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

Monteil, Antonin

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

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.

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

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