Minimum energy problems with external fields on locally compact spaces

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 $κ$ 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 $λ_{A,f}$ minimizing the Gauss functional $\intκ(x,y)\,d(μ\otimesμ)(x,y)+2\int f\,dμ$ over all positive Radon measures $μ$ with $μ(X)=1$, 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 $|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$, and are illustrated by some examples. Furthermore, we provide various alternative characterizations of the minimizer $λ_{A,f}$, and as a by-product we analyze the strong and vague continuity of $λ_{A,f}$ 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.

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