Nonlinear Equations of Monge-Ampere Type

Monge-Ampere型非线性方程组

• 批准号: 0070648
• 负责人: Cristian Gutierrez
• 金额: $ 7.8万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070648&HistoricalAwards=false
• 关键词: Nonlinear; Equations; Monge; Ampere; Type

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.

本文主要研究Monge-Ampere型非线性方程组的数学问题以及线性和非线性方程组的均匀化问题。对于Monge-Ampere型方程，问题主要集中在研究其解的几何性质和正则性。特别地，本文提出了一个问题，即建立几何光学中反射面天线综合方程的广义解的导数的保持器正则性。将被用来解决这一组问题的方法是使用适当的maximumprinciples，本地化和非线性变量的theCalderon-Zygmund分解。关于均匀化，我们开发了一个抽象的计划，产生收敛在L^p$空间的校正一个大类的方程和系统。我们建议调查的有效性结果的性质相同的问题的均匀化在perforated domains.This mathematical research is in the field of partial differential equations，linear and nonlinear.这些方程是将数学应用于物理世界的主要经典工具。该项目与欧几里德空间中的调和分析有很强的联系，欧几里德空间中的调和分析是一个在二十世纪后半期蓬勃发展的学科，已经成为提供关于偏微分方程解的定性和定量信息的不可缺少的工具。该项目的第二部分与描述具有周期性结构的材料（例如聚合物、晶体和分层介质）中的热或电传输行为有关。特别是，我们有兴趣获得描述这些现象的解的准确近似值。