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 $|x-y{|}^{\alpha -n}$, $0<\alpha <n$, on ${\mathbb{R}}^n$, $n⩾2$. For quite a general (not necessarily lower semicontinuous) external field $f$, we obtain necessary and/or sufficient conditions for the existence of ${\lambda }_{A,f}$ minimizing the Gauss functional \[\int|x-y|^{α-n}\,d(μ\otimesμ)(x,y)+2\int f\,dμ\] over all positive Radon measures $\mu$ with $\mu ({\mathbb{R}}^n)=1$, concentrated on quite a general (not necessarily closed) $A\subset {\mathbb{R}}^n$. We also provide various alternative characterizations of the minimizer ${\lambda }_{A,f}$, analyze the continuity of both ${\lambda }_{A,f}$ and the modified Robin constant for monotone families of sets, and give a description of the support of ${\lambda }_{A,f}$. 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.

28 pages, 2 figures. arXiv admin note: text overlap with arXiv:2207.14342