Bernstein's theorem for variational integrals of linear growth and radial structure

Abstract

We consider entire solutions $u:{\mathbb{R}}^2\to \mathbb{R}$ of the Euler-Lagrange equation associated to the variational integral ${\int }_{\Omega }g(|\nabla u|)dx$ with a strictly convex density $g:[0,\infty )\to \mathbb{R}$ being of linear growth. We show that the condition ${\int }_0^{\infty }t{g}^{\prime \prime }(t)dt<\infty$ implies the Bernstein property, which means that $u$ must be an affine function. If this condition on g is weakened, we still have some partial Bernstein results.