Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\partial U$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 31A05, 31B05

Retrieve articles in all journals with MSC: 31A05, 31B05

Additional Information

Keywords: Harmonic functions, mean value property, balayage measures
Article copyright: © Copyright 1995 American Mathematical Society