On the strict concavity of the harmonic radius in dimension \(N\geq 3\)
type: Item
instance of: scholarly article
title: On the strict concavity of the harmonic radius in dimension \(N\geq 3\)
zbMATH Open document ID: 1027.31003
DOI: 10.1016/S0021-7824(01)01246-6
author: Pierre Cardaliaguet
author: Rabah Tahraoui
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2002-07-24T00:00:00Z
review text: The paper provides a result on the strict concavity of the harmonic radius of a convex domain in \({\mathbb R}^N\) for \(N\geq 3.\) Precisely, given a domain \(\Omega\subset {\mathbb R}^N\) and denote by \(G_x(\cdot)\) the Green function for the Laplacian in \(\Omega.\) Indeed, \(G_x\) is decomposable into \[ G_x(\cdot)=F_x(\cdot)-H_x(\cdot) \] with singular part \(F_x\) given by the fundamental solution of the Laplacian (\(\forall y\in\Omega\) \(F_x(y)=F(|x-y|)=c_N |x-y|^{2-N},\) \(c_N=\text{const}\)) and \(H_x(\cdot)\) is a harmonic function in \(\Omega\) attaining the values of \(F_x(\cdot)\) on \(\partial\Omega.\) The Robin function \(t(\cdot)\colon\;\Omega\to {\mathbb R}\) is then given by \[ t(x)=H_x(x)\quad \forall x\in \Omega, \] the harmonic radius is \[ r(x)=F^{-1}(t(x))\quad \forall x\in \Omega \] and the harmonic center of \(\Omega\) is a point where \(r(\cdot)\) attains its maximum/minimum. (In the two-dimensional case \(r(\cdot)\) is known as conformal radius.) These notions play a main role when studying elliptic boundary value problems involving concentration of energy and the question of uniqueness of the harmonic center is crucial therein. Employing refined results on capacity theory (Borell's inequality, Brunn-Minkowski inequality, properties of capacity potentials, etc.), the authors establish strict concavity of the harmonic radius when \(N\geq 3\) unless \(\Omega\) is a cylinder, a cone or the translate of a cone. That result implies in particular that the harmonic center of a convex bounded set is unique and the Robin function is strictly convex.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 1770855
zbMATH Keywords: harmonic radius
zbMATH Keywords: harmonic center
zbMATH Keywords: Green function
zbMATH Keywords: strict concavity
zbMATH Keywords: capacity
MaRDI profile type: Publication
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full work available at URL: s0021-7824(01)01246-6
OpenAlex ID: W2059287301
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description: scientific article; zbMATH DE number 1770855