| Title: A free boundary problem for the Laplace equation
An exceptional domain is an unbounded domain that admits a roof function, a positive harmonic function with zero Dirichlet data and constant Neumann data. In physical terms, viewing the roof function as a potential, the free boundary is simultaneously a set of constant potential and constant magnitude of force. I will discuss joint work with D. Khavinson and R. Teodorescu on a problem posed by L. Hauswirth, F. Helein, and F. Pacard to characterize exceptional domains in the plane. I will also explain how to reinterpret the problem within various settings, namely, null quadrature domains, fluid dynamics, and minimal surfaces.
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