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


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 \({W^{1,1}_{\rm loc}}\).


Positive Measure Hausdorff Dimension Nonlinear Elasticity Oxford Mathematical Monograph Good Vector 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

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

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