Calculus of Variations and Geometric Measure Theory

A. Dall'Acqua - A. Pluda

Some minimization problems for planar networks of elastic curves

created by pluda on 16 Oct 2017
modified on 15 Mar 2019


Published Paper

Inserted: 16 oct 2017
Last Updated: 15 mar 2019

Journal: Geom. flows
Volume: 3
Pages: 105-124
Year: 2017
Doi: 10.1515/geofl-2017-0005


In this note we announce some results that will appear in $[6]$ 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.