Regularity and structure of non-planar $p$-elasticae

TL;DR

The paper establishes optimal regularity and structural classifications for $p$-elasticae in $\mathbb{R}^n$ for all $p\in(1,\infty)$ and $n\ge2$. It shows that non-planar $p$-elasticae are analytic and 3D for $p\le2$, while for $p>2$ flat-core solutions arise, yielding a dichotomy between analytic 3D curves and $d$-dimensional flat-core elasticae with $d\in\{3,\dots,n\}$; the curvature-torsion system is reduced to strong ODEs, and a precise regularity threshold $M_p=\lceil 2/(p-2)\rceil$ governs joint behavior. The work extends pinned-$p$-elastica classifications to higher dimensions, derives a generalized Li–Yau inequality for the normalized $p$-bending energy, and uses these results to show existence of minimal $p$-elastic networks and embeddedness-preserving flows, unifying and extending known results for $p=2$ and for planar cases. Key tools include a detailed Euler–Lagrange analysis for curvature and torsion, positivity-interval bootstrap regularity, and careful treatment of joints and flat-core constructions. Overall, the results deepen the understanding of $p$-elasticae, connect to $\,\infty$-elastica intuitions, and enable applications to networks and geometric flows with robust energy bounds.

Abstract

We prove regularity and structure results for $p$-elasticae in $\mathbb{R}^n$, with arbitrary $p\in (1,\infty)$ and $n\geq2$. Planar $p$-elasticae are already classified and known to lose regularity. In this paper, we show that every non-planar $p$-elastica is analytic and three-dimensional, with the only exception of flat-core solutions of arbitrary dimensions. Subsequently, we classify pinned $p$-elasticae in $\mathbb{R}^n$ and, as an application, establish a Li-Yau type inequality for the $p$-bending energy of closed curves in $\mathbb{R}^n$. This extends previous works for $p=2$ and $n\geq2$ as well as for $p\in (1,\infty)$ and $n=2$.

Figures (4)

• Figure 1: A three dimensional flat-core $p$-elastica ($p=4$): two loops connected by a straight line part.
• Figure 2: A $5$-leafed elastica constructed from a $3$-leafed elastica concatenated with a figure-eight elastica. Starting from the point of multiplicity $m=5$ at the origin, the loops are traversed in their numerical order.
• Figure 3: The sets $P_m$ for different $m$, the values $p_m^*$ are emphasized.
• Figure 4: Construction in the case $m=7$ and $p\in (p_m^*,\infty)\setminus P_m$

Theorems & Definitions (70)

• Definition 1.1: $p$-elastica
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Corollary 1.9
• Corollary 1.10: miura_pinned_p