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The critical points of the elastic energy among curves pinned at endpoints

The author was supported by Grant-in-Aid for JSPS Fellows 19J2074
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  • In this paper we find curves minimizing the elastic energy among curves whose length is fixed and whose ends are pinned. Applying the shooting method, we can identify all critical points explicitly and determine which curve is the global minimizer. As a result we show that the critical points consist of wavelike elasticae and the minimizers do not have any loops or interior inflection points.

    Mathematics Subject Classification: Primary: 49K15, 53A04; Secondary: 34A05.


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  • Figure 1.  Critical points of $ \mathcal{W} $ in $ \mathcal{A}_{l, L} $ are given by Theorem 1.1. According to [41], these curves are called wavelike elasticae

    Figure 2.  The relation between critical curves and the ratio $ l/L $. The number of inflection points (where the sign of the curvature changes) in $ (0, l) $ is given by $ n\in \mathbb{N}\cup\{0\} $. The curve $ \hat{ \gamma}^{\pm}_n $ ($ n\in \mathbb{N} $) can be constructed from $ \hat{ \gamma}^{\pm}_0 $.

    Figure 3.  For any $ n\in \mathbb{N}\cup\{0\} $ the curves $ \check{ \gamma}^+_n $ and $ \check{ \gamma}^-_n $ has a loop

    Figure 4.  The red curve is the graph of $ 2\frac{ \mathrm{E}(p)}{ \mathrm{K}(p)}-1 $; $ 0\leq p \leq p_0 $ and $ -2\frac{ \mathrm{E}(p)}{ \mathrm{K}(p)}+1 $; $ p_0< p <1 $

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