# The "Wrong Minimal Surface Equation" does not have the Bernstein property

## TL;DR

This paper demonstrates that a specific nonlinear PDE, similar to the minimal surface equation, does not possess the Bernstein property, meaning not all solutions over the entire plane are affine linear functions.

## Contribution

It provides a simple proof that the equation
ef{toll} does not have the Bernstein property, contrasting with classical minimal surface equations.

## Key findings

- The equation
ef{toll} admits non-affine solutions over the entire plane.
- The classical Bernstein theorem does not extend to this nonlinear PDE.
- A straightforward argument shows the failure of the Bernstein property for this equation.

## Abstract

A celebrated result of S. Bernstein states that every solution of the minimal surface equation over the entire plane has to be an affine linear function. Since the paper of Bernstein appeared in 1927, many different proofs and generalizations of this beautiful theorem were given, namely to higher dimensions and to more general equations, for a careful account we refer to the paper by Simo and to the monograph by Dierkes-Hildebrandt-Tromba. In his paper Simon posed the question whether the equation \begin{equation} (1+{u_x}^2)u_{xx}+2u_x u_y u_{xy}+ (1+{u_y}^2)u_{yy} = 0 \label{toll} \end{equation} has the Bernstein property i.e. whether every $C^2$-solution defined over the entire plane necessarily has to be affine. We here show by a very simple argument that this is not the case.