Riesz spherical potentials with external fields and minimal energy points separation (Q869271)

Let \( S^d = \{x \in \mathbb R^{d+1}: | x| = 1\}\), \(K_s (x,y) = | y - x| ^{-s}\), \(J_s (\mu) = \iint K_s (x, y) \,d \mu (x) \,d \mu (y)\). A non-negative lower semi-continuous function \(Q: S^d \to [0, \infty]\) such that the set \(\{x \in S^d: Q (x) < \infty\}\) has positive Lebesgue measure is refered as external field. The authors investigate the extremal problem \(J_Q (\mu) \to \inf\) under the conditions \(\sup \mu \subset E \subset S^d\), \(\mu (S^d) = 1,\) where \[ J_Q (\mu ) = J_s (\mu ) + 2 \int Q(x) \,d \mu (x). \] There are a survey of previous results and many new ones in the paper. The authors study the discrete extremal problem as well. They give the separation constant explicitly which is new even for the classical case \(s=d-1\).