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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 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 , showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem and optimal compliance are also discussed.
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Matthieu Bonnivard, Antoine Lemenant, Vincent Millot, On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers. Interfaces Free Bound. 20 (2018), no. 1, pp. 69–106DOI 10.4171/IFB/397