Bucur C., Valdinoci E. Nonlocal Diffusion and Applications

Springer International Publishing, Switzerland, 2016. — 165 p. — (Lecture Notes of the Unione Matematica Italiana 20) — ISBN: 9783319287386Working 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.A Probabilistic Motivation
An Introduction to the Fractional Laplacian
Extension Problems
Nonlocal Phase Transitions
Nonlocal Minimal Surfaces
A Nonlocal Nonlinear Stationary Schrödinger Type Equation