Inverse mean value property of solutions to the modified Helmholtz equation

N. Kuznetsov

Algebra i analiz, Tome 33 (2021) no. 6, pp. 71-77.

Résumé

A theorem characterizing analytically balls in the Euclidean space $\mathbb{R}^m$ is proved. For this purpose positive solutions of the modified Helmholtz equation are applied instead of harmonic functions used in previous results. The resulting Kuran type theorem involves the volume mean value property of solutions to this equation. Other plausible inverse mean value properties of these solutions are discussed.

Détail

Keywords: inverse mean value theorem, characterization of balls, modified Helmholtz equation.

Bibliographie

[1] Adams, D. R., Hedberg, L. I., Function spaces and potential theory, Grundlehren Math. Wiss., 314, Springer-Verlag, Berlin, 1996

[2] Armitage D. H., Goldstein M., “Quadrature and harmonic $L^1$-approximation in annuli”, Trans. Amer. Math. Soc., 312 (1989), 141–154

[3] Armitage D. H., Goldstein M., “The volume mean-value property of harmonic functions”, Complex Variables Theory Appl., 13:3-4 (1990), 185–193

[4] Armitage D. H., Goldstein M., “Characterizations of balls and strips via harmonic functions”, Approximation by solutions of partial differential equations (Hanstholm, 1991), Kluwer Acad. Publ., 1992, 1–9

[5] Burckel R. B., “Three secrets about harmonic functions”, Amer. Math. Monthly, 104 (1997), 52–56

[6] Epstein B., “On the mean-value property of harmonic functions”, Proc. Amer. Math. Soc., 13 (1962), 830

[7] Goldstein M., Haussmann W., Rogge L., “On the mean value property of harmonic functions and best harmonic $L^1$-approximation”, Trans. Amer. Math. Soc., 305:2 (1988), 505–515

[8] Hansen W., Netuka I., “Inverse mean value property of harmonic functions”, Math. Ann., 297:1 (1993), 147–156

[9] Hansen W., Netuka I., “Inverse mean value property of harmonic functions, Corrigendum”, Math. Ann., 303:2 (1995), 373–375

[10] Hedberg L. I., “Approximation in the mean by solutions of elliptic equations”, Duke Math. J., 40 (1973), 9–16

[11] Kosmodemyanskii A. A. (ml.), “Obraschenie teoremy o srednem znachenii dlya garmonicheskikh funktsii”, Uspekhi mat. nauk, 36:5 (1981), 175–176

[12] Kuran Ü., “On the mean value property of harmonic functions”, Bull. London Math. Soc., 4 (1972), 311–312

[13] Kuznetsov N., “Metaharmonic functions: mean flux theorem, its converse and related properties”, Algebra i analiz, 33:2 (2021), 82–97

[14] Kuznetsov N., “Mean value properties of solutions to the Helmholtz and modified Helmholtz equations”, J. Math. Sci., 257 (2021), 673–683

[15] Netuka I., Veselý J., “Mean value property and harmonic functions”, Classical and Modern Potential Theory and Applications (Chateau de Bonas, 1993), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 430, Kluwer, Dordrecht, 1994, 359–398

[16] Neumann C., Allgemeine Untersuchungen über das Newtonsche Prinzip der Fernwirkungen, Teubner, Leipzig, 1896

[17] Nikiforov A. F., Uvarov V. B., Spetsialnye funktsii matematicheskoi fiziki, Nauka, M., 1984

[18] Payne L. E., Schaefer P. W., “Duality theorems in some overdetermined boundary value problems”, Math. Methods Appl. Sci., 11:6 (1989), 805–819

[19] Poritsky H., “Generalizations of the Gauss law of the spherical mean”, Trans. Amer. Math. Soc., 43:2 (1938), 199–225

[20] Sakai M., Quadrature domains, Lecture Notes in Math., 934, Springer-Verlag, Berlin, 1982

[21] Watson G. N., A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1944