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

TL;DR

This paper investigates the regularity and structure of non-planar p-elasticae in higher dimensions, proving they are generally analytic and classifying their forms, with applications to energy inequalities.

Contribution

It extends the classification and regularity results of p-elasticae to non-planar cases in higher dimensions, including a Li-Yau type inequality.

Findings

Non-planar p-elasticae are analytic and three-dimensional.

Flat-core solutions are the only exceptions.

Established a Li-Yau type inequality for p-bending energy.

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 $n=2$.