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

Laboratoire de Mathématiques d’Orsay, Université Paris-Sud 11, Bât. 425, 91405 Orsay, France.
antonin.monteil@gmail.com

Received: 11 March 2015
Accepted: 30 September 2015

Abstract

Models for branched networks are often expressed as the minimization of an energy M^α over vector measures concentrated on 1-dimensional rectifiable sets with a divergence constraint. We study a Modica–Mortola type approximation M^α_ε, introduced by Edouard Oudet and Filippo Santambrogio, which is defined over H¹ vector measures. These energies induce some pseudo-distances between L² functions obtained through the minimization problem min {M^α_ε(u): ∇·u = f⁺ − f⁻ }. We prove some uniform estimates on these pseudo-distances which allow us to establish a Γ-convergence result for these energies with a divergence constraint.