Equations of Monge-Ampere Type and Fully Nonlinear Equations

Monge-Ampere型方程和完全非线性方程

• 批准号: 0201599
• 负责人: Qingbo Huang
• 金额: $ 6.05万
• 依托单位: Wright State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0201599&HistoricalAwards=false
• 关键词: Equations; Monge; Ampere; Type; Fully

Proposal: DMS-0201599Principal Investigator: Qingbo Huang, Wright State UniversityABSTRACTThis mathematical research focuses on equations of Monge-Ampere type and fully nonlinear elliptic equations with geometric or physical motivation. The first part of this projectis devoted to several equations of Monge-Ampere type. In particular, we propose to study regularity of solutions and other quantitative properties such as the large time behavior of solutions and characterization of associated nonlinear semigroups for several parabolic Monge-Ampere equations arising in the deformation of surfaces, to develop regularity theory forweak solutions of degenerate Monge-Ampere equations and study its interaction with Monge-Kantorovich optimal mass transfer problem stemming from economics and other areas of science,and to consider an equation arising in geometric optics for the synthesis of reflector antennas.The second part of the project is concerned with fully nonlinear elliptic equationswithout concavity condition, regularity theory of weak solutions of Hessian equations, and the infinity Laplacian equation as the Euler equation of minimizing Lipschitz extension.This research is in the area of nonlinear partial differential equations. These equations play a crucial role in the application of mathematics to real world. Most phenomena in nature and society, such as heat transfer, flows in porous media, construction of reflectors, optimal disposition of air masses, are described by nonlinear partial differential equations.Study of these equations will greatly help understand the nature of these phenomena and develop practical, fast, and reliable numerical algorithms. One of the proposed problems is related to Monge-Kantorovich optimal mass transfer problem appearing in economic, physics, and meteorology. Another problem arises from the engineering problem ofconstruction of reflector antennas. The parabolic Monge-Ampere equations are motivated from differential geometry and appear in the model of worn stones. The proposed projecthas a strong connection with real harmonic analysis and differential geometry. We expect that this research will stimulate more interplay among these areas.

建议：DMS-0201599主要研究员：黄庆波，莱特州立大学[摘要]本文主要研究具有几何或物理动机的Monge-Ampere型方程和完全非线性椭圆型方程。这个投影的第一部分致力于几个Monge-Ampere型方程。特别地，我们建议研究几个由表面变形引起的抛物型Monge-Ampere方程的解的正则性和其它定量性质，如解的大时间行为和相关的非线性半群的特征，发展退化的Monge-Ampere方程的弱解的正则性理论，并研究它与源于经济学和其他科学领域的Monge-Kantorovich最优传质问题的相互作用，以及考虑几何光学中出现的用于反射面天线综合的方程。项目的第二部分涉及无凹条件的完全非线性椭圆型方程，Hessian方程弱解的正则性理论，而无穷拉普拉斯方程作为极小化Lipschitz扩张的欧拉方程。这些方程在将数学应用到现实世界中起着至关重要的作用。自然界和社会中的大多数现象，如热传递、多孔介质中的流动、反射体的构造、空气质量的优化配置等，都是用非线性偏微分方程组来描述的，研究这些方程将有助于理解这些现象的本质，并开发实用、快速和可靠的数值算法。提出的问题之一与经济学、物理学和气象学中出现的Monge-Kantorovich最优传质问题有关。另一个问题来自反射面天线的建造工程问题。抛物型Monge-Ampere方程从微分几何出发，出现在磨石模型中。所提出的方案与实调和分析和微分几何有很强的联系。我们预计，这项研究将促进这些领域之间更多的相互作用。