# Linear peridynamics Fourier multipliers and eigenvalues

Bacim Alali and Nathan Albin

Department of Mathematics, Kansas State University, Manhattan, KS

###### Abstract

A characterization for the Fourier multipliers and eigenvalues of linear
peridynamic operators is provided. The analysis is presented for state-based
peridynamic operators for isotropic homogeneous media in any spatial
dimension. We provide explicit formulas for the eigenvalues in terms of the
space dimension, the nonlocal parameters, and the material properties.

The approach we follow is based on the Fourier multiplier analysis developed
in [2]. The Fourier multipliers of linear peridynamic operators are second-
order tensor fields, which are given through integral representations. It is
shown that the eigenvalues of the peridynamic operators can be derived
directly from the eigenvalues of the Fourier multiplier tensors. We reveal a
simple structure for the Fourier multipliers in terms of hypergeometric
functions, which allows for providing integral representations as well as
hypergeometric representations of the eigenvalues. These representations are
utilized to show the convergence of the eigenvalues of linear peridynamics to
the eigenvalues of the Navier operator of linear elasticity in the limit of
vanishing nonlocality. Moreover, the hypergeometric representation of the
eigenvalues is utilized to compute the spectrum of linear peridynamic
operators.

Keywords: Fourier multipliers, tensor multipliers, eigenvalues, peridynamics.

##  1 Introduction

In this work, we study the Fourier multipliers of linear state-based
peridynamic operators. The main goals are to find explicit representations for
the multipliers, when the operator is defined on $\mathbb{R}^{n}$, and to find
explicit representations for the eigenvalues of the peridynamic operator, when
it is defined on periodic domains. The formulas that we derive for the Fourier
multipliers and the eigenvalues are of two types: nonlocal (integral)
representations and representations in terms of hypergeometric functions. As
have been demonstrated in [2] and [1], such explicit representations can be
exploited to rigorously characterize the behavior of nonlocal operators and
develop regularity theory for nonlocal equations, as well as to devise
efficient and accurate spectral methods for the numerical solutions of
nonlocal equations. The current work focuses on the derivations of these
representations, while the regularity of peridynamic equations and spectral
methods for peridynamics based on the approach presented here will be pursued
in forthcoming works.

There has been a recent increased interest in spectral methods for
peridynamics and nonlocal equations as these methods provide efficient and
accurate solvers. One of the features of these spectral solvers is that the
nonlocality parameters do not scale with the grid size, thus providing
computational accuracy and efficiency [1]. Spectral methods have been
developed for nonlocal and peridynamic equations in periodic domain, bounded
domains, and for problems on surfaces, as well as problems involving fracture
[4], [10], [6], and [5]. Spectral and Fourier multipliers approaches provide
analysis techniques for studying the regularity of solutions of nonlocal
equations, see for example [3] and [2]. The work in [8] follows a Fourier
multipliers approach to study a fractional Lamé-Navier operator, its
connection to state-based peridynamics, and to establish analysis results for
this operator and certain associated fractional equations, see also [7].

The approach presented in this work to uncover explicit formulas for the
multipliers and the eigenvalues is based on two indirect connections; the
first is a connection between the multipliers of the peridynamic operator,
which are second-order tensor fields, and the scalar multipliers of the
nonlocal Laplace operator. The second connection is between the multipliers of
the peridynamic operator, defined on $\mathbb{R}^{n}$, and the eigenvalues of
the peridynamic operator, when it is defined on periodic domains. Throughout
this article we refer to the Fourier multipliers of the nonlocal Laplacian as
the scalar multipliers, whereas the tensor multipliers refer to the Fourier
multipliers of the peridynamic operator.

