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Algebra i Analiz, 2023, Volume 35, Issue 3, Pages 44–51 (Mi aa1867)  
Research Papers

Weighted means and an analytic characterization of discs

N. G. Kuznetsov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences
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Abstract: Weighted means are obtained for solutions of the two-dimensional Helmholtz and modified Helmholtz equations and also for harmonic functions. The presence of a logarithmic weight diminishes the coefficient in the last two mean value identities. A new theorem characterizing analytically discs in the Euclidean plane $\mathbb{R}^2$ is proved. The weighted mean value property of solutions to the modified Helmholtz equation is used for this purpose.
Keywords: disc, weighted mean value property, logarithmic weight, harmonic function, modified Helmholtz equation, analytic characterization.
Received: 22.09.2022
English version:
St. Petersburg Mathematical Journal, 2024, Volume 35, Issue 3, Pages 467–472
DOI: https://doi.org/10.1090/spmj/1813
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: N. G. Kuznetsov, “Weighted means and an analytic characterization of discs”, Algebra i Analiz, 35:3 (2023), 44–51; St. Petersburg Math. J., 35:3 (2024), 467–472
Citation in format AMSBIB
\Bibitem{Kuz23}
\by N.~G.~Kuznetsov
\paper Weighted means and an analytic characterization of discs
\jour Algebra i Analiz
\yr 2023
\vol 35
\issue 3
\pages 44--51
\mathnet{http://mi.mathnet.ru/aa1867}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=717386}
\transl
\jour St. Petersburg Math. J.
\yr 2024
\vol 35
\issue 3
\pages 467--472
\crossref{https://doi.org/10.1090/spmj/1813}
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