Thesis title: 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.