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Equality of the capacity and the modulus of a condenser in Finsler spaces. (English. Russian original) Zbl 1184.53076
Math. Notes 85, No. 4, 566-573 (2009); translation from Mat. Zametki 85, No. 4, 594-602 (2009).
The equality of capacity and modulus is of great importance in the geometrical theory of functions. It relates the functional-theoretic and geometric properties of the set. Special cases of the equality was proved by L. V. Ahlfors and A. Beurling [Acta Math. 83, 101–129 (1950; Zbl 0041.20301)]. Further, this result was improved in the papers of B. Fuglede [Acta Math. 98, 171–219 (1957; Zbl 0079.27703)] and W. P. Ziemer [Mich. Math. J. 16, 43–51 (1969; Zbl 0172.38701)]. Hesse extended this result to $$p$$-capacity and $$p$$-modulus for the case in which the plates of the condenser do not intersect the boundary of the domain. For the case of Euclidean metric, the equality of the capacity and the modulus was proved by V. A. Shlyk [Sib. Math. J. 34, No. 6, 1196–1200 (1993; Zbl 0810.31004) under the most general assumptions. The results of V. V. Aseev [Dokl. Akad. Nauk SSSR 200, 513–514 (1971; Zbl 0233.46059)] on the completeness of the system of continuous admissible functions was significantly used in Shlyk’s proof.
The notion of Finsler space was first introduced by P. Finsler as a generalization of variational problems [Über Kurven und Flächen in allgemeinen Räumen. Göttingen, Zürich: O. Füssli, 120 p. (1918; JFM 46.1131.02)]. The fundamental function of the Finsler space is a function of position and direction. If the fundamental tensor of the Finsler space is a function of position only, the Finsler space reduces to a Riemannian one. In the present paper, the author proves the equality of the capacity and the modulus of a condenser in Finsler spaces under the most general assumptions. This result allows one to extend many results for Lebesgue function spaces to the case of function spaces with Finsler metric.

##### MSC:
 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics) 31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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##### References:
 [1] L. Ahlfors and A Beurling, ”Conformal invariants and functions-theoretic null-sets,” Acta Math. 83(1), 101–129 (1950). · Zbl 0041.20301 · doi:10.1007/BF02392634 [2] B. Fuglede, ”Extremal length and functional completion,” Acta Math. 98(3), 171–219 (1957). · Zbl 0079.27703 · doi:10.1007/BF02404474 [3] W. P. Ziemer, ”Extremal length and p-capacity,” Michigan Math. J. 16(1), 43–51 (1969). · Zbl 0172.38701 · doi:10.1307/mmj/1029000164 [4] J. Hesse, ”A p-extremal length and p-capacity equality,” Ark. Mat. 13(1), 131–144 (1975). · Zbl 0302.31009 · doi:10.1007/BF02386202 [5] V. A. Shlyk, ”The equality between p-capacity and p-modulus,” Sibirsk. Mat. Zh. 34(6), 216–221 (1993) [SiberianMath. J. 34 (6), 1196–1200 (1993)]. · Zbl 0810.31004 [6] V. V. Aseev, ”On a modulus property,” [J] Sov. Math., Dokl. 12, 1409–1411 (1971 Dokl. Akad. Nauk SSSR 200 (3), 513–514 (1971) [Soviet Math. Dokl. 12, 1409–1411 (1971)]. · Zbl 0233.46059 [7] H. Rund, The Differential Geometry of Finsler Spaces (Springer-Velag, Berlin-Göttingen-Heidelberg, 1959; Nauka, Moscow, 1981). · Zbl 0087.36604 [8] M. Ohtsuka, Extremal Length and Precise Functions, in GAKUTO Internat. Ser. Math. Sci. Appl. (Gakkötosho, Tokyo, 2003), Vol. 19. [9] Yu. V. Dymchenko, ”The equality of the condenser capacity and the condenser modulus on a surface,” Zap. Nauchn. Sem. POMI 276, 112–133 (2001) [J. Math. Sci., 118 (1) 4795–4807 (2003)]. · Zbl 1066.31005
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