*Published Paper*

**Inserted:** 30 aug 2010

**Last Updated:** 11 jan 2013

**Journal:** J. Reine Angew. Math.

**Volume:** 674

**Pages:** 113-194

**Year:** 2013

**Links:**
Link to the published version

**Abstract:**

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 data. Moreover, we provide several related examples. In the case of the model integral \[ \int_\Omega \sqrt{1+\lvert\nabla w\rvert^2} \, dx \qquad \text{for } w\colon\mathbb{R}^n\supset \Omega \to \mathbb{R}^N \] 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 $N \in \mathbb{N}$.

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