A brief description of the main steps in our approach and the organization of
the article are as follows. The definition of the linear peridynamic operator
in $\mathbb{R}^{n}$ and the specific integral kernels are provided in Section
2. In order to find explicit representations in terms of the nonlocality
parameters and the space dimension, we focus on integral kernels of the form
(1), which can be singular or integrable. However, we emphasize that the
results in this work can be generalized to other types of integral kernels.
Section 3.1 presents the nonlocal Laplacian and its multipliers given by the
integral and hypergeometric representations (9) and (10), respectively. The
multipliers of the Navier operator of linear elasticity and the integral
formula for the tensor multipliers of the peridynamic operator are derived in
Section 3.2. Each entry of the $n\times n$ tensor multiplier is written as an
integral in $\mathbb{R}^{n}$. A key step in our approach is to reveal a simple
structure for this tensor. This is accomplished in Section 3.3, where we show
in Section 3.3.2 that the tensor multipliers can be recovered using the
derivatives of the scalar multipliers. By combining this relationship with the
hypergeometric formula of the scalar multipliers together with the aid of some
facts about hypergeometric functions as presented in Section 3.3.1, we arrive
at a simple structure for the tensor multipliers in terms of hypergeometric
functions as demonstrated in Section 3.3.3. An immediate consequence of this
result is the convergence of the tensor multipliers of the peridynamic
operator to the tensor multipliers of the Navier operator for two kinds of
local limits. In Section 3.3.4, the tensor multiplier at any vector in
$\mathbb{R}^{n}$ is shown to be a real symmetric matrix with $n$ orthonormal
eigenvectors and two distinct associated eigenvalues. Using the hypergeometric
representation for the tensor multipliers, we derive explicit formulas for
these eigenvalues in terms of hypergeometric functions. Using these eigenvalue
formulas, we derive integral representations for the eigenvalues in Section
3.3.5. In Section 4, we consider the peridynamic operator defined for periodic
vector-fields. We show how the eigenvector fields and the eigenvalues for the
peridynamic operator on periodic domains can be derived from the tensor
multipliers’ eigenvectors and eigenvalues.

##  2 Overview

Linear peridynamic operators defined in a domain
$\Omega\subseteq\mathbb{R}^{n}$ have the form [9]

$\mathcal{L}{\bf u}(x)=\int_{\Omega}C(x,y)({\bf u}(y)-{\bf u}(x))\,dy,$

where $C(x,y)$ is a second-order tensor and ${\bf
u}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a vector field. For a
homogeneous isotropic solid, the linear operator takes the form

$\displaystyle\mathcal{L}{\bf u}(x)$ $\displaystyle=$
$\displaystyle\rho\int_{\Omega}\frac{\gamma(\|y-x\|)}{\|y-x\|^{2}}(y-x)\otimes(y-x)\big{(}{\bf
u}(y)-{\bf u}(x)\big{)}\,dy$
$\displaystyle+\rho^{\prime}\int_{\Omega}\int_{\Omega}\gamma(\|y-x\|)\gamma(\|z-x\|)(y-x)\otimes(z-x)\big{(}{\bf
u}(z)-{\bf u}(x)\big{)}\,dzdy$
$\displaystyle+\rho^{\prime}\int_{\Omega}\int_{\Omega}\gamma(\|y-x\|)\gamma(\|z-y\|)(y-x)\otimes(z-y)\big{(}{\bf
u}(z)-{\bf u}(y)\big{)}\,dzdy,$

where $\gamma$ is a scalar field, and $\rho$ and $\rho^{\prime}$ are scaling
constants that include the material properties. Taking
$\Omega=\mathbb{R}^{n}$, and due to symmetry, the operator reduces to

$\displaystyle\mathcal{L}{\bf u}(x)$ $\displaystyle=$
$\displaystyle\rho\int_{\mathbb{R}^{n}}\frac{\gamma(\|y-x\|)}{\|y-x\|^{2}}(y-x)\otimes(y-x)\big{(}{\bf
u}(y)-{\bf u}(x)\big{)}\,dy$
$\displaystyle+\rho^{\prime}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\gamma(\|y-x\|)\gamma(\|z-y\|)(y-x)\otimes(z-y){\bf
u}(z)\,dzdy.$

In this work, we focus on radially symmetric kernels with compact support of
the form

$\gamma(\|y-x\|)=c^{\delta,\beta}\frac{1}{\|y-x\|^{\beta}}\;\chi_{B_{\delta}(x)}(y),$
(1)

where $c^{\delta,\beta}$ is given by (3), $\chi_{B_{\delta}(x)}$ is the
indicator function of the ball of radius $\delta>0$ centered at $x$, and the
exponent satisfies $\beta<n+2$. In this case, the linear peridynamic operator,
parametrized by the horizon (nonlocality parameter) $\delta$ and the integral
kernel exponent $\beta$, can be written as

