Degenerate elastic networks

TL;DR

This paper studies the minimization of combined Willmore and length functionals on elastic networks, introducing degenerate elastic networks to handle non-compactness and providing explicit characterizations and algorithms.

Contribution

It introduces the concept of degenerate elastic networks and characterizes their properties without relying on curvature, advancing the understanding of elastic network minimization.

Findings

Characterization of limits of bounded energy networks

Explicit representation of the relaxed problem

Finite algorithm for validating degenerate networks

Abstract

We minimize a linear combination of the Willmore and the length functional among networks in $\mathbb{R}^d$ belonging to a given class determined by the number of curves, the order of the junctions and the angles between curves at the junctions. Since this class lacks compactness, we characterize the set of limits of sequences of networks bounded in energy, providing an explicit representation of the relaxed problem. This is expressed in terms of the new notion of degenerate elastic networks that, rather surprisingly, involves only the properties of the given class, without reference to the curvature. In the case of $d=2$ we also give an equivalent description of degenerate elastic networks by means of a combinatorial definition easy to validate by a finite algorithm. Moreover we provide examples, counterexamples, and additional results that motivate our study and show the sharpness of our characterization.