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 ∫ κ (x , y) d (μ ⊗ μ) (x , y) + 2 ∫ f d μ over all positive Radon measures μ with μ (X) = 1 , concentrated on quite a general (not necessarily closed or bounded) set A ⊂ 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 R n , n ⩾ 2 , and are illustrated by some examples. Furthermore, we provide various alternative characterizations of the minimizer λ A , f , and as a by-product we prove the continuity of both λ A , f and the modified Robin constant under the exhaustion of A by compact K ⊂ A . Along with the advantages of using perfect kernels, the analysis performed is substantially based on the author's recent theory of inner balayage on locally compact spaces (Potential Anal., 2022). The results obtained do hold and are new for many interesting kernels in classical and modern potential theory. [ABSTRACT FROM AUTHOR]