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

TL;DR

This paper proves that in two dimensions, stationary points of certain linear growth functionals with specific ellipticity conditions are affine functions if they grow at most linearly at infinity, extending classical results.

Contribution

It establishes Liouville-type theorems for vector-valued stationary points under linear growth and ellipticity, extending Bernstein's theorem to broader classes of functionals.

Findings

Stationary points with linear growth are affine in 2D under growth conditions.

Extension of Bernstein's theorem to vector-valued cases with linear growth.

Results depend on $ta$-ellipticity and growth conditions at infinity.

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.