General rigidity principles for stable and minimal elastic curves

Abstract

For a wide class of curvature energy functionals defined for planar curves under the fixed-length constraint, we obtain optimal necessary conditions for global and local minimizers. Our results extend Maddocks’ and Sachkov’s rigidity principles for Euler’s elastica by a new, unified and geometric approach. This in particular leads to complete classification of stable closed p-elasticae for all p ∊ (1; ∞) and of stable pinned p-elasticae for p ∊ (1; 2]. Our proof is based on a simple but robust “cut-and-paste” trick without computing the energy nor its second variation, which works well for planar periodic curves but also extends to some non-periodic or non-planar cases. An analytically remarkable point is that our method is directly valid for the highly singular regime p ∊ (1; 32 ] in which the second variation may not exist even for smooth variations.