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Smooth compactness of elasticae

Tatsuya Miura

TL;DR

This work addresses the problem of smooth compactness for elasticae under natural boundedness constraints for bending energy $\\mathcal{B}$ and arclength $\\mathcal{L}$. The authors develop a multiplier-based dichotomy: if the Lagrange multipliers $\\lambda_j$ stay bounded, the sequence converges smoothly to an elastica; if $\\lambda_j$ is unbounded, the limit is a straight segment, explained via the Langer--Singer framework and Jacobi elliptic functions. These results yield rigorous smooth stability for clamped boundary-value problems: minimizers depend continuously and converge smoothly with perturbations of boundary data away from straight-segment limits, and similar stability holds for the length-penalized problem with precise caveats. The findings clarify when elastica minimizers vary continuously with parameters and boundary data, with implications for the physical stability of elastic rods under boundary perturbations.

Abstract

We prove a smooth compactness theorem for the space of elasticae, unless the limit curve is a straight segment. As an application, we obtain smooth stability results for minimizers with respect to clamped boundary data.

Smooth compactness of elasticae

TL;DR

This work addresses the problem of smooth compactness for elasticae under natural boundedness constraints for bending energy and arclength . The authors develop a multiplier-based dichotomy: if the Lagrange multipliers stay bounded, the sequence converges smoothly to an elastica; if is unbounded, the limit is a straight segment, explained via the Langer--Singer framework and Jacobi elliptic functions. These results yield rigorous smooth stability for clamped boundary-value problems: minimizers depend continuously and converge smoothly with perturbations of boundary data away from straight-segment limits, and similar stability holds for the length-penalized problem with precise caveats. The findings clarify when elastica minimizers vary continuously with parameters and boundary data, with implications for the physical stability of elastic rods under boundary perturbations.

Abstract

We prove a smooth compactness theorem for the space of elasticae, unless the limit curve is a straight segment. As an application, we obtain smooth stability results for minimizers with respect to clamped boundary data.
Paper Structure (9 sections, 12 theorems, 70 equations, 3 figures)

This paper contains 9 sections, 12 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Smooth compactness
  3. Smooth stability for boundary value problems
  4. Compactness
  5. Smooth convergence
  6. Counterexamples
  7. Stability
  8. Fixed-length case
  9. Length-penalized case

Key Result

Theorem 1.1

Let $\{\gamma_j\}_{j=1}^\infty\subset W^{2,2}(I;\mathbf{R}^n)$ be a sequence of elasticae such that Let $\bar{\gamma}_j\in C^\omega(\bar{I};\mathbf{R}^n)$ denote the constant-speed reparametrization of $\gamma_j$. Then there are translation vectors $b_j\in\mathbf{R}^n$ such that the sequence $\{\bar{\gamma}_j+b_j\}$ contains a subsequence $\{\bar{\gamma}_{j'}+b_{j'}\}$ converging to some constant

Figures (3)

  • Figure 1: Counterexample of curvature oscillation type.
  • Figure 2: Counterexample of curvature concentration type.
  • Figure 3: Discontinuous transition of minimizers of $\mathcal{E}_\lambda$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 23 more