LEADER 04910nam 22007695 450 001 9910254075003321 005 20220413183558.0 010 $a3-319-28739-7 024 7 $a10.1007/978-3-319-28739-3 035 $a(CKB)3710000000636344 035 $a(EBL)4501074 035 $a(SSID)ssj0001666025 035 $a(PQKBManifestationID)16455509 035 $a(PQKBTitleCode)TC0001666025 035 $a(PQKBWorkID)14999870 035 $a(PQKB)11066359 035 $a(DE-He213)978-3-319-28739-3 035 $a(MiAaPQ)EBC4501074 035 $a(PPN)193444704 035 $a(EXLCZ)993710000000636344 100 $a20160408d2016 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aNonlocal diffusion and applications /$fby Claudia Bucur, Enrico Valdinoci 205 $a1st ed. 2016. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2016. 215 $a1 online resource (165 p.) 225 1 $aLecture Notes of the Unione Matematica Italiana,$x1862-9113 ;$v20 300 $aDescription based upon print version of record. 311 $a3-319-28738-9 320 $aIncludes bibliographical references. 327 $aPreface; 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 Dislocation 327 $a3.3 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 Schro?dinger 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; References 330 $aWorking in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance. 410 0$aLecture Notes of the Unione Matematica Italiana,$x1862-9113 ;$v20 606 $aPartial differential equations 606 $aCalculus of variations 606 $aIntegral transforms 606 $aOperational calculus 606 $aFunctional analysis 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aCalculus of Variations and Optimal Control; Optimization$3https://scigraph.springernature.com/ontologies/product-market-codes/M26016 606 $aIntegral Transforms, Operational Calculus$3https://scigraph.springernature.com/ontologies/product-market-codes/M12112 606 $aFunctional Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12066 615 0$aPartial differential equations. 615 0$aCalculus of variations. 615 0$aIntegral transforms. 615 0$aOperational calculus. 615 0$aFunctional analysis. 615 14$aPartial Differential Equations. 615 24$aCalculus of Variations and Optimal Control; Optimization. 615 24$aIntegral Transforms, Operational Calculus. 615 24$aFunctional Analysis. 676 $a515.53 700 $aBucur$b Claudia$4aut$4http://id.loc.gov/vocabulary/relators/aut$0756016 702 $aValdinoci$b Enrico$4aut$4http://id.loc.gov/vocabulary/relators/aut 712 02$aUnione matematica italiana. 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254075003321 996 $aNonlocal Diffusion and Applications$91983097 997 $aUNINA