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Some overdetermined boundary value problems for harmonic functions. (English) Zbl 0767.35047

Summary: We study various overdetermined problems involving harmonic functions. In particular, we show that if the second eigenfunction \(u_ 2\) of the Stekloff eigenvalue problem in a bounded simply connected plane domain \(\Omega\) has a constant value of \(|\nabla u_ 2|\) on \(\partial\Omega\), then \(\Omega\) is a disk.

MSC:

35N05 Overdetermined systems of PDEs with constant coefficients
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
35P05 General topics in linear spectral theory for PDEs
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