Fourier multipliers for nonlocal Laplace operators

Abstract

Fourier multiplier analysis is developed for peridynamic Laplace operators, which are defined for scalar fields in R n . The focus is on operators L δ,β with compactly supported integral kernels of the form χ B δ (0) (x) 1x β , which are commonly used in peridynamic models [6]. The Fourier multipliers m(ν) of L δ,β are given through an integral representation, which is shown to be well-defined for β < n + 2. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function 2 F 3 in any spatial dimension n and for any β < n + 2. The new general formula for the multipliers is well-defined for any β ∈ R \ {n + 4, n + 6, n + 8, . . .} and is utilized to extend the definition of the peridynamic Laplacians to the case when β ≥ n + 2 (with β = n + 4, n + 6, n + 8, . . .). Some special cases are presented. We show that the Fourier multipliers of L δ,β converge to the Fourier multipliers of ∆ in the limit as β → n + 2. In addition, we show that in the limit as β → n + 2 − , L δ,β u converges to ∆u for sufficiently regular u. Moreover, we identify the limit of the Fourier multipliers of L δ,β as β → −∞ and, furthermore, recognize the limit operator L δ,−∞ as an integral operator with kernel supported in the (n − 1)-sphere.Asymptotic analysis of 2 F 3 is utilized to identify the asymptotic behavior of the Fourier multipliers as ν → ∞. We show that the multipliers are bounded when the peridynamic Laplacian has an integrable kernel (i.e., when β < n), diverge to −∞ at a rate proportional to − log( ν ) when β = n, and diverge to −∞ at a rate proportional to − ν β−n when β > n. The bounds and decay rates are presented explicitly in terms of n, β, and the nonlocality δ.In periodic domains, the Fourier multipliers are the eigenvalues of L δ,β and therefore the presented results provide characterization formulas and asymptotic behavior for the eigenvalues. The asymptotic analysis is applied in the periodic setting to prove a regularity result for the peridynamic Poisson equation.

Fourier Spectral Methods for Nonlocal Models

“…The hypergeometric representation of the Fourier multipliers (6) has shown to be a useful tool for the analysis of nonlocal equations [3]. The usefulness of this representation for computations of nonlocal equations has been demonstrated in this paper.…”

“…For completeness of the presentation, we include a summary of the Fourier multipliers' results, developed in [3], relevant to the computational spectral methods presented in Sections 3 and 4. The multipliers m of L δ,β are defined through the Fourier transform by…”

“…This work concerns Fourier spectral methods for nonlocal equations involving peridynamictype nonlocal Laplace operators. The focus is on nonlocal operators L δ,β , parametrized by a spatial nonlocality parameter δ and an integral kernel exponent β, of the form [3]…”

“…In this work, we present efficient and accurate spectral solvers for nonlocal models in n dimensions. We pursue the Fourier multipliers' approach introduced in [3]. The Fourier multipliers of L δ,β are given by the integral representation…”

Fractional Laplacians : A short survey

“…The peridynamic model was originally proposed as a reformulation of the classical solid mechanics. We refer the interested reader to the pioneering article [75], as well as to [35,55,42,7,62] and references therein.…”