Continuity of logarithmic capacity

Continuity of logarithmic capacity

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect conditio...

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Veröffentlicht in:Journal of mathematical analysis and applications 2022-01, Vol.505 (1), p.125585, Article 125585
Hauptverfasser: Kalmykov, Sergei, Kovalev, Leonid V.
Format: Artikel
Sprache:eng
Schlagworte:
Green's function
Hausdorff metric
Logarithmic capacity
Uniformly perfect sets
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Zusammenfassung:We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2021.125585