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In this paper we consider variational problems involving 1-dimensionalconnected sets in the Euclidean plane, such as the classical Steiner treeproblem and the irrigation (Gilbert-Steiner) problem. We relate them to optimalpartition problems and provide a variational approximation throughModica-Mortola type energies proving a $\Gamma$-convergence result. We alsointroduce a suitable convex relaxation and develop the corresponding numericalimplementations. The proposed methods are quite general and the results weobtain can be extended to $n$-dimensional Euclidean space or to more generalmanifold ambients, as shown in the companion paper [11].

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