Reflector Problem, Equations of Monge-Ampere Type and Fully Nonlinear Equations
Abstract of Proposed Research
Principal Investigator: Qingbo Huang
This 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.
