Minimum energy problems with external fields on locally compact spaces

TL;DR

This paper investigates minimum energy problems with external fields on locally compact spaces, establishing existence conditions for minimizers under general kernels and external fields, and analyzing their properties and continuity.

Contribution

It provides new existence criteria for energy minimizers with external fields on general sets and kernels, answering a question from 1961 and extending classical potential theory results.

Findings

Established necessary and sufficient conditions for minimizer existence.

Provided characterizations and continuity analysis of the minimizers.

Applied results to Riesz kernels and classical potential theory.

Abstract

The paper deals with minimum energy problems in the presence of external fields on a locally compact space $X$ with respect to a function kernel $\kappa$ satisfying the energy and consistency principles. For quite a general (not necessarily lower semicontinuous) external field $f$, we establish sufficient and/or necessary conditions for the existence of $\lambda_{A,f}$ minimizing the Gauss functional \[\int\kappa(x,y)\,d(\mu\otimes\mu)(x,y)+2\int f\,d\mu\] over all positive Radon measures $\mu$ with $\mu(X)=1$, concentrated on quite a general (not necessarily closed or bounded) $A\subset X$, thereby giving an answer to a question raised by M. Ohtsuka (J. Sci. Hiroshima Univ., 1961). Such results are specified for the Riesz kernels $|x-y|^{\alpha-n}$, $0<\alpha<n$, on $\mathbb R^n$, $n\geqslant2$, and are illustrated by some examples. Furthermore, we provide various alternative characterizations of the minimizer $\lambda_{A,f}$, and as a by-product we analyze the strong and vague continuity of $\lambda_{A,f}$ under the exhaustion of $A$ by compact $K\subset A$. The results obtained hold true and are new for many interesting kernels in classical and modern potential theory.