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Nonlocal diffusion and applications

Bucur, Claudia.

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Nonlocal diffusion and applications

Record Type:	Electronic resources : Monograph/item
Title/Author:	Nonlocal diffusion and applications/ by Claudia Bucur, Enrico Valdinoci.
Author:	Bucur, Claudia.
other author:	Valdinoci, Enrico.
Published:	Cham :Springer International Publishing : : 2016.,
Description:	xii, 155 p. :ill., digital ;24 cm.
[NT 15003449]:	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 Schrodinger 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.
Contained By:	Springer eBooks
Subject:	Harmonic functions. -
Online resource:	http://dx.doi.org/10.1007/978-3-319-28739-3
ISBN:	9783319287393

Nonlocal diffusion and applications
Bucur, Claudia.

Nonlocal diffusion and applications[electronic resource] /by Claudia Bucur, Enrico Valdinoci. - Cham :Springer International Publishing :2016. - xii, 155 p. :ill., digital ;24 cm. - Lecture notes of the Unione Matematica Italiana,201862-9113 ;. - Lecture notes of the Unione Matematica Italiana ;10..

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 Schrodinger 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.

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 Schrodinger 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.

ISBN: 9783319287393

Standard No.: 10.1007/978-3-319-28739-3doiSubjects--Topical Terms:

596179
Harmonic functions.

LC Class. No.: QA405

Dewey Class. No.: 515.53

Nonlocal diffusion and applications
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245 1 0 $a Nonlocal diffusion and applications $h [electronic resource] / $c by Claudia Bucur, Enrico Valdinoci.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2016.
300 $a xii, 155 p. : $b ill., digital ; $c 24 cm.
490 1 $a Lecture notes of the Unione Matematica Italiana, $x 1862-9113 ; $v 20
505 0 $a 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 Schrodinger 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.
520 $a 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 Schrodinger 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.
650 0 $a Harmonic functions. $3 596179
650 0 $a Laplace transformation. $3 540531
650 1 4 $a Mathematics. $3 515831
650 2 4 $a Partial Differential Equations. $3 890899
650 2 4 $a Calculus of Variations and Optimal Control; Optimization. $3 898674
650 2 4 $a Integral Transforms, Operational Calculus. $3 897325
650 2 4 $a Functional Analysis. $3 893943
700 1 $a Valdinoci, Enrico. $3 2191804
710 2 $a SpringerLink (Online service) $3 836513
773 0 $t Springer eBooks
830 0 $a Lecture notes of the Unione Matematica Italiana ; $v 10. $3 1535986
856 4 0 $u http://dx.doi.org/10.1007/978-3-319-28739-3
950 $a Mathematics and Statistics (Springer-11649)