We prove regularity and structure results for p-elasticae in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textbf{R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">R</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, with arbitrary <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p\in (1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">R</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and, as an application, establish a Li–Yau type inequality for the p-bending energy of closed curves in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textbf{R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">R</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. This extends previous works for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> as well as for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p\in (1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Regularity and structure of non-planar p-elasticae

期刊论文
发表日期：

2025年10月22日

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Florian Gruen,
Tatsuya Miura

摘要

We prove regularity and structure results for p-elasticae in $\textbf{R}^n$ $R^{n}$ , with arbitrary $p\in (1,\infty )$ $p \in (1, \infty)$ and $n\ge 2$ $n \geq 2$ . 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 $\textbf{R}^n$ $R^{n}$ and, as an application, establish a Li–Yau type inequality for the p-bending energy of closed curves in $\textbf{R}^n$ $R^{n}$ . This extends previous works for $p=2$ $p = 2$ and $n\ge 2$ $n \geq 2$ as well as for $p\in (1,\infty )$ $p \in (1, \infty)$ and $n=2$ $n = 2$ .