DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps ${\left\{u_k\right\}}_{k\in \mathbb{N}}\subset L^p(\Omega ;{ℝ}^m)$ satisfying a linear differential constraint $𝒜u_k=0$. Applications to sequential weak lower semicontinuity of integral functionals on $𝒜$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla {\varphi }_k\stackrel{*}{\rightharpoonup }\det \nabla \varphi$ in measures on the closure of $\Omega \subset {ℝ}^n$ if ${\varphi }_k\rightharpoonup \varphi$ in $W^{1,n}(\Omega ;{ℝ}^n)$. This convergence holds, for example, under...

We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla u_k\right\}$, bounded in $L^p(Ø;{ℝ}^{m\times n})$ if $p\>1$ and $\Omega \subset {ℝ}^n$ is a bounded domain with the extension property in $W^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega$ are required and links with lower semicontinuity results...

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla u_k\right\}$, bounded in $L^p(\Omega ;{ℝ}^{m\times n})$ if p > 1 and $\Omega \subset {ℝ}^n$ is a bounded domain with the extension property in $W^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of Ω are required and links with lower semicontinuity...