Liouville-type results in two dimensions for stationary points of functionals with linear growth

Abstract

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.