The Linearized Monge-Ampere Equation and Applications in Nonlinear, Geometric Partial Differential Equations

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This research project focuses on fine properties of solutions to the linearized Monge-Ampere equation and their applications to nonlinear, geometric partial differential equations (PDEs) of great importance in geometric, mechanical, and economic contexts. The Monge-Ampere type equations arise naturally in optimal transportation problems in economics and in traffic network planning in cities, in the design of reflector antennae in geometric optics, and in the semi-geostrophic equations of meteorology. The project covers a broad class of PDEs where key structural quantities could be possibly extremely small (degenerate) or extremely large (singular). The main goal of the project aims at discovering new underlying principles and correct perspectives on these equations in order to develop innovative tools and methodologies to tackle them. Understanding deeper properties of affine maximal surface equations studied in this project will help design faster algorithms in architectural free-form structures, in computer graphics, and in visualization where convex objects are involved. In addition to applications, the successful analysis of PDEs investigated in this project will reveal deep and interesting connections between different areas of mathematics such as analysis, PDEs, the calculus of variations, geometry, and fluid mechanics, thereby augmenting the fruitful interaction among them.This project, in the field of analysis and partial differential equations (PDEs), focuses on regularity properties of solutions to the linearized Monge-Ampere (LMA) equation and their applications to nonlinear, geometric PDEs. The linearized Monge-Ampere equation arises in several fundamental problems of current interest in computer graphics, affine geometry, complex geometry, fluid mechanics, and economics. The purpose of this project is to obtain fine and higher order boundary regularity properties of the LMA equation and apply them to tackle several outstanding problems in analysis, geometry, and PDEs. More specifically, the objectives of the project are to: investigate sharp boundary regularity for the LMA equation; apply these regularity results to understand qualitative properties of several interesting but highly challenging nonlinear, fourth-order geometric equations such as the second boundary value problems for the affine maximal surface and Abreu's equations, and finally resolve the outstanding open problem on global smoothness of eigenfunctions to the Monge-Ampere operator. The techniques used in attacking the problems under study in this project include perturbation arguments, localization techniques, covering arguments, partial Legendre transform, geometry of the Monge-Ampere equation, and also harmonic analysis on homogeneous spaces.