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

Abstract

We consider elliptic systems generated by variational integrals of linear growth satisfying the condition of \(\mu\)-ellipticity for some exponent \(\mu >1\) and prove that stationary points \(u\colon\mathbb{R}^2\to\mathbb{R}^N\) 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.