Nonlocal Diffusion and Applications

✍ Scribed by Claudia Bucur, Enrico Valdinoci (auth.)

• Publisher: Springer International Publishing
• Year: 2016
• Tongue: English
• Leaves: 165
• Series: Lecture Notes of the Unione Matematica Italiana 20
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xii
A Probabilistic Motivation....Pages 1-5
An Introduction to the Fractional Laplacian....Pages 7-37
Extension Problems....Pages 39-65
Nonlocal Phase Transitions....Pages 67-95
Nonlocal Minimal Surfaces....Pages 97-126
A Nonlocal Nonlinear Stationary Schrödinger Type Equation....Pages 127-138
Back Matter....Pages 139-157

✦ Subjects

Partial Differential Equations; Calculus of Variations and Optimal Control; Optimization; Integral Transforms, Operational Calculus; Functional Analysis