Geodesic
The final version of the mean value theorem for harmonic functions
Matematičeskie zametki, Tome 59 (1996) no. 3, pp. 351-358.

Voir la notice de l'article provenant de la source Math-Net.Ru

We construct examples of nonharmonic functions satisfying the mean value equation for some set of spheres. These results permit us to obtain the two-circle theorem in its definitive form. 
@article{MZM_1996_59_3_a3,
     author = {V. V. Volchkov},
     title = {The final version of the mean value theorem for harmonic functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {351--358},
     publisher = {mathdoc},
     volume = {59},
     number = {3},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_3_a3/}
}
TY  - JOUR
AU  - V. V. Volchkov
TI  - The final version of the mean value theorem for harmonic functions
JO  - Matematičeskie zametki
PY  - 1996
SP  - 351
EP  - 358
VL  - 59
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_1996_59_3_a3/
LA  - ru
ID  - MZM_1996_59_3_a3
ER  - 
%0 Journal Article
%A V. V. Volchkov
%T The final version of the mean value theorem for harmonic functions
%J Matematičeskie zametki
%D 1996
%P 351-358
%V 59
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_1996_59_3_a3/
%G ru
%F MZM_1996_59_3_a3
V. V. Volchkov. The final version of the mean value theorem for harmonic functions. Matematičeskie zametki, Tome 59 (1996) no. 3, pp. 351-358. http://geodesic.mathdoc.fr/item/MZM_1996_59_3_a3/

[1] Flatto L., “The converse of Gauss's theorem for harmonic functions”, J. Diff. Equat., 1:4 (1965), 483–490 | DOI | MR | Zbl

[2] Zalcman L., “Offbeat integral geometry”, Amer. Math. Monthly, 87:3 (1980), 161–175 | DOI | MR | Zbl

[3] Zalcman L., “Mean values and differential equations”, Isr. J. Math., 14 (1973), 339–352 | DOI | MR | Zbl

[4] Berenstein C. A., Zalcman L., “Pompeiu's problem on symmetric spaces”, Comment. Math. Helv., 55 (1980), 593–621 | DOI | MR | Zbl

[5] Berenstein C. A., Gay R., “A local version of the two-circler theorem”, Isr. J. Math., 55 (1986), 267–288 | DOI | MR | Zbl

[6] Volchkov V. V., “Novye teoremy o dvukh radiusakh v teorii garmonicheskikh funktsii”, Izv. RAN. Ser. matem., 58:1 (1994), 41–49 | MR

[7] Berenstein K. A., Struppa D., “Kompleksnyi analiz i uravneniya v svertkakh”, Itogi nauki i tekhn. Sovrem. probl. matem. Fundament. napravleniya, 54, VINITI, M., 1989, 5–111 | MR

[8] Vilenkin N. Ya., Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1991 | Zbl

[9] Korenev B. G., Vvedenie v teoriyu besselevykh funktsii, Nauka, M., 1971 | Zbl

[10] Kurant R., Uravneniya s chastnymi proizvodnymi, Mir, M., 1964

[11] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[12] Khelgason S., Gruppy i geometricheskii analiz, Mir, M., 1987

[13] Leng S., $\operatorname{SL}_2(\mathbb R)$, Mir, M., 1977