Confined elastic wires of large length: the sharp second-order asymptotics
Abstract
We study the least bending energy mL of a closed embedded wire of prescribed length L confined to the closed unit disk. It is classical that the bending energy of any closed curve in the disk is at least its length, with equality only for multiply covered unit circles. Such a circle is the energy limit of a stack of disjoint embedded circles, but not of a single embedded loop, and the size of the resulting defect mL−L has remained open. We prove that mL = L+c1 L4/9+O(L1/3) as L→∞ , with a sharp constant c1=5.223049… given in closed form by two one-dimensional model problems. In particular mL−Lasymp L4/9, which improves the previously known upper bound of order √L and provides the first lower bound beyond the trivial one. This paper was produced with substantial assistance from large language models; Section 7 sets out in detail what they contributed and what I verified.
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BibTeX
@misc{mugnai2026confined,
title = {{Confined elastic wires of large length: the sharp second-order asymptotics}},
author = {Luca Mugnai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10605},
eprint = {2609.10605},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10605},
}
RIS
TY - UNPB
TI - Confined elastic wires of large length: the sharp second-order asymptotics
AU - Luca Mugnai
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.10605
UR - https://arxiv.org/abs/2609.10605
AB - We study the least bending energy m_L of a closed embedded wire of prescribed length L confined to the closed unit disk. It is classical that the bending energy of any closed curve in the disk is at least its length, with equality only for multiply covered unit circles. Such a circle is the energy limit of a stack of disjoint embedded circles, but not of a single embedded loop, and the size of the resulting defect m_L−L has remained open. We prove that m_L = L+c_1 L^4/9+O(L^1/3) as L→∞ , with a sharp constant c_1=5.223049… given in closed form by two one-dimensional model problems. In particular m_L−Lasymp L^4/9, which improves the previously known upper bound of order √L and provides the first lower bound beyond the trivial one. This paper was produced with substantial assistance from large language models; Section 7 sets out in detail what they contributed and what I verified.
ER -