Calculus of Variations and Geometric Measure Theory

G. Del Nin - A. Pluda - M. Pozzetta

Degenerate elastic networks

created by pluda on 31 Jul 2019
modified by pozzetta1 on 16 Jul 2021


Published Paper

Inserted: 31 jul 2019
Last Updated: 16 jul 2021

Journal: J. Geom. Anal.
Volume: 31
Pages: 6128-6170
Year: 2021


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 of 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.