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Uniqueness and minimality of Euler's elastica with monotone curvature

Tatsuya Miura, Glen Wheeler

TL;DR

The paper addresses global minimality of planar Euler's elastica under clamped boundary data, focusing on elasticae with non-constant monotone curvature. It develops a free-boundary framework on parallel support lines, leverages a no-flux condition, and uses the elliptic-function classification of elasticae to obtain explicit parametrisations and energy relations that pinpoint a unique minimiser. The main results prove global minimality for fixed length under monotone curvature and extend to the length-penalised problem under a non-subcriticality condition, with applications to straightening for generic boundary angles, and they establish a rigorous link between fixed-length and length-penalised formulations, including convergence to borderline elastica in the straightening limit. This work provides a comprehensive global minimality criterion for planar elasticae, clarifies when local stability implies global minimality, and informs boundary-value problems and straightening applications in elasticity.

Abstract

For an old problem of Euler's elastica we prove the novel global property that every planar elastica with non-constant monotone curvature is uniquely minimal subject to the clamped boundary condition. We also partly extend this unique minimality to the length-penalised case; this result is new even in view of local minimality. As an application we prove uniqueness of global minimisers in the straightening problem for generic boundary angles.

Uniqueness and minimality of Euler's elastica with monotone curvature

TL;DR

The paper addresses global minimality of planar Euler's elastica under clamped boundary data, focusing on elasticae with non-constant monotone curvature. It develops a free-boundary framework on parallel support lines, leverages a no-flux condition, and uses the elliptic-function classification of elasticae to obtain explicit parametrisations and energy relations that pinpoint a unique minimiser. The main results prove global minimality for fixed length under monotone curvature and extend to the length-penalised problem under a non-subcriticality condition, with applications to straightening for generic boundary angles, and they establish a rigorous link between fixed-length and length-penalised formulations, including convergence to borderline elastica in the straightening limit. This work provides a comprehensive global minimality criterion for planar elasticae, clarifies when local stability implies global minimality, and informs boundary-value problems and straightening applications in elasticity.

Abstract

For an old problem of Euler's elastica we prove the novel global property that every planar elastica with non-constant monotone curvature is uniquely minimal subject to the clamped boundary condition. We also partly extend this unique minimality to the length-penalised case; this result is new even in view of local minimality. As an application we prove uniqueness of global minimisers in the straightening problem for generic boundary angles.
Paper Structure (5 sections, 21 theorems, 75 equations)

This paper contains 5 sections, 21 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Free boundary problems with fixed length
  4. Free boundary problems with variable length
  5. Straightening problem

Key Result

Theorem 1.1

Let $\gamma:[0,L]\to\mathbf{R}^2$ be an arc-length parametrised elastica with non-constant and monotone signed curvature. Then $\gamma$ is the unique minimiser of the bending energy $\mathsf{B}$ in the set $\mathcal{A}(\gamma)$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1: No flux condition
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 38 more