Seminari del Dipartimento
Fisica Matematica The Allen-Cahn and the cahn-Hilliard equation: classification and construction of solutions.
Matteo Rizzi 06-04-2017 - 14:00 AULA 311 (SEMINARI) Largo San L. Murialdo,1
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. |