Mathematical Sciences: Partial Differential Equations, GroupRepresentations and Number Theory

数学科学：偏微分方程、群表示和数论

• 批准号: 8805909
• 负责人: Leon Ehrenpreis
• 金额: $ 6.05万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8805909&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Partial; Differential; Equations

Ehrenpreis 8805909 This project will encompass three areas of mathematical analysis, each of which represents a continuation of earlier research. The first concerns the extensions of solutions of partial differential equations from domains which have the shape of convex wedges to surrounding domains when the wedges share common boundary elements. In its general form the goal is to determine the relations among the Cauchy data on the boundary which imly the extendability of solutions. Solutions have been obtained for constant coefficient equations in low dimentions. Work will also be done on problems of scattering theory. A recent concept of nonabelian mechanics in which the poisson bracket of space variables makes the space and momentum into seperate Lie algebras (rather than vanishing), when suitably paired. This approach leads to a new notion of completely integrable systems which can be applied to scattering problems on nonabelian groups. Work planned seeks to develop this research in higher dimensions. Efforts will also continue on the question of determining when a left-invariant operater on a Lie group can be surjective on the infinitely differentiable functions. This work derives from earlier research on the so-called Lewy operater - a partial differential equation with no solutions. Initial results using the Husenberg group suggest that a general framework for deciding on the surjectivity of differential operators and hence the conditions under which one can expect solutions of the corresponding inhomogeneous equations.

这个项目将包括三个数学分析领域，每个领域都代表了早期研究的延续。第一部分是关于在凸楔形状的域上的偏微分方程的解在凸楔有共同边界元的情况下向周围域的扩展。其一般形式的目标是确定边界上的柯西数据之间的关系，这些关系决定了解的可扩展性。得到了低维常系数方程的解。我们还将研究散射理论的问题。非阿贝尔力学的一个新概念，其中空间变量的泊松括号使空间和动量在适当配对时成为单独的李代数（而不是消失）。这种方法提出了完全可积系统的新概念，可应用于非abel群上的散射问题。计划开展的工作旨在在更高的维度上开展这项研究。我们还将继续研究李群上的左不变算子在无穷可微函数上何时是满射的问题。这项工作源于早期对所谓的路易算子的研究——一个没有解的偏微分方程。利用Husenberg群的初步结果提出了决定微分算子满射性的一般框架，从而确定了可以期望相应非齐次方程解的条件。