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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Densities with the mean value property
for harmonic functions in a Lipschitz domain

Author: Hiroaki Aikawa
Journal: Proc. Amer. Math. Soc. 125 (1997), 229-234
MSC (1991): Primary 31A05, 31B05
MathSciNet review: 1363444
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Abstract: Let $ D$ be a bounded domain in $ \mathbb {R}^{n}$, $ n\ge 2$, and let $ x_{0}\in D$. We consider positive functions $w$ on D$ such that $h( x_{0}) = (\int _{D}w dx)^{-1} \int _{D} h w dx$ for all bounded harmonic functions $h$ on $ D$. We determine Lipschitz domains $ D$ having such $w$ with $\inf _{D} w>0$.

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Hiroaki Aikawa
Affiliation: Department of Mathematics, Shimane University, Matsue 690, Japan

Keywords: Harmonic functions, mean value property, gradient of harmonic function
Received by editor(s): July 13, 1995
Received by editor(s) in revised form: August 1, 1995
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society