Nonlinear equations of Monge-Ampere type

Monge-Ampere型非线性方程

• 批准号: 0901430
• 负责人: Cristian Gutierrez
• 金额: $ 20万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901430&HistoricalAwards=false
• 关键词: Nonlinear; equations; Monge; Ampere; type

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.

本计画研究Monge-Ampere型的非线性偏微分方程（即，涉及映射的雅可比行列式的方程）。大部分的研究是关于透镜和反射器天线的设计。透镜基本上是一个光学表面，它将两种具有不同折射率的材料分开。两种类型的情况被认为是：远场问题，其中光或辐射需要在一组规定的方向接收;和近场问题，其中目标或屏幕需要以规定的方式被照亮（或辐射）。 在这两种情况下，辐射都是从源点发出的。 由于折射和反射的现象总是同时发生，当能量被折射（或透射）时，总有一部分能量在内部反射中损失。在应用中，重要的是优化折射的能量，因此我们对考虑这种能量损失的模型的开发和处理感兴趣。对于这些问题的数学处理，出现了一个根本的区别：远场问题可以在最佳运输的框架内（一个处理资源最佳分配的数学领域）。相比之下，由于近场问题不是变分的，它们不能用这些术语来描述。这使得近场问题更加困难。该项目中的问题范围从各种方程的解的存在性和唯一性问题，这些问题的几何和规律性的研究模型。它们提供了不同程度的难度。最近蒙格-安培型方程的重大突破使这些问题在数学上变得合理和具有挑战性。他们中的很大一部分有实际的兴趣，此外，还有美学上的美丽。为解决这些问题而提出的想法将提高对完全非线性偏微分方程的理论理解，并将对几何光学的应用产生影响。该项目的研究源于对众多光学、声学和电磁应用以及全球定位系统（GPS）的数学描述。如果成功的话，这将对工程设计和制造带来巨大的好处。该项目在数学和外部的几个领域内有联系，互动和应用。除了前面提到的问题，大众运输中的问题还可以应用于微分几何和凸几何、优化、经济学和质量控制。对最优映射性质的理解也可能对数值计算产生影响。这项工作将涉及与美国和国外的数学家合作，并将有助于培养研究生。