Estimates of Jacobians by subdeterminants.
type: Item
instance of: scholarly article
title: Estimates of Jacobians by subdeterminants.
zbMATH Open document ID: 1053.42024
DOI: 10.1007/BF02922041
author: Flavia Giannetti
author: Tadeusz Iwaniec
author: Jani Onninen
author: Anna Verde
published in: The Journal of Geometric Analysis
publication date: 2002-08-08T00:00:00Z
review text: Let \(\Omega\) be a nonempty open subset in \(\mathbb{R}^{n}\), \(n\geq 2\). A mapping \(f=(f^{1},\dots,f^{2}):\Omega\to\mathbb{R}^{n}\) in the Sobolev space \(W^{1,n-1}(\Omega,\mathbb{R}^{n})\) is studied. The differential matrix is \(Df=[\partial f^{i}/\partial x_{j}]\) and the corresponding cofactor matrix is \(D^{\sharp}f\). The determinant of the differential matrix \(Df(x)\) is assumed to be nonnegative and the cofactor matrix is assumed to satisfy \(| D^{\sharp}f| ^{n/(n-1)}\in L^{P}(\Omega)\), where \(L^{P}(\Omega)\) is an Orlicz space. The assumptions on \(P\) are \[ \int_{1}^{\infty}{P(s)ds\over s^{2}}=\infty \] and \[ \left[t^{-1}P(t)\right]'\leq 0\leq \left[t^{-s}P(t)\right]',\quad s={n^{2}-2n+1\over n^{2}-n-1}. \] The main result of this paper is that, under the above assumptions, the Jacobian determinant of \(f\) is locally integrable.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1777202
zbMATH Keywords: integrability of Jacobians
zbMATH Keywords: distributional Jacobian
zbMATH Keywords: cofactor matrix
zbMATH Keywords: Orlicz space
MaRDI profile type: Publication
Wikidata QID: Q110031917
cites work: Q4105853
cites work: Convexity conditions and existence theorems in nonlinear elasticity
cites work: Integrability for the Jacobian of orientation preserving mappings
cites work: Q4039741
cites work: Compensated compactness and Hardy spaces
cites work: Direct methods in the calculus of variations
cites work: Q5565773
cites work: A remark on the equality \(\text{det} Df=\text{Det} Df\)
cites work: Q4235447
cites work: Limits of the improved integrability of the volume forms
cites work: A note on weak approximation of minors
cites work: Mappings of finite distortion: Monotonicity and continuity
cites work: Q4415601
cites work: Geometric function theory and nonlinear analysis
cites work: \({\mathcal H}^1\)-estimates of Jacobians by subdeterminants
cites work: On the integrability of the Jacobian under minimal hypotheses
cites work: A study of Jacobians in Hardyd–Orlicz spaces
cites work: Minimal assumptions for the integrability of the Jacobian
cites work: Higher integrability of determinants and weak convergence in L1.
cites work: Q5748433
cites work: Invertibility and a Topological Property of Sobolev Maps
cites work: On a new class of elastic deformations not allowing for cavitation
cites work: Some geometrical properties of functions and mappings with generalized derivatives
cites work: Q5607484
cites work: Q4933737
cites work: Regularity properties of deformations with finite energy
description: scientific article; zbMATH DE number 1777202