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Sobolev homeomorphisms with gradients of low rank via laminates

  • Daniel Faraco , Carlos Mora-Corral and Marcos Oliva EMAIL logo
Published/Copyright: August 30, 2016

Abstract

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.

MSC 2010: 46E35; 25B25; 25B35

Communicated by Frank Duzaar


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

Award Identifier / Grant number: RYC-2010-06125

Award Identifier / Grant number: 307179

Funding statement: 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).

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Received: 2016-3-2
Revised: 2016-7-7
Accepted: 2016-7-21
Published Online: 2016-8-30
Published in Print: 2018-4-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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