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In this article, we consider and analyse a small variant of a functionaloriginally introduced in \cite{BLS,LS} to approximate the (geometric) planarSteiner problem. This functional depends on a small parameter $\varepsilon>0$and resembles the (scalar) Ginzburg-Landau functional from phase transitions.In a first part, we prove existence and regularity of minimizers for thisfunctional. Then we provide a detailed analysis of their behavior as$\varepsilon\to0$, showing in particular that sublevel sets Hausdorff convergeto optimal Steiner sets. Applications to the average distance problem andoptimal compliance are also discussed.

On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers