Oscillations and Concentrations in Sequences of Gradients up to the Boundary

Oscillations and concentrations in sequences of gradients $\{\nabla u_k\}$, bounded in $L^p(\Omega; \mathbb{R}^{M\times N})$ if $p>1$ and $\Omega\subset\mathbb{R}^n$ is a bounded domain with the extension property in $W^{1,p}$, and their interaction with local integral functionals can be described by a generalization of Young measures due to DiPerna and Majda. We characterize such DiPerna-Majda measures, thereby extending a result by A. Ka{ł}amajska and M. Kruž{í}k [``Oscillations and concentrations in sequences of gradients'', ESAIM, Control Optim. Calc. Var. 14(1) (2008) 71–104], where the full characterization was possible only for sequences subject to a fixed Dirichlet boundary condition. As an application we state a relaxation result for noncoercive multiple-integral functionals.