A Bernstein- Type result for the minimal surface equation

Abstract

We prove the following Bernstein- Type theorem: if u is an entire solution to the minimal surface equation, such that N - 1 partial derivatives are bounded on one side (not necessarily the same), then u is an affine function. Its proof relies only on the Harnack inequality on minimal surfaces proved in [4] thus, besides its novelty, our theorem also provides a new and self-contained proof of celebrated results of Moser and of Bombieri ancfGiusti.