© 2016

Nonlocal Diffusion and Applications

  • Gives a rich introduction to the fractional Laplacian and its applications

  • Well explained, self-contained and easy to follow, even for those who are not familiar with the subject

  • Contains brand new and interesting research trends on the fractional Laplacian


Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 20)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Claudia Bucur, Enrico Valdinoci
    Pages 1-5
  3. Claudia Bucur, Enrico Valdinoci
    Pages 7-37
  4. Claudia Bucur, Enrico Valdinoci
    Pages 39-65
  5. Claudia Bucur, Enrico Valdinoci
    Pages 67-95
  6. Claudia Bucur, Enrico Valdinoci
    Pages 97-126
  7. Claudia Bucur, Enrico Valdinoci
    Pages 127-138
  8. Back Matter
    Pages 139-157

About this book


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.


35S05; 47G10; 35S30; 47B34; 35R11; 49Q05; 82D25; 74E15; 35Q40. Fractional diffusion Fractional Laplacian Nonlocal minimal surfaces Nonlocal phase transitions Nonlocal quantum mechanics

Authors and affiliations

  1. 1.DipartimentodiMatematicaFederigoEnriquesUniversità degli Studi di MilanoMilanoItaly
  2. 2.DipartimentodiMatematicaFederigoEnriquesUniversità degli Studi di MilanoMilanoItaly

Bibliographic information

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“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.” (Łukasz Płociniczak, Mathematical Reviews, March, 2017)