## 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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### References:

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