# American Institute of Mathematical Sciences

2013, 2013(special): 565-585. doi: 10.3934/proc.2013.2013.565

## Representation formula for the plane closed elastic curves

 1 Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194, Japan, Japan 2 Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194

Received  September 2012 Revised  April 2013 Published  November 2013

Let $\Gamma$ be a plane closed elastic curve with length $L>0.$ Let $M$ be the signed area of the domain bounded by $\Gamma$. We are interested in the following variational problem. Find a curve $\Gamma$ (the curvature $\kappa(s)$) which minimizes the elastic energy subject to $L^{2}-4 \pi M >0$ and $L^{2} \neq 4 \pi \omega M$, where $\omega$ is the winding number. This variational problem was first studied in the case $\omega=1$ and the Euler-Lagrange equation was derived. The existence of the minimizer was showed and the profile near the disk was investigated by using the Euler-Lagrange equation. As the first step to investigate the structure of solutions of this equation, we show all the solutions to an auxiliary second order boundary value problem. Moreover, we obtain the representation of the integral of $\kappa(s)$.
Citation: Minoru Murai, Waichiro Matsumoto, Shoji Yotsutani. Representation formula for the plane closed elastic curves. Conference Publications, 2013, 2013 (special) : 565-585. doi: 10.3934/proc.2013.2013.565
##### References:
 [1] J. V. Armitage and W. F. Eberlein, "Elliptic Fucntions ", Cambridge University Press, Cambridge, 2006. [2] H.Ikeda, K.Kondo, H.Okamoto and S.Yotsutani, On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows, Commun. Pure Appl. Anal. 2 (2003), no.3, 381-390. [3] S.Kosugi, Y.Morita and S.Yotsutani, A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions, Commun. Pure Appl. Anal. 4 (2005), no.3, 665-682. [4] Y.Lou, W-M.Ni and S.Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Partial differential equations and applications, Discrete Contin. Dyn. Syst. 10 (2004), no.1-2, 435-458. [5] V.I. Smirnov, "A Course of Higher Mathematics", vol.3, part2, Pergamon Press, Oxford, 1964. [6] K.Watanabe, Plane domains which are spectrally determined, Ann. Global Anal. Geom. 18(2000), no.5, 447-475. [7] K.Watanabe, Plane domains which are spectrally determined. II, J. Inequal. Appl. 7(2002), no.1, 25-47.

show all references

##### References:
 [1] J. V. Armitage and W. F. Eberlein, "Elliptic Fucntions ", Cambridge University Press, Cambridge, 2006. [2] H.Ikeda, K.Kondo, H.Okamoto and S.Yotsutani, On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows, Commun. Pure Appl. Anal. 2 (2003), no.3, 381-390. [3] S.Kosugi, Y.Morita and S.Yotsutani, A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions, Commun. Pure Appl. Anal. 4 (2005), no.3, 665-682. [4] Y.Lou, W-M.Ni and S.Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Partial differential equations and applications, Discrete Contin. Dyn. Syst. 10 (2004), no.1-2, 435-458. [5] V.I. Smirnov, "A Course of Higher Mathematics", vol.3, part2, Pergamon Press, Oxford, 1964. [6] K.Watanabe, Plane domains which are spectrally determined, Ann. Global Anal. Geom. 18(2000), no.5, 447-475. [7] K.Watanabe, Plane domains which are spectrally determined. II, J. Inequal. Appl. 7(2002), no.1, 25-47.
 [1] Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449 [2] Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511 [3] Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure and Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51 [4] Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 609-629. doi: 10.3934/dcds.2007.19.609 [5] Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577 [6] Francisco Crespo, Sebastián Ferrer. On the extended Euler system and the Jacobi and Weierstrass elliptic functions. Journal of Geometric Mechanics, 2015, 7 (2) : 151-168. doi: 10.3934/jgm.2015.7.151 [7] Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 [8] Yuan Xu, Xin Jin, Saiwei Wang, Yang Tang. Optimal synchronization control of multiple euler-lagrange systems via event-triggered reinforcement learning. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1495-1518. doi: 10.3934/dcdss.2020377 [9] Patrizia Pucci, Marco Rigoli. Entire solutions of singular elliptic inequalities on complete manifolds. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 115-137. doi: 10.3934/dcds.2008.20.115 [10] Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053 [11] Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227 [12] Claudianor Oliveira Alves, Paulo Cesar Carrião, Olímpio Hiroshi Miyagaki. Signed solution for a class of quasilinear elliptic problem with critical growth. Communications on Pure and Applied Analysis, 2002, 1 (4) : 531-545. doi: 10.3934/cpaa.2002.1.531 [13] Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048 [14] Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335 [15] Evelyn Herberg, Michael Hinze. Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022013 [16] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [17] Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks and Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361 [18] Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883 [19] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [20] Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583

Impact Factor: