Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-15T02:53:19.289Z Has data issue: false hasContentIssue false

On the Balayage for Logarithmic Potentials

Published online by Cambridge University Press:  22 January 2016

Nobuyuki Ninomiya*
Affiliation:
Department of Mathematics, Osaka City University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper, we shall consider the logarithmic potential

where μ is a positive measure in the plane, P and Q are any points and PQ denotes the distance from P to Q. In general, consider the potential

of a positive measure μ taken with respect to a kernel K(P, Q) which is a continuous function in P and Q and may be + ∞ for P = Q. A kernel K (P, Q) is said to satisfy the balayage principle if, given any compact set F and any positive measure μ with compact support, there exists a positive measure μ′ supported by F such that K(P, μ′) = K(P, μ) on F with a possible exception of a set of k-capacity zero and K(P, μ′)≦K(P, μ) everywhere. A kernel K(P, Q) is said to satisfy the equilibrium principle if, given any compact set F there exists a positive measure λ supported by F such that K(P, λ) = V (a constant) on F with a possible exception of a set of K-capacity zero and K(p, λ)≦V everywhere.

Information

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1967