Three counterexamples for a question concerning Green's functions and circular symmetrization期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Three counterexamples for a question concerning Green's functions and circular symmetrization

Authors:	Alexander R Pruss

Institution:	University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

Abstract:	We construct domains in the plane such that if $G(re^{i\theta })$ is the Green's function of with pole at zero, while $\tilde G(r e^{i\theta })$ is the symmetric non-increasing rearrangement of $G(re^{i\theta })$ for each fixed and $G^{}$ is the Green's function of the circular symmetrization $U^{}$ , again with pole at zero, then there are positive numbers and $\varepsilon$ such that $\begin{equation}G^{}(r e^{i\theta }) < \tilde G(r e^{i\theta }), \end{equation*}$ whenever $0<\|\pi -\theta \|<\varepsilon$ . One of our constructions will have simply connected. We also consider the case where the poles of the Green's functions do not lie at the origin. Our work provides a negative answer to a question of Hayman (1967).

Keywords:	Green's functions circular symmetrization

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