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Archive for Rational Mechanics and Analysis

, Volume 219, Issue 1, pp 183–202 | Cite as

Sobolev Homeomorphism that Cannot be Approximated by Diffeomorphisms in W1,1

  • Stanislav Hencl
  • Benjamin Vejnar
Article

Abstract

We construct a Sobolev homeomorphism in dimension \({n \geqq 4,\,f \in W^{1,1}((0, 1)^n,\mathbb{R}^n)}\) such that \({J_f = {\rm det} Df > 0}\) on a set of positive measure and J f  < 0 on a set of positive measure. It follows that there are no diffeomorphisms (or piecewise affine homeomorphisms) f k such that \({f_k\to f}\) in \({W^{1,1}_{\rm loc}}\).

Keywords

Positive Measure Hausdorff Dimension Nonlinear Elasticity Oxford Mathematical Monograph Good Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisCharles UniversityPrague 8Czech Republic

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