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Register a new userOn a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers
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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 $varepsilonto0$, 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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