NonLinear Equations of Monge-Ampere Type

Pi：Cristian E.Gutierrez，Temple University DMS-0300004ABSTRACT：这项数学研究集中在Monge-Ampere类型的非线性方程问题上，代表了Pi在以前拨款下所做工作的自然延续。这些问题集中于研究Monge-Ampere型方程解的几何性质和正则性。特别是，提出的一个问题是关于几何光学中出现的反射面天线综合方程的广义解的正则性。一个更一般的Monge-Ampere型方程将被研究，它自然地出现在质量运输问题中。我们建议发展一种关于此类方程的广义解和正则性的理论。我们计划用来解决这一组问题的一般方法包括适当的最大值原理，因为标准方法不适用于与矢量场相关的非散度形式运算符。我们提出了一种基于部分积分的新方法，并在Heisenberg群的模型实例中被证明是成功的，该模型实例出现在对人类视觉模型的应用中。拟议问题的更广泛影响包括它在数学和外部的几个领域内的联系和应用。大众运输问题涉及从一个地点到另一个地点的最优运输问题，其中最优性取决于问题的背景。这些问题以多种形式出现，出现在数学及其应用的各个领域：经济学、概率论、最优化、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，以及对经济指标结构的分析。而设备的作业分配、播种面积的最佳利用、复杂资源的利用、运输流的分配等几个不同的问题，有着相似的数学形式。对最优映射的性质的理解也可能在数值计算中产生影响。这项拟议的工作涉及与美国和国外的数学家合作，它将为研究生的培养做出巨大贡献。

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.