Nonlinear Equations of Monge-Ampere type

Monge-Ampere型非线性方程研究建议摘要 本文主要研究Monge-Ampere型非线性方程解及其导数的几何性质和光滑性质。一般的方法是基于使用适当的最大值原则，本地化，先验估计，和非线性变量的卡尔德龙-Zygmund分解。根据该资助研究的方程将是 可以使用非常相似的方法和技术来分析的类。这类方程出现在几种情况下，包括反射天线的建设和大规模运输问题。质量运输问题关注的是质量从一个位置到另一个位置的最优运输，其中最优性取决于问题的上下文。这些问题以几种形式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、整个国民经济水平的规划问题以及经济指标结构的分析中。而几个不同的问题如设备的工作分配、播种面积的最佳利用、复杂资源的利用、运输流的分配等，都有类似的数学形式。

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.