Asymptotics of a reduced logarithmic capacity

The homogeneous Dirichlet problem for the Laplace operator in a layer with a hole $G$ is considered. Periodicity conditions are imposed on the planes of the layer. A solution is sought in the class of functions that increase logarithmically at infinity. The reduced logarithmic capacity of the closed domain $\overline G$ is defined as a generalization of the logarithmic capacity (the outer conformal radius) of a closed plane domain. Formal asymptotics are constructed for the following shapes of $G$: an almost cylindrical domain, a thin cylinder of low height, a domain of small diameter, and a narrow cylinder of small thickness.