NonLinear Equations of Monge-Ampere Type

Temple University, Philadelphia PA

Investigators

Abstract

PI: Cristian E. Gutierrez, Temple University DMS-0300004 ABSTRACT: 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.