Generalized elastica problems under area constraint

TL;DR

This paper extends the known result that circles minimize elastic energy among curves with fixed area to a broader class of curvature-dependent functionals, including p-elastic energies for p>1.

Contribution

It generalizes the area-constrained minimization problem to other curvature functionals, demonstrating circles remain optimal for p-elastic energies when p>1.

Findings

Circles are minimizers for the p-elastic energy with p>1 under area constraint.

The result extends previous proofs for elastic energy to a wider class of curvature functionals.

Conditions are identified under which the minimal shape remains a circle.

Abstract

It was recently proved that the elastic energy $E(\gamma)=\tfrac{1}{2}\int_\gamma\kappa^2 ds$ of a closed curve $\gamma$ with curvature $\kappa$ has a minimizer among all plane, simple, regular and closed curves of given enclosed area $A(\gamma)$, and that the minimum is attained only for circles. Here we show under which hypothesis the result can be extended to other functionals involving the curvature. As an example we show that the optimal shape remains a circle for the $p$-elastic energy $\int_\gamma|\kappa|^p ds$, whenever $p>1$.