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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 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.

Harmonic functions. --- Laplace transformation. --- Transformation, Laplace --- Calculus, Operational --- Differential equations --- Transformations (Mathematics) --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Harmonic functions --- Laplace transformation --- Fonctions harmoniques --- Laplace, Transformation de --- Differential equations, partial. --- Mathematical optimization. --- Integral Transforms. --- Functional analysis. --- Partial Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Integral Transforms, Operational Calculus. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Transform calculus --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Partial differential equations --- Partial differential equations. --- Calculus of variations. --- Integral transforms. --- Operational calculus. --- Operational calculus --- Electric circuits --- Isoperimetrical problems --- Variations, Calculus of