We consider Jordan domains Ω with piece-wise smooth boundaries such that all arcs α ⊂ ∂Ω having fixed length l, 0 < l < length(∂Ω), have equal harmonic measures ω(z0, α, Ω) evaluated at some point z0 ∈ Ω. It is proved that Ω is a disk centered at z0 if the ratio l/length(∂Ω) is irrational and that Ω possesses rotational symmetry by some angle 2π/n, n ≥ 2, around the point z0, if this ratio is rational.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 491, 2020, pp. 145–152.
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Samarasiri, S., Solynin, A.Y. Harmonic Measure of Arcs of Fixed Length. J Math Sci 261, 826–831 (2022). https://doi.org/10.1007/s10958-022-05791-2
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DOI: https://doi.org/10.1007/s10958-022-05791-2