Oscillations and concentrations up to the boundary

Abstract

Oscillations and concentrations in sequences of gradients $\{\nabla u_k\}$, 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 $W^{1,p}$, 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.