Fractional Laplace Operator and Meijer G-function

Dyda, Bartłomiej; Kuznetsov, Alexey; Kwaśnicki, Mateusz

doi:10.1007/s00365-016-9336-4

Cited by 57 publications

(48 citation statements)

References 21 publications

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“…For one-dimensional or radial domains, these regularity estimates can be further sharpened by deriving explicit expressions for the map w → (−∆) s [dist(·, ∂Ω) s w] in terms of expansions in bases consisting of special functions, see [3,19]. Of importance in the design of optimally convergent finite element schemes is [2], where regularity in spaces similar to those introduced in Definition 2.5 (weighted fractional Sobolev spaces) was derived.…”

Section: The Linear Problemmentioning

confidence: 99%

“…We first describe how to construct a non-trivial solution to (1.3) in the unit ball of R n . For this domain, reference [19] explicitly expresses eigenfunctions of an operator closely related to the fractional Laplacian in terms of Jacobi polynomials and an s-dependent weight. For example, in dimension n = 2 and using the Jacobi polynomial P We now consider a smooth obstacle χ that coincides with u in Λ = B 1/5 and modifyf in B 1/5 so that within this contact set the strict inequality (−∆) s u > f holds.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

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Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

Borthagaray

Nochetto

Salgado

2019

Math. Models Methods Appl. Sci.

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We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian (−∆) s . The weight is a power of the distance to the boundary that accounts for the singular boundary behavior of the solution for any 0 < s < 1. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.definitions of the operator (−∆) s , motivated by applications, here we choose the so-called integral one; that is, for a sufficiently smooth function v : R n → R we setOur choice of definition is justified by the fact that, unlike the regional or the spectral ones, the integral fractional Laplacian of order s is the infinitesimal generator of a 2s-stable Lévy process. These processes have been widely employed for modeling market fluctuations, both for risk management and option pricing purposes. It is in this context that, as mentioned above, the fractional obstacle problem arises as a pricing model for American options. More precisely, if u represents the rational price of a perpetual American option, modeling the assets prices by a Lévy process X t and denoting by χ the payoff function, then u solves (1.3). We refer the reader to [16] for an overview of the use of jump processes in financial modeling.Taking into account their applications in finance, it is not surprising that numerical schemes for integrodifferential inequalities have been proposed and analyzed in the literature; we refer the reader to [26] for a survey on these methods. These applications aim to approximate the price of a number of assets; therefore, the consideration of a logarithmic price leads to problems posed in the whole space R n . For the numerical solution, it is usual to perform computations on a sufficiently large tensor-product domain. Among the schemes based on Galerkin discretizations, reference [46] utilizes piecewise linear Lagrangian finite elements, while [29] proposes the use of wavelet bases in space. As for approximations of variational inequalities involving integral operators on arbitrary bounded domains, an a posteriori error analysis is performed in [36].Since the seminal work of Silvestre [45], the fractional obstacle problem started to draw the attention of the mathematical community. Using potential theoretic methods, reference [45] shows that if the obstacle is of class C 1,s , then the solution to the fractional obstacle problem is of class C 1,α for all α ∈ (0, s); optimal C 1,s regularity of solutions was derived assuming convexity of the contact set. The pursuit of the optimal regularity of solutions without a convexity hypothesis, in turn, motivated the celebrated extension by Caffarelli and Silvestre [11] for the fractional Laplacian in R n . Using this extension technique, Caffarelli, Salsa and Silvestre proved, in [10], the optimal regularity of solutions (cf. Proposition 3.4 below). It is important to notice, however, that this is only an interior regularity result. Nothing is said about the boundary behavior o...

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Section: The Linear Problemmentioning

confidence: 99%

Section: Numerical Illustrationsmentioning

confidence: 99%

Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

Borthagaray

Nochetto

Salgado

2019

Math. Models Methods Appl. Sci.

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“…Besides its impact on the study of fractional differential equations, inequality (1.3) might have an independent interest. For instance, it is strictly related to Bochner's relations and to the results in [10,14]. Inequality (1.3) and its generalizations below provide quite useful technical tools.…”

Section: Introductionmentioning

confidence: 99%

A tool for symmetry breaking and multiplicity in some nonlocal problems

Musina

Nazarov

2020

Math Methods in App Sciences

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We prove some basic inequalities relating the Gagliardo-Nirenberg seminorms of a symmetric function u on R n and of its perturbation uϕ µ , where ϕ µ is a suitably chosen eigenfunction of the Laplace-Beltrami operator on the sphere S n−1 , thus providing a technical but rather powerful tool to detect symmetry breaking and multiplicity phenomena in variational equations driven by the fractional Laplace operator. A concrete application to a problem related to the fractional Caffarelli-Kohn-Nirenberg inequality is given.

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“…Proposition 4 gives a precise statement of what is outlined in a recent work of Dyda, Kuznetsov, and Kwaśnicki, see [DKK,Remark 1]. One should, however, observe that the Green's functions mentioned there were obtained before only for s ∈ (0, 1).…”

Section: Green Function With Pole At the Originmentioning

confidence: 63%