# A Bernstein-type result for the minimal surface equation

created by farina on 14 Jan 2016
modified on 15 Jan 2016

[BibTeX]

Published Paper

Inserted: 14 jan 2016
Last Updated: 15 jan 2016

Journal: Ann. Sc. Norm. Super. Pisa Cl. Sci.
Volume: Vol. XIV
Pages: 1231-1237
Year: 2015

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 $\frac{\partial u }{\partial {x_j}}$ 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 \cite{BG} thus, besides its novelty, our theorem also provides a new and self-contained proof of celebrated results of Moser and of Bombieri $\&$ Giusti.