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We completely characterize generalized Young measures generated by sequencesof 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 whereconcentrations of the sequence of gradients at the boundary of $\Omega$ wereexcluded. We apply our results to relaxation of non-quasiconvex variationalproblems with linear growth at infinity. We also link our characterization toSou\v{c}ek spaces \cite{soucek}, an extension of $W^{1,1}(\Omega;\R^M)$ wheregradients are considered as measures on $\bar\Omega$.

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