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

反射器问题、Monge-Ampere型方程和完全非线性方程研究计划摘要主要研究者：黄庆波本项目主要研究完全非线性二阶方程和Monge-Ampere型方程。第一个问题是分析控制光学和电气工程中反射器天线设计的方程。其他问题包括退化Monge-Ampere方程、线性化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这些方程的解决方案的各种估计的推导。这些估计将被用来证明存在定理在各种功能空间。这将需要使用的方法，从几何，功能和真实的分析，以及那些偏微分方程，并涉及许多具有挑战性的问题。这些研究成果将有助于天线设计及相关光学、声学和电磁场理论问题的更好算法的发展，并将促进工程与数学的进一步交流。

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.