# On separation of gradient Young measures

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## 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## Preview

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