A simple phase-field approximation of the Steiner problem in dimension two

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 $(\{F_\epsilon\}_{\epsilon>0})$ which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the $\Gamma$-convergence of $\{F_\epsilon\}$ as $\epsilon \downarrow 0$. 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.