Bernstein type results for stationary points of variational integrals with subquadratic growth in two dimensions

Abstract

We discuss entire solutions $u\in C^2({\mathbb{R}}^2)$ of the equation $\mathrm{div}(\nabla f(\nabla u))=0$ with strictly convex density $f:{\mathbb{R}}^2\to \mathbb{R}$ of subquadratic growth and prove that $u$ is an affine function provided that at least one partial derivative is bounded from one side. Further results concern the behaviour of non-affine entire solutions.