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

研究摘要项目负责人：黄青波本项目主要研究涉及全非线性二阶方程和蒙日-安培型方程的若干课题。第一个问题是分析控制光学和电气工程中反射天线设计的方程。其他问题包括退化蒙日-安培方程、线性化蒙日-安培方程和仿射极大面方程的正则性和定性。许多提出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性定理。这将需要使用几何、泛函和实分析以及偏微分方程的方法，并涉及许多具有挑战性的问题。期望这些结果可以帮助发展更好的天线设计算法以及光学、声学和电磁场理论中的相关问题。它还应该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.