Weighted means and an analytic characterization of discs

Kuznetsov, N.

doi:10.1090/spmj/1813

Weighted means and an analytic characterization of discs
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by N. Kuznetsov;
Translated by: the author

St. Petersburg Math. J. 35 (2024), 467-472

DOI: https://doi.org/10.1090/spmj/1813

Published electronically: July 30, 2024

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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 reduces the coefficient in the last two mean value identities. A new theorem characterizing disks in the Euclidean plane $\mathbb {R}^2$ analytically is proved; it is based on the weighted mean value property of solutions to the modified Helmholtz equation.

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