$\displaystyle\mathcal{L}^{\delta,\beta}{\bf u}(x)$ $\displaystyle=$
$\displaystyle(n+2)\,\mu\,c^{\delta,\beta}\int_{B_{\delta}(x)}\frac{(y-x)\otimes(y-x)}{\|y-x\|^{\beta+2}}\big{(}{\bf
u}(y)-{\bf u}(x)\big{)}\,dy$ (2)
$\displaystyle+(\lambda^{*}-\mu)\,\frac{(c^{\delta,\beta})^{2}}{4}\int_{B_{\delta}(x)}\int_{B_{\delta}(y)}\frac{y-x}{\|y-x\|^{\beta}}\otimes\frac{z-y}{\|z-y\|^{\beta}}\,{\bf
u}(z)\,dzdy,$

where $\mu$ and $\lambda^{*}$ are Lamé parameters, and the scaling constant
$c^{\delta,\beta}$ is defined by

$\displaystyle c^{\delta,\beta}$ $\displaystyle:=$
$\displaystyle\left(\frac{1}{2n}\int_{B_{\delta}(0)}\frac{\|w\|^{2}}{\|w\|^{\beta}}\;dw\right)^{-1},$
(3) $\displaystyle=$
$\displaystyle\frac{2(n+2-\beta)\Gamma\left(\frac{n}{2}+1\right)}{\pi^{n/2}\delta^{n+2-\beta}}.$

######  Remark 1.

The second Lamé parameter is usually denoted by $\lambda$, but we choose to
use $\lambda^{*}$ instead in order to keep $\lambda$ to denote an eigenvalue.

It is convenient to use the following decomposition of
$\mathcal{L}^{\delta,\beta}$

$\mathcal{L}^{\delta,\beta}=\mathcal{L}_{b}+\mathcal{L}_{s},$

where, after changing variables,

$\mathcal{L}_{b}{\bf
u}(x)=(n+2)\,\mu\,c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{w\otimes
w}{\|w\|^{\beta+2}}\big{(}{\bf u}(x+w)-{\bf u}(x)\big{)}\,dw,$ (4)

and

$\mathcal{L}_{s}{\bf
u}(x)=(\lambda^{*}-\mu)\,\frac{(c^{\delta,\beta})^{2}}{4}\int_{B_{\delta}(0)}\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}\otimes\frac{q}{\|q\|^{\beta}}\,{\bf
u}(x+q+w)\,dqdw.$ (5)

We note that $\mathcal{L}_{b}$ is the linear operator for bond-based
peridynamics.

We denote by ${\mathcal{N}}$ the Navier operator of linear elasticity. For a
homogeneous isotropic medium, it is given by

$\displaystyle{\mathcal{N}}{\bf u}$ $\displaystyle=$
$\displaystyle(\lambda^{*}+\mu)\nabla(\nabla\cdot{\bf u})+\mu\Delta{\bf u}.$
(6)

##  3 Fourier multipliers

###  3.1 Multipliers for the nonlocal Laplacian

For scalar fields $u:\mathbb{R}^{n}\rightarrow\mathbb{R}$, the analogue to the
peridynamic operator $\mathcal{L}^{\delta,\beta}$, is the nonlocal Laplacian,
which in this case is given by

$L^{\delta,\beta}u(x)=c^{\delta,\beta}\int_{B_{\delta}(x)}\frac{u(y)-u(x)}{\|y-x\|^{\beta}}\;dy,$
(7)

with $c^{\delta,\beta}$ given by (3).

