Robin functions and energy functionals of multiply connected domains
type: Item
instance of: scholarly article
title: Robin functions and energy functionals of multiply connected domains
zbMATH Open document ID: 0682.30015
author: Peter Duren
author: M. M. Schiffer
published in: Pacific Journal of Mathematics
publication date: 1991-01-01T00:00:00Z
review text: The Robin function of a finitely connected domain \(\Omega\) is the function R(z,\(\zeta)\) harmonic in \(\Omega\) except for a logarithmic singularity at \(\zeta\), with \(R=0\) on one set of boundary components and \(\partial R/\partial n=0\) on the others. The Robin function generalizes Green's function and can be used similarly to solve mixed boundary-value problems for harmonic functions. The existence of the Robin function is proved by displaying it as a solution to a certain extremal problem involving transfinite diameter. The Robin function generates a Robin matrix analogous to the classical Riemann matrix of periods of harmonic conjugates of harmonic measures of boundary components. Some comparison theorems for the Robin function and the Robin matrix are established. Finally, the Robin matrix is used together with a variational method to solve an extremal problem for the energy functional of \(\Omega\). The energy functional arose and was given a physical interpretation in the authors' previous study of conformal mappings onto nonoverlapping regions [Ann. Acad. Sci. Fenn., Ser. A I, to appear].
reviewed by: Peter Duren
Mathematics Subject Classification ID: 30C99
Mathematics Subject Classification ID: 30C70
zbMATH DE Number: 4116883
zbMATH Keywords: multiply connected domains
zbMATH Keywords: quadratic differentials
zbMATH Keywords: transfinite diameter
zbMATH Keywords: Robin function
zbMATH Keywords: Riemann matrix
zbMATH Keywords: harmonic measures
zbMATH Keywords: energy functional
MaRDI profile type: Publication
full work available at URL: pjm.1991.148.251
OpenAlex ID: W1977508699
DOI: 10.2140/PJM.1991.148.251
description: scientific article; zbMATH DE number 4116883