Regularity Estimates for the Linearized Monge-Ampere and Degenerate Monge-Ampere Equations and Applications in Nonlinear Partial Differential Equations

线性蒙日安培方程和简并蒙日安培方程的正则估计及其在非线性偏微分方程中的应用

基本信息

批准号：
1764248
负责人：
Nam Le
金额：
$ 16.88万
依托单位：
Indiana University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764248&HistoricalAwards=false
关键词：
Regularity Estimates Linearized Monge Ampere

项目摘要

This research project studies fine quantitative behaviors of solutions to several classes of nonlinear partial differential equations (PDEs) that have connections and applications in several areas of mathematics such as analysis, PDEs, the calculus of variations, convex geometry, shape optimization, and fluid mechanics. They also appear in many areas of sciences and engineering such as economics, urban planning, meteorology, and geometric optics. For example, the Monge-Ampere type equations investigated in this project arise naturally in the optimal transportation problems (which consist of finding the least expensive way to transport a distribution of mass from one location to another) in economics and in traffic network planning in cities, in the design of reflector antennae in geometric optics and in the weather forecast models used in meteorology. The PDEs investigated in this project have two distinguished features: their key structural quantities could be possibly extremely small (degenerate) or extremely large (singular) and their settings frequently involve irregular geometries. Classical methods are usually inadequate in handling these equations and thus, their analysis calls for new methods, fresh perspectives and advancing knowledge in many fields of mathematics. The main goal of the project aims at providing deep insights into these problems, discovering novel methodologies to tackle them as well as revealing unexpected connections with other areas of mathematics. The results of this project will be widely disseminated via publications of research papers and lecture notes, via presentations at national and international venues, and via training of graduate students.This project, in the field of analysis and partial differential equations (PDEs), focuses on regularity properties of solutions to the linearized Monge-Ampere (LMA) and degenerate Monge-Ampere equations and their applications in nonlinear PDEs arising from convex geometry, optimal transportation, and meteorology. The purpose of this project is to obtain fine and higher order regularity properties of some important classes of LMA and degenerate Monge-Ampere equations and apply them to several interesting problems in analysis, geometry, and PDEs. More specifically, the objectives of the project are to: (i) investigate higher order derivatives estimates for LMA equations with lower order terms having low regularity and their applications in the semigeostrophic equations as well as polar factorizations; (ii) study the sharp Sobolev estimates for the Monge-Ampere equation and its related maximal functions; (iii) settle a shape optimization problem concerning the minimum of the Monge-Ampere eigenvalue on convex domains subject to a volume constraint; and (iv) establish global regularity for degenerate Monge-Ampere equations on nonsmooth domains and the second boundary value problem for degenerate Monge-Ampere equations. The principal investigator and his collaborators have recently developed new methods and techniques including the Green function estimates, localization technique, iteration argument, sliding paraboloids method and geometry of the Monge-Ampere equation to solve some open problems related to these proposed problems. These techniques are expected to be further developed and strengthened to successfully attack the problems proposed in this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该研究项目研究了对几类非线性偏微分方程（PDE）的溶液的精细定量行为，这些方程（PDE）在数学的多个领域（例如分析，PDE，PDE，pDES，均变异的计算，凸几何，形状优化和流体力学）中具有连接和应用。它们还出现在许多科学和工程领域，例如经济学，城市规划，气象学和几何光学。例如，该项目中研究的Monge-Ampere类型方程式自然出现在最佳运输问题（包括找到最便宜的方式，将质量分布从一个位置转移到另一个位置和城市的交通网络规划中，在几何学光学和天气模型中在气象学中使用反射型天线的设计。该项目中研究的PDE具有两个杰出的特征：它们的关键结构数量可能非常小（退化）或极大（单数），其设置经常涉及不规则的几何形状。经典方法通常在处理这些方程式时不足，因此，他们的分析要求在许多数学领域中使用新的方法，新的观点和提高知识。该项目的主要目标旨在为这些问题提供深入的见解，发现解决方案的新方法以及与其他数学领域的意外联系。该项目的结果将通过研究论文和讲座注释的出版物，国家和国际地点的介绍以及通过培训研究生进行广泛分散。该项目在分析和部分微分方程（PDES）领域（PDES），重点介绍解决方案的规则性属性，介绍解决方案对线性化的Monge-Monge-pampere（LMA）和其他典范的应用程序，以及他们的应用程序，以及他们的临时培养基，以实施范围，以实施培养基，并涉及他们的临时培养基，以实现跨越的范围，以实施范围和培养基。几何，最佳运输和气象。该项目的目的是获得一些重要类别的LMA和脱名的Monge-Ampere方程的罚款和高阶规则性能，并将其应用于分析，几何和PDE中的几个有趣的问题。更具体地说，该项目的目标是：（i）研究具有低规律性及其在半神经际方程中的较低阶段的LMA方程的高阶衍生物估计值以及极性因素化；（ii）研究Monge-Ampere方程及其相关最大功能的尖锐Sobolev估计；（iii）在凸面域上的最小值蒙格 - 安培特征值的最小值解决了形状优化问题；（iv）在非平滑域上建立退化蒙格 - 安培方程的全球规律性，而退化的monge-ampere方程的第二个边界值问题。首席研究人员及其合作者最近开发了新的方法和技术，包括绿色功能估算，本地化技术，迭代参数，滑动抛物面方法和Monge-Ampere方程的几何形状，以解决与这些提议的问题相关的一些开放问题。预计这些技术将得到进一步开发和加强，以成功地攻击该项目中提出的问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来支持的。