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）资助的。本专题将研究在有角点的区域上柯西-黎曼方程的解。 Cauchy-Riemann方程是多复变量的基本微分方程，其研究涉及到位势理论、复几何和偏微分方程边值问题等工具。 这个项目将解决定性问题，如解决方案运营商的紧凑性，以及定量问题，如Sobolev空间的正则性。 当边界是光滑的时，我们对这些问题已经有了很多了解（尽管仍有许多未解决的问题），因此本项目将关注角点附近的解的行为。因为现代物理中的许多现象都是用复分析的语言来描述的，所以柯西-黎曼方程是数学物理的重要工具。 除了物理应用外，它们也是重要的偏微分方程模型，这些模型在区域内部表现出很大的规律性，但在边界处失去了规律性。 因此，柯西-黎曼方程的结果提供了一个自然的起点，更大的一类方程，甚至更大的适用性。 当域的边界是光滑的时，这些方程已经被广泛研究，但是在数学中，就像在物理世界中一样，许多域都有角，所以理解解的性质如何在非光滑域的角附近变化是很重要的。 由于这些问题涉及调和分析、复几何、偏微分方程和数学物理，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.