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

本研究项目聚焦于线性化蒙日-安培方程解的精细性质及其在非线性几何偏微分方程（PDEs）中的应用，这些方程在几何、力学和经济领域具有重要意义。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天线设计中，以及在气象学的半地转方程中，都会自然地出现蒙日-安培型方程。该项目涵盖了广泛的pde类别，其中关键结构数量可能极小（简并）或极大（奇异）。该项目的主要目标是发现新的基本原则和正确的观点，以开发创新的工具和方法来解决这些问题。本项目研究的仿射最大曲面方程的更深层性质将有助于在建筑自由形式结构、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，本项目对偏微分方程的成功分析将揭示分析、偏微分方程、变分法、几何和流体力学等不同数学领域之间深刻而有趣的联系，从而增加它们之间富有成效的互动。本项目在分析和偏微分方程（PDEs）领域，重点研究线性化蒙日-安培（LMA）方程解的正则性及其在非线性几何偏微分方程中的应用。线性化的蒙日-安培方程出现在当前计算机图形学、仿射几何、复杂几何、流体力学和经济学的几个基本问题中。本课题的目的是获得LMA方程的精细和高阶边界正则性，并将其应用于分析、几何和偏微分方程中的几个突出问题。更具体地说，该项目的目标是：研究LMA方程的锐边界正则性；应用这些正则性结果来理解一些有趣但极具挑战性的非线性四阶几何方程的定性性质，如仿射极大曲面的第二次边值问题和Abreu方程，并最终解决了Monge-Ampere算子的特征函数全局光滑性的突出开放问题。在这个项目中用于解决问题的技术包括摄动参数，局部化技术，覆盖参数，部分勒让德变换，蒙日-安培方程的几何，以及齐次空间的谐波分析。

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.