Abstract
In this article, we investigate the behaviour of conformal capacities of several plane condensers and logarithmic capacities of some compact sets under the following types of geometric transformations: a gap moving in one or two plates, translation, and rotation of plates and breaking plates. Main motivating examples are provided by parallel plate condensers although the majority of our results are true for more general condensers. We prove monotonicity theorems in most cases and find extremal configurations with maximal capacity in more difficult ones. Our main tools are polarization and direct application of the extended Dirichlet principle. The last section of this article contains several open problems and conjectures.
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