Nonlinear Equations of Monge-Ampere Type

本文主要研究Monge-Ampere型非线性方程以及线性和非线性方程的齐次化问题。对于Monge-Ampere型方程，这些问题集中于研究其解的几何性质和正则性。特别地，提出的问题是建立几何光学中用于反射面天线综合的方程的广义解的导数的Holder正则性。将用来解决这一组问题的方法是使用Calderon-Zygmund分解的适当的最大值原理、局部化和非线性变体。关于齐化，我们发展了一种抽象格式，它在一大类方程和系统的校正子的$L^p$空间中收敛。我们建议研究同样性质的结果对于穿孔区域中的齐化问题的有效性。这项数学研究是在线性和非线性偏微分方程组的领域中进行的。这些方程是将数学应用于物理世界的主要经典工具。该项目与欧几里得空间的调和分析有很强的联系，欧几里德空间的调和分析是二十世纪后半叶蓬勃发展的一门学科，它已经成为提供有关偏微分方程解的定性和定量信息的不可或缺的工具。第一部分提出的问题源于反射面天线的构造的工程问题。该项目的第二部分与描述热或电在具有周期性结构的材料中的传递行为有关，例如我们的聚合物、晶体和层状介质。我们特别感兴趣的是获得描述这些现象的解的精确近似。

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.