Minimum Riesz energy problems with external fields

Abstract

The paper deals with minimum energy problems in the presence of external fields with respect to the Riesz kernels |x−y|α−n, 0<α<n, on Rn, n⩾2. For quite a general (not necessarily lower semicontinuous) external field f, we obtain necessary and/or sufficient conditions for the existence of λA,f minimizing the Gauss functional∫|x−y|α−nd(μ⊗μ)(x,y)+2∫fdμ over all positive Radon measures μ with μ(Rn)=1, concentrated on quite a general (not necessarily closed) A⊂Rn. We also provide various alternative characterizations of the minimizer λA,f, analyze the continuity of both λA,f and the modified Robin constant for monotone families of sets, and give a description of the support of λA,f. The significant improvement of the theory in question thereby achieved is due to a new approach based on the close interaction between the strong and the vague topologies, as well as on the theory of inner balayage, developed recently by the author.