发表机构
Institute of Science Tokyo; Japan Society for the Promotion of Science（东京科学研究所; 日本学术振兴会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对分支输运问题，通过修改Brakke变分逼近格式构造输运网络的几何流，证明极限关于平坦范数Hölder连续且成本递减，并分析1-varifold逼近流的可求长性与运动定律。
AI 中文摘要
分支输运问题是在具有给定边界的 $\mathbb{R}^n$ 中正规 $1$-流上的非凸且非光滑的变分优化问题。其最优性与某个非递减、下半连续且次可加的函数 $\tau:\mathbb{R}_+\to\mathbb{R}_+$（满足 $\tau(0)=0$）有关，该函数描述了每单位距离移动质量为 $m$ 的物质的成本 $\tau(m)$。次可加性导致（次优）解的支撑中出现复杂的分层分支模式。这些网络状集合似乎具有与所谓的 Brakke 流（平均曲率流的弱推广）中出现的奇异曲面相似的规则性性质。我们通过修改 Brakke 的变分逼近格式，构造了对应于正规实 $1$-可求长流的输运网络的几何流。我们证明了存在一个关于平坦范数 Hölder 连续的极限，并且其分支输运成本沿几何演化递减。我们进一步分析了由 $1$-varifold 逼近的密切相关的几何流，其权重测度模拟分支输运成本，并建立了其 $1$-可求长性以及类似于 Brakke 不等式的运动定律。
英文摘要
The branched transport problem is a nonconvex and nonsmooth variational optimization problem on normal $1$-currents in $\mathbb{R}^n$ with prescribed boundary. The optimality is with respect to some non-decreasing, lower semicontinuous, and subadditive function $τ:\mathbb{R}_+\to\mathbb{R}_+$ with $τ(0)=0$ describing the cost $τ(m)$ to move an amount of mass $m$ per unit distance. The subadditivity leads to complicated, hierarchically ramified patterns in the support of (suboptimal) solutions. These network-like sets appear to have regularity properties similar to those of the singular surfaces arising in the so-called Brakke flow, a weak generalization of the mean curvature flow. We construct a geometric flow of transportation networks, which correspond to normal real $1$-rectifiable currents, by modifying Brakke's variational approximation scheme. We prove the existence of a limit that is Hölder continuous with respect to the flat norm and whose branched transport cost decreases along the geometric evolution. We further analyze a closely related geometric flow approximated by $1$-varifolds, whose weight measures model the branched transport cost, and establish its $1$-rectifiability together with a motion law analogous to Brakke's inequality.
