Nonlinear Equations of Monge-Ampere type

Monge-Ampere型非线性方程组

• 批准号: 0610374
• 负责人: Cristian Gutierrez
• 金额: $ 11.5万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0610374&HistoricalAwards=false
• 关键词: Nonlinear; Equations; Monge; Ampere; type

Nonlinear Equations of the Monge-Ampere TypeAbstract of Proposed ResearchCristian E Gutierrez This mathematical research focuses on problems concerning geometric and smoothness properties of the solutions and their derivatives of nonlinear equations of Monge-Ampere type. The general methodology is based on the use of appropriate maximum principles, localization, a priori estimates, and nonlinear variants of the Calderon-Zygmund decomposition. The equations to be studied under this grant will be a class which can be analyzed using very similar methods and techniques. This class of equations appears in several contexts, including the construction of reflector antennae and in mass transportation problems. Mass transportation problems are concerned with the optimal transport of masses from one location to another, where the optimality depends upon the context of the problem. The problems appear in several forms and in various areas of mathematics and its applications: economics, probability theory, optimization, meteorology, and computer graphics. For example, in economics they appear in planning problems at the level of an industry, a region, the whole national economy as well as the analysis of the structure of economic indices. And several different problems such as work distribution for equipment, the best use of sowing area, use of complex resources, distribution of transport flows, have a similar mathematical form.

Monge-Ampere 型非线性方程拟议研究摘要 Cristian E Gutierrez 这项数学研究的重点是有关 Monge-Ampere 型非线性方程的解及其导数的几何和平滑特性的问题。一般方法基于使用适当的最大原理、局部化、先验估计和 Calderon-Zygmund 分解的非线性变体。这笔资助下要研究的方程将是一类可以使用非常相似的方法和技术进行分析的方程。此类方程出现在多种情况下，包括反射器天线的构造和公共交通问题。大众运输问题涉及大众从一个地点到另一个地点的最优运输，其中最优性取决于问题的背景。这些问题以多种形式出现在数学及其应用的各个领域：经济学、概率论、优化、气象学和计算机图形学。例如，在经济学中，它们出现在一个行业、一个地区、整个国民经济层面的规划问题以及经济指标结构的分析中。设备的工作分配、播种面积的最佳利用、复杂资源的利用、运输流量的分配等几个不同的问题都有类似的数学形式。