  We consider variational integrals of linear growth satisfying the condition
of $\mu$-ellipticity for some exponent $\mu >1$ and prove that stationary
points $u$: $\mathbb{R}^2 \to \mathbb{R}^N$ with the property \[
\limsup_{|x|\to \infty} \frac{|u(x)|}{|x|} < \infty \] must be affine
functions. The latter condition can be dropped in the scalar case together with
appropriate assumptions on the energy density providing an extension of
Bernstein's theorem.
