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

本研究项目主要研究线性化的Monge-Ampere方程的解的精细性质及其在几何、力学和经济领域具有重要意义的非线性几何偏微分方程解的应用。Monge-Ampere型方程自然地出现在经济学中的最优交通问题和城市交通网络规划中，在几何光学中的反射面天线的设计中，在气象学的半地转方程中。该项目涵盖了一大类PDE，其中关键的结构量可能是极小的(简并的)或极大的(奇异的)。该项目的主要目标是发现关于这些方程式的新的基本原则和正确的观点，以便开发解决这些方程式的创新工具和方法。了解本课题中研究的仿射极大曲面方程的更深层次的性质，将有助于在建筑自由结构、计算机图形学和涉及凸对象的可视化中设计更快的算法。除了应用之外，本项目中研究的偏微分方程解的成功分析将揭示不同数学领域如分析、偏微分方程组、变分、几何和流体力学之间的深刻而有趣的联系，从而加强它们之间卓有成效的相互作用。本项目在分析和偏微分方程组(PDE)领域，专注于线性化Monge-Ampere(LMA)方程解的正则性及其在非线性、几何偏微分方程组中的应用。线性化的Monge-Ampere方程产生于当前计算机图形学、仿射几何、复杂几何、流体力学和经济学中的几个基本问题。这个项目的目的是得到LMA方程的精细和高阶边界正则性，并将它们应用于解决分析、几何和偏微分方程组中的几个突出问题。更具体地说，该项目的目标是：研究LMA方程的精确边界正则性；应用这些正则性结果来理解几个有趣但极具挑战性的四阶几何方程的定性性质，例如仿射极大曲面的第二边值问题和Abreu方程，并最终解决Monge-Ampere算子特征函数的全局光滑性的未决问题。本项目所研究的问题所使用的技术包括扰动变元、局部化技术、覆盖变元、部分勒让德变换、Monge-Ampere方程的几何以及齐次空间上的调和分析。

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.