We study minimum energy problems relative to the $$\alpha $$α-Riesz kernel $$|x-y|^{\alpha -n}$$|x-y|α-n, $$\alpha \in (0,2]$$α∈(0,2], over signed Radon measures $$\mu $$μ on $${\mathbb {R}}^n$$Rn, $$n\geqslant 3$$n⩾3, associated with a generalized condenser $$(A_1,A_2)$$(A1,A2), where $$A_1$$A1 is a relatively closed subset of a domain D and $$A_2={\mathbb {R}}^n\setminus D$$A2=Rn\D. We show that although $$A_2\cap {\mathrm {Cl}}_{{\mathbb {R}}^n}A_1$$A2∩ClRnA1 may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to $$\mu $$μ with $$\mu ^+\leqslant \xi $$μ+⩽ξ, where a constraint $$\xi $$ξ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted $$\alpha $$α-Riesz potentials of the solutions, single out their characteristic properties, and analyze their supports. The approach developed is mainly based on the establishment of an intimate relationship between the constrained minimum $$\alpha $$α-Riesz energy problem over signed measures associated with $$(A_1,A_2)$$(A1,A2) and the constrained minimum $$\alpha $$α-Green energy problem over positive measures carried by $$A_1$$A1. The results are illustrated by examples.

中文翻译：

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带接触板的冷凝器和受限最小 Riesz 和绿色能源问题

提供解的加权 $$\alpha $$α-Riesz 势的描述，挑出它们的特性，并分析它们的支持。所开发的方法主要基于在与 $$(A_1,A_2)$$(A1,A2) 相关的带符号度量上的约束最小 $$\alpha $$α-Riesz 能量问题与约束条件之间建立密切关系。最小 $$\alpha $$α-绿色能源问题超过 $$A_1$$A1 进行的积极措施。结果通过实例说明。A2) 和约束最小 $$\alpha $$α-绿色能源问题超过 $$A_1$$A1 携带的积极措施。结果通过实例说明。A2) 和约束最小 $$\alpha $$α-绿色能源问题超过 $$A_1$$A1 携带的积极措施。结果通过实例说明。