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.

The fractional Laplacian, also called the Riesz fractional derivative, describes an unusual diffusion process associated with random excursions. The Fractional Laplacian explores applications of the fractional Laplacian in science, engineering, and other areas where long-range interactions and conceptual or physical particle jumps resulting in an irregular diffusive or conductive flux are encountered. Presents the material at a level suitable for a broad audience of scientists and engineers with rudimentary background in ordinary differential equations and integral calculus Clarifies the concept of the fractional Laplacian for functions in one, two, three, or an arbitrary number of dimensions defined over the entire space, satisfying periodicity conditions, or restricted to a finite domain Covers physical and mathematical concepts as well as detailed mathematical derivations Develops a numerical framework for solving differential equations involving the fractional Laplacian and presents specific algorithms accompanied by numerical results in one, two, and three dimensions Discusses viscous flow and physical examples from scientific and engineering disciplines Written by a prolific author well known for his contributions in fluid mechanics, biomechanics, applied mathematics, scientific computing, and computer science, the book emphasizes fundamental ideas and practical numerical computation. It includes original material and novel numerical methods.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This second volume collects authoritative chapters covering the mathematical theory of fractional calculus, including ordinary and partial differential equations of fractional order, inverse problems, and evolution equations.

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.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This first volume collects authoritative chapters covering the mathematical theory of fractional calculus, including fractional-order operators, integral transforms and equations, special functions, calculus of variations, and probabilistic and other aspects.

This volume collects contributions from the speakers at an INdAM Intensive period held at the University of Bari in 2017. The contributions cover several aspects of partial differential equations whose development in recent years has experienced major breakthroughs in terms of both theory and applications. The topics covered include nonlocal equations, elliptic equations and systems, fully nonlinear equations, nonlinear parabolic equations, overdetermined boundary value problems, maximum principles, geometric analysis, control theory, mean field games, and bio-mathematics. The authors are trailblazers in these topics and present their work in a way that is exhaustive and clearly accessible to PhD students and early career researcher. As such, the book offers an excellent introduction to a variety of fundamental topics of contemporary investigation and inspires novel and high-quality research.

The material presented here corresponds to Fermi lectures that I was invited to deliver at the Scuola Normale di Pisa in the spring of 1998. The obstacle problem consists in studying the properties of minimizers of the Dirichlet integral in a domain D of Rn, among all those configurations u with prescribed boundary values and costrained to remain in D above a prescribed obstacle F. In the Hilbert space H1(D) of all those functions with square integrable gradient, we consider the closed convex set K of functions u with fixed boundary value and which are greater than F in D. There is a unique point in K minimizing the Dirichlet integral. That is called the solution to the obstacle problem.

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

The fractional Laplacian, also called the Riesz fractional derivative, describes an unusual diffusion process associated with random excursions. The Fractional Laplacian explores applications of the fractional Laplacian in science, engineering, and other areas where long-range interactions and conceptual or physical particle jumps resulting in an irregular diffusive or conductive flux are encountered. Presents the material at a level suitable for a broad audience of scientists and engineers with rudimentary background in ordinary differential equations and integral calculus Clarifies the concept of the fractional Laplacian for functions in one, two, three, or an arbitrary number of dimensions defined over the entire space, satisfying periodicity conditions, or restricted to a finite domain Covers physical and mathematical concepts as well as detailed mathematical derivations Develops a numerical framework for solving differential equations involving the fractional Laplacian and presents specific algorithms accompanied by numerical results in one, two, and three dimensions Discusses viscous flow and physical examples from scientific and engineering disciplines Written by a prolific author well known for his contributions in fluid mechanics, biomechanics, applied mathematics, scientific computing, and computer science, the book emphasizes fundamental ideas and practical numerical computation. It includes original material and novel numerical methods.

Due to its ubiquity across a variety of fields in science and engineering, fractional calculus has gained momentum in industry and academia. While a number of books and papers introduce either fractional calculus or numerical approximations, no current literature provides a comprehensive collection of both topics. This monograph introduces fundamental information on fractional calculus, provides a detailed treatment of existing numerical approximations, and presents an inclusive review of fractional calculus in terms of theory and numerical methods and systematically examines almost all existing numerical approximations for fractional integrals and derivatives. The authors consider the relationship between the fractional Laplacian and the Riesz derivative, a key component absent from other related texts, and highlight recent developments, including their own research and results. The core audience spans several fractional communities, including those interested in fractional partial differential equations, the fractional Laplacian, and applied and computational mathematics. Advanced undergraduate and graduate students will find the material suitable as a primary or supplementary resource for their studies.