In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert-Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in R-n. Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, SIAM J. Math. Anal. 50 (2018), no. 6, 6307-6332], we provide a variational approximation through Ginzburg-Landau type energies proving a Gamma-convergence result for n >= 3.
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