Nonlinear Equations of Monge-Ampere Type

Temple University, Philadelphia PA

Investigators

Abstract

ABSTRACT This mathematical research focuses on problems for nonlinear equations of Monge-Ampere type and also for homogenization of linear and nonlinear equations. For Monge-Ampere type equations, the problems concentrate on the study of geometric properties of their solutions and regularity. In particular, a question proposed is to establish Holder regularity of derivatives of generalized solutions for an equation that appears in geometric optics for the synthesis of reflector antennas. The methodology that will be used to solve this set of problems is using appropriate maximum principles, localization and nonlinear variants of the Calderon-Zygmund decomposition. Concerning homogenization, we developed an abstract scheme that yields convergence in $L^p$ spaces of correctors for a large class of equations and systems. We propose to investigate the validity of results of the same nature for problems of homogenization in perforated domains. This mathematical research is in the field of partial differential equations, linear and nonlinear. These equations are the principal classical tool of the applications of mathematics to the physical world. The project has a strong connection with Harmonic Analysis in Euclidean space, a subject that has flourished during the second half of the twentieth century and that has become an indispensable tool to provide qualitative and quantitative information about the solutions of partial differential equations. The problems proposed in the first part are motivated from the engineering problem of construction of reflector antennas. The second part of the project is related with the description of the behavior of transmission of heat or electricity in materials with periodic structure such us polymers, crystals and layered media. In particular, we are interested in obtaining accurate approximations of the solutions that describe these phenomena.