Nonlocal diffusion and applications

Abstract

We consider the fractional Laplace framework and provide models and theorems related to nonlocal diffusion phenomena. Some applications are presented, including: a simple probabilistic interpretation, water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations. Furthermore, an example of an $s$-harmonic function, the harmonic extension and some insight on a fractional version of a classical conjecture formulated by De Giorgi are presented. Although this book aims at gathering some introductory material on the applications of the fractional Laplacian, some proofs and results are original. Also, the work is self contained, and the reader is invited to consult the rich bibliography for further details, whenever a subject is of interest.

132 pages, 26 figures

References in corpus (11)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Gamma-convergence of nonlocal perimeter functionals
• Some observations on the Green function for the ball in the fractional Laplace framework
• Regularity for parabolic integro-differential equations with very irregular kernels
• Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one
• Crystal dislocations with different orientations and collisions
• Nonlocal Minimal Lawson Cones
• Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
• Nonlocal elliptic equations in bounded domains: a survey
• Fractional Perimeter and Nonlocal Minimal Surfaces
• Regularity of nonlocal minimal cones in dimension 2

Cited by in corpus (104)

• Ten equivalent definitions of the fractional Laplace operator
• Some observations on the Green function for the ball in the fractional Laplace framework
• The Calderón problem for the fractional Schrödinger equation
• Fractional Integrable Nonlinear Soliton Equations
• What Is the Fractional Laplacian?
• Regularity estimates for elliptic nonlocal operators
• Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces
• Nonlinear commutators for the fractional p-Laplacian and applications
• Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials
• Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
• Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation
• Exponential instability in the fractional Calderón problem
• Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation
• One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels
• Integro-Differential Elliptic Equations
• Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth
• The higher order fractional Calderón problem for linear local operators: uniqueness
• A simple mathematical model inspired by the Purkinje cells: from delayed travelling waves to fractional diffusion
• On the maximum principle for higher-order fractional Laplacians
• Should I stay or should I go? Zero-size jumps in random walks for Lévy flights
• Fractional Laplacians in bounded domains: Killed, reflected, censored and taboo Lévy flights
• An Asymptotically Compatible Formulation for Local-to-Nonlocal Coupling Problems without Overlapping Regions
• General self-similar solutions of diffusion equation and related constructions
• A general framework for nonlocal Neumann problems
• Global regularity of the three-dimensional fractional micropolar equations
• Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations
• A System of Local/Nonlocal $p$-Laplacians: The Eigenvalue Problem and Its Asymptotic Limit as $p\to \infty$
• Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces
• The fractional Schrödinger equation with singular potential and measure data
• An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems
• Local density of Caputo-stationary functions in the space of smooth functions
• A gradient estimate for nonlocal minimal graphs
• Homogenization of random convolution energies in heterogeneous and perforated domains
• Non-local control in the conduction coefficients: well posedness and convergence to the local limit
• Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent
• Symmetrization for fractional elliptic problems: a direct approach
• A review of Local-to-Nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics
• Fractional powers of vector operators and fractional Fourier's law in a Hilbert space
• The nonlinear fractional relativistic Schrödinger equation: existence, multiplicity, decay and concentration results
• On fractional Schrödinger equations with Hartree type nonlinearities
• Fractional Dissipative PDEs
• Null controllability from the exterior of a one-dimensional nonlocal heat equation
• Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schrödinger Equations
• An Asymptotically Compatible Approach For Neumann-Type Boundary Condition On Nonlocal Problems
• Ground state solutions for the nonlinear fractional Schrodinger-Poisson system
• Nonlocal phase transitions in homogeneous and periodic media
• A reduced conjugate gradient basis method for fractional diffusion
• Energy asymptotics of a Dirichlet to Neumann problem related to water waves
• Fractional Calderón problem on a closed Riemannian manifold
• Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
• Nonlocal complement value problem for a global in time parabolic equation
• Delaunay hypersurfaces with constant nonlocal mean curvature
• Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension
• Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian
• A reaction coefficient identification problem for fractional diffusion
• Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation
• Lamellar phase solutions for diblock copolymers with nonlocal diffusions
• Singular $p$-biharmonic problems involving the Hardy-Sobolev exponent
• Helmholtz Solutions for the Fractional Laplacian and Other Related Operators
• Propagating front solutions in a time-fractional Fisher-KPP equation
• A quantitative rigidity result for a two-dimensional Frenkel-Kontorova model
• A framework for non-local, non-linear initial value problems
• A fractional elliptic problem in ${\mathbb{R}}^n$ with critical growth and convex nonlinearities
• Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation
• Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications
• Interior and boundary regularity results for strongly nonhomogeneous $p,q$-fractional problems
• Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form
• Concentration phenomena for fractional magnetic NLS equations
• Fractional Poincaré Inequality for Unbounded Domains with Finite Ball Condition: Counter Example
• Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential
• Minimization problems involving nonlocal functionals: nonlocal minimal surfaces and a free boundary problem
• Potential theoretic approach to Schauder estimates for the fractional Laplacian
• Stability of solutions for a parabolic problem involving fractional p-Laplacian with logarithmic nonlinearity
• Fractional Laplacians and Levy flights in bounded domains
• Ultrarelativistic bound states in the shallow spherical well
• Long-range connections and mixed diffusion in fractional networks
• Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media
• Long-range connections, real-world networks and rates of diffusion
• Fractional powers of the backward heat operator and Carleman type inequalities
• Fractional boundary Hardy inequality for the critical cases
• Toeplitz matrices for the study of the fractional Laplacian on a bounded interval
• Asymptotic method of moving planes for fractional parabolic equations
• Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations
• Fractional Laplace operator and Meijer G-function
• $H$-compactness for nonlocal linear operators in fractional divergence form
• Unique weak solutions of the d-dimensional micropolar equation with fractional dissipation
• Nonlinear problem involving the fractional p(x)-Laplacian operator by Topological degree
• The set of $p$-harmonic functions in $B_1$ is total in $C^k({\bar{B}}_1)$
• Concentrating phenomenon for fractional nonlinear Schrödinger-Poisson system with critical nonlinearity
• Strong comparison principle for the fractional $p$-Laplacian and applications to starshaped rings
• A Carleman estimate for the fractional heat equation and its application in final state observability
• Uniqueness of the inverse reaction coefficient problems for nonlocal diffusion models
• A class of elliptic equations involving a general nonlocal integrodifferential operators with sign-changing weight functions
• Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space
• The dual approach to optimal control in the coefficients of nonlocal nonlinear diffusion
• Nonlocal degenerate parabolic hyperbolic equations on bounded domains
• Liouville results for semilinear integral equations with conical diffusion
• Surfactants in a Non-local Model for Phase Transitions
• Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights
• Integral geometry and unique continuation principles
• Five solutions for the fractional p-Laplacian with noncoercive energy
• Global existence and blow-up of solutions for a parabolic equation involving the fractional $p(x)$-Laplacian
• On some nonlinear fractional equations involving the Bessel operator
• The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions