基于多边缘最优输运的DeGiorgi-type系统解耦
Decoupling of DeGiorgi-type systems via multi-marginal optimal transport
摘要：
我们展示了一个令人惊讶的椭圆梯度方程组的偏微分方程，多边际Monge-Kantorovich最优输运问题，和多变量Hardy-Littlewood不等式之间的关系。我们证明了[6]中提出的可定向椭圆系统的概念，它意味着（在低维情况下）具有某些单调性质的解本质上是1维的，等价于相容费用函数的定义，已知它意味着某些Monge-Kantorovich问题的最优测度的唯一性和结构结果[11]。定向非线性和兼容的费用函数原来也与次模函数，这出现在重排不等式的Hardy-Littlewood型。我们使用这种等价性来建立椭圆型偏微分方程某些解的解耦结果，并表明在可定向条件下，解耦具有额外的属性，由于连接到最优输运。
英文摘要：
We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal Monge-Kantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an orientable elliptic system, conjectured in [6] to imply that (in low dimensions) solutions with certain monotonicity properties are essentially 1-dimensional, is equivalent to the definition of a compatible cost function, known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems [11]. Orientable nonlinearities and compatible cost functions turned out to be also related to submodular functions, which appear in rearrangement inequalities of Hardy-Littlewood type. We use this equivalence to establish a decoupling result for certain solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.
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