Nonlinear Equations of Monge-Ampere type

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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.