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Minimal harmonic functions on Denjoy domains


Author: Stephen J. Gardiner
Journal: Proc. Amer. Math. Soc. 107 (1989), 963-970
MSC: Primary 31B25
DOI: https://doi.org/10.1090/S0002-9939-1989-0991695-8
MathSciNet review: 991695
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Abstract: Let $ \Omega = {\mathbb{R}^n}\backslash E$, where $ E$ is a closed subset of the hyperplane $ \left\{ {{x_n} = 0} \right\}$ and every point of $ E$ is regular for the Dirichlet problem on $ \Omega $. Further, let $ {\alpha _k}$. denote the $ (n - 1)$-dimensional measure of the set $ \{ X \in \Omega :{x_n} = 0,{e^k} < \vert X\vert < {e^{k + 1}}\} $. It is known that the cone, $ {\mathcal{P}_E}$, of positive harmonic functions on $ \Omega $ which vanish on $ E$ has dimension 1 or 2. In this paper it is shown that if $ \sum {{e^{ - nk}}\alpha _k^{n/(n - 1)} < + \infty } $ then $ \dim {\mathcal{P}_E} = 2$. This result, which in the case $ n = 2$ implies a recent theorem of Segawa, is also shown to be sharp.


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

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DOI: https://doi.org/10.1090/S0002-9939-1989-0991695-8
Article copyright: © Copyright 1989 American Mathematical Society

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