| Abstract: |
| Let $\Gamma$ be a plane closed elastic curve with length $L>0.$ We denote the arc-length and the curvature by $s$ and $\kappa(s)$, respectively.
Let ${\mathcal A}(\Gamma)$ be the signed area of the domain bounded by $\Gamma$. We consider the variational problem:
Find a curve $\Gamma$ (the curvature $\kappa(s)$) which minimize
the functional
$\displaystyle
{1 \over 2} \int_{0}^{L} \kappa(s)^{2} ds+p{\mathcal A}(\Gamma)$
subject to $\displaystyle \int_{0}^{L} \kappa(s)ds=2 \pi$, where $p$ is a given positive constant.
This variational problem was first considered by Tadjbakhsh-Odeh to study the
equilibrium states of an elastic
inextensible ring under a uniform pressure $p>0$.
S.Okabe showed the asymptotic form of solutions of
the Euler-Lagrange equation
for sufficient large $p$ and derived the second variation.
He showed the uniqueness and the profile of the minimizer
for sufficiently large $p$.
In this talk, we will give new proofs of the asymptotic form
of solutions of the Euler-Lagrange equation for sufficient large $p$. |
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