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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 Schroedinger 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.
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