Pinned planar p-elasticae

TL;DR

The paper completes a comprehensive classification of pinned planar $p$-elasticae for all $p>1$, revealing arc, loop, and flat-core structures and identifying a transition to uncountably many pinned solutions when $p>2$ and $r>1/(p-1)$. It establishes the existence and uniqueness of global minimizers under pinned length constraints, and proves a Li--Yau type inequality for the normalized $p$-bending energy, uncovering a unique exponent $p_3\approx1.5728$ that yields full optimality across all multiplicities. The work further constructs and analyzes minimal $p$-elastic networks, proving existence of energy-minimizing $\Theta$-networks for suitable angles and p-values, including a symmetric $\alpha=2\pi/3$ case for $p>p_3$. Central to the results are the $p$-elliptic framework and the crossing-angle monotonicity $\phi^*(p)$ of half-fold figure-eights, which drive both the classification and optimization phenomena. The results open avenues for gradient-flow limits, network dynamics, and refined geometric characterizations of minimizing elasticae in the planar setting.

Abstract

Building on our previous work, we classify all planar $p$-elasticae under the pinned boundary condition, and then obtain uniqueness and geometric properties of global minimizers. As an application we establish a Li--Yau type inequality for the $p$-bending energy, and in particular discover a unique exponent $p \simeq 1.5728$ for full optimality. We also prove existence of minimal $p$-elastic networks, extending a recent result of Dall'Acqua--Novaga--Pluda.

Figures (7)

• Figure 1: The left $\gamma^n_{\rm arc}$ represents an $\frac{n}{2}$-fold figure-eight $p$-elastica, the middle $\gamma^n_{\rm arc}$ a $(p,r,n)$-arc, and the right $\gamma^n_{\rm loop}$ a $(p,r,n)$-loop, where $p=4$ and $r=\tfrac{1}{5}$.
• Figure 2: The left $\gamma^n_{\rm flat}$ represents a $(p,r,n, \boldsymbol{\sigma}, \boldsymbol{L})$-flat-core with $p=4$ and $r=\frac{2}{5}$. The right two curves are also $(4,\frac{2}{5},2, \boldsymbol{\sigma}, \boldsymbol{L})$-flat-cores, with different choices of the direction of the loops $\boldsymbol{\sigma}$ and the length of the connecting segments $\boldsymbol{L}$.
• Figure 3: The angle $2\phi^*(p)$, and three examples of the half-fold figure-eight $p$-elasticae with $p=\frac{6}{5}$, $2$, $10$ (from left to right).
• Figure 4: The $3$-leafed $p_3$-elastica and $5$-leafed $p_5$-elastica.
• Figure 5: The graph of $p\mapsto \phi^*(p)/\pi$.

Theorems & Definitions (99)

• Theorem 1.1: Classification of pinned $p$-elasticae
• Remark 1.2: Countability-uncountability transition
• Remark 1.3: Loss of regularity
• Theorem 1.4: Unique existence of global minimizers
• Theorem 1.5: Monotonicity of the crossing angle
• Theorem 1.6: Li--Yau type inequality and rigidity
• Theorem 1.7: Optimality for even multiplicity
• Theorem 1.8: Unique exponent for full optimality
• Theorem 1.9: Existence of minimal $p$-elastic $\Theta$-networks
• Corollary 1.10