Novaga, M., Pluda, A.: Minimal elastic networks, preprint. arXiv:1712.09589
Deckelnick, K., Grunau, H.-C.: Boundary value problems for the one-dimensional Willmore equation. Calc. Var. Part. Differ. Equ. 30(3), 293–314 (2007)
Deckelnick, K., Grunau, H.-C.: Stability and symmetry in the Navier problem for the one-dimensional Willmore equation. SIAM J. Math. Anal. 40(5), 2055–2076 (2009)
Djondjorov, P., Hadzhilazova, M. Ts., Mladenov, I. M., Vassilev, V. M.: Explicit parameterization of Euler’s elastica, In: Geometry, integrability and quantization, Softex, Sofia, pp. 175–186 (2008)
Dondl, P.W., Lemenant, A., Wojtowytsch, S.: Phase field models for thin elastic structures with topological constraint. Arch. Ration. Mech. Anal. 223(2), 693–736 (2017)
Dondl, P.W., Mugnai, L., Röger, M.: Confined elastic curves. SIAM J. Appl. Math. 71(6), 2205–2226 (2011)
Eichmann, S., Koeller, A.: Symmetry for Willmore surfaces of revolution. J. Geom. Anal. 27(1), 618–642 (2017)
Euler, L.: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimitrici latissimo sensu accepti, Marcum-Michaelem Bousquet & socios, Lausanne, Geneva, (1744)
Ferone, V., Kawohl, B., Nitsch, C.: The elastica problem under area constraint. Math. Ann. 365(3–4), 987–1015 (2016)
Fraser, C.G.: Mathematical technique and physical conception in Euler’s investigation of the elastica. Centaurus 34(3), 211–246 (1991)
Gabutti, B., Lepora, P., Merlo, G.: A bifurcation problem involving elastica. Meccanica 15, 154–165 (1980)
Gage, M.E.: An isoperimetric inequality with applications to curve shortening. Duke Math. J. 50(4), 1225–1229 (1983)
Gerlach, H., Reiter, P., von der Mosel, H.: The Elastic Trefoil is the Doubly Covered Circle. Arch. Ration. Mech. Anal. 225(1), 89–139 (2017)
Gonzalez, O., Maddocks, J.H., Schuricht, F., von der Mosel, H.: Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Part. Diff. Equ. 14(1), 29–68 (2002)
Grunau, H.-C.: The asymptotic shape of a boundary layer of symmetric Willmore surfaces of revolution, In: Inequalities and applications 2010, International Series of Numerical Mathematics 161, Springer, Basel, pp. 19–29 (2012)
Hutchinson, J.E., Tonegawa, Y.: Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. Part. Differ. Equ. 10(1), 49–84 (2000)
Jin, M., Bao, Z.B.: An improved proof of instability of some Euler elasticas. J. Elast. 121(2), 303–308 (2015)
Keller, L.G.A., Mondino, A., Rivière, T.: Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint. Arch. Ration. Mech. Anal. 212(2), 645–682 (2014)
Kemmochi, T.: Numerical analysis of elastica with obstacle and adhesion effects. Appl. Anal. https://doi.org/10.1080/00036811.2017.1416100 (in press)
Koiso, N.: Elasticae in a Riemannian submanifold. Osaka J. Math. 29(3), 539–543 (1992)
Kuwert, E., Schätzle, R.: The Willmore functional, In: Topics in modern regularity theory, CRM Series, 13, Ed. Norm., Pisa, pp. 1–115 (2012)
Langer, J., Singer, D.A.: The total squared curvature of closed curves. J. Differ. Geom. 20(1), 1–22 (1984)
Langer, J., Singer, D.A.: Knotted elastic curves in \({\mathbb{R}}^{3}\). J. Lond. Math. Soc. (2) 30(3), 512–520 (1984)
Langer, J., Singer, D.A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24(1), 75–88 (1985)
Lawden, D.F.: Elliptic functions and applications. Springer-Verlag, New York (1989)
Levien, R.: The elastica: a mathematical history, Technical Report No. UCB/EECS-2008-10, University of California, Berkeley, (2008)
Linnér, A.: Existence of free nonclosed Euler-Bernoulli elastica. Nonlinear Anal. 21(8), 575–593 (1993)
Linnér, A.: Unified representations of nonlinear splines. J. Approx. Theory 84(3), 315–350 (1996)
