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

TL;DR

This paper studies a phase field approximation of the planar Steiner problem, proving existence, regularity, and analyzing the asymptotic behavior of minimizers as the approximation parameter tends to zero.

Contribution

It introduces a new phase field functional for the Steiner problem, proves existence and regularity of minimizers, and analyzes their asymptotic convergence to optimal Steiner sets.

Findings

Minimizers exist and are regular for the functional.

Sublevel sets of minimizers converge to Steiner sets as epsilon approaches zero.

Applications to average distance and optimal compliance are discussed.

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.