JournalsifbVol. 20, No. 1pp. 69–106

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

  • Matthieu Bonnivard

    Université Denis Diderot – Paris 7, France
  • Antoine Lemenant

    Université Paris Diderot – Paris 7, France
  • Vincent Millot

    Université Denis Diderot - Paris 7, France
On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers cover
Download PDF

A subscription is required to access this article.

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\epsilon > 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 ϵ0\epsilon \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.

Cite this article

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. 69–106

DOI 10.4171/IFB/397