General rigidity principles for stable and minimal elastic curves

Abstract

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 , ∞ ) {p\in(1,\infty)} and of stable pinned p-elasticae for p ∈ ( 1 , 2 ] {p\in(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 , 3 2 ] {p\in(1,\frac{3}{2}]} in which the second variation may not exist even for smooth variations.