This is a unique book that provides a comprehensive understanding of nonlinear equations involving the fractional Laplacian as well as other nonlocal operators. Beginning from the definition of fractional Laplacian, it gradually leads the readers to the frontier of current research in this area. The explanations and illustrations are elementary enough so that first year graduate students can follow easily, while it is advanced enough to include many new ideas, methods, and results that appeared recently in research literature, which researchers would find helpful. It focuses on introducing direct methods on the nonlocal problems without going through extensions, such as the direct methods of moving planes, direct method of moving spheres, direct blowing up and rescaling arguments, and so on. Different from most other books, it emphasizes on illuminating the ideas behind the formal concepts and proofs, so that readers can quickly grasp the essence.Contents: Introduction to Fractional LaplacianThe Green's FunctionsMaximum Principles for the Fractional LaplacianPoisson RepresentationsLiouville Type Theorems for α-Harmonic FunctionsA Direct Method of Moving Planes for the Fractional Order EquationsMethod of Moving Planes in Integral FormsA Method of Moving Spheres for the Fractional LaplacianA Priori Estimates and the Existence of SolutionsVariational Methods and Pohozaev IdentitiesHigher Order Fractional LaplaciansThe Regularity for Fractional Equations
Readership: Graduate students and researchers interested in analysis and differential equations.Fractional Laplacian;Nonlocal Operators;Method of Moving Planes;Moving Spheres;Liouville Theorems;A Priori Estimate;Existence;Symmetry;Regularity0Key Features:The subjects discussed here are currently hot in the research communityDifferent from most other books, it emphasizes on illuminating the ideas behind the formal concepts and proofs, so that the readers can quickly grasp the essenceIt accumulates the authors as well as other researchers' experience and understanding in their research process and tries to present these in a vivid and intuitive way, so that the readers can follow easily