Degenerate Elastic Networks
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Degenerate Elastic Networks Giacomo Del Nin1
· Alessandra Pluda2
· Marco Pozzetta2
Received: 4 February 2020 / Accepted: 15 September 2020 © The Author(s) 2020
Abstract We minimize a linear combination of the length and the L 2 -norm of the curvature among networks in Rd 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. Keywords Networks · Relaxation · Elastic energy · Singular structures Mathematics Subject Classification Primary 49J45 · 35A15; Secondary 49Q10 · 53A04
1 Introduction A regular network N is a connected set in Rd composed of N regular curves γ i of class H 2 that meet at their endpoints in junctions of possibly different order. Moreover the angles at the junctions are assigned by a fixed set of directions D as we will define
B
Alessandra Pluda [email protected] Giacomo Del Nin [email protected] Marco Pozzetta [email protected]
1
Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7HP, UK
2
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
123
G. Del Nin et al.
more precisely in Definition 2.11. The elastic energy functional E for a network N is given by E (N ) :=
N i=1
γi
|ki |2 ds + (γ i )
,
(1.1)
where ki is the curvature, s the arclength parameter and (γ i ) is the length of the curve γi. The elastic energy functional has a long history. Already at the times of Galileo scientists tried to model elastic rods and strings, looking for equations for equilibrium of moments and forces. The idea to relate the curvature of the fiber of the beam to the bending moment came only later when, in 1691, Jacob Bernoulli proposed to model the bending energy of thin inextensible elastic rods with a functional involving the curvature. Several authors refer to the functional (1.1) as Euler Elastic energy in honor of Euler (while we will simply call it elastic energy) who solved the problem of minimizing the potential energy of the elastic laminae using variational techniques. Even nowadays the elastic energy appears in several mechanical and physical models (c.f. [21]) and in imaging sciences, see for instance [18]. We are interested in the minimization of the functional E among networks with fixed topology and with fixed angles at the junction
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