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Recent Developments in Nonlocal Theory
This edited volume aims at giving an overview of recent advances in the theory and applications of Partial Differential Equations and energy functionals related to the fractional Laplacian operator as well as to more general integro-differential operators with singular kernel of fractional differentiability.After being investigated firstly in Potential Theory and Harmonic Analysis, fractional operators defined via singular integral are nowadays riveting great attention in different research fields related to Partial Differential Equations with nonlocal terms, since they naturally arise in many different contexts, as for instance, dislocations in crystals, nonlocal minimal surfaces, the obstacle problem, the fractional Yamabe problem, and many others.Much progress has been made during the last years, and this edited volume presents a valuable update to a wide community interested in these topics. List of contributorsClaudia Bucur, Zhen-Qing Chen, Francesca Da Lio, Donatella Danielli, Serena Dipierro, Rupert L. Frank, Maria del Mar Gonzalez, Moritz Kassmann, Tuomo Kuusi, Giuseppe Mingione, Giovanni Molica Bisci, Stefania Patrizi, Xavier Ros-Oton, Sandro Salsa, Yannick Sire, Enrico Valdinoci, Xicheng Zhang.
- Cover
- Half-Title
- Titlepage
- Copyright
- Contents
- Preface
- Claudia Bucur
- Essentials of Nonlocal Operators
- 0.1 Fractional Sobolev Spaces
- 0.2 The Fractional Laplacian
- 0.2.1 The harmonic extension
- 0.2.2 Maximum Principle and Harnack inequality
- 0.3 More General Nonlocal Operators
- 0.3.1 Some remarks on weak and viscosity solutions
- Bibliography
- Zhen-Qing Chen and Xicheng Zhang
- Heat Kernels for Non-symmetric Non-local Operators
- 1.1 Introduction
- 1.2 Lévy Process
- 1.3 Stable-Like Processes and their Heat Kernels
- 1.3.1 Approach
- 1.3.2 Upper bound estimates
- 1.3.3 Lower bound estimates
- 1.3.4 Strong stability
- 1.3.5 Applications to SDE driven by stable processes
- 1.4 Diffusion with Jumps
- 1.4.1 Approach
- 1.4.2 Application to SDE
- 1.5 Other Related Work
- Bibliography
- Francesca Da Lio
- Fractional Harmonic Maps
- 2.1 Overview
- 2.2 3-Commutators Estimates
- 2.3 Regularity of Horizontal 1/2-harmonic Maps and Applications
- 2.3.1 Case of 1/2-harmonic maps with values into a sphere
- 2.3.2 Case of 1/2-harmonic maps into a closed manifold
- 2.3.3 Case of horizontal 1/2-harmonic maps
- 2.3.4 Applications
- Bibliography
- Donatella Danielli and Sandro Salsa
- Obstacle Problems Involving the Fractional Laplacian
- 3.1 Introduction
- 3.2 The Obstacle Problem for the Fractional Laplacian
- 3.2.1 Construction of the solution and basic properties
- 3.2.2 Lipschitz continuity and semiconvexity and C1,α estimates
- 3.2.3 Thin obstacle for the operator La. Local C1,α estimates
- 3.2.4 Minimizers of the weighted Rayleigh quotient and a monotonicity formula
- 3.2.5 Optimal regularity for tangentially convex global solutions
- 3.2.6 Frequency formula
- 3.2.7 Blow-up sequences and optimal regularity
- 3.2.8 Nondegenerate case. Lipschitz continuity of the free boundary
- 3.2.9 Boundary Harnack principles and C1,α regularity of the free boundary
- 3.2.10 Non regular points on the free boundary
- 3.2.11 Non zero obstacle
- 3.2.12 A global regularity result (fractional Laplacian)
- 3.3 Comments and Further Reading
- 3.4 Parabolic Obstacle Problems
- 3.4.1 The parabolic fractional obstacle problem
- 3.4.2 The parabolic Signorini problem
- Bibliography
- Serena Dipierroand Enrico Valdinoci
- Nonlocal Minimal Surfaces: Interior Regularity, Quantitative Estimates and Boundary Stickiness
- 4.1 Introduction
- 4.2 Proof of Lemma 4.1.1
- 4.3 Proof of Theorem 4.1.4
- 4.4 Proof of Theorem 4.1.8
- 4.5 Sketch of the Proof of Theorem 4.1.12
- A A Short Discussion on the Asymptotics of the s-perimeter
- B A Short Discussion on the Asymptotics of the s-mean Curvature
- C Second Variation Formulas and Graphs of Zero Nonlocal Mean Curvature
- Bibliography
- Rupert L. Frank
- Eigenvalue Bounds for the Fractional Laplacian: A Review
- 5.1 Introduction
- 5.2 Bounds on Single Eigenvalues
- 5.2.1 The fractional Faber–Krahn inequality
- 5.2.2 The fractional Keller inequality
