Classification and stability of penalized pinned elasticae

TL;DR

The paper delivers a complete classification of penalized pinned elasticae, i.e., critical points of the length-penalized bending energy \\mathcal{E}_\\lambda[\\gamma] = B[\\gamma] + \\lambda L[\\gamma] under pinned endpoints, by expressing nontrivial solutions via explicit elliptic-function parametrizations and a modulus determined by the key threshold \\hat{\\lambda}. It proves precise stability results: for 0 < \\lambda \\ell^2 < \\hat{\\lambda} there are two local minimizers (the line segment and the longer arc), whereas for \\lambda \\ell^2 \\ge \\hat{\\lambda} only the line segment remains stable; higher modes and loops are unstable. The authors perform rigorous energy comparisons, establishing monotonicity in the mode number and identifying the unique minimal nontrivial PPE in each regime: the longer arc for small \\lambda\\ell^2 and the loop for large. These findings inform the \\lambda-elastic flow under Navier boundary conditions, proving an optimal energy threshold below which the flow becomes embedded in finite time and delineating the asymptotic stationary states. Collectively, the results illuminate the energy landscape, stability, and dynamical behavior of pinned elastic rods with length penalization.

Abstract

This paper considers critical points of the length-penalized elastic bending energy among planar curves whose endpoints are fixed. We classify all critical points with an explicit parametrization. The classification strongly depends on a special penalization parameter $\hatλ\simeq 0.70107$. Stability of all the critical points is also investigated, and again the threshold $\hatλ$ plays a decisive role. In addition, our explicit parametrization is applied to compare the energy of critical points, leading to uniqueness of minimal nontrivial critical points. As an application we obtain eventual embeddedness of elastic flows.

Figures (4)

• Figure 1: The left $\gamma_{\rm sarc}^{\lambda,\ell,n}$ represents a $(\lambda,\ell,n)$-shorter arc, the middle $\gamma_{\rm larc}^{\lambda,\ell,n}$ a $(\lambda,\ell,n)$-longer arc, the right $\gamma_{\rm loop}^{\lambda,\ell,n}$ a $(\lambda,\ell,n)$-loop, where $\lambda=1/2$ and $\ell=1$.
• Figure 2: Case $\ell=1$. The penalization parameter $\lambda$ increases from left to right; $\lambda=1/5, 1/2, \hat{\lambda}, 1$.
• Figure 3: The graph of $f$ (left) and the graph of $g$ (right).
• Figure 4: Plots of $\lambda \mapsto \mathcal{E}_\lambda[\gamma_{\rm sarc}^{\lambda,1,1}]$ (blue), $\lambda \mapsto \mathcal{E}_\lambda[\gamma_{\rm larc}^{\lambda,1,1}]$ (yellow) and $\lambda \mapsto \mathcal{E}_\lambda[\gamma_{\rm loop}^{\lambda,1,1}]$ (green) for $\lambda \in (0.02,0.7)$

Theorems & Definitions (64)

• Theorem 1.1: Classification for penalized pinned elasticae
• Theorem 1.2: Stability of penalized pinned elasticae
• Theorem 1.3: Uniqueness of nontrivial minimal penalized pinned elasticae
• Theorem 1.4: Eventual embeddedness of elastic flow
• Definition 2.1
• Lemma 2.2: The Euler--Lagrange equation and boundary condition
• proof
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5