Abstract
The concept of consistent kernels introduced by Fuglede in 1960 is widely used in extremal problems of the theory of potential on classes of positive measures. In the present paper, we show that this concept is also efficient for the investigation of extremal problems on fairly broad classes of signed measures. In particular, for an arbitrary consistent kernel in a locally compact space, we prove a theorem on the strong completeness of fairly general subspaces E of all measures with finite energy. (Note that, according to the well-known Cartan counterexample, the entire space E is strongly incomplete even in the classical case of the Newton kernel in ℝn Using this theorem, we obtain new results for the Gauss variational problem, namely, in the non-compact case, we give a description of vague and (or) strong limiting measures of minimizing sequences and obtain sufficient solvability conditions.
Similar content being viewed by others
REFERENCES
B. Fuglede, “On the theory of potentials in locally compact spaces,” Acta Math., 103, No.3–4, 139–215 (1960).
B. Fuglede, “Caracterisation des noyaux consistants en theorie du potentiel,” Comptes Rendus., 255, 241–243 (1962).
M. Ohtsuka, “On potentials in locally compact spaces,” J. Sci. Hiroshima Univ. Ser. A1, 25, No.2, 135–352 (1961).
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, Berlin (1997).
N. Zorii, “On the solvability of the Gauss variational problem,” Comput. Meth. Funct. Theor., 2, No.2, 427–448 (2002).
N. V. Zorii, “Equilibrium potentials with external fields,” Ukr. Mat. Zh., 55, No.9, 1178–1195 (2003).
N. V. Zorii, “Problems of equilibrium for potentials with external fields,” Ukr. Mat. Zh., 55, No.10, 1315–1339 (2003).
N. Bourbaki, Integration. Measures, Integration of Measures [Russian translation], Nauka, Moscow (1967).
R. E. Edwards, Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969).
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).
H. Cartan, “Theorie du potentiel newtonien: energie, capacite, suites de potentiels,” Bull. Soc. Math. France, 73, 74–106 (1945).
J. Deny, “Les potentiels d’energie finie,” Acta Math., 82, 107–183 (1950).
J. Deny, “Sur la definition de l’energie en theorie du potentiel,” Ann. Inst. Fourier, 2, 83–99 (1950).
N. S. Landkof, Foundations of Modern Potential Theory [in Russian], Nauka, Moscow (1966).
N. V. Zorii, “Extremal problems in the theory of capacities of condensers in locally compact spaces. II, ” Ukr. Mat. Zh., 53, No.4, 466–488 (2001).
N. Bourbaki, General Topology. Main Structures [Russian translation], Nauka, Moscow (1968).
N. V. Zorii, “On one extremal problem on energy minimum for space condensers,” Ukr. Mat. Zh., 38, No.4, 431–437 (1986).
N. V. Zorii, “Problem on energy minimum for space condensers and Riesz kernels,” Ukr. Mat. Zh., 41, No.1, 34–41 (1989).
N. V. Zorii, “On one variational problem in the theory of Green potentials. I, II,” Ukr. Mat. Zh., 42, No.4, 494–500 (1990); No. 11, 1475–1480 (1990).
N. V. Zorii, “On one noncompact variational problem in the theory of Riesz potentials. I, II,” Ukr. Mat. Zh., 47, No.10, 1350–1360 (1995); 48, No. 5, 603–613 (1996).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1513–1526, November, 2004.
Rights and permissions
About this article
Cite this article
Zorii, N.V. Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials. Ukr Math J 56, 1796–1812 (2004). https://doi.org/10.1007/s11253-005-0152-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-005-0152-4