Giovedi’ 06 Aprile 2017 alle ORE 14:00 presso il Dipartimento di Matematica e Fisica dell’Universita’ degli Studi Roma Tre – AULA 311 (SEMINARI) Largo San L. Murialdo,1 – il prof. Matteo Rizzi (UNAM Mexico City) terrà il seguente seminario di Fisica Matematica:
The Allen-Cahn and the cahn-Hilliard equation: classification and construction of solutions
Abstract. 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.