Problem of the minimum energy for space condensers and Riesz kernels (Q1262974)

Let \(J_{\alpha}(\nu)\) be the energy of a charge \(\nu\) for a Riesz kernel \(k_{\alpha}(x)=| x|^{\alpha -p}\), \(0<\alpha <p\) in \({\mathbb{R}}^ p\), \(p\geq 3,\) \(V_{\alpha}(E)=\inf \{J_{\alpha}(\nu):\quad \nu \in N^ 1(E)\},\) where \(E=(E^+,E^-)\) is a space condenser and \[ N^ 1(E)=\{\nu =\nu^+-\nu^-:\quad \sup p(\nu^+)=E^+,\quad \sup p(\nu^-)=E^-\}. \] If the equality \(V_{\alpha}(E)=J_{\alpha}(\lambda)\) holds, then the charge \(\lambda =\lambda_{E,\alpha}\) is called an extremal one. The questions of existence and uniqueness and some properties of the extremal charge are studied.