## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- International Conference on Harmonic Analysis and Related Topics. Xuan Thinh Duong and Alan Pryde, eds. Proceedings of the Centre for Mathematics and its Applications, v. 41. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2003), 161 - 169

### 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 $\int_Omega dist(Du_j, \barB_epsilon(B))dx \rightarrow 0$.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416322435

**Mathematical Reviews number (MathSciNet)**

MR1994523

**Zentralblatt MATH identifier**

1104.49016

#### Citation

Zhang, Kewei. Separation of gradient Young measures and the BMO. International Conference on Harmonic Analysis and Related Topics, 161--169, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 2003. https://projecteuclid.org/euclid.pcma/1416322435