The elastica problem under area constraint

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.