Variational approximations and relaxations of the Steiner problem

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Mauro Bonafini
Universita' di Trento
Abstract. In this talk we focus our attention on problems involving one dimensional sets, using as guiding example the Euclidean Steiner tree problem. We first reformulate the problem over a suitable family of rank one tensor valued measures and then we take two different perspectives. In the first case we focus on the planar setting and provide a variational approximation via Gamma convergence by means of functionals of phase transition type. In the second case we describe a convex framework associated to the problem which provides relevant tools extensively used from a numerical point of view to identify optimal 1d structures. This is joint work with Giandomenico Orlandi (Verona) and Edouard Oudet (Grenoble).