Quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

Quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

Fri, 18/06/2010
11:00 		Martin Kruzik (Academy of Sciences, Prague) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Quasiconvexity at the boundary and weak lower semicontinuity of integral functionals
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;\mathbb{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;\mathbb{R}^{m\times n}) $,$ 1\le p $ <$ +\infty $, and a complete separable subring $ {\cal R} $ of continuous bounded functions on $ \mathbb{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}\mathbb{R}^{m\times n} $ (the metrizable compactification of $ \mathbb{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} $
$$ \lim_{k\to\infty}\int_\Omega g(x)v(y_k(x))d x\ = \int_{\bar\Omega}\int_{\beta_{\cal R}\R^{m\times n}}g(x)v_0(s)\hat\nu_x(d s)\sigma(d x)\ , $$
where $ v(s)=v_0(s)(1+|s|^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;\mathbb{R}^m) $ We are going to state necessary and sufficient conditions. The notion of quasiconvexity at the boundary due to Ball and Marsden plays a crucial role in this characterization. Based on this result, we then find sufficient and necessary conditions ensuring sequential weak lower semicontinuity of $ I:W^{1,p}(\Omega;\mathbb{R}^m)\to\mathbb{R} $,
$$ I(u)=\int_\Omega v(\nabla u(x))\,\md x\ ,$$
where $ v:\mathbb{R}^{m\times n}\to\mathbb{R} $ satisfies $ |v|\le C(1+|\cdot|^p) $, $ C $>$ 0 $.