Uniform estimates for a Modica–Mortola type approximation of branched transportation

Abstract

Models for branched networks are often expressed as the minimization of an energyMαover vector measures concentrated on 1-dimensional rectifiable sets with a divergence constraint. We study a Modica–Mortola type approximationMαε, introduced by Edouard Oudet and Filippo Santambrogio, which is defined overH1vector measures. These energies induce some pseudo-distances betweenL2functions obtained through the minimization problem min {Mαε(u): ∇·u=f+−f−}. We prove some uniform estimates on these pseudo-distances which allow us to establish aΓ-convergence result for these energies with a divergence constraint.