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

TL;DR

This paper establishes uniform estimates and a b3-convergence result for a Modica-Mortola type approximation of branched transportation models, connecting $H^1$ vector measures with classical energy minimization problems.

Contribution

It provides the first uniform estimates for the approximation energies and proves b3-convergence under divergence constraints, bridging discrete branched networks and smooth approximations.

Findings

Uniform estimates on pseudo-distances between $L^2$ functions

b3-convergence of the energies with divergence constraints

Connection between $H^1$ approximations and classical branched transportation models

Abstract

Models for branched networks are often expressed as the minimization of an energy $M^\alpha$ over vector measures concentrated on $1$-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation $M^\alpha_\varepsilon$, introduced by Edouard Oudet and Filippo Santambrogio, which is defined over $H^1$ vector measures. These energies induce some pseudo-distances between $L^2$ functions obtained through the minimization problem $\min \{M^\alpha_\varepsilon (u)\;:\;\nabla\cdot u=f^+-f^-\}$. We prove some uniform estimates on these pseudo-distances which allow us to establish a $\Gamma$-convergence result for these energies with a divergence constraint.