Regularity and structure of non-planar $p$-elasticae
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914110196678656 |
|---|---|
| author | Gruen, Florian Miura, Tatsuya |
| author_facet | Gruen, Florian Miura, Tatsuya |
| contents | 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 $n=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07987 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity and structure of non-planar $p$-elasticae Gruen, Florian Miura, Tatsuya Analysis of PDEs Differential Geometry 49Q10, 53A04, and 53E40 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 $n=2$. |
| title | Regularity and structure of non-planar $p$-elasticae |
| topic | Analysis of PDEs Differential Geometry 49Q10, 53A04, and 53E40 |
| url | https://arxiv.org/abs/2501.07987 |