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On the characterization of harmonic and subharmonic functions via mean-value properties. (English) Zbl 1187.31003
The paper studies the relation between mean-value properties on the one side and harmonicity and subharmonicity on the other side. If $$G$$ is a bounded domain with sufficiently smooth boundary and $$u$$ is a continuous function on its closure $$\overline G$$, denote by $$B_u(G)$$ the mean value of $$u$$ over $$G$$, and by $$S_u(G)$$ the mean value of $$u$$ over $$\partial G$$. The first result is the following: Let $$u$$ be a continuous function on a domain $$\Omega$$. Then $$u$$ is harmonic (or subharmonic) in $$\Omega$$ if and only if $$B_u(B)=S_u(B)$$ (or $$B_u(B)\leq S_u(B)$$) for every ball $$B$$ whose closure $$\overline B$$ is contained in $$\Omega$$. The second result is the following: Let $$\Omega$$ be a bounded domain with sufficiently smooth boundary. Then there exist constants $$0<c_1 \leq 1\leq c_2 <\infty$$ such that $$c_1S_u (\Omega ) \leq B_u (\Omega ) \leq c_2 S_u(\Omega )$$ for all non-negative harmonic functions $$u\in C^1(\overline \Omega )$$. If $$c_1 =1$$ or $$c_2=1$$, then $$\Omega$$ is a ball.

MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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References:
 [1] Armitage, D.H., Goldstein, M.: The volume mean-value property of harmonic functions. Complex Var. Theory Appl. 13, 185–193 (1990) · Zbl 0652.31004 [2] Axler, S., Bourdon, P., Ramey, W.: Harmonic function theory. GTM, vol. 137. Springer, New York (1992) · Zbl 0765.31001 [3] Beardon, A.F.: Integral means of subharmonic functions. Proc. Camb. Philol. Soc. 69, 151–152 (1971) · Zbl 0207.11004 · doi:10.1017/S0305004100046491 [4] Beckenbach, E.F., Radó, T.: Subharmonic functions and surfaces of negative curvature. Trans. Am. Math. Soc. 35, 662–674 (1933) · Zbl 0007.13001 · doi:10.1090/S0002-9947-1933-1501708-X [5] Bennett, A.: Symmetry in an overdetermined fourth order elliptic boundary value problem. SIAM J. Math. Anal. 17, 1354–1358 (1986) · Zbl 0612.35039 · doi:10.1137/0517095 [6] Beckenbach, E.F., Reade, M.: Mean values and harmonic polynomials. Trans. Am. Math. Soc. 51, 240–245 (1945) · Zbl 0063.00270 [7] Epstein, B., Schiffer, M.M.: On the mean–value property of harmonic functions. J. d’Analyse Math. 14, 109–111 (1965) · Zbl 0131.10003 · doi:10.1007/BF02806381 [8] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, Berlin (1983) · Zbl 0562.35001 [9] Kosmodem’yanskiĭ, A.A.: Converse of the mean value theorem for harmonic functions (Russian). Uspekhi Mat. Nauk 36, 175–176 (1981) [10] Kuran, Ü.: On the mean–value property of harmonic functions. Bull. Lond. Math. Soc. 4, 311–312 (1972) · Zbl 0257.31006 · doi:10.1112/blms/4.3.311 [11] Netuka, I., Veselý, J.: Mean value property and harmonic functions. In: Classical and modern potential theory and applications (Chateau de Bonas, 1993), pp. 359–398. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 430. Kluwer, Dordrecht (1994) · Zbl 0863.31012 [12] Payne, L.E., Schaefer, P.W.: Duality theorems in some overdetermined boundary value problems. Math. Methods Appl. Sci. 11, 805–819 (1989) · Zbl 0698.35051 · doi:10.1002/mma.1670110606 [13] Radó, T.: Subharmonic functions. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 5. Springer, Berlin (1937) · JFM 63.0458.05 [14] Rao, M.: Integral Harnack inequality. Glasgow Math. J. 26, 115–120 (1985) · Zbl 0577.31001 · doi:10.1017/S0017089500005875 [15] Reade, M.: Some remarks on subharmonic functions. Duke Math. J. 10, 531–536 (1943) · Zbl 0063.06444 · doi:10.1215/S0012-7094-43-01045-2 [16] Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971) · Zbl 0222.31007 · doi:10.1007/BF00250468
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