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On separation of gradient Young measures

  • K. Zhang

Abstract.

Let \(E\subset M^{N\times n}\) be a linear subspace of real \(N\times n\) matrices without rank-one matrices and let \(K = \{A_i\}_{i = 1}^m\subset E\) be a finite set. Suppose \(\Omega \subset \mathbb R^n\) is a bounded arcwise connected Lipschitz domain and \(u_j:\Omega \to \mathbb R^N\) is a sequence of bounded vector-valued mappings in \(W^{1,1}(\Omega, \mathbb R^N)\) such that \({\rm dist}(Du_j,K_\epsilon)\to 0\) in \(L^1(\Omega)\) as \(j\to \infty\), where \(K_\epsilon = \cup_{i = 1}^m\bar B_\epsilon(A_i)\) is the closed \(\epsilon\)-neighbourhood and \({\rm dist}(\cdot,K_\epsilon)\) the distance function to \(K_\epsilon\). We give estimates for \(\epsilon \gt 0\) such that up to a subsequence, \({\rm dist}(Du_j,B_\epsilon(A_{i_0})) \to 0\) in \(L^1(\Omega)\) for some fixed \(A_{i_0}\in K\). In other words, we give estimates on \(\epsilon \gt 0\) such that \(K_\epsilon\) separates gradient Young measure. The two point set \(K = \{ A_1, A_2\}\subset M^{N\times n}\) with \({\rm rank}(A_2-A_1) \gt 1\) is a special case of such sets up to a translation.

Keywords

Distance Function Linear Subspace Lipschitz Domain Real Matrice Young Measure 
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 2003

Authors and Affiliations

  • K. Zhang
    • 1
  1. 1.School of Mathematical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH (e-mail: k.zhang@sussex.ac.uk) GB

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