The elastica problem under area constraint

TL;DR

This paper proves that among all simple closed curves with a fixed enclosed area, the circle minimizes elastic energy, using a geometric deformation approach.

Contribution

It establishes the existence of energy minimizers under area constraints and characterizes the circle as the optimal shape.

Findings

The circle uniquely minimizes elastic energy for fixed area.

A geometric deformation method is used to prove the minimization.

The proof constructs convex sets with lower energy through finite steps.

Abstract

We show that the elastic energy $E(\gamma)$ of a closed curve $\gamma$ has a minimizer among all plane simple regular closed curves of given enclosed area $A(\gamma)$, and that the minimum is attained for a circle. The proof is of a geometric nature and deforms parts of $\gamma$ in a finite number of steps to construct some related convex sets with smaller energy.