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

Abstract:

We construct domains $U$ in the plane such that if $G(re^{i\theta })$ is the Green’s function of $U$ 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 $r$ and $G^{*}$ is the Green’s function of the circular symmetrization $U^{*}$, again with pole at zero, then there are positive numbers $r$ 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 $U$ 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).