The Cauchy-Riemann Complex on Non-Smooth Domains

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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.