Volume densities with the mean value property for harmonic functions

Hansen, W.; Netuka, I.

doi:10.1090/S0002-9939-1995-1213859-3

Volume densities with the mean value property for harmonic functions
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by W. Hansen and I. Netuka PDF

Proc. Amer. Math. Soc. 123 (1995), 135-140 Request permission

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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