Natalia Zorii (Ukrainian Academy of Sciences)
Necessary and sufficient condition for the solvability of the Gauss variational problem
Abstract. We discuss the well-known Gauss variational problem on the minimum of energy in the presence of an external field, the infimum being taken over rather general classes of signed Radon measures associated with a system of sets in 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 noncompact case the Gauss variational problem is in general nonsolvable, and give necessary and sufficient conditions for the problem to be solvable. The results obtained are also specified for the Newton, Green, and Riesz kernels in an Euclidean space. Certain generalizations of the Gauss variational problem are also discussed.


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