Abstract
This is a continuation of [1]. Under study are the differentiability properties of the logarithmic potential determined for some class of complex measures distributed on Van Koch’s curves. Unlike the classical case of regular curves, the potential is shown to be of class C 1 on the whole plane ℂ. We also study a related analog of Robin’s problem. The proofs are based on some results of [1].
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References
Ponomarev S. P., “Some properties of Van Koch’s curves,” Siberian Math. J., 48, No. 6, 1046–1059 (2007).
Ponomarev S. P., “On Hausdorff dimensions of quasiconformal curves,” Siberian Math. J., 34, No. 4, 717–722 (1993).
Stoïlow S., Theory of Functions of Complex Variable. Vol. 2 [Russian translation], Izdat. Inostr. Lit., Moscow (1962).
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To Yu. G. Reshetnyak on his 80th birthday.
Original Russian Text Copyright © 2009 Ponomarev S. P.
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Slupsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 5, pp. 1137–1147, September–October, 2009.
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Ponomarev, S.P. On the logarithmic potential defined for a Van Koch curve. Sib Math J 50, 898–906 (2009). https://doi.org/10.1007/s11202-009-0100-x
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DOI: https://doi.org/10.1007/s11202-009-0100-x