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Representation formula for the plane closed elastic curves

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  • 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 $\kappa(s)$.
    Mathematics Subject Classification: Primary: 53A04, 34A05; Secondary: 34B15.


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