An extremal problem on the minimum of energy for space condensers (Q1115989)

Let \(E^+\), \(E^-\) be closed sets in \({\mathbb{R}}^ p\), \(p\geq 3\) and one of them bounded. The author considers the problem of the minimisation of the energy \[ \iint_{{\mathbb{R}}^ p\times {\mathbb{R}}^ p}| x- y|^{2-p} d\nu (x) d\nu (y), \] where the charge \(\nu\) has a representation \(\nu =\nu^+-\nu^-.\) The charges \(\nu^{\pm}\) satisfy the following conditions: \(\sup p \nu^{\pm}\subset E^{\pm},\quad \nu^{\pm}(1)=1.\) Necessary and sufficient conditions for the existence of the extremal charge are stated.