  In this article, we consider and analyse a small variant of a functional
originally introduced in \cite{BLS,LS} to approximate the (geometric) planar
Steiner 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 this
functional. Then we provide a detailed analysis of their behavior as
$\varepsilon\to0$, showing in particular that sublevel sets Hausdorff converge
to optimal Steiner sets. Applications to the average distance problem and
optimal compliance are also discussed.
