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A harmonic quadrature formula characterizing open strips

Published online by Cambridge University Press:  24 October 2008

D. H. Armitage  and
C. S. Nelson
D. H. Armitage
Affiliation:
The Queen's University of Belfast
C. S. Nelson
Affiliation:
The Queen's University of Belfast
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Extract

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, then

A converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such that

for every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 1 , January 1993 , pp. 147 - 151
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Aemitage, D. H. and Gardiner, S. J.. On the growth of the hyperplane mean of a subharmonic function. J. London Math. Soc. (2) 36 (1987). 501–512.CrossRefGoogle Scholar
[2]Armitage, D. H. and Goldstein, M.. The volume mean-value property of harmonic functions. Complex Variables 13 (1990), 185–193.Google Scholar
[3]Akmitage, D. H. and Goldstein, M.. Characterizations of balls and strips via harmonic quadrature. In Approximation by Solutions of Partial Differential Equations (Fuglede, B. et al. eds.), (Kluwer, 1992), pp. 1–9.Google Scholar
[4]Aemitage, D. H. and Goldstein, M.. Quadrature and harmonic approximation of sub-harmonic functions in strips. J. London Math. Soc. to appear.Google Scholar
[5]Doob, J. L.. Classical Potential Theory and its Probabilistic Counterpart (Springer-Verlag. 1984).Google Scholar
[6]Goldstein, M., Haussmann, W. and Rogge, L.. Characterization of open strips by harmonic quadrature. In Approximation by Solutions of Partial Differential Equations (Fuglede, B. et al. eds.), (Kluwer, 1992), pp. 87–92.CrossRefGoogle Scholar
[7]Goldstein, M., Haussmann, W. and Rogge, L.. On the inverse mean value property of harmonic functions on strips. Bull. London Math. Soc. 24 (1992), to appear.CrossRefGoogle Scholar
[8]Kuran, Ü.. On the mean value property of harmonic functions. Bull. London Math. Soc. 4 (1972), 311–312.CrossRefGoogle Scholar
[9]Nualtaranee, S.. On least harmonic majorants in half-spaces. Proc. London Math. Soc. (3) 27 (1973), 243–260.CrossRefGoogle Scholar
[10]Saks, S.. Theory of the Integral, 2nd ed. (Hafner, 1937).Google Scholar