Estimates for Jacobian and dilatation coefficients of open discrete mappings with controlled \(p\)-module
type: Item
instance of: scholarly article
title: Estimates for Jacobian and dilatation coefficients of open discrete mappings with controlled \(p\)-module
zbMATH Open document ID: 1380.30017
author: A. Golberg
author: R. R. Salimov
author: Evgeny Sevost'yanov
published in: Complex Analysis and Operator Theory
publication date: 2017-12-01T00:00:00Z
review text: Given a family \(\Gamma\) of paths in \(\mathbb{R}^n\) and some \(p>1\), the \(p\)-modulus is the quantity \[ \mathcal{M}_p(\Gamma) = \inf\left\{ \int_{\mathbb{R}^n}\rho^p(x) \, dm(x)\right\} \] where the infimum is taken over all Borel functions \(\rho:\mathbb{R}^n \to [0,\infty]\) such that \(\int_{\gamma}\rho \geq 1\) for all \(\gamma \in \Gamma\). Such functions \(\rho\) are called admissible functions for \(\Gamma\). In the paper under review, the authors consider two classes of maps which, although weaker than quasiregular maps, still retain many nice analytic properties that quasiregular maps satisfy. Fix a domain \(D\subset \mathbb{R}^n\), a Lebesgue-measurable map \(Q:D \to [0,\infty]\) and some \(p\geq 1\). A discrete open map \(f:D\to\overline{\mathbb{R}^n}\) is a \((p,Q)\)-map if for all path families \(\Gamma\) in \(D\) and all admissible \(\rho\) for \(\Gamma\), \[ \mathcal{M}_p(f(\Gamma)) \leq \int_D Q(x) \rho^p(x) \, dm(x). \] Slightly weaker, a discrete open map \(f:D\to\overline{\mathbb{R}^n}\) is a ring \((p,Q)\)-map if for all annuli \(A=A(x_0,r,R)\subset D\) and for all path families \(\Gamma\) with curves in \(A\) joining the two components of \(\partial A\), there exists an admissible \(\rho(x) = \eta(|x-x_0|)\) for \(\Gamma\) such that \[ \mathcal{M}_p(f(\Gamma)) \leq \int_A Q(x)\eta^p(|x-x_0|) \, dm(x). \] It is known that both classes are ACL and therefore differentiable almost everywhere. For those \(x\in D\) that \(f'(x)\) is defined, \(\ell(f'(x))\) (resp. \(\|f'(x)\|\)) denotes the smallest (resp. biggest) modulus of the complex eigenvalues of \(f'(x)\) while \(J(x,f)\) denotes the Jacobian of \(f\) at \(x\). For the \((p,Q)\)-maps with \(Q\in L^1_{\text{loc}}(D)\), the authors prove in Theorems 1.2 and 4.1 that if \(1 < p < n\), then \[ C^{-1} Q(x)^{\frac{n}{n-p}} \leq |J(x,f)| \leq C Q(x)^{\frac{n}{n-p}} \quad\text{ a.e. in }D \] for some \(C=C(n,p)\) while if \(n-1 < p\leq n\), then \[ 0 \neq |J(x,f)| \leq Q(x) \ell(f'(x))^{p} \quad\…
reviewed by: Vyron Vellis
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 26B30
zbMATH DE Number: 6814624
zbMATH Keywords: \(p\)-module of families of curves
zbMATH Keywords: weighted \(p\)-module
zbMATH Keywords: ring \((p, Q)\)-mappings
zbMATH Keywords: \(p\)-inner and \(p\)-outer dilatations
zbMATH Keywords: bilipschitz mappings
zbMATH Keywords: mappings of finite distortion
MaRDI profile type: Publication
full work available at URL: s11785-016-0628-2
OpenAlex ID: W2564112979
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DOI: 10.1007/S11785-016-0628-2
description: scientific article; zbMATH DE number 6814624