The Cauchy-Riemann Complex on Non-Smooth Domains

非光滑域上的柯西-黎曼复形

• 批准号: 1002332
• 负责人: Phillip Harrington
• 金额: $ 8.54万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1002332&HistoricalAwards=false
• 关键词: Cauchy; Riemann; Complex; Non; Smooth

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project will study solutions to the Cauchy-Riemann equations on domains with corners. The Cauchy-Riemann equations are the fundamental differential equations in several complex variables, and their study involves tools from potential theory, complex geometry, and boundary value problems in partial differential equations. This project will address qualitative questions, such as compactness for the solution operator, and quantitative questions, such as regularity in Sobolev spaces. Much is known about these questions when the boundary is smooth (although many open problems remain), so this project will focus on the behavior of the solution near corners.Because so many phenomena in modern physics are best described using the language of complex analysis, the Cauchy-Riemann equations are a crucial tool for mathematical physics. In addition to physical applications, they are also significant as models of partial differential equations which behave with great regularity in the interior of a domain but lose regularity at the boundary. Hence, results for the Cauchy-Riemann equations provide a natural starting place for a much larger class of equations with even greater applicability. These equations have been studied extensively when the boundary of a domain is smooth, but in mathematics as in the physical world, many domains have corners, so it is important to understand how the properties of a solution change near the corners of a non-smooth domain. Because these problems are related to harmonic analysis, complex geometry, partial differential equations, and mathematical physics, the potential for collaboration is high.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本课题将研究柯西-黎曼方程在带角域上的解。柯西-黎曼方程是几种复杂变量的基本微分方程，它的研究涉及到势理论、复杂几何和偏微分方程边值问题的工具。该项目将解决定性问题，如解算子的紧性，以及定量问题，如Sobolev空间的正则性。当边界是光滑的时候，我们对这些问题已经有了很多了解（尽管仍然存在许多悬而未决的问题），所以这个项目将重点关注解决方案在拐角附近的行为。因为现代物理学中的许多现象都是用复杂分析的语言来描述的，所以柯西-黎曼方程是数学物理学的一个重要工具。除了物理应用外，它们还可以作为偏微分方程的模型，这些模型在区域内部表现出很大的规律性，但在边界处失去规律性。因此，柯西-黎曼方程的结果为更大的一类方程提供了一个自然的起点，这些方程具有更大的适用性。当一个域的边界是光滑的时候，这些方程已经被广泛地研究过，但是在数学和物理世界中，许多域都有角，所以了解解的性质在非光滑域的角附近是如何变化的是很重要的。由于这些问题与谐波分析、复杂几何、偏微分方程和数学物理有关，因此合作的潜力很大。