Generalized ${\mathbf{W}}^{1,1}$-Young measures and relaxation of problems with linear growth

Abstract

We completely characterize generalized Young measures generated by sequences of gradients of maps from $W^{1,1}(\Omega ;{\mathbb{R}}^M)$ where $\Omega \subset {\mathbb{R}}^N$. This extends and completes previous analysis by Kristensen and Rindler where concentrations of the sequence of gradients at the boundary of $\Omega$ were excluded. We apply our results to relaxation of non-quasiconvex variational problems with linear growth at infinity. We also link our characterization to Souček spaces \cite{soucek}, an extension of $W^{1,1}(\Omega ;{\mathbb{R}}^M)$ where gradients are considered as measures on $\bar{\Omega }$.