Variational problems in mathematics and the sciences

这篇论文讨论了在数学和科学中应用的一系列变分问题的分析。其中的中心问题是Monge和Kantorovich的最优运输问题***，可以将其讽刺为：***给定一组铁矿和一组消耗铁矿石的工厂***，决定哪个矿山应该供应***哪个工厂，以使总运输成本最小化。***当矿山和工厂连续分布在***弯曲的景观中，其几何形状反映在运输成本中，***问题与***几何，拓扑，动力系统和非线性偏微分方程有着深刻的联系。尽管最近取得了很大进展，但一些基本问题仍有待解决。我们的目标是这些，以及这个问题的显著应用***在经济学和物理学中的模型。******在过去的十年里，我们对这些问题的理解取得了巨大的进步。答案在很大程度上取决于成本的选择。成本的拓扑限制保证了只有一个解决方案，每个矿山供应一个工厂；在进一步的几何限制下，这种对应关系是光滑的。然而，对于紧凑的流形，如地球球面上的平滑运输成本，一个或两个条件必须失败。表征不连续成为一个有趣的问题，由此产生的传输模式要复杂得多：我们计划研究解的唯一性，以及它们支持的维数、分支和平滑性。******与此同时，我们还解决了诸如：拥堵如何影响情况？一般来说，无论在哪里发生传输，该解决方案都会使拥塞边界饱和。因此，它完全是由***活跃和非活跃的运输路线的分离决定的。我们计划研究这种分离的解析性质和几何性质。******最后，我们的目标是选择应用到物理学和经济学。上述不连续点***对应于大气运动半匀转理论中的压力锋***。我们试图分析它们的形成和演变。从半导体中的电子输运到薄膜动力学的非线性扩散模型现象；我们试图使用关于运输成本的梯度流动力学公式来分析它们的长期行为。稳定婚姻、最优定价和团队建设问题也涉及到连续线性规划的求解。***我们寻求对模型的分析，以提高我们对潜在自然现象的理解，并在资源分配、匹配问题、团队建设以及影响、解释、利用和控制经济现象的能力方面得出更好的决策原则

***This proposal addresses the analysis of a collection of variational ***problems with applications in mathematics and the sciences.***Central among them is an optimal transportation problem***of Monge and Kantorovich, which can be caricatured as follows:***given a collection of iron mines and a collection of factories ***consuming iron ore, decide which mine should supply***which factory in order to minimize the total transportation costs.***When the mines and factories are distributed continuously throughout***a curved landscape, with geometry reflected in the transport cost,***the problem has deep connections to***geometry, topology, dynamical systems, and nonlinear partial differential equations.***Despite much recent progress, fundamental questions remain to be resolved. ***We target these, and striking applications of this problem ***to models in economics and in physics.******The last decade has seen enormous advances in our understanding of such***issues. The answers depend heavily on***the choice of cost. Topological restrictions on the cost guarantee***that there is only one solution and every mine supplies a single factory; ***under further geometric restrictions this correspondence is smooth. ***However for smooth transportation costs on compact manifolds such as the***spherical surface of the earth one or both conditions must fail.***Characterizing discontinuities then becomes an interesting question,***The resulting transport patterns are much more complicated: we plan to investigate***uniqueness of solutions, and what can be said about dimension,***branching, and smoothness of their support. ******At the same time, we tackle variants such as: how does congestion affect the situation? ***Generically, the solution saturates the congestion bound***wherever transport occurs. Thus it is completely determined by a separation of***active from inactive transportation routes. We plan to investigate the analytic and geometric***properties of this separation. ******Finally, we target select applications to physics and to economics. The discontinuities***described above correspond to pressure fronts in the semigeostrophic theory***of atmospheric motion. We seek to analyze their formation and evolution.***Nonlinear diffusions model phenomena ranging from electronic transport***in semiconductors to the dynamics of thin films; we seek to analyze their long time***behaviour using a gradient flow formulation of the dynamics with respect to***transportation costs. Stable marriage, optimal-pricing and team-building***problems also involve the solution of continuous linear programs.***We seek an analysis of the models which improves our understanding***of the underlying natural phenomena, and leads to better decision-making principles***concerning allocation of resources, matching problems, team-building, and***the ability to influence, explain, exploit, and control economic phenomena.**