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VOL. 41 | 2003 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 $\int_Omega dist(Du_j, \barB_epsilon(B))dx \rightarrow 0$.

Information

Published: 1 January 2003
First available in Project Euclid: 18 November 2014

zbMATH: 1104.49016
MathSciNet: MR1994523

Rights: Copyright © 2003, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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