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Representation formula for the plane closed elastic curves. (English) Zbl 1310.53004

Summary: 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 shown 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 \(\kappa(s)\).

MSC:

53A04 Curves in Euclidean and related spaces
34A05 Explicit solutions, first integrals of ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
49Q10 Optimization of shapes other than minimal surfaces
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