On the Dirichlet problem for variational integrals in BV

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 integral(Omega) root 1 + vertical bar del w vertical bar(2) dx for w : R-n superset of Omega -> R-N our results extend classical results from the scalar case N = 1-where the problem coincides with the non-parametric least area problem-to the general vectorial setting N is an element of N.