Reflector Problem, Equations of Monge-Ampere Type and Fully Nonlinear Equations

反射面问题、Monge-Ampere型方程和完全非线性方程建议的研究摘要主要研究人员：黄庆波本项目是研究涉及完全非线性二阶方程和Monge-Ampere型方程的若干课题。第一个问题是光学和电气工程中控制反射面天线设计的方程的分析。其他问题还包括关于退化的Monge-Ampere方程、线性化的Monge-Ampere方程和仿射极大曲面方程的正则性和定性性质。许多拟议的研究涉及对这些方程的解的各种估计的推导。这些估计将被用来证明各种函数空间中的存在定理。这将需要使用几何、泛函和实分析以及偏微分方程式的方法，并涉及许多具有挑战性的问题。期望这些结果能够帮助开发更好的算法来设计天线以及光学、声学和电磁场理论中的相关问题，也将促进工程和数学之间的进一步互动。

Reflector Problem, Equations of Monge-Ampere Type andFully Nonlinear EquationsAbstract of Proposed ResearchPrincipal Investigator: Qingbo HuangThis project is to study a number of topics involving fully nonlinear second order equations and equations of Monge-Ampere type. The first question is the analysis of the equation that governs the design of reflector antennae in optics and electrical engineering. Other questions include some about the regularity and qualitative properties of the degenerate Monge-Ampere equation, the linearized Monge-Ampere equation and the affine maximal surface equation. Much of the proposed research involves the derivation of various estimates on the solutions of these equations. These estimates will then be used to prove existence theorems in various function spaces. This will require the use of methods from geometry, functional and real analysis as well as those of partial differential equations and involves many challenging questions. It is expected that these results could help the development of better algorithms for antenna design and related problems in optics, acoustics and electromagnetic field theory.It should also promote further interaction between engineering and mathematics.