Necessary and sufficient conditions for the solvability of the Gauss variational problem for infinite dimensional vector measures

Abstract: We continue our investigation of the Gauss variational problem for infinite dimensional vector measures associated with a condenser $(A_i){i\in I}$. It has been shown in Potential Anal., DOI:10.1007/s11118-012-9279-8 that, if some of the plates (say $A\ell$ for $\ell\in L$) are noncompact then, in general, there exists a vector $\mathbf a=(a_i){i\in I}$, prescribing the total charges on $A_i$, $i\in I$, such that the problem admits no solution. Then, what is a description of all the vectors $\mathbf a$ for which the Gauss variational problem is nevertheless solvable? Such a characterization is obtained for a positive definite kernel satisfying Fuglede's condition of perfectness; it is given in terms of a solution to an auxiliary extremal problem intimately related to the operator of orthogonal projection onto the cone of all positive scalar measures supported by $\bigcup{\ell\in L}A_\ell$. The results are illustrated by examples pertaining to the Riesz kernels.