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Advances in Calculus of Variations

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Volume 11, Issue 2


Sobolev homeomorphisms with gradients of low rank via laminates

Daniel Faraco
  • Department of Mathematics, Faculty of Sciences, Universidad Autónoma de Madrid, E-28049 Madrid; and ICMAT CSIC-UAM-UCM-UC3M, E-28049 Madrid, Spain
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/ Carlos Mora-Corral / Marcos Oliva
  • Corresponding author
  • Department of Mathematics, Faculty of Sciences, Universidad Autónoma de Madrid, E-28049 Madrid; and ICMAT CSIC-UAM-UCM-UC3M, E-28049 Madrid, Spain
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Published Online: 2016-08-30 | DOI: https://doi.org/10.1515/acv-2016-0009


Let Ωn be a bounded open set. Given 2mn, we construct a convex function u:Ω whose gradient f=u is a Hölder continuous homeomorphism, f is the identity on Ω, the derivative Df has rank m-1 a.e. in Ω and Df is in the weak Lm space Lm,w. The proof is based on convex integration and staircase laminates.

Keywords: Sobolev homeomorphisms; low rank; laminates; Luzin condition

MSC 2010: 46E35; 25B25; 25B35


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About the article

Received: 2016-03-02

Revised: 2016-07-07

Accepted: 2016-07-21

Published Online: 2016-08-30

Published in Print: 2018-04-01

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2014-57769-C3-1-P

Award identifier / Grant number: RYC-2010-06125

Funding Source: European Research Council

Award identifier / Grant number: 307179

The authors have been supported by Project MTM2014-57769-C3-1-P of the Spanish Ministry of Economy and Competitivity and the ERC Starting grant no. 307179. The second author has also been supported by the “Ramón y Cajal” grant RYC-2010-06125 (Spanish Ministry of Economy and Competitivity).

Citation Information: Advances in Calculus of Variations, Volume 11, Issue 2, Pages 111–138, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2016-0009.

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