Working in the fractional Laplace framework this book provides models and theorems related to
nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation some
applications to water waves crystal dislocations nonlocal phase transitions nonlocal minimal
surfaces and Schrödinger equations are given. Furthermore an example of an s-harmonic function
its harmonic extension and some insight into a fractional version of a classical conjecture due
to De Giorgi are presented. Although the aim is primarily to gather some introductory material
concerning applications of the fractional Laplacian some of the proofs and results are new.
The work is entirely self-contained and readers who wish to pursue related subjects of
interest are invited to consult the rich bibliography for guidance.
