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

TL;DR

This paper characterizes the vectors of total charges for which the Gauss variational problem for infinite dimensional vector measures is solvable, especially when some plates are noncompact, using potential theory and orthogonal projections.

Contribution

It provides necessary and sufficient conditions for solvability of the Gauss variational problem under perfect kernels, extending previous results to more general settings.

Findings

Characterization of solvable charge vectors for the variational problem.

Solution expressed via auxiliary extremal problem and orthogonal projection.

Illustrations with examples involving Riesz kernels.

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.