Topics in fractional Laplacian and dynamical systems

Abstract

Abstract In this thesis, we consider problems involving the $n$-dimensional fractional Laplacians including elliptic equations and parabolic equations. We also consider the problems involving fractional Monge-Amp\'ere operators. The thesis is mostly devoted to presenting our original work on the progress obtained in the development of direct methods that can effectively deal with the above problems. ¶

In the second part of the work, we are interested in the length of a few consecutive long free flights in infinite horizon Lorentz Gas. In dimension D=2, it is well known that a flight of length T>>1 is typically followed by a flight of length $C\sqrt{T}$. Here, we extend this result to any dimension $D$.