The Optimal Partial Transport Problem
The Optimal Partial Transport Problem
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DOI：
10.1007/s00205-008-0212-7
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发表时间：
2010-02-01
影响因子：
2.5
通讯作者：
Figalli, Alessio
中科院分区：
文献类型：
--
作者：
Figalli, Alessio
给定两个密度F和g，我们认为传输分数m的问题是[0，min {平行于f平行于（l1）的f，平行于f的g平行于f的g（l1）}] G最小化运输成本。如果每单位质量成本由垂直条x -y垂直条（2）给出，我们将看到m的唯一性对于m的独特性是[平行于f boolean，g和g平行于（l1），min {平行于平行于（L1）的F，平行于G平行于（L1）}]。这扩展了Caffarelli和McCann在Ann Math（印刷）中的结果，作者考虑了两种具有不相交支撑的密度。活跃区域的自由边界显示为（n -1） - 可旋转（前提是F和G的支撑具有Lipschitz边界），并且在支撑的几何形状上的某些弱规则性假设下，它们也是局部半核心。此外，假设F和G在两个有界的严格凸组欧米茄，R-N的lambda子集中，并在其各自的支持上脱离零和无穷大的界限，C-LOC（0，Alpha）的最佳运输图和局部C-的规律性显示了远离欧米茄布尔和兰伯达的自由边界的规律性。最后，最佳运输图扩展到活跃区域之间的全球同构形态。
Given two densities f and g, we consider the problem of transporting a fraction m is an element of [0, min{parallel to f parallel to(L1), parallel to g parallel to(L1)}] of the mass of f onto g minimizing a transportation cost. If the cost per unit of mass is given by vertical bar x - y vertical bar(2), we will see that uniqueness of solutions holds for m is an element of [parallel to f boolean AND g parallel to(L1), min{parallel to f parallel to(L1), parallel to g parallel to(L1)}]. This extends the result of CAFFARELLI and MCCANN in Ann Math (in print), where the authors consider two densities with disjoint supports. The free boundaries of the active regions are shown to be (n - 1)-rectifiable (provided the supports of f and g have Lipschitz boundaries), and under some weak regularity assumptions on the geometry of the supports they are also locally semiconvex. Moreover, assuming f and g supported on two bounded strictly convex sets Omega, Lambda subset of R-n, and bounded away from zero and infinity on their respective supports, C-loc(0, alpha) regularity of the optimal transport map and local C-1 regularity of the free boundaries away from Omega boolean AND Lambda are shown. Finally, the optimal transport map extends to a global homeomorphism between the active regions.