March 26, 2012 -
4:00 pm to 5:00 pm
Location: Cupples I, Room 199 |
Host: Prof. John McCarthy
Abstract: The mean-value property for harmonic functions states that the integral of any harmonic function over a disk (resp. n-ball) equals a constant times point-evaluation at the center. More generally, a so-called quadrature domain admits a formula for integration of any harmonic function in terms of a sum of weighted point-evaluations. We state equivalent definitions that bring together perspectives from potential theory, holomorphic PDE, and function theory. As an application, we describe a moving-boundary problem from fluid dynamics (Hele-Shaw flows) for which quadrature domains give exact solutions. Our main goal is to address the lack of explicit examples in higher dimensions compared to the abundance of examples in the plane.
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