On the fractional Laplacian and nonlocal operators

Abstract

The aim of this work is to study certain type of operators that have adquired a renewed relevance in the last decade. This wide class of operators, genericaly called nonlocal operators, appears naturally in many applications in physics, probability, biology and economics. Throughout this work our attention will be focused in a very important nonlocal operator known as the fractional Laplacian. This operator plays a similar role in the theory of nonlocal operators as the one played by the Laplacian in the classical theory of elliptic partial differential equations, i.e. the fractional Laplacian constitutes the most simple and, at the same time, the “model” nonlocal operator and therefore it is considered as the benchmark in the study of these kind of operators. Thus, by analogy, most of the relevant questions associated to the Laplacian (e.g. comparison principles, linear and nonlinear boundary value problems, and regularity results) have an equivalent for the fractional Laplacian. Also, in the same spirit of the multiplicity results for the classical semilinear partial differential equations we prove a multiplicity result for semilinear problems involving the fractional Laplacian using standard variational methods.