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arXiv · 1611.07875

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

Matthieu Bonnivard·Antoine Lemenant·Vincent Millot
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Abstract

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.

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Matthieu Bonnivard, Antoine Lemenant, Vincent Millot. 2016-11-23. On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers. https://arxiv.org/abs/1611.07875

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