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| In the early 20th century, Bernstein proved that if $f \in C^2(\mathbb{R}^2)$ and the graph of $z=f(x,y)$ is a minimal surface in $\mathbb{R}^3$, then $f$ is necessarily a linear function of $x$ and $y$. For Monge-Amp\^^ ere equation, Pogolerov proved the Bernstein type theorem of the following form: Let $u \in C^4(\mathbb{R}^n)$ be a convex function of $\det D^2u=1$ in $\mathbb{R}^n$. Then $u$ is necessarily a quadratic polynomial.
In this talk, we shall obtain the Bernstein type theorem for some fully nonlinear equations. |
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