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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board Member: Armstrong, Scott N. / Astala, Kari / Colding, Tobias / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Finster, Felix / Gianazza, Ugo / Gursky, Matthew / Hardt, Robert / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / McCann, Robert / Mingione, Giuseppe / Nystrom, Kaj / Pacard, Frank / Preiss, David / Riviére, Tristan / Schaetzle, Reiner / Silvestre, Luis

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1864-8266
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Volume 7, Issue 3 (Jul 2014)

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,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, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2012-0026.

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

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