Oscillations and concentrations up to the boundary

TL;DR

This paper characterizes DiPerna-Majda measures for sequences of gradients in Sobolev spaces, extending previous results to boundary cases and applying this to relaxation problems in calculus of variations.

Contribution

It provides a full characterization of DiPerna-Majda measures for boundary-including gradient sequences, generalizing earlier fixed-boundary results.

Findings

Extended DiPerna-Majda measure characterization to boundary cases.

Connected measure characterization to relaxation of noncoercive functionals.

Applied results to relaxation problems in calculus of variations.

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.