Rectifiability of Optimal Transportation Plans

分支输运网络的几何流

Geometric flows of branched transportation networks

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。

具有DMO型系数的椭圆算子的层位势：大块$Tb$定理、定量可求长性及自由边界问题

Layer potentials for elliptic operators with DMO-type coefficients: big pieces $Tb$ theorem, quantitative rectifiability, and free boundary problems

本文研究了具有DMO型系数的椭圆算子的层位势，建立了测度的可求长性准则与压制$T_μ$的$Tb$定理，并以此解决了有界Wiener正则域中椭圆测度的单相及双相自由边界问题。

Local regularity result for an optimal transportation problem with rough measures in the plane

We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that such convex functions cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a $C^1$ regularity result u…

单侧 Rellich 不等式、正则性问题与一致可整流性

One-sided Rellich inequalities, Regularity problem and uniform rectifiability

本文证明在边界一致 n-可整流的有界 corkscrew 域上，单侧 Rellich 不等式成立，并证明该不等式在平面情形下刻画了边界的一致可整流性；同时证明此类域上正则性问题在弱 L^1 中可解，并由此说明一般椭圆算子的 Dirichlet 问题可解性不蕴含正则性问题可解性。

Ramified optimal transportation with payoff on the boundary

This paper studies a variant of ramified/branched optimal transportation problems. Given the distributions of production capacities and market sizes, a firm looks for an allocation of productions over factories, a distribution of sales across markets, and a transport path that delivers the product to maximize its profit. Mathematically, given any two measures $μ$ and $ν$ on $X$, and a payoff function $h$, the planner wants to minimize $\mathbf{M}_{α}(T)-\int_{X}hd(\partial T)$ among all transport paths $T$ from $\tildeμ$ to $\tildeν$ with $\tildeμ\leq μ$ and $\tildeν\leq ν$, where $\mathbf{M}_…

$L^\infty$ estimates in optimal mass transportation

We show that in any complete metric space the probability measures $μ$ with compact and connected support are the ones having the property that the optimal tranportation distance to any other probability measure $ν$ living on the support of $μ$ is bounded below by a positive function of the $L^\infty$ transportation distance between $μ$ and $ν$. The function giving the lower bound depends only on the lower bound of the $μ$-measures of balls centered at the support of $μ$ and on the cost function used in the optimal transport. We obtain an essentially sharp form of this function. In the case of…