Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

IMPACT FACTOR 2017: 1.676

CiteScore 2017: 1.30

SCImago Journal Rank (SJR) 2017: 2.045
Source Normalized Impact per Paper (SNIP) 2017: 1.138

Mathematical Citation Quotient (MCQ) 2017: 1.15

See all formats and pricing
More options …
Volume 7, Issue 3


A generalized Stoilow decomposition for pairs of mappings of integrable dilatation

Andrew Lorent
Published Online: 2013-03-26 | DOI: https://doi.org/10.1515/acv-2012-0026


We prove a rigidity result for pairs of mappings of integrable dilatation whose gradients pointwise deform the unit ball to similar ellipses. Our result implies as corollaries a version of the generalized Stoilow decomposition provided by Theorem 5.5.1 of [Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, 2009] and the two-dimensional rigidity result of [Calc. Var. Partial Differential Equations (2012), DOI 10.1007/s00526-012-0566-4] for mappings whose symmetric part of gradient agrees.

Specifically let u,vW1,2(Ω,2) where det(Du)>0, det(Dv)>0 a.e. and u is a mapping of integrable dilatation. Suppose for a.e. zΩ we have Du(z)TDu(z)=λDv(z)TDv(z)forsomeλ>0. Then there exists a meromorphic function ψ and a homeomorphism wW1,1(Ω:2) such that Du(z)=𝒫ψ(w(z))Dv(z) where 𝒫(a+ib)=a-bba.

We show by example that this result is sharp in the sense that there can be no continuous relation between the gradients of Du and Dv on a dense open connected subset of Ω unless one of the mappings is of integrable dilatation.

Keywords: Symmetric part of gradient; stability and rigidity of differential inclusions; Beltrami equation

MSC: 30C65; 26B99

About the article

Received: 2012-11-12

Revised: 2013-01-25

Accepted: 2013-03-13

Published Online: 2013-03-26

Published in Print: 2014-07-01

Citation Information: Advances in Calculus of Variations, Volume 7, Issue 3, Pages 327–351, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2012-0026.

Export Citation

© 2014 by Walter de Gruyter Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Andrew Lorent
Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2016, Volume 33, Number 1, Page 23

Comments (0)

Please log in or register to comment.
Log in