Nonlinear Equations of Monge-Ampere type

Monge-Ampere型非线性方程建议研究的摘要Cristian E Gutierrez这项数学研究集中于Monge-Ampere型非线性方程及其导数的几何性质和光滑性问题。一般的方法是基于适当的最大值原理、局部化、先验估计和Calderon-Zygmund分解的非线性变体。在这项资助下要研究的方程将是一类可以使用非常类似的方法和技术进行分析的类别。这类方程出现在几种情况下，包括反射器天线的构造和大众运输问题。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。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。

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.