Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-7

On mean value properties involving a logarithm-type weight

Nikolay Kuznetsov

Received May 05, 2023. Published online October 12, 2023.

Abstract: Two new assertions characterizing analytically disks in the Euclidean plane $\mathbb{R}^2$ are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier.

Keywords: harmonic function; Helmholtz equation; modified Helmholtz equation; mean value property; logarithmic weight; characterization of balls

Classification MSC: 31A10, 35B05, 35J05

DOI: 10.21136/MB.2023.0072-23

PDF available at: Institute of Mathematics CAS

References:
[1] R. J. Duffin: Yukawan potential theory. J. Math. Anal. Appl. 35 (1971), 105-130. DOI 10.1016/0022-247X(71)90239-3 | MR 0277743 | Zbl 0214.36501
[2] M. A. Evgrafov: Asymptotic Estimates and Entire Functions. Nauka, Moscow (1979). (In Russian.) MR 0552276 | Zbl 0447.30016
[3] W. Hansen, I. Netuka: Inverse mean value property of harmonic functions. Math. Ann. 297 (1993), 147-156. DOI 10.1007/BF01459493 | MR 1238412 | Zbl 0794.31001
[4] W. Hansen, I. Netuka: Corrigendum: "Inverse mean value property of harmonic functions". Math. Ann. 303 (1995), 373-375. DOI 10.1007/BF01460995 | MR 1348805 | Zbl 0839.31001
[5] Ü. Kuran: On the mean-value property of harmonic functions. Bull. Lond. Math. Soc. 4 (1972), 311-312. DOI 10.1112/blms/4.3.311 | MR 0320348 | Zbl 0257.31006
[6] N. Kuznetsov: Characterization of balls via solutions of the modified Helmholtz equation. C. R., Math., Acad. Sci. Paris 359 (2021), 945-948. DOI 10.5802/crmath.250 | MR 4322995 | Zbl 1477.35077
[7] N. Kuznetsov: Mean value properties of solutions to the Helmholtz and modified Helmholtz equations. J. Math. Sci., New York 257 (2021), 673-683. DOI 10.1007/s10958-021-05509-w | MR 4313622 | Zbl 1477.35075
[8] N. Kuznetsov: Inverse mean value property of metaharmonic functions. J. Math. Sci., New York 264 (2022), 603-608. DOI 10.1007/s10958-022-06019-z | MR 4466320 | Zbl 1497.35109
[9] N. Kuznetsov: Metaharmonic functions: Mean flux theorem, its converse and related properties. St. Petersbg Math. J. 33 (2022), 243-254. DOI 10.1090/spmj/1699 | MR 4445758 | Zbl 1485.35129
[10] N. Kuznetsov: Panharmonic functions: Mean value properties and related topics. J. Math. Sci., New York 269 (2023), 53-76. DOI 10.1007/s10958-023-06254-y | MR 4546947 | Zbl 07676279
[11] N. Kuznetsov: Weighted means and characterization of balls. J. Math. Sci., New York 269 (2023), 853-858. DOI 10.1007/s10958-023-06323-2 | MR 4558666
[12] I. Netuka: Harmonic functions and mean value theorems. Čas. Pěst. Mat. 100 (1975), 391-409. (In Czech.) DOI 10.21136/CPM.1975.117893 | MR 0463461 | Zbl 0314.31007
[13] I. Netuka, J. Veselý: Mean value property and harmonic functions. Classical and Modern Potential Theory and Applications NATO ASI Series, Ser. C: Mathematical and Physical Sciences 430. Kluwer Academic, Dordrecht (1994), 359-398. MR 1321628 | Zbl 0863.31012
[14] A. F. Nikiforov, V. B. Uvarov: Special Functions of Mathematical Physics: A Unified Introduction with Applications. Birkhäuser, Basel (1988). DOI 10.1007/978-1-4757-1595-8 | MR 0922041 | Zbl 0624.33001

Affiliations: Nikolay Kuznetsov, Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, 199178, V.O., Bol'shoy pr. 61, Russian Federation, e-mail: nikolay.g.kuznetsov@gmail.com

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