Some minimization problems for planar networks of elastic curve

TL;DR

This paper explores the minimization of an elastic energy functional for planar networks of three curves with fixed angles, including explicit calculations for the 'Figure Eight' elastica, and discusses extensions to network variants.

Contribution

It introduces new results on elastic energy minimization for planar networks with fixed angles, including explicit energy calculations for the 'Figure Eight' elastica.

Findings

Explicit elastic energy of the 'Figure Eight' elastica computed.

Penalizing length is equivalent to fixing it in the energy functional.

Discussion on extending the problem from curves to networks.

Abstract

In this note we announce some results that will appear in [6] (joint work with also Matteo Novaga) on the minimization of the functional $F(\Gamma)=\int_\Gamma k^2+1\,\mathrm{d}s$, where $\Gamma$ is a network of three curves with fixed equal angles at the two junctions. The informal description of the results is accompanied by a partial review of the theory of elasticae and a diffuse discussion about the onset of interesting variants of the original problem passing from curves to networks. The considered energy functional $F$ is given by the elastic energy and a term that penalize the total length of the network. We will show that penalizing the length is tantamount to fix it. The paper is concluded with the explicit computation of the penalized elastic energy of the 'Figure Eight', namely the unique closed elastica with self--intersections.