Nonlocal Diffusion and Applications
Published on 8. April 2016
XII, 155 pages
978-3-319-28739-3 (ISBN)
Description
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.
Reviews / Votes
"The book under review is a result of a series of lectures given in various places throughout the world. It gives an introduction to the analysis of nonlocal operators, most notably the fractional Laplacian. . the book does a great job of introducing the topic of nonlocal analysis for every newcomer in the field. It provides a good starting point for doing research and therefore is highly recommended." (Lukasz Plociniczak, Mathematical Reviews, March, 2017)More details
Series
Edition
1st ed. 2016
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
Professional and scholarly
Product notice
Reflowable
Illustrations
XII, 155 p. 26 illus., 23 illus. in color.
File size
1,98 MB
ISBN-13
978-3-319-28739-3 (9783319287393)
DOI
10.1007/978-3-319-28739-3
Schweitzer Classification
Other editions
Additional editions
Claudia Bucur | Enrico Valdinoci
Nonlocal Diffusion and Applications
Book
04/2016
1st Edition
Springer
€69.54
Shipment within 10-15 days
Content
Introduction.- 1 A probabilistic motivation.-1.1 The random walk with arbitrarily long jumps.- 1.2 A payoff model.-2 An introduction to the fractional Laplacian.-2.1 Preliminary notions.- 2.2 Fractional Sobolev Inequality and Generalized Coarea Formula.- 2.3 Maximum Principle and Harnack Inequality.- 2.4 An s-harmonic function.- 2.5 All functions are locally s-harmonic up to a small error.- 2.6 A function with constant fractional Laplacian on the ball.- 3 Extension problems.- 3.1 Water wave model.- 3.2 Crystal dislocation.- 3.3 An approach to the extension problem via the Fourier transform.- 4 Nonlocal phase transitions.- 4.1 The fractional Allen-Cahn equation.- 4.2 A nonlocal version of a conjecture by De Giorgi.- 5 Nonlocal minimal surfaces.- 5.1 Graphs and s-minimal surfaces.- 5.2 Non-existence of singular cones in dimension 2 5.3 Boundary regularity.- 6 A nonlocal nonlinear stationary Schrödinger type equation.- 6.1 From the nonlocal Uncertainty Principle to a fractional weighted inequality.- Alternative proofs of some results.- A.1 Another proof of Theorem A.2 Another proof of Lemma 2.3.- References.
