Variational problems in mathematics and the sciences

该建议涉及分析一系列变化的 数学和科学中的应用问题。 其中的核心问题是最优运输问题 蒙日和康托洛维奇，这可以漫画如下： 因为这里有很多铁矿和工厂 消耗铁矿石，决定哪个矿应该供应 为了使总运输成本最小化。 当矿山和工厂不断分布在 一个弯曲的景观，几何形状反映在运输成本中， 这个问题与 几何学、拓扑学、动力系统和非线性偏微分方程。 尽管最近取得了很大进展，但一些根本问题仍有待解决。 我们的目标是这些，以及这个问题的惊人应用 经济学和物理学中的模型。 在过去的十年里，我们对这种现象的理解有了巨大的进步。 问题.答案很大程度上取决于 成本的选择。对成本保证的拓扑限制 只有一个解决方案，每个矿山都为一家工厂提供服务; 在进一步的几何限制下，这种对应是平滑的。 然而，对于紧凑型歧管的平稳运输成本，例如 地球表面的一个或两个条件必须失败。 描述不连续性就变成了一个有趣的问题， 由此产生的运输模式要复杂得多：我们计划调查 解的唯一性，以及维数的定义 分支和光滑的支撑。 与此同时，我们还可以解决各种问题，例如：拥堵如何影响情况？ 一般来说，该解决方案使拥塞边界饱和 无论在哪里运输。因此，它完全是由一个分离的 从不活跃的运输路线。我们计划研究 这种分离的特性。 最后，我们的目标选择应用物理学和经济学。间断部 对应于半地转理论中的压力锋 大气运动。我们试图分析它们的形成和演变。 非线性扩散模型的现象，从电子输运 在半导体薄膜的动力学;我们试图分析他们的长时间 行为使用梯度流公式的动态相对于 运输费用。稳定的婚姻、最优定价和团队建设 问题还涉及连续线性规划的解。 我们寻求对模型的分析，以提高我们的理解 的基本自然现象，并导致更好的决策原则 关于资源分配，匹配问题，团队建设， 影响、解释、利用和控制经济现象的能力。

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.