Nonlinear equations of Monge-Ampere type

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This project investigates nonlinear partial differential equations of Monge-Ampere type (i.e., equations involving the Jacobian determinant of a map). A large portion of the research is concerned with lens and reflector antenna design. A lens is basically an optical surface that separates two materials with different indices of refraction. Two types of situations are considered: the far field problem, in which light or radiation needs to be received in a prescribed set of directions; and the near field problem, in which a target or screen needs to be illuminated (or radiated) in a prescribed way. In both cases, radiation emanates from a source point. Since the phenomena of refraction and reflection always occur simultaneously, when energy is refracted (or transmitted) there is always a fraction of this energy that is lost in internal reflection. It is important in the applications to optimize the energy refracted and so we are interested in the development and treatment of models that take into account this loss of energy. For the mathematical treatment of these problems, a fundamental difference appears: far field problems can be cast in the frame of optimal transportation (an area of mathematics dealing with the optimal allocation of resources). By contrast, since near field problems are not variational, they cannot be cast in those terms. This makes near field problems more difficult. The problems in the project range from questions of existence and uniqueness of solutions to various equations that model these problems to the study of their geometric and regularity properties. They offer various degrees of difficulty. Recent major breakthroughs for Monge-Ampere-type equations make these problems mathematically sound and challenging. A large portion of them have practical interest and, in addition, are aesthetically beautiful. The ideas proposed for their solution will improve the theoretica lunderstanding of fully nonlinear partial differential equations and will have an impact on applications in geometric optics.The research in this project arises in the mathematical description of numerous optical, acoustic, and electromagnetic applications, as well as in global positioning systems (GPS). If successful, it could be of great benefit for engineering design and manufacturing. The project has connections, interactions, and applications within several areas in mathematics and outside. In addition to what was mentioned earlier, questions in mass transportation have applications to differential and convex geometry, optimization, economics, and quality control. The understanding of the properties of optimal maps also has possible implications for numerical computations. The work will involve collaborations with mathematicians in the US and abroad and will contribute to the training of graduate students.