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

线性蒙日-安培方程及其在非线性几何偏微分方程中的应用

• 批准号: 1500400
• 负责人: Nam Le
• 金额: $ 15万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500400&HistoricalAwards=false
• 关键词: Linearized; Monge; Ampere; Equation; Applications

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.

该研究项目侧重于线性化蒙格 - 安培方程的良好特性及其在几何，机械和经济环境中非常重要的非线性，几何偏微分方程（PDE）中的应用。 Monge-Ampere类型方程自然出现在经济学和城市的交通网络规划，几何光学中反射器天线的设计以及气象学的半地球形方程中。该项目涵盖了一类PDE，其中关键结构量可能非常小（退化）或极大（单数）。该项目的主要目标旨在发现这些方程式的新基本原则和正确的观点，以开发针对它们的创新工具和方法。了解该项目中研究的仿射最大表面方程的更深特性将有助于在涉及凸面对象的架构自由形式结构，计算机图形和可视化中设计更快的算法。除应用外，对该项目研究的PDE的成功分析还将揭示数学不同领域之间的深厚而有趣的联系，例如分析，PDE，PDES，变化的计算，几何学，几何学和流体机制，从而增强它们之间的富有成果的互动，从而在它们之间进行富有成果（LMA）方程及其在非线性，几何PDE中的应用。线性化的Monge-Ampere方程是在当前对计算机图形，仿射几何，复杂几何形状，流体力学和经济学兴趣的几个基本问​​题出现的。该项目的目的是获得LMA方程的罚款和高阶边界规则性属性，并将其应用于解决分析，几何和PDE中的几个出色问题。更具体地说，项目的目标是：研究LMA方程的尖锐边界规律性；应用这些规律性结果来了解几种有趣但高度挑战的非线性，四阶几何方程的定性特性，例如仿射最大表面和Abreu方程的第二个边界价值问题，并最终解决了对Monge-Ampere操作员特征函数的全球平滑性的出色开放问题。用于攻击该项目中研究的问题的技术包括扰动论点，本地化技术，涵盖参数，部分勒金德变换，Monge-Ampere方程的几何形状以及对均质空间的谐波分析。