Oscillations and Concentrations in Sequences of Gradients up to the Boundary

\renewcommand{\O}{\Omega} Oscillations and concentrations in sequences of gradients $\{\nabla u_k\}$, bounded in $L^p(\O; \R^{M\times N})$ if $p>1$ and $\O\subset\R^n$ 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{\l}amajska and M. Kru\v{z}{\'\i}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.