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 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.