Carnegie Mellon
Department of Mathematical 
Institute of Information Theory and Automation
Academy of Sciences of the Czech Republic
Pod vodárenskou vezí 4, CZ-182 08 Praha 8, Czech Republic.

"Oscillations and Concentrations in Sequences of Gradients"


It is well-known that Morrey's quasiconvexity is closely related to gradient Young measures, i.e., Young measures generated by sequences of gradients in $L^p(\Omega;{I\!\! R}^{m\times n})$. Concentration effects, however, cannot be treated by Young measures. One way how to describe both oscillation and concentration effects in a fair generality are the so-called DiPerna-Majda measures.

DiPerna and Majda showed that having a sequence $\{y_k\}$ bounded in $L^p(\Omega;{I\!\! R}^{m\times n})$, $1\le p<+\infty$, and a complete separable subring ${\cal R}$ of continuous bounded functions on ${I\!\! R}^{m\times n}$ then there exists a subsequence of $\{y_k\}$ (not relabeled), a positive Radon measure $\sigma$ on $\bar\Omega$, and a family of probability measures on $\beta_{\cal R}{I\!\! R}^{m\times n}$ (the metrizable compactification of ${I\!\! R}^{m\times n}$ corresponding to ${\cal R}$), $\{\hat\nu_x\}_{x\in\bar\Omega}$, such that for all $g\in C(\bar\Omega)$ and all $v_0\in{\cal R}$

\lim_{k\to\infty}\int_\Omega g(x)v(y_k(x))d x\ =
...}{I\!\! R}^{m\times n}}g(x)v_0(s)\hat\nu_x(d
s)\sigma(d x)\ ,

where $v(s)=v_0(s)(1+\vert s\vert^p)$. Our talk will address the question: What conditions must $(\sigma,\hat\nu)$ satisfy, so that $y_k=\nabla u_k$ for $\{u_k\}\subset W^{1,p}(\Omega;{I\!\! R}^m)$ with $u_k=u_0$ on $\partial\Omega$? We are going to state necessary and sufficient conditions. We will mainly work with $p>1$, however, special cases for $p=1$ will be discussed, as well.

TUESDAY, April 25, 2006
Time: 1:30 P.M.
Location: PPB 300