Dottorato in MATEMATICA

Titolo della tesi: Variational methods for coarse graining with long range interactions

In this thesis we use variational methods in order to study crystallization problems and the asymptotic behavior for the s-fractional heat flows as s -> 0+ and s -> 1−. The main link between the proposed results is the asymptotic analysis and the common variational approach used in these frameworks. In the first part of the thesis we propose a vectorial crystallization model, a variant of the classical Heitmann-Radin crystallization model, that is a first proposal to predict collective behavior by using a variational approach. In the second part we analyze the crystallization problem for suitable variants of the Lennard-Jones potential in the real line, obtaining that crystallization can not be true in any reasonable sense. In the last part we study the asymptotic behavior of the s-fractional heat flow as s -> 0+ and s -> 1− and we show that it converges to the standard heat flow as s -> 1− and to a degenerate ODE type flow as s -> 0−. Moreover, after suitable renormalization procedures, the limit as s -> 0+ gives back the gradient flow of a sort of 0−Gagliardo seminorm.