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
Academic Journal
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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