Some Remarks on the (Non-) Attainment of the Boundary Data for Variational Problems in the Space BV

We discuss the standard relaxed version of a minimization problem for variational integrals of linear growth together with prescribed Dirichlet boundary data $u_0$ and give estimates for the size of the set $\{x \in \partial \Omega: u (x) \not= u_0 (x)\}$ for BV-minimizers $u$ which imply $${\cal{H}}^{n -1} \left(\left\{x \in \partial \Omega: u (x) < u_0 (x)\right\}\right) = {\cal{H}}^{n - 1} \left(\left\{x \in \partial \Omega: u (x) > u_0 (x) \right\}\right)$$ in the case of minimal surfaces $u$ not attaining the boundary values $u_0$ on a subset of $\partial \Omega$ with positive measure.