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.

Nonlocal Diffusion and Applications

This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.

/pdf/ten-equivalent-definitions-of-the-fractional-laplace-4r7dvvvzfi.pdf

Ten equivalent definitions of the fractional laplace operator

What is the fractional Laplacian? A comparative review with new results

Calculus of Variations and Partial Differential Equations attracts and collects many of the important top-quality contributions to this field of research, and stresses the interactions between analysts, geometers, and physicists.  Coverage in the journal includes:  Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory  Variational methods for partial differential equations, linear and nonlinear eigenvalue problems, bifurcation theory  Variational problems in differential and complex geometry  Variational methods in global analysis and topology  Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems  Variational methods in mathematical physics, nonlinear elasticity, crystals, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions  Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.

Calculus of Variations and Partial Differential Equations

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford–Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.

Numerical methods for fractional diffusion

We consider a fractional Laplace equation and we give a self-contained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new, however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.

Some observations on the Green function for the ball in the fractional Laplace framework

We consider a fractional Laplace equation and we give a self-contained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new (see for instance [2, 9]), however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.

https://www.aimsciences.org/article/exportPdf?id=fb3fb062-4952-4a67-a69b-1987a6af1044

Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter

We consider the Caputo fractional derivative and say that a function is Caputo-stationary if its Caputo derivative is zero. We then prove that any C k ( [ 0,1 ]) function can be approximated in [0,1] by a function that is Caputo-stationary in [0,1], with initial point a C k loc (R).

Claudia Bucur

Papers

Nonlocal Diffusion and Applications

Some observations on the Green function for the ball in the fractional Laplace framework

Some observations on the Green function for the ball in the fractional Laplace framework

Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter

Local density of Caputo-stationary functions in the space of smooth functions