- 5.2.3 Comparing eigenvalues of and
- 5.3 Eigenvalue Asymptotics
- 5.3.1 Eigenvalue asymptotics for the fractional Laplacian
- 5.3.2 Eigenvalue asymptotics for fractional Schrödinger operators
- 5.4 Bounds on Sums of Eigenvalues
- 5.4.1 Berezin–Li–Yau inequalities
- 5.4.2 Lieb–Thirring inequalities
- 5.5 Some Further Topics
- A Proof of (5.1.1)
- B Lieb–Thirring Inequality in the Critical Case
- Bibliography
- María del Mar González
- Recent Progress on the Fractional Laplacian in Conformal Geometry
- 6.1 Introduction
- 6.2 Scattering Theory and the Conformal Fractional Laplacian
- 6.3 The Extension and the s-Yamabe Problem
- 6.4 The Conformal Fractional Laplacian on the Sphere
- 6.5 The Conformal Fractional Laplacian on the Cylinder
- 6.6 The Non-compact Case
- 6.7 Uniqueness
- 6.8 An Introduction to Hypersurface Conformal Geometry
- Bibliography
- Moritz Kassmann
- Jump Processes and Nonlocal Operators
- 7.1 Prerequisites and Lévy Processes
- 7.2 Lévy-Khintchine Representation
- 7.3 Generators of Lévy Processes
- 7.4 Nonlocal Operators and Jump Processes
- 7.4.1 References for the martingale problem for nonlocal operators
- 7.4.2 The path space of càdlàg paths
- 7.4.3 Uniqueness of the martingale problem
- 7.5 Regularity Estimates in Hölder Spaces
- 7.5.1 Probabilistic approach
- 7.5.2 Analytic approach
- Bibliography
- Tuomo Kuusi, Giuseppe Mingione, and Yannick Sire
- Regularity Issues Involving the Fractional p-Laplacian
- 8.1 Introduction
- 8.2 The Basic Existence Theorem and SOLA
- 8.3 The De Giorgi-Nash-Moser Theory for the Fractional p-Laplacian
- 8.3.1 Some recent results on nonlocal fractional operators
- 8.4 The “Harmonic Replacement", a Crucial Estimate and the Proof of Theorem 8.3
- 8.4.1 The crucial inequality
- 8.4.2 Basic estimates in the case p ≥ 2
- 8.4.3 Basic estimates in the case 2 > p > 2 – s/n
- 8.4.4 Proof of Theorem 8.3
- 8.5 Pointwise Behaviour of SOLA Solutions
- 8.5.1 Proofs of Theorems 8.12 and 8.13
- 8.6 A Lower Bound via Wolff Potentials
- 8.7 Continuity Conditions for SOLA
- Bibliography
- Xavier Ros-Oton
- Boundary Regularity, Pohozaev Identities and Nonexistence Results
- 9.1 Introduction
- 9.2 Boundary Regularity
- 9.2.1 Higher order boundary regularity estimates
- 9.2.2 Sketch of the proof of Theorem 9.2(a)
- 9.2.3 Comments, remarks, and open problems
- 9.3 Pohozaev Identities
- 9.3.1 Sketch of the proof
- 9.3.2 Comments and further results
- 9.4 Nonexistence Results and Other Consequences
- Bibliography
- Giovanni Molica Bisci
- Variational and Topological Methods for Nonlocal Fractional Periodic Equations
- 10.1 Introduction
- 10.2 Nonlocal Periodic Setting
- 10.2.1 Fractional Sobolev spaces
- 10.2.2 Weak solutions
- 10.2.3 Spectral properties of (–Δ + m2)s
- 10.3 Existence Results
- 10.3.1 A Mountain Pass solution
- 10.3.2 Ground state solutions
- 10.3.3 A local minimum result
- 10.3.4 A Morse theoretical approach
- 10.3.5 Some localization theorems
- 10.4 Multiple Solutions
- 10.4.1 Periodic equations sublinear at infinity
- 10.4.2 On the Ambrosetti–Rabinowitz condition
- 10.4.3 Three solutions for perturbed equations
- 10.4.4 Periodic equations with bounded primitive
- 10.4.5 Resonant periodic equations
- 10.5 Critical and Supercritical Nonlinearities
- 10.5.1 A periodic critical equation
- 10.5.2 Supercritical periodic problems
- Bibliography
- Stefania Patrizi
- Change of Scales for Crystal Dislocation Dynamics
- 11.1 Introduction
- 11.2 The Peierls-Nabarro Model
- 11.3 From the Peierls-Nabarro Model to the Discrete Dislocation Dynamics Model
- 11.3.1 Heuristics of the dynamics
- 11.3.2 Dislocation dynamics after the collision time
- 11.3.3 The case of two transition layers
- 11.3.4 The case of three transition layers
- 11.4 From the Peierls-Nabarro Model to the Dislocation Density Model
- 11.4.1 Viscosity solutions for non-local operators
- 11.4.2 Mechanical interpretation of the homogenization
- 11.4.3 The Orowan’s law
- 11.4.4 Heuristic for the proof of Orowan’s law
- 11.4.5 Homogenization and Orowan’s law for anisotropic fractional operators of any order
- 11.5 Non-local Allen-Cahn Equation
- 11.6 Some Open Problems
- Bibliography
- Back Cover
- 出版地 ： 德國
- 語言 ： 德文
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