Nonlocal diffusion and applications

TL;DR

This paper explores the fractional Laplace framework, presenting models, theorems, and applications across various fields such as water waves, phase transitions, and Schrödinger equations, including original results and insights.

Contribution

It provides a comprehensive, self-contained overview of nonlocal diffusion phenomena with some original proofs and results, connecting theory with diverse applications.

Findings

Probabilistic interpretation of nonlocal diffusion

Applications to water waves, crystal dislocations, and phase transitions

Insights into fractional versions of classical conjectures

Abstract

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. Also, the work is self contained, and the reader is invited to consult the rich bibliography for further details, whenever a subject is of interest.