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

We completely characterize generalized Young measures generated by sequences of gradients of maps from $W^{1,1}(\Omega;\R^M)$ where $\Omega\subset\R^N$. 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 Sou\v{c}ek spaces \cite{soucek}, an extension of $W^{1,1}(\Omega;\R^M)$ where gradients are considered as measures on $\bar\Omega$.