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

By C. Bucur. 2015

143 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/07adbc13a93938955052bfc9867a57cd40bdfec5

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

A new look at the fractional Poisson problem via the logarithmic Laplacian (2019, unknown): “RN \B 1. By Lemma 3.2, u 2 has a Poisson kernel representation given by u 2 = R RN\B1 v(z)P s(·,z)dz, where P s(x,z)=−(−∆)s zG s(x,z)=τ N,s (1−|x|2)s (|z|2 −1)s |x−z|−N for x ∈B 1,z∈RN \B 1 (see e.g. [7,11] and the references therein). Next, let x,y ∈B 1/2. We then have the estimates (1−|x|2)s −(1−|”
All functions are locally $s$-harmonic up to a small error (2014, unknown): “[1,7]) to write, for any x ∈ B1, ψ(x) = c ∫”
What is the fractional Laplacian? A comparative review with new results (2018, background): “11 of [108], a solution formula for (64) with f ≡ 0 in the case that is a ball Br of radius r is u(x) = { ∫ Rd\ Pr(y, x)g(y)dy x ∈ Br, g(x) x ∈ Rd \ Br .”
Blow-up analysis of a nonlocal Liouville-type equation (2015, unknown): “, [Blumenthal et al. 1961; Bucur 2015]), namely G(x, y)= 1 2π ∫ r0(x,y)”
Getting Acquainted with the Fractional Laplacian (2017, background): “2 in [15] (and related formulas hold true also for higher-order fractional operators, see [3, 4, 35]).”