Nonlinear Equations of Monge-Ampere Type

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ABSTRACTThis mathematical research focuses on problems for nonlinearequations of Monge-Ampere type and also for homogenization oflinear and nonlinear equations. For Monge-Ampere type equations,the problems concentrate on the study of geometric properties oftheir solutions and regularity. In particular, a question proposedis to establish Holder regularity of derivatives of generalizedsolutions for an equation that appears in geometric optics for thesynthesis of reflector antennas. The methodology that will be usedto solve this set of problems is using appropriate maximumprinciples, localization and nonlinear variants of theCalderon-Zygmund decomposition. Concerning homogenization, wedeveloped an abstract scheme that yields convergence in $L^p$spaces of correctors for a large class of equations andsystems. We propose to investigate the validity of results of thesame nature for problems of homogenization in perforated domains.This mathematical research is in the field of partial differentialequations, linear and nonlinear. These equations are the principalclassical tool of the applications of mathematics to the physicalworld. The project has a strong connection with Harmonic Analysisin Euclidean space, a subject that has flourished during thesecond half of the twentieth century and that has become anindispensable tool to provide qualitative and quantitativeinformation about the solutions of partial differential equations.The problems proposed in the first part are motivated from theengineering problem of construction of reflector antennas. Thesecond part of the project is related with the description of thebehavior of transmission of heat or electricity in materials withperiodic structure such us polymers, crystals and layered media.In particular, we are interested in obtaining accurateapproximations of the solutions that describe these phenomena.