A small remark on Bernstein's theorem

Abstract

We investigate splitting-type variational problems with some linear growth conditions. For balanced solutions of the associated Euler-Lagrange equation we receive a result analogous to Bernstein's theorem on non-parametric minimal surfaces. Without assumptions of this type, Bernstein's theorem cannot be carried over to the splitting case, which follows from an elementary counterexample. We also include some modifications of our main theorem.