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For the Riesz kernel κα(x, y) := |x − y|α−n of order 0 < α < n on Rn, n ⩾ 2, we introduce the so-called inner pseudo-balayage ˆωA of a (Radon) measure ω on Rn to a set A ⊂ Rn as the (unique) measure minimizing the Gauss functional
∫ κα(x, y) d(μ ⊗ μ)(x, y) − 2 ∫ κα(x, y) d(ω ⊗ μ)(x, y)
over the class ε+(A) of all positive measures μ of finite energy, concentrated on A. For quite general signed ω (not necessarily of finite energy) and A (not necessarily closed), such ˆωA does exist, and it maintains the basic features of inner balayage for positive measures (defined when α ⩽ 2), except for those implied by the domination principle. (To illustrate the latter, we point out that, in contrast to what occurs for the balayage, the inner pseudo-balayage of a positive measure may increase its total mass.) The inner pseudo-balayage ˆωA is further shown to be a powerful tool in the problem of minimizing the Gauss functional over all μ ∈ ε+(A) with μ(Rn) = 1, which enables us to improve substantially many recent results on this topic, by strengthening their formulations and/or by extending the areas of their applications. For instance, if A is a quasiclosed set of nonzero inner capacity c∗(A), and if ω is a signed measure, compactly supported in Rn \ ClRnA, then the problem in question is solvable if and only if either c∗(A) < ∞, or ˆωA(Rn) ⩾ 1. In particular, if c∗(A) = ∞, then the problem has no solution whenever ω+(Rn) < 1/Cn,α, where Cn,α := 1 if α ⩽ 2, and Cn,α := 2n−α otherwise; whereas ω−(Rn), the total amount of the negative charge, has no influence on this phenomenon. The results obtained are illustrated by some examples.
2010 Mathematics Subject Classification: Primary 31C15.