On the Dirichlet problem for variational integrals in $BV$

created by beck on 30 Aug 2010
modified by schmidt on 11 Jan 2013

[BibTeX]

Published Paper

Inserted: 30 aug 2010
Last Updated: 11 jan 2013

Journal: J. Reine Angew. Math.
Volume: 674
Pages: 113-194
Year: 2013
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 $N \in \mathbb{N}$.