The Cauchy-Riemann Complex on Non-Smooth Domains

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。本课题将研究带角域上柯西-黎曼方程的解。柯西-黎曼方程是多个复变量的基本微分方程，它的研究涉及到位势理论、复几何和偏微分方程组的边值问题。这个项目将解决定性问题，如解算子的紧性，以及定量问题，如索博列夫空间中的正则性。当边界光滑时(尽管仍有许多未解决的问题)，人们对这些问题已有了很多了解，因此本项目将重点研究解在角点附近的行为。由于现代物理中的许多现象都可以用复分析语言来描述，所以柯西-黎曼方程是数学物理的重要工具。除了物理应用，它们作为偏微分方程的模型也很重要，它在区域内部表现出很大的规律性，但在边界处失去了规则性。因此，柯西-黎曼方程的结果为具有更大适用性的更大类别的方程提供了一个自然的起点。当区域的边界光滑时，这些方程已被广泛研究，但在数学中，就像在物理世界中一样，许多区域都有角，因此了解解的性质在非光滑区域的角附近如何变化是很重要的。由于这些问题与调和分析、复杂几何、偏微分方程式和数学物理有关，因此合作的潜力很大。

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.