Quasiconformal maps and Poincaré domains
type: Item
instance of: scholarly article
title: Quasiconformal maps and Poincaré domains
zbMATH Open document ID: 0874.30015
author: Ritva Hurri-Syrjänen
author: Susan G. Staples
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-11-13T00:00:00Z
full work available at URL: CONT26-4.html
review text: Let \(D\) be a domain in \(\mathbb{R}^m\) and \(1\leq p<\infty\). If for some constant \(C <\infty\) \(\inf_{a\in\mathbb{R}} |u- a|_{L^p(D)} \leq C|\nabla u|_{L^p(D)}\) for all \(u \in C^1(D)\), then \(D\) is called a \(p\)-Poincaré domain, Poincaré domains are invariant under locally bi-Lipschitz maps [\textit{R. Hurri}: Ann. Acad. Sci. Fenn., Ser. A I Math. Dissertationes 71, 1-42 (1988; Zbl 0667.46023)] but not necessarily under quasiconformal maps. Some extra conditions are needed: Let \(D\) be a John domain in \(\mathbb{R}^n\) [\textit{O. Martio}, \textit{J. Sarvas}: Ann. Acad. Sci. Fenn., Ser. AI 4, 383-401 (1979; Zbl 0427.30021)] and \(f\) a quasiconformal map of \(D\) onto \(D^*\) with (*) \(|f'|_{L^{n+ \varepsilon} (D)}< \infty\). Then \(D^*\) is \(p\)-Poincaré for \(p\geq n\) and for \(p_0<p<n\) where \(p_0\) essentially depends on the higher integrability condition (*). The method of the proof gives explicit bounds for the Poincaré constant \(C\) for \(D^*\).
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1036051
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: John domain
reviewed by: O. Martio
MaRDI profile type: Publication
OpenAlex ID: W2001687365
cites work: Quasiconformal mappings and global integrability of the derivative
cites work: Hölder continuity of conformal mappings and non-quasiconformal Jordan curves
cites work: Q3811180
cites work: The L\(^p\)-integrability of the partial derivatives of a quasiconformal mapping
cites work: Quasiconformally homogeneous domains
cites work: Lipschitz classes and quasiconformla mappings
cites work: Elliptic Partial Differential Equations of Second Order
cites work: Q4693180
cites work: Q3803399
cites work: Poincaré domains and quasiconformal mappings
cites work: Q3994780
cites work: Unbounded Poincaré domains
cites work: Q3797452
cites work: Injectivity theorems in plane and space
cites work: Global Lp -integrability of the derivative of a quasiconformal mapping
cites work: Q4000499
cites work: Functions of bounded mean oscillation and quasiconformal mappings
cites work: Exponential integrability of the quasi-hyperbolic metric on Hölder domains
cites work: Holder Domains and Poincare Domains
cites work: Maximal Functions, A ∞ -Measures, and Quasiconformal Maps
cites work: Global integrability of the Jacobian and quasiconformal maps
cites work: Q5607484
DOI: 10.1216/RMJM/1181071995
description: scientific article; zbMATH DE number 1036051