On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers
On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers
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关于平面斯坦纳问题的相场近似：极小值的存在性、正则性和渐近性
DOI：
10.4171/ifb/397
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
V. Millot
中科院分区：
文献类型：
--
作者：
M. Bonnivard;A. Lemenant;V. Millot
在这篇文章中，我们考虑和分析一个小的变体的功能最初介绍了\cite{BLS，LS}近似的（几何）平面斯坦纳问题。这个功能依赖于一个小参数$\vareps>0$，类似于（标量）金-朗道功能从相变。在第一部分中，我们证明了该泛函极小解的存在性和正则性。然后，我们提供了一个详细的分析，他们的行为为$\varepsilon\to0$，特别是显示子级集Hausdorff收敛到最佳施泰纳集。应用的平均距离问题和最佳的遵守进行了讨论。
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.