Nonlocal Diffusion and Applications

1. Front Matter

   Pages i-xii

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2. A Probabilistic Motivation

   • Claudia Bucur, Enrico Valdinoci

   Pages 1-5

3. An Introduction to the Fractional Laplacian

   • Claudia Bucur, Enrico Valdinoci

   Pages 7-37

4. Extension Problems

   • Claudia Bucur, Enrico Valdinoci

   Pages 39-65

5. Nonlocal Phase Transitions

   • Claudia Bucur, Enrico Valdinoci

   Pages 67-95

6. Nonlocal Minimal Surfaces

   • Claudia Bucur, Enrico Valdinoci

   Pages 97-126

7. A Nonlocal Nonlinear Stationary Schrödinger Type Equation

   • Claudia Bucur, Enrico Valdinoci

   Pages 127-138

8. Back Matter

   Pages 139-157

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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
• partial differential equations

“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)

• DipartimentodiMatematicaFederigoEnriques, Università degli Studi di Milano, Milano, Italy

  Claudia Bucur, Enrico Valdinoci

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