Geodesic
Asymptotics of a reduced logarithmic capacity
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 1, pp. 126-144

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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. 
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     author = {I. I. Argatov},
     title = {Asymptotics of a~reduced logarithmic capacity},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {126--144},
     publisher = {mathdoc},
     volume = {45},
     number = {1},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_1_a8/}
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I. I. Argatov. Asymptotics of a reduced logarithmic capacity. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 1, pp. 126-144. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_1_a8/