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

Abstract

In this article, we consider and analyse a variant of a functional originally introduced in [9, 27] to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter $ϵ>0$ 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 $ϵ\to 0$, showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem and optimal compliance are also discussed.