Nonlocal Diffusion and Applications

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Springer, Apr 8, 2016 - Mathematics - 155 pages
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.
 

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Contents

1 A Probabilistic Motivation
1
2 An Introduction to the Fractional Laplacian
7
3 Extension Problems
38
4 Nonlocal Phase Transitions
67
5 Nonlocal Minimal Surfaces
96
6 A Nonlocal Nonlinear Stationary Schrödinger Type Equation
127
A Alternative Proofs of Some Results
139
References
149
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