Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a $\Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure

Minimal Steiner Trees for 2k×2kSquare Lattices

AbstractWe prove a conjecture of Chung, Graham, and Gardner (Math. Mag.62(1989), 83–96), giving the form of the minimal Steiner trees for the set of points comprising the vertices of a 2k×2ksquare lattice. Each full component of these minimal trees is the minimal Steiner tree for the four vertices of a square

Numerical Calibration of Steiner trees

In this paper we propose a variational approach to the Steiner tree problem, which is based on calibrations in a suitable algebraic environment for polyhedral chains which represent our candidates. This approach turns out to be very efficient from numerical point of view and allows to establish whether a given Steiner tree is optimal. Several examples are provided

Elusive multiquark spectroscopy

A review is presented of past and recent attempts to build multiquark states within current models already describing ordinary mesons and baryons. This includes: coherence in the chromomagnetic interaction, tetraquarks with two heavy quarks, Steiner-tree models of confinement, and hadronic molecules, in particular in the hidden-charm sector. Some emphasis is put on the difficulties encountered when extrapolating toward higher configurations the dynamics of confining forces, starting from the simple case of a quark and an antiquark, or three quarks in a colour singlet