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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.

Information

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

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