Oscillations and concentrations up to the boundary

Oscillations and concentrations in sequences of gradients $\{\mathrm{\nabla }{u}_k\}$, bounded in $L^p(\mathrm{\Omega };{\mathbb{R}}^{M\times N})$ if $p>1$ and $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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 Ka{\l}amajska and Kru\v{z}'{\i}k (2008), 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.