分支输运的Modica-Mortola型近似的一致估计
Uniform estimates for a Modica-Mortola type approximation of branched transportation
摘要：
分支网络的模型通常表示为能量最小化$M^α$集中在$1$维可求长集的向量测度上，具有发散约束。研究了Edouard Oudet和Filippo Santambrogio在$H^1$向量测度上定义的Modica-Mortola型逼近$M^α_\varepsilon$.这些能量导致通过最小化问题$\min \{M^α_\varepsilon (u)\;:\;\nabla\cdot u=f^+-f^-\}$获得的$L^2$函数之间的一些伪距离。我们证明了这些伪距离，使我们能够建立一个$Γ$的收敛结果，这些能量与发散约束的一些统一的估计。
英文摘要：
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 $M^α_\varepsilon$, 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 $\min \{M^α_\varepsilon (u)\;:\;\nabla\cdot u=f^+-f^-\}$. 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.
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