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. Contents 1. Introduction 1 2. The Euler-Lagrange equations 9 3. p-Elliptic functions 11 4. Classification, representation, and regularity of p-elasticae 25 5. Closed p-elasticae 39 Appendix A. First variation of the p-bending energy 45 References 47

Polar tangential angles and free elasticae

“…The Euler-Lagrange equation for p-elastica is formally given by The degeneracy as well as singularity leads to generic loss of regularity and hence the analytical treatment of p-elastica is much more delicate than the classical one. Building on Watanabe's previous work [54], the authors recently obtained a complete classification of planar p-elasticae with explicit formulae and optimal regularity [34]. This in particular shows that in the non-degenerate regime p ∈ (1,2] the zero set of the curvature k is always discrete, while in the degenerate regime p ∈ (2, ∞) the zero set may contain nontrivial intervals, which are also called flat core in this context.…”

“…In this section we first recall from [34] the definitions and fundamental properties of p-elliptic integrals and functions, and also some known facts for planar p-elasticae.…”

“…In fact, even for p ̸ = 2, there are also many studies on p-elasticae (e.g. [26,[34][35][36]54]) as well as related problems on the p-bending energy (e.g. [1-3, 5-9, 11, 12, 14, 15, 17, 25, 29, 30, 37-41, 46]).…”

“…This choice turns out to be very well suited: On one hand, it is so relaxed that we can use additional natural boundary conditions for restricting all critical points in "computable" forms. Here we heavily rely on our previous classification theory [34,35], which describes p-elasticae in terms of p-elliptic functions introduced by the authors and of p-elliptic integrals introduced by Watanabe [54] with reference to Takeuchi's work [50]. Armed with those tools and resulting monotonicity properties, we can explicitly obtain unique global minimizers and represent their energy by complete p-elliptic integrals (Theorem 3.10).…”

Closed 1/2-Elasticae in the 2-Sphere

“…In a recent paper [35] Miura-Yoshizawa obtained a complete classification of p-elasticae in the plane R 2 , for every real number p > 1. For p ∈ (0, 1), since the Lagrangian is merely continuous at the origin, the Bernoulli's functionals are defined for convex curves.…”