Abstract
We continue the investigation of the problem of energy minimum for condensers began in the first part of the present work. Condensers are treated in a certain generalized sense. The main attention is given to the case of classes of measures noncompact in the vague topology. In the case of a positive-definite kernel, we develop an approach to this minimum problem based on the use of both strong and vague topologies in the corresponding semimetric spaces of signed Radon measures. We obtain necessary and (or) sufficient conditions for the existence of minimal measures. We describe potentials for properly determined extremal measures.
Similar content being viewed by others
REFERENCES
N. V. Zorii, “Extremal problems in the theory of capacities of condensers in locally compact spaces. I,” Ukr. Mat. Zh., 53, No. 2, 168–189 (2001).
N. V. Zorii, “On one noncompact variational problem in the theory of Riesz potentials. I,” Ukr. Mat. Zh., 47, No. 10, 1350–1360 (1995).
N. V. Zorii, “On one noncompact variational problem in the theory of Riesz potentials. II,” Ukr. Mat. Zh., 48, No. 5, 603–613 (1996).
N. V. Zorii, “On one variational problem in the theory of Green potentials. I,” Ukr. Mat. Zh., 42, No. 4, 494–500 (1990).
N. V. Zorii, “On one variational problem in the theory of Green potentials. II,” Ukr. Mat. Zh., 42, No. 11, 1475–1480 (1990).
N. V. Zorii, “On one extremal problem on energy minimum for space condensers,” Ukr. Mat. Zh., 38, No. 4, 431–437 (1986).
H. Cartan, “Théorie du potentiel newtinien: énergie, capacité, suites de potentiels,” Bull. Soc. Math. France, 73, 74–106 (1945).
B. Fuglede, “On the theory of potentials in locally compact spaces,” Acta Math., 103, No. 3–4, 139–215 (1960).
R. E. Edwards, Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969).
N. Bourbaki, Integration. Measures, Integration of Measures [Russian translation], Nauka, Moscow (1967).
M. Brélot, Eléments de la Théorie Classique du Potentiel [Russian translation], Mir, Moscow (1964).
N. S. Landkof, Foundations of Modern Potential Theory [in Russian], Nauka, Moscow (1966).
M. Ohtsuka, “On potentials in locally compact spaces,” J. Sci. Hiroshima Univ. Ser. A-1, 25, No. 2, 135–352 (1961).
N. Bourbaki, General Topology. Main Structures [Russian translation], Nauka, Moscow (1968).
J. L. Kelley, General Topology [Russian translation], Mir, Moscow (1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zorii, N.V. Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. II. Ukrainian Mathematical Journal 53, 528–554 (2001). https://doi.org/10.1023/A:1012370419990
Issue Date:
DOI: https://doi.org/10.1023/A:1012370419990