Regularity for dual solutions and for weak cluster points of minimizing sequences of variational problems with linear growth

The minimum problem $\int_{\Omega}f(\nabla u)dx\longrightarrow\min$ among mappings $u:\mathbb R^n\supset\Omega\to\mathbb R^N$ with prescribed Dirichlet boundary data and for integrands $f$ of linear growth in general fails to have solutions in the Sobolev space $W^1_1$. We therefore concentrate on the dual variational problem which admits a unique maximizer $\sigma$ and prove partial Hölder continuity of $\sigma$. Moreover, we study smoothness properties of $L^1$-limits of minimizing sequences of the original problem.