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首页> 外文期刊>Journal of Differential Equations >Mass transport problems for the Euclidean distance obtained as limits of ρ-Laplacian type problems with obstacles
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Mass transport problems for the Euclidean distance obtained as limits of ρ-Laplacian type problems with obstacles

机译：作为具有障碍物的ρ-Laplacian型问题的极限而获得的欧几里德距离的质量输运问题

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摘要

In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying an extra cost of - gi(.v) for extra production of one unit at location x and an extra cost of g2(y) for creating one unit of demand at j. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p →∞ to a double obstacle problem (with obstacles g_1, g2) for the ρ-Laplacian. In fact, under a certain natural constraint on the extra costs (that is equivalent to impose that the total optimal cost is bounded) we prove that this limit gives the extra material and extra demand needed for optimality and a Kantorovich potential for the mass transport problem involved. We also show that this problem can be interpreted as an optimal mass transport problem in which one can make the transport directly (paying a cost given by the Euclidean distance) or may hire a courier that cost g2(y) - g1(x) to pick up a unit of mass at у and deliver it to x. For this different interpretation we provide examples and a decomposition of the optimal transport plan that shows when we have to use the courier.
机译：在本文中，我们分析了一种大众运输问题，该问题包括最优地移动（支付由欧几里得距离给出的运输成本）一种商品的数量大于或等于固定数量以满足也大于或等于固定数量的需求第一种，有义务支付额外的-gi（.v）成本，以在x位置额外生产一个单位，并支付g2（y）的额外成本，以在j位置创建一个需求单位。多余的质量（商品/需求）是问题的未知数。我们对这个问题的解决方法是将ρ-拉普拉斯算子的极限设为p→∞作为双障碍问题（障碍g_1，g2）。实际上，在对额外成本有一定自然约束的情况下（相当于施加总最优成本的约束），我们证明了该限制给出了优化所需的额外材料和额外需求，以及大规模运输问题的Kantorovich潜力参与。我们还表明，这一问题可以解释为一种最优的大规模运输问题，在该问题中，人们可以直接进行运输（支付欧几里得距离所给的成本），或者可以雇用成本为g2（y）-g1（x）的快递在у处拾取一个质量单位并将其传递给x。对于这种不同的解释，我们提供示例和最佳运输计划的分解图，以显示何时需要使用快递。

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