Nonlocal diffusion and applications

Understand

We consider the fractional Laplace framework and provide models and theorems related to nonlocal diffusion phenomena.

• Some applications are presented, including: a simple probabilistic interpretation, water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schr\"{o}dinger equations.
• Furthermore, an example of an $s$-harmonic function, the harmonic extension and some insight on a fractional version of a classical conjecture formulated by De Giorgi are presented.
• Although this book aims at gathering some introductory material on the applications of the fractional Laplacian, some proofs and results are original.