Oscillations and concentrations generatedby ${\mathcal A}$ -freemappings and weak lower semicontinuityof integral functionals

Abstract

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps $\{u_k\}_{k\in{\mathbb N}} \subset L^p(\Omega;{\mathbb R}^m)$ satisfying a linear differential constraint ${\mathcal A}u_k=0$ . Applications to sequential weak lower semicontinuity of integral functionals on ${\mathcal A}$ -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det $\nabla\varphi_k\stackrel{*}{\rightharpoonup}{\rm det}\nabla\varphi$ in measures on the closure of $\Omega\subset{\mathbb R}^n$ if $\varphi_k\rightharpoonup\varphi$ in $W^{1,n}(\Omega;{\mathbb R}^n)$ . This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemmaprecisely stating which subsets $\Omega_j\subset \Omega$ must be removed to obtain weak lower semicontinuity of $u\mapsto\int_{\Omega\setminus\Omega_j} v(u(x))\,{\rm d}x$ along $\{u_k\}\subset L^p(\Omega;{\mathbb R}^m)\cap{\rm ker}\ {\mathcal A}$ . Specifically, $\Omega_j$ are arbitrarily thin “boundary layers”.