arXiv 1302.28862013-02math.AP

对称Monge-Kantorovich问题与向量场的极式分解

Symmetric Monge-Kantorovich problems and polar decompositions of vector fields

Nassif Ghoussoub, Abbas Moameni

备注：: 29 pages, Further updated version - if any - can be downloaded at http://birs.ca/~nassif/

摘要：

对任意给定的整数$N\geq 2$，证明了从有界域$Ω$到$\R^d$的所有有界可测向量场都是$N$ -循环单调的，直到一个保$N$ -对合的测度.证明涉及的多维对称Monge-Kantorovich问题的解决方案，我们首先研究的情况下，产品域$Ω^N$的一般成本函数。上述极分解对应于从所讨论的向量场（实际上是它们中的$N-1$）导出的特殊代价函数。在这种情况下，我们证明了$Ω^N$上所有概率测度的上确界，这些概率测度在循环置换下是不变的，并且具有给定的第一边缘$μ$，在支持$x\to (x, Sx, S^2x,..., S^{N-1}x)$形式的函数的图上的概率测度上获得，其中$S$是$Ω$上的$μ$ -测度保持变换，使得$S^N=I$ a.e.证明利用了显着的对偶之间的对合和这些哈密顿是$N$ -循环反对称。

英文摘要：

For any given integer $N\geq 2$, we show that every bounded measurable vector field from a bounded domain $Ω$ into $\R^d$ is $N$-cyclically monotone up to a measure preserving $N$-involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich problem, which we first study in the case of a general cost function on a product domain $Ω^N$. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually $N-1$ of them). In this case, we show that the supremum over all probability measures on $Ω^N$ which are invariant under cyclic permutations and with a given first marginal $μ$, is attained on a probability measure that is supported on the graph of a function of the form $x\to (x, Sx, S^2x,..., S^{N-1}x)$, where $S$ is a $μ$-measure preserving transformation on $Ω$ such that $S^N=I$ a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are $N$-cyclically antisymmetric.

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英文摘要：

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