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 classical

more »

... ts from the scalar case N ¼ 1-where the problem coincides with the non-parametric least area problem-to the general vectorial setting N A N. 11) In the proof of Lemma 2.10-as in the whole paper-we have suppressed an explicit notation for the continuous linear trace operator T. However, it should be noted that working withũ u on qW we are implicitly using T maxfu; vg ¼ maxfTu; Tvg on qW. To establish this equality one first proves Tju À vj ¼ jTðu À vÞj (approximating u À v with continuous functions on W) and then writes 2 maxfu; vg ¼ u þ v þ ju À vj.