大长度受限弹性线：尖锐的二阶渐近
Confined elastic wires of large length: the sharp second-order asymptotics
中文总结 AI 辅助
本文研究圆盘内固定长度封闭嵌入线的最小弯曲能量，证明其与长度的偏差以 $L^{4/9}$ 阶增长，并给出尖锐常数，改进已知上界并提供首个非平凡下界。
中文摘要 AI 辅助
我们研究封闭单位圆盘内、具有给定长度 $L$ 的封闭嵌入线的最小弯曲能量 $m_L$。经典结论是，圆盘内任意封闭曲线的弯曲能量至少等于其长度，当且仅当该曲线为多重覆盖的单位圆时取等。这样的圆是若干不相交嵌入圆堆叠的能量极限，但并非单个嵌入环所能达到，由此产生的缺陷大小 $m_L-L$ 此前一直未解。我们证明当 $L\to\infty$ 时，$m_L = L + c_1 L^{4/9} + O(L^{1/3})$，其中尖锐常数 $c_1=5.223049\ldots$ 由两个一维模型问题以闭式给出。特别地，$m_L-L\asymp L^{4/9}$，这改进了先前已知的 $\sqrt{L}$ 阶上界，并提供了超越平凡下界的首个非平凡下界。
英文摘要
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 \[ m_L \;=\; L+c_1\,L^{4/9}+O\bigl(L^{1/3}\bigr)\qquad\text{as }L\to\infty , \] with a sharp constant $c_1=5.223049\ldots$ given in closed form by two one-dimensional model problems. In particular $m_L-L\asymp L^{4/9}$, which improves the previously known upper bound of order $\sqrt{L}$ 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.
