Equations of Monge-Ampere Type and Fully Nonlinear Equations

提案：DMS-0201599首席研究员：黄庆波，赖特州立大学摘要本数学研究的重点是Monge-Ampere型方程和完全非线性椭圆方程的几何或物理动机。本计划的第一部分致力于几个Monge-Ampere型方程。特别是，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相关的非线性半群的表征几个抛物型Monge-Ampere方程产生的变形表面，发展退化Monge-Ampere方程弱解的正则性理论，研究其与Monge-Kantorovich最优传质问题源于经济学和其他科学领域，并考虑几何光学中反射面天线综合的一个方程，第二部分是完全非线性椭圆方程，条件，Hessian方程弱解的正则性理论，以及无穷Laplacian方程作为极小化Lipschitz扩张的Euler方程，这是非线性偏微分方程的研究领域。这些方程在数学应用于真实的世界中起着至关重要的作用。 自然界和社会生活中的许多现象，如传热、多孔介质中的流动、反射器的构造、空气质量的优化配置等，都是由非线性偏微分方程来描述的，对这些方程的研究将有助于理解这些现象的本质，并有助于发展实用、快速、可靠的数值算法。提出的问题之一是有关Monge-Kantorovich最优传质问题出现在经济，物理和气象学。另一个问题来自反射面天线的工程问题。抛物型Monge-Ampere方程是从微分几何出发的，并出现在磨损的石头模型中。所提出的方案与真实的调和分析和微分几何有很强的联系。我们希望这项研究能够促进这些领域之间的相互作用。

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.