TITLE

# On separation of gradient Young measures

AUTHOR(S)
K. Zhang
PUB. DATE
May 2003
SOURCE
Calculus of Variations & Partial Differential Equations;May2003, Vol. 17 Issue 1, p85
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
Abstract. Let $E\subset M^{N\/FORMULA> be a linear subspace of real$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\/FORMULA> with ${\rm rank}(A_2-A_1) \gt 1$ is a special case of such sets up to a translation.
ACCESSION #
12008080

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