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

Bucur, Claudia

doi:10.3934/cpaa.2016.15.657

Cited by 107 publications

(118 citation statements)

References 6 publications

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“…, and y ∈ K, then |x − y| ∼ |x|. Therefore using the estimate of G P from Lemma 2.1 and Hölder inequality, we have 8) since N > 2s and γ ∈ (0, (N − 2s)/2). For x ∈ B R (0), x = 0, using (2.7) and Lemma 2.1, we have…”

Section: 2mentioning

confidence: 86%

Integral representation of solutions using Green function for fractional Hardy equations

Bhakta

Biswas

Ganguly

et al. 2020

Journal of Differential Equations

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Section: 2mentioning

confidence: 86%

Integral representation of solutions using Green function for fractional Hardy equations

Bhakta

Biswas

Ganguly

et al. 2020

Journal of Differential Equations

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“…In the following an equivalent integral formulation of (2.4) will be useful. Using the Green's function for the Dirichlet problem in the unit ball ( [72,10]), we see that (2.4) is equivalent to…”

Section: 3mentioning

confidence: 99%

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

Ao¹,

Chan²,

DelaTorre³

et al. 2019

Duke Math. J.

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We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard ([63]) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

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“…The parameters for this algorithm are the coursest level 0 , the finest level L, and the tolerance ε. As test cases, we recall two examples on the unit ball where exact solutions are known (e.g., [Buc15]).…”

Section: Numerical Experimentsmentioning

confidence: 99%

A Walk Outside Spheres for the fractional Laplacian: Fields and first eigenvalue

Shardlow

2019

Math. Comp.

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The solution of the exterior-value problem for the fractional Laplacian can be computed by a walk-outside-spheres algorithm. This involves sampling α-stable Levy processes on their exit from maximally inscribed balls and sampling their occupation distribution. Kyprianou, Osojnik, and Shardlow (2017) developed this algorithm, providing a complexity analysis and an implementation, for approximating the solution at a single point in the domain. This paper shows how to efficiently sample the whole field by generating an approximation in L 2 (D), for a domain D. The method takes advantage of a hierarchy of triangular meshes and uses the multilevel Monte Carlo method for Hilbert space-valued quantities of interest. We derive complexity bounds in terms of the fractional parameter α and demonstrate that the method gives accurate results for two problems with exact solutions. Finally, we show how to couple the method with the variable-accuracy Arnoldi iteration to compute the smallest eigenvalue of the fractional Laplacian. A criteria is derived for the variable accuracy and a comparison is given with analytical results of Dyda (2012).We will make use of the following bounds on the unique solution to Eq. (1.1). We use C r (D) to denote the Banach space of r-times differentiable functions with the supremum norm on derivatives up to order r ∈ N. For s ∈ (0, 1), C r+s (D) denotes the Hölder-continuous subspace of u ∈ C r (D) with norm u C r + sup |r|=r [D r u] s < ∞, for [u] s := sup x,y∈D |u(x) − u(y)| x − y s and D r is the partial derivative defined by the multi-index r.

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

Cited by 107 publications

References 6 publications

Integral representation of solutions using Green function for fractional Hardy equations

Integral representation of solutions using Green function for fractional Hardy equations

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

A Walk Outside Spheres for the fractional Laplacian: Fields and first eigenvalue

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