04937nam 22007695 450 991025407500332120220413183558.03-319-28739-710.1007/978-3-319-28739-3(CKB)3710000000636344(EBL)4501074(SSID)ssj0001666025(PQKBManifestationID)16455509(PQKBTitleCode)TC0001666025(PQKBWorkID)14999870(PQKB)11066359(DE-He213)978-3-319-28739-3(MiAaPQ)EBC4501074(PPN)193444704(EXLCZ)99371000000063634420160408d2016 u| 0engur|n|---|||||txtccrNonlocal diffusion and applications[electronic resource] /by Claudia Bucur, Enrico Valdinoci1st ed. 2016.Cham :Springer International Publishing :Imprint: Springer,2016.1 online resource (165 p.)Lecture Notes of the Unione Matematica Italiana,1862-9113 ;20Description based upon print version of record.3-319-28738-9 Includes bibliographical references.Preface; Acknowledgments; Contents; 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; 3.1.1 Application to the Water Waves; 3.2 Crystal Dislocation3.3 An Approach to the Extension Problem via the Fourier Transform4 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 Schrödinger Type Equation; 6.1 From the Nonlocal Uncertainty Principle to a Fractional Weighted Inequality; A Alternative Proofs of Some Results; A.1 Another Proof of Theorem 2.4.1; A.2 Another Proof of Lemma 2.3; ReferencesWorking 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.Lecture Notes of the Unione Matematica Italiana,1862-9113 ;20Partial differential equationsCalculus of variationsIntegral transformsOperational calculusFunctional analysisPartial Differential Equationshttps://scigraph.springernature.com/ontologies/product-market-codes/M12155Calculus of Variations and Optimal Control; Optimizationhttps://scigraph.springernature.com/ontologies/product-market-codes/M26016Integral Transforms, Operational Calculushttps://scigraph.springernature.com/ontologies/product-market-codes/M12112Functional Analysishttps://scigraph.springernature.com/ontologies/product-market-codes/M12066Partial differential equations.Calculus of variations.Integral transforms.Operational calculus.Functional analysis.Partial Differential Equations.Calculus of Variations and Optimal Control; Optimization.Integral Transforms, Operational Calculus.Functional Analysis.515.53Bucur Claudiaauthttp://id.loc.gov/vocabulary/relators/aut756016Valdinoci Enricoauthttp://id.loc.gov/vocabulary/relators/autUnione matematica italiana.MiAaPQMiAaPQMiAaPQBOOK9910254075003321Nonlocal Diffusion and Applications1983097UNINA