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We investigate the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. We prove uniqueness of minimizers up to additive constants and deduce additional assertions about these constants and the possible (non-)attainment of the boundary values. Moreover, we provide several related examples. In the case of the model integral dx for w : R n I W ! R N our results extend classicaldoi:10.1515/crelle.2011.188 fatcat:v6b2bcnrw5bsrnnkmzhrri7cou