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

Representation formula for the plane closed elastic curves

Minoru Murai 1, , Waichiro Matsumoto 1, and  Shoji Yotsutani 2,

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 $\kappa(s)$.
Keywords: elliptic functions, complete elliptic integrals, exact solution., Variational problem, Euler-Lagrange equation.
Mathematics Subject Classification: Primary: 53A04, 34A05; Secondary: 34B1.
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.

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