---
abstract: 'In this paper we consider the branched transportation problem in 2D associated with a cost per unit length of the form $1+\alpha m$ where $m$ denotes the amount of transported mass and $\alpha>0$ is a fixed parameter (notice that the limit case $\alpha=0$ corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ($\{{\mathcal{F}}_{\varepsilon}\}_{{\varepsilon}>0}$) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the $\Gamma$-convergence of $\{{\mathcal{F}}_{\varepsilon}\}$ as ${\varepsilon}\downarrow0$. Our functionals are modeled on the Ambrosio-Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.'
author:
- 'A. Chambolle'
- 'B. Merlet'
- 'L. Ferrari'
bibliography:
- 'bib.bib'
title: 'A simple phase-field approximation of the Steiner problem in dimension two'
---

Introduction
============

In this paper, we introduce a phase-field approximation of a branched transportation energy for lines in the plane. Our main goal is to derive a computationally tractable approximation of the Steiner problem (of minimizing the length of lines connecting a given set of points) in a phase-field setting. Similar results have recently be obtained by [@Bon_Lem_San], however we 