The book deals with fractional Laplace operator properties, and it is distributed in six chapters. The first chapter concentrates on two probabilistic motivations, one on the random walk with arbitrarily long jumps and the second one on a payoff model. The second chapter centralizes on the following subjects: The fractional Sobolev inequality, the generalized co-area formula, the maximum principle, the Harnack inequality, \(s\)-harmonic function, and suitable function such that its fractional Laplacian is a constant. The third chapter focuses on reducing a fractional Laplacian operator to a local operator, this reduction is due to L. Caffarelli and L. Silvestre; besides the chapter is endowed with a few models arising in physics as crystal dislocation and the water wave.
The rest of chapters are on current research subjects. Therefore, the fourth chapter contains the extension of the Allen-Cahn equation (a reaction-diffusion equation) to a nonlocal setting and the fractional counterpart of the conjecture by De Giorgi. The fifth chapter copes with two meaningful results one on the Bernstein-type theorem (on solutions properties to a minimal surface equation) in high dimension and the second one on the non-existence of nontrivial \(s\)-minimal cones in two-dimension. The sixth chapter deals with a nonlocal nonlinear stationary Schrödinger type equation. The book is endowed with an appendix including alternative proofs of a few results, e.g., the existence of a positive bounded \(s\)-harmonic function such that its infimum, on the unit ball, vanishes. The book contains some pleasant plots.