Dirichlet principle for the Royden compactification
type: Item
instance of: scholarly article
title: Dirichlet principle for the Royden compactification
zbMATH Open document ID: 0639.30042
author: Moses Glasner
published in: Kodai Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let R be a hyperbolic Riemann surface, and denote by \(\Delta\) its Royden harmonic boundary. For a compact subset K of \(\Delta\) there is the important notation of Dirichlet capacity, c(K), developed by the author and \textit{M. Nakai} [Isr. J. Math. 55, 15-32 (1986; Zbl 0605.30045)]. Given a Dirichlet function f on R, in the sense of Constantinescu-Cornea, the author establishes the following generalization of the Dirichlet principle: For a compact set \(K\subseteq \Delta\) with \(c(K)>0\) and the family \[ {\mathcal F}_ f=\{\phi: \phi \text{ is a Dirichlet function, }\phi =f \text{ q.e. on } K\}, \] there exists a unique harmonic function \(h_ f\in {\mathcal F}_ f\) satisfying \[ D(h_ f)=\inf \{D(\phi): \phi \in {\mathcal F}_{\phi}\}. \] Here \(D(\phi)=\int_{R}d\phi \wedge *d\phi\), and ``q.e.'' means up to a set of Dirichlet capacity zero.
Mathematics Subject Classification ID: 30F20
zbMATH DE Number: 4042265
zbMATH Keywords: hyperbolic Riemann surface
zbMATH Keywords: Royden harmonic boundary
zbMATH Keywords: Dirichlet capacity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037461
OpenAlex ID: W1964469615
cites work: Étude et extensions du principe de Dirichlet
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DOI: 10.2996/KMJ/1138037461
description: scientific article; zbMATH DE number 4042265