Equations of Monge-Ampere Type and Fully Nonlinear Equations

摘要本文主要研究具有几何或物理动机的蒙日-安培型方程和全非线性椭圆型方程。本课题的第一部分主要讨论了几个蒙日-安培型方程。特别地，我们建议研究解的正则性和其他定量性质，如解的大时间行为和在曲面变形中产生的几个抛物型蒙日-安培方程的相关非线性半群的表征。发展退化蒙日-安培方程弱解的正则性理论，研究其与源自经济学和其他科学领域的蒙日-坎托洛维奇最优传质问题的相互作用，并考虑反射面天线合成几何光学中的一个方程。第二部分研究了无凹性条件的完全非线性椭圆方程，Hessian方程弱解的正则性理论，以及作为最小化Lipschitz扩展的Euler方程的无穷拉普拉斯方程。本研究属于非线性偏微分方程领域。这些方程在数学应用于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，如传热、多孔介质中的流动、反射器的构造、气团的最佳配置等，都可以用非线性偏微分方程来描述。对这些方程的研究将极大地帮助理解这些现象的本质，并开发实用、快速、可靠的数值算法。其中一个被提出的问题与Monge-Kantorovich最优传质问题有关，该问题出现在经济学、物理学和气象学中。另一个问题来自于反射天线结构的工程问题。抛物型蒙日-安培方程由微分几何推导而来，并出现在磨损石模型中。所提出的方案与实谐波分析和微分几何有很强的联系。我们期望这项研究将激发这些领域之间更多的相互作用。

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.