Computational Methods and Function Theory 2 (2002), No. 2, 427--448
Copyright Heldermann Verlag 2002
Natalia Zorii
zorii@uprotel.net.ua, zorii@imath.kiev.ua, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str., 01601, Kyiv-4, Ukraine.
The work deals with the well-known Gauss variational problem considered over certain rather general classes of signed Radon measures on a locally compact space. In the compact case, sufficient conditions for the solvability of that problem were obtained by M.~Ohtsuka. We show that, in contrast to the compact case, in the non-compact case the Gauss variational problem is in general non-solvable, and give a criterion (necessary and sufficient conditions) for the problem to be solvable. The results obtained are also specified for the Newtonian, Green, and Riesz kernels in~$\mathbb{R}^n$.
Keywords: Radon measure, potential, external field, minimum energy problem.
MSC 2000: 31C15.
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