Representation formula for the plane closed elastic curves
Minoru Murai, Waichirô Matsumoto, Shoji Yotsutani · Conference Publications · 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 minimizesthe elastic energy subject to$L^{2}-4 \pi M >0$ and $ L^{2} eq 4 \pi \omega M$,where $\omega$ is the winding number.This variational problem was first studiedin the case $\omega=1$ andthe Euler-Lagrange equation was derived.The existence of the minimizer was showed andthe profile near the disk was investigatedby using the Euler-Lagrange equation.As the first step to investigate the structure of solutions of this equation,we show all the solutions toan auxiliary second order boundary value problem.Moreover, we obtain the representation of the integral of $\kappa(s)$.