Geodesic
On a Problem of Dubinin for the Capacity of a Condenser with a Finite Number of Plates
Matematičeskie zametki, Tome 103 (2018) no. 6, pp. 841-852.

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It is proved that, in Euclidean $n$-space, $n\ge 2$, the weighted capacity (with Muckenhoupt weight) of a condenser with a finite number of plates is equal to the weighted modulus of the corresponding configuration of finitely many families of curves. For $n=2$, in the conformal case, this equality solves a problem posed by Dubinin.
Keywords: capacity of a condenser, Muckenhoupt weight, generalized condenser
Mots-clés : modulus of a configuration. 
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Yu. V. Dymchenko; V. A. Shlyk. On a Problem of Dubinin for the Capacity of a Condenser with a Finite Number of Plates. Matematičeskie zametki, Tome 103 (2018) no. 6, pp. 841-852. http://geodesic.mathdoc.fr/item/MZM_2018_103_6_a3/

[1] A. V. Sychev, Moduli i prostranstvennye kvazikonformnye otobrazheniya, Novosibirsk, Novosibirsk. gos. un-t, 1983 | MR

[2] M. Ohtsuka, Extremal Length and Precise Functions, Gakkōtosho, Tokyo, 2003 | MR | Zbl

[3] L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York, 1973 | MR | Zbl

[4] V. N. Dubinin, “Metody geometricheskoi teorii funktsii v klassicheskikh i sovremennykh zadachakh dlya polinomov”, UMN, 67:4 (406) (2012), 3–88 | DOI | MR | Zbl

[5] V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory, Birkhäuser, Basel, 2014 | MR | Zbl

[6] V. Dubinin, “Some unsolved problems about condenser capacities on the plane”, Complex Analysis and Dynamical Systems, Trends in Math., Birkhäuser, Cham, 2018, 81–92 | DOI

[7] J. A. Jenkins, “Some problems in conformal mapping”, Trans. Amer. Math. Soc., 67:2 (1949), 327–350 | DOI | MR | Zbl

[8] G. V. Kuzmina, “Obschaya teorema koeffitsientov Dzhenkinsa i metod modulei semeistv krivykh”, Analiticheskaya teoriya chisel i teoriya funktsii. 29, Zap. nauchn. sem. POMI, 429, POMI, SPb., 2014, 140–156

[9] V. V. Aseev, B. Yu. Sultanov, Moduli polikondensatorov i izomorfizmy prostranstv sledov nepreryvnykh funktsii klassa $W^1_n$, Preprint No 31, IM SO AN SSSR, Novosibirsk, 1989

[10] G. Federer, Geometricheskaya teoriya mery, Nauka, M., 1987 | MR | Zbl

[11] J. Väisälä, Lectures on $n$-Dimensional Quasiconformal Mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin, 1971 | DOI | MR | Zbl

[12] V. M. Goldshtein, Yu. G. Reshetnyak, Vvedenie v teoriyu funktsii s obobschennymi proizvodnymi i kvazikonformnye otobrazheniya, Nauka, M., 1983 | MR | Zbl

[13] J. Hesse, “A $p$-extremal length and $p$-capacity equality”, Ark. Mat., 13:1 (1975), 131–144 | DOI | MR | Zbl

[14] Yu. V. Dymchenko, V. A. Shlyk, “Dostatochnost semeistva lomanykh v metode modulei i ustranimye mnozhestva”, Sib. matem. zhurn., 51:6 (2010), 1298–1315 | MR | Zbl

[15] V. A. Shlyk, A. A. Yakovlev, “Moduli prostranstvennykh konfiguratsii i ustranimye mnozhestva”, Analiticheskaya teoriya chisel i teoriya funktsii. 32, Zap. nauchn. sem. POMI, 449, POMI, SPb., 2016, 275–288 | MR