Linnér, A.: Curve-straightening and the Palais-Smale condition. Trans. Am. Math. Soc. 350(9), 3743–3765 (1998)
Linnér, A.: Explicit elastic curves. Ann. Global Anal. Geom. 16(5), 445–475 (1998)
Linnér, A., Jerome, J.W.: A unique graph of minimal elastic energy. Trans. Am. Math. Soc. 359(5), 2021–2041 (2007)
Love, A.E.H.: A treatise on the mathematical theory of elasticity, 4th edn. Dover Publications, New York (1944)
Maddocks, J.H.: Stability of nonlinearly elastic rods. Arch. Ration. Mech. Anal. 85(4), 311–354 (1984)
Mandel, R.: Boundary value problems for Willmore curves in \({\mathbb{R}}^2\). Calc. Var. Part. Differ. Equ. 54(4), 3905–3925 (2015)
Mandel, R.: Explicit formulas and symmetry breaking for Willmore surfaces of revolution. Ann. Global Anal. Geom. 54(2), 187–236 (2018)
Manning, R.S.: A catalogue of stable equilibria of planar extensible or inextensible elastic rods for all possible Dirichlet boundary conditions. J. Elast. 115(2), 105–130 (2014)
Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. (2) 179(2), 683–782 (2014)
Matsutani, S.: Euler’s elastica and beyond. J. Geom. Symmetry Phys. 17, 45–86 (2010)
Miura, T.: Singular perturbation by bending for an adhesive obstacle problem. Calc. Var. Part. Differ. Equ. https://doi.org/10.1007/s00526-015-0941-z (in press)
Miura, T.: Overhanging of membranes adhering to periodic graph substrates. Phys. D 355, 34–44 (2017)
Mladenov, I.M., Hadzhilazova, M.: The many faces of elastica. Springer, Cham (2017)
Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)
Modica, L., Mortola, S.: Un esempio di \(\varGamma ^-\)-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)
Ni, W.-M., Pan, X.-B., Takagi, I.: Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents. Duke Math. J. 67(1), 1–20 (1992)
Ni, W.-M., Takagi, I.: On the shape of least-energy solutions to a semilinear Neumann problem. Commun. Pure Appl. Math. 44(7), 819–851 (1991)
Ni, W.-M., Takagi, I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70(2), 247–281 (1993)
Nitsche, J.C.C.: Boundary value problems for variational integrals involving surface curvatures. Q. Appl. Math. 51, 363–387 (1993)
Novaga, M., Okabe, S.: Convergence to equilibrium of gradient flows defined on planar curves. J. Reine Angew. Math. 733, 87–119 (2017)
Röger, M., Schätzle, R.: On a modified conjecture of De Giorgi. Math. Z. 254(4), 675–714 (2006)
Sachkov, YuL: Maxwell strata in the Euler elastic problem. J. Dyn. Control Syst. 14(2), 169–234 (2008)
Sachkov, YuL: Conjugate points in the Euler elastic problem. J. Dyn. Control Syst. 14(3), 409–439 (2008)
Sachkov, YuL: Closed Euler elasticae. Proc. Steklov Inst. Math. 278(1), 218–232 (2012)
Sachkov, YuL, Sachkova, E.F.: Exponential mapping in Euler’s elastic problem. J. Dyn. Control Syst. 20(4), 443–464 (2014)
Sachkov, YuL, Levyakov, S.V.: Stability of inflectional elasticae centered at vertices or inflection points. Proc. Steklov Inst. Math. 271(1), 177–192 (2010)
Schätzle, R.: The Willmore boundary problem. Calc. Var. Part. Differ. Equ. 37(3–4), 275–302 (2010)
Singer, D. A.: Lectures on elastic curves and rods, In: Curvature and variational modeling in physics and biophysics, Vol. 1002, Amer. Inst. Phys., Melville, NY, (pp. 3–32) (2008)
Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101(3), 209–260 (1988)
Truesdell, C.: The influence of elasticity on analysis: the classic heritage. Bull. Amer. Math. Soc. (N.S.) 9(3), 293–310 (1983)
van der Waals, J.D.: The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J Stat. Phys. 20, 200–244 (1979). (Translated version of: J. D. van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung Stetiger Dichteänderung, Zeitschrift für Physikalische Chemie 13 (1894), 657–725.)
