Complete classification of planar p-elasticae

Abstract

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.