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

本研究项目研究了几类非线性偏微分方程（PDEs）解的精细定量行为，这些方程在数学的几个领域有联系和应用，如分析、偏微分方程、变分法、凸几何、形状优化和流体力学。它们也出现在许多科学和工程领域，如经济学、城市规划、气象学和几何光学。例如，本项目研究的Monge-Ampere型方程自然出现在经济学和城市交通网络规划中的最优运输问题（包括寻找从一个地方到另一个地方运输质量分布的最便宜的方式）、几何光学中的反射天线设计以及气象学中使用的天气预报模型中。本项目研究的偏微分方程有两个显著特征：它们的关键结构量可能极小（简并）或极大（奇异），它们的设置经常涉及不规则几何。经典的方法通常不足以处理这些方程，因此，它们的分析需要新的方法，新的视角和在许多数学领域的先进知识。该项目的主要目标是提供对这些问题的深刻见解，发现解决这些问题的新方法，并揭示与其他数学领域的意想不到的联系。该项目的成果将通过出版研究论文和课堂讲稿、在国家和国际场合发表演讲以及培训研究生等方式广泛传播。本项目在分析和偏微分方程（PDEs）领域，重点研究线性化蒙日-安培方程（LMA）和退化蒙日-安培方程解的正则性及其在非线性偏微分方程中的应用，这些非线性偏微分方程由凸几何、最优运输和气象学引起。本项目的目的是获得一些重要的LMA和退化蒙日-安培方程的精细和高阶正则性，并将其应用于分析、几何和偏微分方程中的几个有趣问题。更具体地说，该项目的目标是：(i)研究具有低正则性的低阶项的LMA方程的高阶导数估计及其在半转方程和极性分解中的应用；（ii）研究Monge-Ampere方程及其相关极大函数的尖锐Sobolev估计；（iii）在体积约束下求解凸域上蒙日-安培特征值最小值的形状优化问题；（iv）建立了退化Monge-Ampere方程在非光滑区域上的全局正则性和退化Monge-Ampere方程的第二边值问题。主要研究者和他的合作者最近开发了新的方法和技术，包括格林函数估计、定位技术、迭代论证、滑动抛物面方法和蒙日-安培方程的几何来解决与这些问题相关的一些开放问题。这些技术有望得到进一步发展和加强，以成功地解决本项目提出的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.