In complex analysis of one variable, Liouville's theorem says that a bounded entire function is constant. Bernstein (1915-17) proved an analogous result in differential geometry, namely, if the graph of a function $f:\mathbf R^2\to\mathbf R$ of class $C^2$ is a minimal surface in $\mathbf R^3$, then the graph a plane. He then posed the classical Bernstein problem, namely, whether the same result also holds for real functions of $n>2$ variables. In terms of differential equations:

(Classical) Bernstein problem: Let the function $f:\mathbf R^n\to\mathbf R$ of class $C^2$ be a solution of $$\sum_{i=1}^nD_i\left(\frac{D_i f}{\sqrt{1+|D f|^2}}\right)=0.$$ Must $f$ be a linear function?

Recall that a hypersurface in $\mathbf R^{n+1}$ is defined to be minimal if its mean curvature vanishes, where its mean curvature is simply the sum of the principal curvatures (sometimes divided by $n$). Equivalently, the hypersurface is a critical point for the $n$-volume with respect to compactly supported variations. The equation above is the condition that the mean curvature of the graph of $f$ vanishes everywhere.

Part of the importance of the Bernstein problem is that it has a direct bearing on the existence of minimal cones and singularities of minimal hypersurfaces in $\mathbf R^{n+1}$. The answer to the problem was proved to be affirmative in the cases $n=3$ by de Giorgi (1965), $n=4$ by Almgren (1966), and $n\leq7$ by Simons (1968), and apparently there was some hope to extend the result to all dimensions.

However, in 1969 Bombieri, de Giorgi and Giusti constructed a counter-example for $n=8$, which yields a counter-example in each dimension $n>8$ by a standard construction, closing the problem. The complete solution of the Bernstein problem turned out to involve a good deal of geometric measure theory and non-linear analysis.