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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Volume densities with the mean value property for harmonic functions


Authors: W. Hansen and I. Netuka
Journal: Proc. Amer. Math. Soc. 123 (1995), 135-140
MSC: Primary 31A05; Secondary 31B05
MathSciNet review: 1213859
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Abstract: On a bounded domain U in $ {\mathbb{R}^d}$ containing the origin, probability measures $ \mu $ which have a density w with respect to Lebesgue measure and satisfy $ h(0) = \smallint h\;d\mu $ for every bounded harmonic function on U are studied. A domain U is constructed such that $ \inf w(U) = 0$ for any such measure. (This solves a problem proposed by A. Cornea.) If, however, U has smooth boundary, then $ \mu $ having a density $ w \in {\mathcal{C}^\infty }(U)$ which is bounded away from zero on U can be constructed. On the other hand, for arbitrary U it is always possible to choose a strictly positive $ w \in {\mathcal{C}^\infty }(U)$ tending to zero at $ \partial U$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1213859-3
PII: S 0002-9939(1995)1213859-3
Keywords: Harmonic functions, mean value property, balayage measures
Article copyright: © Copyright 1995 American Mathematical Society