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A nongraphical obstacle problem for elastic curves

Marius Müller, Kensuke Yoshizawa

TL;DR

The paper investigates obstacle problems for the length-penalized bending energy $\mathcal{E}_\lambda[\gamma]=B[\gamma]+\lambda L[\gamma]$ on open planar curves pinned at endpoints, combining a detailed classification of penalized pinned elasticae, figure-eight elasticae, and free elasticae via elliptic-function formulas with variational-inequality analysis. It establishes a sharp $\lambda$-dependent dichotomy: for small $\lambda$ the energy-minimizing curves in the symmetric class avoid the obstacle and are given by explicit longer-arc elastica profiles, while for large $\lambda$ any minimizer necessarily touches the obstacle (with a universal threshold $\hat{\lambda}\approx0.70107$). A rhomb obstacle constraint is introduced to recover compactness and enable sharp regularity results; the analysis yields $W^{3,\infty}$-regularity above the obstacle under Lipschitz obstacles and clarifies how nongraphical curves alter the minimizer landscape compared with graphical-obstacle problems. Overall, the work reveals a nuanced transition between non-touching and touching minimizers driven by $\lambda$ and obstacle geometry, and provides explicit critical profiles that govern the obstacle-interaction regime. The explicit constants, energy formulas, and elasticity classifications offer precise tools for understanding high-order obstacle problems in elastic curves.

Abstract

We study an obstacle problem for the length-penalized elastic bending energy for open planar curves pinned at the boundary. We first consider the case without length penalization and investigate the role of global minimizers among graph curves in our minimization problem for planar curves. In addition, for large values of the length-penalization parameter $λ>0$, we expose an explicit threshold parameter above which minimizers touch the obstacle, regardless of its shape. On contrary, for small values of $λ>0$ we show that the minimizers do not touch the obstacle, and they are given by an explicit elastica.

A nongraphical obstacle problem for elastic curves

TL;DR

The paper investigates obstacle problems for the length-penalized bending energy on open planar curves pinned at endpoints, combining a detailed classification of penalized pinned elasticae, figure-eight elasticae, and free elasticae via elliptic-function formulas with variational-inequality analysis. It establishes a sharp -dependent dichotomy: for small the energy-minimizing curves in the symmetric class avoid the obstacle and are given by explicit longer-arc elastica profiles, while for large any minimizer necessarily touches the obstacle (with a universal threshold ). A rhomb obstacle constraint is introduced to recover compactness and enable sharp regularity results; the analysis yields -regularity above the obstacle under Lipschitz obstacles and clarifies how nongraphical curves alter the minimizer landscape compared with graphical-obstacle problems. Overall, the work reveals a nuanced transition between non-touching and touching minimizers driven by and obstacle geometry, and provides explicit critical profiles that govern the obstacle-interaction regime. The explicit constants, energy formulas, and elasticity classifications offer precise tools for understanding high-order obstacle problems in elastic curves.

Abstract

We study an obstacle problem for the length-penalized elastic bending energy for open planar curves pinned at the boundary. We first consider the case without length penalization and investigate the role of global minimizers among graph curves in our minimization problem for planar curves. In addition, for large values of the length-penalization parameter , we expose an explicit threshold parameter above which minimizers touch the obstacle, regardless of its shape. On contrary, for small values of we show that the minimizers do not touch the obstacle, and they are given by an explicit elastica.

Paper Structure

This paper contains 15 sections, 23 theorems, 154 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Penalized pinned elasticae
  4. Figure-eight elastica
  5. Free elastica
  6. Fundamental properties for the obstacle problem
  7. Description of local minimizers
  8. Optimal regularity in the case of Lipschitz obstacles
  9. Existence of global minimizers
  10. Analysis of the case of $\lambda = 0$
  11. Stability analysis for symmetric cut-and-glued free-elasticae
  12. Nontouching minimizers for small $\lambda > 0$
  13. Touching the obstacle for large $\lambda$
  14. Elliptic integrals and functions
  15. Technical proofs

Key Result

Theorem 1.1

Assume that $\psi$ is a symmetric cone obstacle and satisfies Assumptions (A1)--(A3). Let $\gamma\in A$ be a unique symmetric cut-and-glued free-elastica. Then the following assertions hold: where $h_*>0$ is the universal constant defined by def:h_*.

Figures (3)

  • Figure 1: Symmetric cut-and-glued free-elasticae with $\psi(\frac{1}{2})=h_*/2$ (top left), $\psi(\frac{1}{2})=h_*$ (top center), $\psi(\frac{1}{2})=2h_*$ (top right), where $h_*\simeq 0.83463$ (cf. \ref{['def:h_*']}). The bottom three figures represent rectangular elasticae $\gamma_{\rm rect}$ of each corresponding top figure.
  • Figure 2: $(\lambda,1,1)$-Longer arcs ($\gamma_{\rm larc}^{\lambda,1,1}$) with $\lambda={1}/{20}$, $\lambda=\hat{\lambda}/2\simeq 0.35355$, $\lambda=\hat{\lambda}\simeq0.70710$ (from left to right), after rescaling.
  • Figure 3: Examples of constructions of the curve $C_n$ as in Remark \ref{['rem:escaping-circular_arcs']}. The energy $\mathcal{E}_{\lambda_n}[C_n]$ can be close to $0$ as $n\to\infty$ while $C_n$ always touch the obstacle.

Theorems & Definitions (55)

  • Theorem 1.1: Destabilization of symmetric cut-and-glued free-elasticae
  • Theorem 1.2: Nontouching symmetric minimizers for small $\lambda > 0$
  • Theorem 1.3: Touching minimizers for large $\lambda$
  • Definition 2.1
  • Proposition 2.2: MYarXiv2409
  • Remark 2.3
  • Proposition 2.4
  • Lemma 3.1: Euler--Lagrange equation above the obstacle
  • proof : Sketch of proof.
  • Lemma 3.2: Variational inequality
  • ...and 45 more