Gevrey regularity for integro-differential operators

Albanese, Guglielmo; Fiscella, Alessio; Valdinoci, Enrico

doi:10.1016/j.jmaa.2015.04.002

Cited by 11 publications

(11 citation statements)

References 29 publications

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We prove that for a certain class of kernels K(y) that viscosity solutions of the integro-differential equationare locally analytic if f is an analytic function. This extends results in [1] in which it was shown that such solutions belong to certain Gevrey classes.

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confidence: 85%

Analyticity for Solution of Integro-Differential Operators

Blatt

2020

Preprint

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confidence: 85%

Analyticity for Solution of Integro-Differential Operators

Blatt

2020

Preprint

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“…The function N s α (x) then equals the series on the right-hand side of (3.15). In view of point (4) we thus see that, at least in the case α = s + n, N s α (x) is analytic with respect to x for |x| < 1 and, further, that the desired relation (3.9) holds.…”

Section: Boundary Singularity and Diagonal Form Of The Single-intervamentioning

confidence: 67%

Regularity theory and high order numerical methods for the (1D)-fractional Laplacian

et al. 2017

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Abstract. This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight ω times a "regular" unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein Ellipse, analyticity in the same Bernstein Ellipse is obtained for the "regular" unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the Fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babuška and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nyström numerical method for the Fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.

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“…For the case r = 1/2, L α 1/2 represents the integral fractional Laplacian operator [2]. The existence, uniqueness and regularity of the solution to the fractional Laplacian equation has been investigated by a number of authors, in R 1 see [3], in R n≥2 [4,11,19,31]. Recently the regularity results for the fractional Laplacian equation was extended by Hao and Zhang in [22,38] Fewer results on the regularity of the solution to the general fractional diffusion, advection, reaction equation have been established.…”

Section: Introductionmentioning

confidence: 99%

“…In the next section we present some preliminary definitions and results. Sections 3,4, and 6 present the three different (but equivalent) definitions of the weighted Sobolev spaces. Section 4 also establishes some useful properties of the space H s (a,b) (I).…”

Section: Introductionmentioning

confidence: 99%

Regularity of the solution to 1-D fractional order diffusion equations

2018

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In this report we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in R 1 . The analysis is performed in the weighted Sobolev spaces, H s (a,b) (I). Three different characterizations of H s (a,b) (I) are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters α and r defining the fractional diffusion operator. Additionally, the analysis shows that for a sufficiently smooth right hand side function, the regularity of the solution to fractional diffusion reaction equation is lower than that of the fractional diffusion equation. Also, the regularity of the solution to fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation.In this article we present the general regularity results for (1.3),(1.4), in appropriately weighted Sobolev spaces. The analysis establishes that the presence of a reaction term (i.e. c(x) = 0) limits the regularity of the solution, regardless of the smoothness of the right hand side function, f (x). This reduction in regularity is greater (by a factor of 2) when an advective term (i.e. b(x) = 0) appears in (1.3). This behavior of the solution is in sharp contrast to that for the integer order (α = 2) diffusion, advection, reaction equation. In that case, assuming b(x) and c(x) are sufficiently regular, for the right hand side function f ∈ H s (I) the solution lies in H s+2 (I).The results we present herein extend those in [13] for the fractional diffusion equation, and those in [22] for the fractional Laplacian equation with a constant advection and reaction term. The proofs given are significantly different that those used in [21,22].The analysis of (1.3),(1.4) is most appropriately performed in weighted Sobolev spaces (due to the singular behavior of the solution at the endpoints). There are different ways to define the weighted Sobolev spaces: (i) using interpolation (Section 3), (ii) using an appropriate basis (Section 4), (iii)

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Gevrey regularity for integro-differential operators

Abstract: We prove for some singular kernels K(x, y) that viscosity solutions of the integro-differential equation

Cited by 11 publications

References 29 publications

Analyticity for Solution of Integro-Differential Operators

Analyticity for Solution of Integro-Differential Operators

Regularity theory and high order numerical methods for the (1D)-fractional Laplacian

Regularity of the solution to 1-D fractional order diffusion equations

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