Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-11T10:40:36.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of Dirichlet finite harmonic measures on Euclidean balls

Published online by Cambridge University Press:  22 January 2016

Mitsuru Nakai
Mitsuru Nakai*
Affiliation:
Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa, Nagoya 466, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Divide the ideal boundary of a noncompact Riemannian manifold M into two parts δ0 and δ1 Viewing that M is surrounded by two conducting electrodes δ0 and δ1 we ask whether (M; δ0, δ1) functions as a condenser in the sense that the unit electrostatic potential difference between two electrodes is produced by putting a charge of finite energy on one electrode when the other is grounded. The generalized condenser problem asks whether there exists a subdivision δ0 ∪ δ1 of the ideal boundary of M such that (M; δ0, δ1 functions as a condenser.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 133 , March 1994 , pp. 85 - 125
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1994

References

[1] Accola, R., The bilinear relation on open Riemann surfaces, Trans. Amer. Math. Soc, 96(1960), 143–161.CrossRefGoogle Scholar
[2] Adams, R. A., Sobolev Spaces, Academic Press, 1978.Google Scholar
[3] Ahlfors, L. and Sario, L., Riemann Surfaces, Princeton, 1960.Google Scholar
[4] Constantinescu, C. und Cornea, A., Ideale Ränder Riemannscher Flächen, Springer, 1963.CrossRefGoogle Scholar
[5] Rham, G. de, Variété, Différentiables, Hermann, 1955.Google Scholar
[6] Dunford, N. and Schwartz, J. T., Linear Operators, Part I, Interscience, 1967.Google Scholar
[7] Glasner, M. and Nakai, M., Riemannian manifolds with discontinuous metrics and the Dirichlet integrals, Nagoya Math. J., 46 (1972), 1–48.CrossRefGoogle Scholar
[8] Heins, M., On the Lindelöf principle, Ann. Math., 61 (1955), 440–473.CrossRefGoogle Scholar
[9] Holopainen, I., Nonlinear potential theory and quasiregular mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn., Ser. AI Math. Dissertation, 74 (1990), 1–45.Google Scholar
[10] Ishida, H., Harmonic Dirichlet functions and the components of harmonic boundaries of Riemann surfaces, J. Math. Kyoto Univ., 29 (1989), 625–641.Google Scholar
[11] Krantz, S., Function Theory of Several Complex Variables, Wiley, 1982.Google Scholar
[12] Kusunoki, Y., Characterizations of canonical differentials, J. Math. Kyoto Univ., 5 (1966), 197–207.Google Scholar
[13] Lelong-Ferrand, J., Etude d’une class d’application liées à des homomorphismes d’algebres de fonctions, et généralisant les quasi conformes, Duke Math. J., 40 (1973), 163–186.CrossRefGoogle Scholar
[14] Lewis, L. G., Quasiconformal mappings and Royden Algebras in space, Trans. Amer. Math. Soc, 158 (1971), 481–492.CrossRefGoogle Scholar
[15] Milnor, J., Topology from the Differentiable Viewpoint, Univ. Press of Va., 1965.Google Scholar
[16] Mizohata, S., The Theory of Partial Differential Equations, Cambridge Univ. Press, 1973.Google Scholar
[17] Maz’ja, V. G., On the continuity at a boundary point of solutions of quasi-linear elliptic equations, Vestnik Leningrad Univ., 3 (1976), 225–242.Google Scholar
[18] Maz’ja, V. G., Sobolev Spaces, Springer, 1985.CrossRefGoogle Scholar
[19] Nakai, M., “On parabolicity and Royden compactifications of Riemannian manifolds” in Proceedings of the International Conference on Functional Analysis and Related Topics, 1969, 316–325.Google Scholar
[20] Nakai, M., Royden algebras and quasi-isometries of Riemannian manifolds, Pacific J. Math., 40 (1972), 397–414.CrossRefGoogle Scholar
[21] Nakai, M., Dirichlet finite harmonic measures on Riemann surfaces, Complex Variables, 17(1991), 123–131.Google Scholar
[22] Nakai, M., Riemannian manifolds with connected Royden harmonic boundaries, Duke Math. J., 67 (1992), 589–625.CrossRefGoogle Scholar
[23] Nakai, M., Existence of Dirichlet finite harmonic measures in nonlinear potential theory, Complex Variables, 21 (1993), 107–114.Google Scholar
[24] Nakai, M. and Tanaka, H., Existence of quasiconformal mappings between Riemannian manifolds, Kodai Math. J., 5 (1982), 122–131.CrossRefGoogle Scholar
[25] Ohtsuka, M., Extremal Length and p-precise Functions (In preparation).Google Scholar
[26] Reshetnyak, Yu. G., Space Mappings with Bounded Distortion, Amer. Math. Soc, 1989.Google Scholar
[27] Rudin, W., Real and Complex Analysis, McGraw-Hill, 1987.Google Scholar
[28] Sario, L. and Nakai, M., Classification Theory of Riemann Surfaces, Springer, 1970.CrossRefGoogle Scholar
[29] Serrin, J., Local behavior of solutions of quasilinear equations, Acta Math., 111 (1964), 247–302.CrossRefGoogle Scholar
[30] Serrin, J., Isolated singularities of solutions of quasilinear equations, Acta Math., 113(1965), 219–240.CrossRefGoogle Scholar
[31] Sonderborg, N., Quasiregular mappings with finite multiplicity and Royden algebras, Indiana Univ. Math. J., 40 (1991), 1143–1167.CrossRefGoogle Scholar
[32] Tanaka, H., Harmonic boundaries of Riemannian manifolds, Nonlinear Anal., 14 (1990), 55–67.CrossRefGoogle Scholar