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Comptes Rendus. Mathématique
Partial differential equations
Characterization of balls via solutions of the modified Helmholtz equation
Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 945-948.

A theorem characterizing analytically balls in the Euclidean space m is proved. For this purpose positive solutions of the modified Helmholtz equation are used instead of harmonic functions applied in previous results. The obtained Kuran type theorem is based on the volume mean value property of solutions to this equation.

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Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.250
Nikolay Kuznetsov 1

1. Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy pr. 61, St Petersburg 199178, Russian Federation
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Nikolay Kuznetsov. Characterization of balls via solutions of the modified Helmholtz equation. Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 945-948. doi : 10.5802/crmath.250. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.250/

[1] David H. Armitage; Myron Goldstein The volume mean-value property of harmonic functions, Complex Variables, Theory Appl., Volume 13 (1990) no. 3-4, pp. 185-193 | Article | MR 1039156 | Zbl 0652.31004

[2] Bernard Epstein On the mean-value property of harmonic functions, Proc. Am. Math. Soc., Volume 13 (1962), p. 830 | MR 140700 | Zbl 0109.07501

[3] Wolfhard Hansen; Ivan Netuka Inverse mean value property of harmonic functions, Math. Ann., Volume 297 (1993) no. 1, pp. 147-156 corrigendum in ibid. 303 (1995), no. 2, p.373-375 | Article | MR 1238412 | Zbl 0794.31001

[4] A. A. Kosmodemʼyanskij Jr A converse of the mean value theorem for harmonic functions, Russ. Math. Surv., Volume 36 (1981) no. 5, pp. 159-160 | MR 637445

[5] Ülkü Kuran On the mean value property of harmonic functions, Bull. Lond. Math. Soc., Volume 4 (1972), pp. 311-312 | Article | MR 320348 | Zbl 0257.31006

[6] Nikolay Kuznetsov Mean value properties of solutions to the Helmholtz and modified Helmholtz equations (2021) https://arxiv.org/abs/2105.09916v1 (to appear in Russian) | Zbl 07394451

[7] Nikolay Kuznetsov Metaharmonic functions: mean flux theorem, its converse and related properties, Algebra Anal., Volume 33 (2021), pp. 82-97 | MR 4240827

[8] Ivan Netuka; Jiří Veselý Mean value property and harmonic functions, Classical and modern potential theory and applications (NATO ASI Series. Series C. Mathematical and Physical Sciences), Volume 430, Kluwer Academic Publishers, 1994, pp. 359-398 | Article | MR 1321628 | Zbl 0863.31012

[9] Carl Neumann Allgemeine Untersuchungen über das Newton’sche Princip der Fernwirkungen, Teubner, 1896 | Zbl 26.0893.02

[10] Arnold F. Nikiforov; Vasilii B. Uvarov Special Functions of Mathematical Physics: A Unified Introduction with Applications, Birkhäuser, 1988 | Zbl 0624.33001

[11] Hillel Poritsky Generalizations of the Gauss law of the spherical mean, Trans. Am. Math. Soc., Volume 43 (1938), pp. 199-225 | Article | MR 1501939 | Zbl 0018.12604

[12] George N. Watson A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944 | Zbl 0063.08184

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