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

29 pages, 14 figures, final version

References in corpus (2)

• A survey of the elastic flow of curves and networks
• Convergence of elastic flows of curves into manifolds

Cited by in corpus (2)

• Migrating elastic flows
• Elastic curves and self-intersections