Mathematical Sciences: Fourier Analysis and Partial Differential Equations

数学科学：傅里叶分析和偏微分方程

• 批准号: 9311692
• 负责人: Robert Fefferman
• 金额: $ 12.38万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9311692&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Fourier; Analysis; Partial

Fefferman 9311692 This project will focus on several ongoing research themes in harmonic analysis and partial differential equations. In particular work will continue on the extension of the theory of singular integrals on product spaces. Here the objective is to analyze the properties of operators which commute with various groups of dilations of Euclidean space. So far, a complete theory is available only for the classical dilations and product dilations. The theory evolving is quite different from the Calderon-Zygmund theory. Work will also continue on problems from elliptic equations with non-smooth coefficients. In particular, the solvability of the Dirichlet problem with Lp coefficients will be treated. An important goal in this research is to determine the extent that one can perturb coefficients of the equation and still guarantee that the solutions still remain in the same Lebesgue space. In the same spirit, elliptic operators in nondivergence form with coefficients which are only assumed to be measurable will be studied. The approach used will be that of approximating solutions by solutions to equations with smooth coefficients and determine whether or not the process leads to unique limits regardless of the smoothing of the original operator. *** Partial differential operators form a basis for studying the generic forms of partial differential equations. The analysis of these operators through deep abstract methods leads to surprisingly complete information about related classes of equations. Present research concentrates on operators in which smoothness assumptions are limited to the bare essentials. In this way, results reflect a more realistic picture of the physical world which the differential equations seek to represent.

9311692 这个项目将集中在调和分析和偏微分方程的几个正在进行的研究主题。 特别是工作将继续对产品空间的奇异积分理论的延伸。 本文的目的是分析与欧氏空间的各种扩张群可交换的算子的性质。 到目前为止，只有经典膨胀和乘积膨胀才有完整的理论。 该理论的发展与卡尔德龙-齐格蒙德理论有很大的不同。还将继续研究非光滑系数椭圆方程的问题。 特别是，可解性的Dirichlet问题的Lp系数将被处理。 在这项研究中的一个重要目标是确定的程度，可以扰动系数的方程，仍然保证解决方案仍然保持在同一个勒贝格空间。 在同样的精神，椭圆算子的非发散形式的系数，仅假设是可测的将进行研究。 所使用的方法将是近似的解决方案的解决方案的方程与平滑系数，并确定是否或不的过程导致唯一的限制，而不管原来的运营商的平滑。 *** 偏微分算子是研究偏微分方程一般形式的基础。通过深入的抽象方法对这些算子进行分析，可以获得关于相关方程类的令人惊讶的完整信息。 目前的研究集中在运营商的平滑性假设仅限于裸露的本质。 通过这种方式，结果反映了微分方程试图表示的物理世界的更现实的画面。