Complete classification of planar p-elasticae

Euler's elastica is defined by a critical point of the total squared curvature under the fixed length constraint, and its \(L^p\)-counterpart is called \(p\)-elastica. In this paper we completely classify all \(p\)-elasticae in the plane and obtain their explicit formulae as well as optimal reg...

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Bibliographic Details
Published inarXiv.org
Main Authors Miura, Tatsuya, Yoshizawa, Kensuke
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 30.03.2024
Subjects
Classification
Critical point
Elastica
Elliptic functions
Mathematics - Analysis of PDEs
Mathematics - Classical Analysis and ODEs
Mathematics - Differential Geometry
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Summary:Euler's elastica is defined by a critical point of the total squared curvature under the fixed length constraint, and its \(L^p\)-counterpart is called \(p\)-elastica. In this paper we completely classify all \(p\)-elasticae in the plane and obtain their explicit formulae as well as optimal regularity. To this end we introduce new types of \(p\)-elliptic functions which streamline the whole argument and result. As an application we also classify all closed planar \(p\)-elasticae.
ISSN:2331-8422
DOI:10.48550/arxiv.2203.08535