The Liouville property for entire local minimizers from the space $B{V}_{\mathrm{l}\mathrm{o}\mathrm{c}}({\mathbb{R}}^2,{\mathbb{R}}^N)$ of variational integrals with linear growth

We consider functions $u:{\mathbb{R}}^2\to {\mathbb{R}}^N$ from the class $\text{Bloc}$ which locally minimize the relaxed version of a variational integral with a strictly convex density being in addition of linear growth. It turns out that the boundedness of entire local minimizers $u$ implies their constancy.