PDEs Seminar – Miles Wheeler

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Title. Overdetermined problems for the Helmholtz equation Abstract. We consider overdetermined problems for the Helmholtz equation where one asks for both constant Dirichlet data and constant Neumann data on the entire boundary. It is easy to find explicit radial solutions when the domain is a ball. For zero Neumann data, there is a famous conjecture, attributed to Schiffer, that balls are the only solutions. For zero Dirichlet data, there is a less famous conjecture, attributed to Berenstein, that balls are the only solutions. When both the Dirichlet data and Neumann data can be nonzero, the…