Acknowledgements
The author would like to thank Professor Yoshikazu Giga, Professor Yasuhito Miyamoto, Professor Michiaki Onodera, and Dr. Olivier Pierre-Louis for their helpful comments and discussion. In particular, Professor Miyamoto indicated to the author that our study is related to the works by Ni and Takagi. The author learned of Audoly and Pomeau’s book from Dr. Pierre-Louis, and found a description related to our study. The author would also like thank anonymous referees for their careful reading and useful comments. This work is mainly carried out at the University of Tokyo and partly at the Max Planck Institute for Mathematics in the Sciences. This work is partially supported by JSPS KAKENHI Grant Numbers JP15J05166, JP18J30004, and also the Program for Leading Graduate Schools, MEXT, Japan.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Giga.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
A Existence of minimizers
Fix \(l>0\) and \(\theta _0,\theta _1\in [-\pi ,\pi ]\). We say that \(\gamma \in H^2(I;{\mathbb {R}}^2)\subset C^1({\bar{I}};{\mathbb {R}}^2)\) is \(H^2\)-admissible if \(\gamma \) is of constant speed and satisfying the boundary condition (2.1). We denote the set of \(H^2\)-admissible curves by \({\mathcal {X}}\). Note that the \(H^2\)-weak topology is stronger than \(C^1\)-topology; hence, in particular, the set \({\mathcal {X}}\) is \(H^2\)-weakly closed in \(H^2(I;{\mathbb {R}}^2)\).
In this \(H^2\)-framework we have an existence theorem of standard type: Let \({\mathcal {X}}'\subset {\mathcal {X}}\) be an \(H^2\)-weakly closed subset. Then the functional \({\mathcal {E}}_\varepsilon =\varepsilon ^2{\mathcal {B}}+{\mathcal {L}}\) defined on \({\mathcal {X}}'\) attains its minimum in \({\mathcal {X}}'\).
The proof is straightforward. Since any \(\gamma \in {\mathcal {X}}'\) is of constant speed, we have the following representations:
By the above relations and the boundary condition, we find that a minimizing sequence is \(H^2\)-bounded. Since \({\mathcal {E}}_\varepsilon \) is lower semicontinuous with respect to the \(H^2\)-weak topology, a standard direct method implies the existence of a minimizer, completing the proof.
Moreover, if \({\mathcal {X}}'\) admits any local perturbation, then we find that any minimizer \(\gamma \in {\mathcal {X}}'\) is of class \(C^\infty \) by a bootstrap argument. In particular, the problem (2.2) admits a smooth minimizer. Using the Lagrange multiplier method to modify the length constraint, we find that the problem (2.4) also admits a smooth minimizer. One may also refer to [52, Theorem 2.2] for a different argument.
