Condensers with Touching Plates and Constrained Minimum Riesz and Green Energy Problems

Content Provider	Semantic Scholar
Author	Dragnev, Peter D. Fuglede, Bent Hardin, Douglas P. Saff, Edward B. Zorii, Natalia
Copyright Year	2017
Abstract	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.
Starting Page	1
Ending Page	33
Page Count	33
File Format	PDF HTM / HTML
DOI	10.1007/s00365-019-09454-5
Alternate Webpage(s)	http://www.math.vanderbilt.edu/saffeb/texts/270.pdf
Alternate Webpage(s)	https://arxiv.org/pdf/1711.05484v2.pdf
Alternate Webpage(s)	https://doi.org/10.1007/s00365-019-09454-5
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article