Planar p-elastic curves and related generalized complete elliptic integrals

Abstract

Planar elastica problem is a classical but has broad connections with various fields, such as elliptic functions, differential geometry, soliton theory, material mechanics, etc. This paper regards classical elastica as a theory corresponding to Lebesgue L² case, and extends it to L^p cases. For the sake of the effect of p-Laplacian, novel curious solutions appear especially for cases p > 2. These solutions never appear in 1 < p ≤ 2 cases and we call them flat-core solutions according to Takeuchi [6, 7].