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# 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

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1864-8266
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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

## Abstract.

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 $u,v\in {W}^{1,2}\left(\Omega ,{ℝ}^{2}\right)$ where $det\left(Du\right)>0$, $det\left(Dv\right)>0$ a.e. and u is a mapping of integrable dilatation. Suppose for a.e. $z\in \Omega$ we have $Du{\left(z\right)}^{T}Du\left(z\right)=\lambda Dv{\left(z\right)}^{T}Dv\left(z\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}\lambda >0\text{.}$ Then there exists a meromorphic function ψ and a homeomorphism $w\in {W}^{1,1}\left(\Omega :{ℝ}^{2}\right)$ such that $Du\left(z\right)=𝒫\left(\psi \left(w\left(z\right)\right)\right)Dv\left(z\right)$ where $𝒫\left(a+ib\right)=\left(\genfrac{}{}{0pt}{}{a\phantom{\rule{1.em}{0ex}}-b}{b\phantom{\rule{1.em}{0ex}}a}\right).$

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.

MSC: 30C65; 26B99

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,

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© 2014 by Walter de Gruyter Berlin/Boston.