Equations of Monge-Ampere Type and Fully Nonlinear Equations

Wright State University, Dayton OH

Investigators

Abstract

Proposal: DMS-0201599 Principal Investigator: Qingbo Huang, Wright State University ABSTRACT This mathematical research focuses on equations of Monge-Ampere type and fully nonlinear elliptic equations with geometric or physical motivation. The first part of this project is 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 for weak 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 equations without 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 of construction of reflector antennas. The parabolic Monge-Ampere equations are motivated from differential geometry and appear in the model of worn stones. The proposed project has a strong connection with real harmonic analysis and differential geometry. We expect that this research will stimulate more interplay among these areas.