NonLinear Equations of Monge-Ampere Type

Monge-Ampere型非线性方程

• 批准号: 0300004
• 负责人: Cristian Gutierrez
• 金额: $ 9万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300004&HistoricalAwards=false
• 关键词: NonLinear; Equations; Monge; Ampere; Type

PI: Cristian E. Gutierrez, Temple UniversityDMS-0300004ABSTRACT:This mathematical research focuses on problems for nonlinear equations of Monge-Ampere type and represents a natural continuation of the work done by the PI under previous grants. The problems concentrate on the study of geometric and regularity properties of solutions to Monge-Ampere type equations. In particular, a question proposed is about the regularity of generalized solutions for an equation that appears in geometric optics for the synthesis of reflector antennae. A more general Monge-Ampere type equation that will be investigated appears naturally from mass transportation problems. We propose to develop a theory of generalized solutions and regularity for such equations. The general methodology that we plan to use to solve this set of problems consists of appropriate maximum principles for non-divergence form operators related to vector fields are of interest due to the fact that standard methods do not apply. We proposed a new approach based on integration by parts that we proved successful in the model example of the Heisenberg group, which appears in the applications to a model of human vision. Broader impacts of the proposed problems include its connections and applications within several areas in mathematics and outside. 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. 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. The understanding of the properties of optimal maps has also possible implications in numerical computations. The work proposed involves collaborations with mathematicians in the US and abroad, and it will contribute a great deal to the training of graduate students.

PI：Cristian E. Gutierrez，天普大学DMS-0300004摘要：这项数学研究重点关注 Monge-Ampere 型非线性方程问题，代表了 PI 在先前资助下所做工作的自然延续。问题集中于 Monge-Ampere 型方程解的几何和正则性质的研究。特别是，提出的问题是关于反射器天线合成的几何光学中出现的方程的广义解的规律性。将要研究的更一般的 Monge-Ampere 型方程自然会从公共交通问题中出现。我们建议发展此类方程的广义解和正则性理论。我们计划用来解决这组问题的通用方法包括与向量场相关的非发散形式算子的适当最大原则，由于标准方法不适用，所以我们很感兴趣。我们提出了一种基于分部集成的新方法，我们在海森堡群的模型示例中证明了该方法的成功，该方法出现在人类视觉模型的应用中。 所提出问题的更广泛影响包括其在数学内外多个领域的联系和应用。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。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。对最优映射属性的理解也可能对数值计算产生影响。拟议的工作涉及与美国和国外数学家的合作，它将为研究生的培养做出巨大贡献。