Minimum energy problems with external fields on locally compact spaces

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.