Quasihyperbolic boundary conditions and Poincaré domains
type: Item
instance of: scholarly article
title: Quasihyperbolic boundary conditions and Poincaré domains
zbMATH Open document ID: 1007.30029
author: Pekka Koskela
author: Jani Onninen
author: Jeremy T. Tyson
published in: Mathematische Annalen
publication date: 2002-12-01T00:00:00Z
review text: A domain \(\Omega\subset \mathbb{R}^n\) of finite volume is said to be a \(p\)-Poincaré domain \((p\geq 1)\) if there exists a constant \(M_p\) such that \[ \int_\Omega|u(x)- u_\Omega|^p dx\leq M_p \int_\Omega|\nabla u(x)|^p dx \] for all \(u\in C^\infty(\Omega)\); here \(u_\Omega\) is the mean value of \(u\) on \(\Omega\). A general characterization for Poincaré domains in terms of a capacity-type estimate was given by Maz'ya, and sufficient conditions in terms of integrability of the quasihyperbolic distance \[ k_\Omega(x,y)= \inf_\gamma \int{ds\over \text{dist}(z,\partial\Omega)} \] were obtained by Jerison, Murri, Smith and Stegenya. In particular, \(\Omega\) is an \(n\)-Poincaré domain if \(k_\Omega\) satisfies the condition \[ k_\Omega(x_0, x)\leq{1\over\beta} \log{\text{dist}(x_0, \partial\Omega)\over \text{dist}(x,\partial\Omega)}+ C_0,\quad \forall x\in\Omega,\tag{1} \] where \(x_0\) is a fixed basepoint, \(C_0= C_0(x_0)< \infty\), and \(\beta> 0\). The main result of the present paper is as follows. If \(\Omega\) satisfies (1) with some \(\beta\leq 1\), then it is a \(p\)-Poincaré domain for all \(p\in [1,\infty)\cap (n- n\beta, n)\). For each \(p< n-n\beta\), there exist domains \(\Omega\) satisfying (1) and which are not \(p\)-Poincaré domains. Similar statements are proved for the so-called \((q,p)\)-Poincaré domains as well. The proof relies on the description of Poincaré domains due to Maz'ya. An application to the Neumann problem on domains with irregular boundaries is given, and relations between condition (1) and the \(s\)-John (``twisted cusp'') condition is discussed.
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1837108
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: quasihyperbolic metric
zbMATH Keywords: Whitney cube
zbMATH Keywords: \(s\)-John domain
MaRDI profile type: Publication
full work available at URL: s00208-002-0331-7
OpenAlex ID: W2038658208
Wikidata QID: Q110032332
DOI: 10.1007/S00208-002-0331-7
description: scientific article; zbMATH DE number 1837108