On minimally thin and rarefied sets in \(R^ p\), p\(\geq 2\)
type: Item
instance of: scholarly article
title: On minimally thin and rarefied sets in \(R^ p\), p\(\geq 2\)
zbMATH Open document ID: 0594.31014
published in: Hiroshima Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Let \(p\geq 2\), \(D=\{x\in R^ p:\) \(x_ 1>0\}\) (where \(x=(x_ 1,...,x_ p))\). If u is subharmonic in D, \(y\in \partial D\), put \(u(y)=\lim \sup u(x),\) \(x\to y\), \(x\in D\). If \(u\leq 0\) on \(\partial D\) and if sup u(x)/x\({}_ 1<\infty\), then it is known that \(u(x)/x_ 1\to \alpha\), \(x\to \infty\), \(x\in D-E\), where the exceptional set E is minimally thin at infinity. If \(p\geq 3\), it is also known that \((u(x)-\alpha x_ 1)/| x| \to 0,\) \(x\to \infty\), \(x\in D-F\), where the exceptional set F is rarefied at infinity. In the present paper precise descriptions of the geometric properties of the exceptional sets E and F are given in terms of a covering by balls.
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 3955375
zbMATH Keywords: subharmonic functions
zbMATH Keywords: minimally thin sets
zbMATH Keywords: rarefied sets
zbMATH Keywords: exceptional set
zbMATH Keywords: geometric properties of the exceptional sets
author: Matts Essén
author: Howard Lawrence Jackson
author: Philip Jonathan Rippon
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3955375