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

2016/11/23 by Bonnivard, Matthieu, Lemenant, Antoine, Millot, Vincent · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.07875

Abstract

In this article, we consider and analyse a small variant of a functional originally introduced in \citeBLS,LS to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter ε>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 ε→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.

Cited by

• A field-road system with a rectifiable set
• Phase field approximation for Plateau's problem: a curve geodesic distance penalty approach

Related

• Variational approximation of functionals defined on 1-dimensional connected sets in ℝⁿ
• The Ginzburg-Landau equation in the Heisenberg group
• Phase field approximation for Plateau's problem: a curve geodesic distance penalty approach
• A simple phase-field approximation of the Steiner problem in dimension two
• Existence of Ginzburg-Landau minimizers with optimal boundary data and applications