A celebrated conjecture by E. De Giorgi asserts that any monotone entire solution
to the Allen-Cahn equation must be one dimensional, that is the level sets must be
hyperplanes, at least in dimension less or equal than 8. I will present some known
results about the conjecture and I will give an idea of what happens in higher dimension.
The behaviour of these solutions is strictly related to the nature of entire minimal graphs.
Then I will discuss some analogue results for the Cahn-Hilliard equation, which is related
to Willmore hypersurfaces. I will show the construction of some particular solutions in
dimension 3, vanishing close to the Clifford Torus, which is known to be a Willmore surface,
and in dimension 2, that are not one-dimensional and almost monotone.