Inserted: 23 nov 2016
Last Updated: 23 nov 2016
In this article, we consider and analyse a small variant of a functional originally introduced in $[$Lemenant-Santambrogio-2014$]$, $[$Bonnivard-Lemenant-Santambrogio-2015$]$ 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.