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On Various Definitions of Capacity and Related Notions

Published online by Cambridge University Press:  22 January 2016

Makoto Ohtsuka
Makoto Ohtsuka*
Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University
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The electric capacity of a conductor in the 3-dimensional euclidean space is defined as the ratio of a positive charge given to the conductor and the potential on its surface. The notion of capacity was defined mathematically first by N. Wiener [7] and developed by C. de la Vallée Poussin, O. Frostman and others. For the history we refer to Frostman’s thesis [2]. Recently studies were made on different definitions of capacity and related notions. We refer to M. Ohtsuka [4] and G. Choquet [1], for instance. In the present paper we shall investigate further some relations among various kinds of capacity and related notions. A part of the results was announced in a lecture of the author in 1962.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 30 , August 1967 , pp. 121 - 127
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[l] Choquet, G.: Diamètre transfini et comparaison de diverses capacités, Sém. Theorie du potentiel, 3 (1958/59), n° 4, 7 pp.Google Scholar
[2] Frostman, O.: Potentiel d’équilibre et capacité des ensembles, Thèse, Lund, 1935, 118 pp.Google Scholar
[3] Fuglede, B.: Le théorème du minimax et la théorie fine du potentiel, Ann. Inst. Fourier, 15 (1965), pp. 65–87.Google Scholar
[4] Ohtsuka, M.: Selected topics in function theory, Tokyo, 1957, in Japanese.Google Scholar
[5] Ohtsuka, M.: An application of the minimax theorem to the theory of capacity, J. Sci. Hiroshima Univ. Ser. A-I Math., 29 (1965), pp. 217–221.Google Scholar
[6] Ohtsuka, M.: Generalized capacity and duality theorem in linear programming, ibid., 30 (1966), pp. 45–56.Google Scholar
[7] Wiener, N.: Certain notions in potential theory, J. Math. Phys. M.I.T., 3 (1924), pp. 24–51.CrossRefGoogle Scholar