Separation of gradient Young measures and the BMO

Abstract

Let $K = \{A,B} \subset M^{Nxn$ with rank$(A − B) \gt 1$ and $\Omega \subset \mathbbR^n$ be a bounded arcwise connected Lipschitz domain. We show that there is a direct estimate of the size of the $\epsilon$-neighborhood $K_\epsilon$ of $K$ such that $K_\epsilon = \barB_\epsilon(A) \bigcup \barB_\epsilon(B)$ separates gradient Young measures, that is, if $(u_j) \subset W^{1,1}(\Omega, \mathbbR^N)$ is bounded and $\int_\Omega$ dist$(Du_j, K_\epsilon)dx \rightarrow 0$ as $j \rightarrow \infty$, then up to a subsequence, either $\int_Omega dist(DU_j, \barB_epsilon(A))dx \rightarrow 0$ or $\int_Omega dist(Du_j, \barB_epsilon(B))dx \rightarrow 0$.