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

Monteil, Antonin¹ 

¹ Laboratoire de Mathématiques d’Orsay, Université Paris-Sud 11, Bât. 425, 91405 Orsay, France.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 309-335

Résumé

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 }$${}_{\epsilon }$, introduced by Edouard Oudet and Filippo Santambrogio, which is defined over $H^1$ vector measures. These energies induce some pseudo-distances between $L^2$ functions obtained through the minimization problem $min\{M^{\alpha }$${}_{\epsilon }\left(u\right)$: $\nabla ·u=f^{+}-f^{-}\}$. 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.

Détail

Reçu le : 2015-03-11
Accepté le : 2015-09-30
DOI : 10.1051/cocv/2015049

Classification : 49J45, 90B06, 90B18
Keywords: Branched transportation networks, Γ-convergence, phase field models