Abstract
We investigate the well-known Gauss variational problem over classes of Radon measures associated with a system of sets in a locally compact space. Under fairly general assumptions, we obtain necessary and sufficient conditions for its solvability. As an auxiliary result, we describe the potentials of vague and (or) strong limit points of minimizing sequences of measures. The results obtained are also specified for the Newton kernel in ℝn.
Similar content being viewed by others
REFERENCES
M. Ohtsuka, “On potentials in locally compact spaces,” J. Sci. Hiroshima Univ. Ser. A-1, 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, “Equilibrium problems 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, Holt, Rinehart and Winston, New York (1965).
B. Fuglede, “On the theory of potentials in locally compact spaces,” Acta Math., 103, No.3-4, 139–215 (1960).
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).
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 noncompact variational problem in the theory of Riesz potentials. I,” Ukr. Mat. Zh., 47, No.10, 1350–1360 (1995); “On one noncompact variational problem in the theory of Riesz potentials. II,” Ukr. Mat. Zh., 48, No. 5, 603-613 (1996).
N. V. Zorii, “Problem on the minimum of Green energy for space condensers,” Dokl. Akad. Nauk SSSR, 307, No.2, 265–269 (1989).
N. V. Zorii, “On one variational problem in the theory of Green potentials. I,” Ukr. Mat. Zh., 42, No.4, 494–500 (1990); “On one variational problem in the theory of Green potentials. II, ” Ukr. Mat. Zh., 42, No. 11, 1475–1480 (1990).
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).
J. L. Kelley, General Topology, van Nostrand, New York (1957).
N. V. Zorii, “Extremal problems in the theory of capacities of condensers in locally compact spaces. III, ” Ukr. Mat. Zh., 53, No.6, 758–782 (2001).
N. V. Zorii, “Theory of potentials with respect to consistent kernels: theorem on completeness and sequences of potentials,” Ukr. Mat. Zh., 56, No.11, 1513–1526 (2004).
N. S. Landkof, Foundations of Modern Potential Theory [in Russian], Nauka, Moscow (1966).
M. Kishi, “Sur l’existence des mesures des condensateurs,” Nagoya Math. J., 30, 1–7 (1967).
N. Zorii, “On existence of a condenser measure,” Mat. Stud., 13, No.2, 181–189 (2000).
M. Brelot, Elements de la Theorie Classique du Potentiel, Sorbonne University, Paris (1959).
M. Brelot, On Topologies and Boundaries in Potential Theory, Springer, Berlin (1971).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 60–83, January, 2005.
Rights and permissions
About this article
Cite this article
Zorii, N.V. Necessary and Sufficient Conditions for the Solvability of the Gauss Variational Problem. Ukr Math J 57, 70–99 (2005). https://doi.org/10.1007/s11253-005-0172-0
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-005-0172-0