In addition, it is also proved that there are infinitely many local minimizers with different winding numbers in a sense. Here \(\gamma \in {\mathcal {X}}\) is a local minimizer of the energy \({\mathcal {E}}_\varepsilon \) if there is \(\delta >0\) such that \({\mathcal {E}}_\varepsilon [\gamma ]\le {\mathcal {E}}_\varepsilon [\gamma ']\) for any \(\gamma '\in {\mathcal {X}}\) with \(\Vert \gamma -\gamma '\Vert _{H^2}\le \delta \). To state the above fact, we use a kind of winding number; for \(\gamma \in {\mathcal {X}}\) we define \({\mathcal {N}}[\gamma ]\in {\mathbb {Z}}\) as
where \(\kappa \) is the counterclockwise signed curvature (\(\kappa =\partial _s\vartheta _{{\tilde{\gamma }}}\)). We notice that the functional \({\mathcal {N}}\) is \({\mathbb {Z}}\)-valued and continuous with respect to the \(H^2\)-weak and -strong topologies. Thus for any \(m\in {\mathbb {Z}}\) the set \({\mathcal {X}}_m=\{\gamma \in {\mathcal {X}} \mid {\mathcal {N}}[\gamma ]=m \}\) is open and closed in \({\mathcal {X}}\) both weakly and strongly. Since \({\mathcal {X}}_m\) is weakly closed, the energies \({\mathcal {E}}_\varepsilon \) defined on \({\mathcal {X}}_m\) and \({\mathcal {B}}\) defined on \({\mathcal {X}}_m\cap {\mathcal {X}}^L\) attain their minimizers, where \(L>l\) and \({\mathcal {X}}^L=\{\gamma \in {\mathcal {X}}\mid {\mathcal {L}}[\gamma ]=L \}\). Moreover, the set \({\mathcal {X}}_m\) is strongly open, and hence such minimizers are local minimizers on \({\mathcal {X}}\) or \({\mathcal {X}}^L\), respectively.
B Uniqueness of minimizers for well-prepared boundary conditions
In this section we briefly show that the uniqueness of global minimizers is easily proved or disproved for some special parameters of the boundary condition, which possess well-prepared symmetry.
We observe that under the generic boundary angle condition (2.3), if circular arcs are admissible, then global minimizers are unique. For the extensible problem with any fixed \(\varepsilon >0\), the inequality \(X^2+Y^2\ge 2XY\) leads to
where the equality is attained if and only if \(|\kappa |=1/\varepsilon \). In addition we easily observe that the right-hand side attains its minimum among admissible curves \(\gamma \in {\mathcal {A}}_{\theta _0,\theta _1,l}\) if and only if a convex curve of rotation angle \(|\theta _0|+|\theta _1|\) is admissible. Therefore, if a circular arc of radius \(1/\varepsilon \) is admissible, i.e., \(\theta _0=-\theta _1\) and \(l=2\cos \theta _0/\varepsilon \), then the circular arc of radius \(1/\varepsilon \) is a unique global minimizer. For the inextensible problem with fixed \(L>0\), the Cauchy–Schwarz inequality leads to
where the equality is attained if and only if \(|\kappa |\) is constant. Then a similar consideration implies that if a circular arc is admissible, i.e., \(\theta _0=-\theta _1\) and \(l=L\sin \theta _0/\theta _0\), then the circular arc of radius \(L/2\theta _0\) is a unique global minimizer.
We shall finally think of some particular critical cases. Both for the extensible problem with fixed \(\varepsilon >0\) and the inextensible problem with fixed \(L>0\), the most trivial case is that \((l,\theta _0,\theta _1)=(l,0,0)\) with \(l>0\); in this case the segment is a unique minimizer. If \((l,\theta _0,\theta _1)=(0,0,0)\) or \((l,|\theta _0|,|\theta _1|)=(0,\pi ,\pi )\), a consideration as in the above paragraph implies there are only two minimizers of suitable circle, thus being not unique in the strict sense but unique up to symmetry. In other critical cases, we are often able to ensure some nonuniqueness by a simple argument on symmetry, but up-to-symmetry uniqueness is a delicate issue (cf. [76]).