MARTIN KRUZ´iK
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"

Abstract

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 $v_0\in{\cal R}$

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

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 on $\partial\Omega$ ? We are going to state necessary and sufficient conditions. We will mainly work with

, however, special cases for

will be discussed, as well.

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

Mathematical Sciences Home Mellon College of Science Carnegie Mellon University