Fourier multipliers for nonlocal Laplace operators

TL;DR

This paper develops Fourier multiplier analysis for nonlocal Laplace operators, providing explicit integral representations and asymptotic behavior, and demonstrates convergence to classical solutions in the periodic setting.

Contribution

It introduces a unified formula for Fourier multipliers of nonlocal Laplacians using hypergeometric functions and analyzes their asymptotics and boundedness.

Findings

Fourier multipliers are explicitly represented via hypergeometric functions.

Boundedness of multipliers depends on kernel integrability.

Solutions to the nonlocal Poisson equation converge to classical solutions.

Abstract

Fourier multiplier analysis is developed for nonlocal peridynamic-type Laplace operators, which are defined for scalar fields in $\mathbb{R}^n$. The Fourier multipliers are given through an integral representation. 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 $_2F_3$ in any spatial dimension $n$. Asymptotic analysis of $_2F_3$ is utilized to identify the asymptotic behavior of the Fourier multipliers $m(\nu)$ as $\|\nu\|\rightarrow \infty$. We show that the multipliers are bounded when the peridynamic Laplacian has an integrable kernel, and diverge when the kernel is singular. The bounds and decay rates are presented explicitly in terms of the dimension $n$, the integral kernel, and the peridynamic Laplacian nonlocality. The asymptotic analysis is applied in the periodic setting to prove a regularity result for the peridynamic Poisson equation and, moreover, show that its solution converges to the solution of the classical Poisson equation.