On geometric properties of harmonic \(\text{Lip}_ 1\)-capacity
type: Item
instance of: scholarly article
title: On geometric properties of harmonic \(\text{Lip}_ 1\)-capacity
zbMATH Open document ID: 0852.31004
published in: Pacific Journal of Mathematics
publication date: 1996-11-07T00:00:00Z
review text: We study geometric properties of the harmonic \(\text{Lip}_1\)-capacity \(\kappa_n' (E)\), \(E \subset \mathbb{R}^n\). It is related to functions which are harmonic outside \(E\) and locally Lipschitzian everywhere. We show that \(\kappa_{n + 1}' (E \times I)\) is comparable to \(\kappa_n' (E)\) for \(E \subset \mathbb{R}^n\) and for intervals \(I \subset \mathbb{R}\). We also show that if \(E\) lies on a Lipschitz graph, then \(\kappa_n' (E)\) is comparable to the \((n - 1)\)-dimensional Hausdorff measure \({\mathcal H}^{n - 1} (E)\). Finally we give some general criteria to guarantee that \(\kappa_n' (E) = 0\) although \({\mathcal H}^{n - 1} (E) > 0\).
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 912156
zbMATH Keywords: harmonic \(\text{Lip}_ 1\)-capacity
zbMATH Keywords: harmonic functions
zbMATH Keywords: capacities
zbMATH Keywords: Hausdorff measures
zbMATH Keywords: geometric properties
author: Pertti Mattila
author: P. V. Paramonov
reviewed by: Pertti Mattila
MaRDI profile type: Publication
full work available at URL: pjm.1995.171.469
OpenAlex ID: W1483707686
DOI: 10.2140/PJM.1995.171.469
description: scientific article; zbMATH DE number 912156