The Fourier multipliers for the nonlocal Laplacian in (7) have been studied in
[2], in which the multiplier $m^{\delta,\beta}$ is defined through the Fourier
transform by

$L^{\delta,\beta}u(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}m^{\delta,\beta}(\nu)\widehat{u}(\nu)e^{i\nu\cdot
x}\;d\nu,$ (8)

where $m^{\delta,\beta}$ has the integral representation

$m^{\delta,\beta}(\nu)=c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{\cos(\nu\cdot
w)-1}{\|w\|^{\beta}}\;dw.$ (9)

The hypergeometric representation of the multipliers is provided by

$m^{\delta,\beta}(\nu)=-\|\nu\|^{2}\,_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right).$
(10)

###  3.2 Integral representations for the peridynamic multipliers

In this section, we extend the approach developed in [2] for the nonlocal
Laplacian $L^{\delta,\beta}$ in (7) to the peridynamic operator
$\mathcal{L}^{\delta,\beta}$ in (2). We begin by deriving integral formulas
for the Fourier multipliers of $\mathcal{L}^{\delta,\beta}$. Express ${\bf u}$
through its Fourier transform as

${\bf u}(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\widehat{{\bf
u}}(\nu)e^{i\nu\cdot x}\;d\nu.$

Since the definition of $\mathcal{L}^{\delta,\beta}$ can be extended to the
space of tempered distributions through the multipliers derived below, it is
sufficient to assume that ${\bf u}$ is a Schwartz vector field. We compute the
multipliers for $\mathcal{L}_{b}$ and $\mathcal{L}_{s}$ separately. Applying
$\mathcal{L}_{b}$ shows that

$\begin{split}\mathcal{L}_{b}{\bf
u}(x)&=(n+2)\,\mu\,c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{w\otimes
w}{\|w\|^{\beta+2}}\big{(}{\bf u}(x+w)-{\bf u}(x)\big{)}\,dw,\\\
&=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\left[(n+2)\,\mu\,c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{w\otimes
w}{\|w\|^{\beta+2}}\left(e^{i\nu\cdot w}-1\right)\;dw\right]\widehat{{\bf
u}}(\nu)e^{i\nu\cdot x}\;d\nu,\end{split}$

providing the representation

$\mathcal{L}_{b}{\bf
u}(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}M_{b}(\nu)\widehat{{\bf
u}}(\nu)e^{i\nu\cdot x}\;d\nu,$ (11)

where

$\displaystyle M_{b}(\nu)$ $\displaystyle=$
$\displaystyle(n+2)\,\mu\,c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{w\otimes
w}{\|w\|^{\beta+2}}\left(e^{i\nu\cdot w}-1\right)\;dw$ (12) $\displaystyle=$
$\displaystyle(n+2)\,\mu\,c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{w\otimes
w}{\|w\|^{\beta+2}}\left(\cos(\nu\cdot w)-1\right)\;dw.$

Similarly, we compute the multipliers of $\mathcal{L}_{s}$,

$\begin{split}\mathcal{L}_{s}{\bf
u}(x)&=(\lambda^{*}-\mu)\,\frac{(c^{\delta,\beta})^{2}}{4}\int_{B_{\delta}(0)}\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}\otimes\frac{q}{\|q\|^{\beta}}\,{\bf
u}(x+q+w)\,dqdw,\\\
&=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\left[(\lambda^{*}-\mu)\,\frac{(c^{\delta,\beta})^{2}}{4}\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}e^{i\nu\cdot
w}\,dw\otimes\int_{B_{\delta}(0)}\frac{q}{\|q\|^{\beta}}e^{i\nu\cdot
q}\,dq\right]\widehat{{\bf u}}(\nu)e^{i\nu\cdot x}\;d\nu,\end{split}$

providing the representation

$\mathcal{L}_{s}{\bf
u}(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}M_{s}(\nu)\widehat{{\bf
u}}(\nu)e^{i\nu\cdot x}\;d\nu,$ (13)

where

$\displaystyle M_{s}(\nu)$ $\displaystyle=$
$\displaystyle(\lambda^{*}-\mu)\,\frac{(c^{\delta,\beta})^{2}}{4}\left(\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}e^{i\nu\cdot
w}\,dw\right)\otimes\left(\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}e^{i\nu\cdot
w}\,dw\right)$ (14) $\displaystyle=$
$\displaystyle-(\lambda^{*}-\mu)\,\frac{(c^{\delta,\beta})^{2}}{4}\left(\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}\sin(\nu\cdot
w)\,dw\right)\otimes\left(\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}\sin(\nu\cdot
w)\,dw\right).$

Combining (12) and (14), we obtain the multipliers for
$\mathcal{L}^{\delta,\beta}$,

$\displaystyle M^{\delta,\beta}=M_{b}+M_{s},$ (15)

which satisfy

$\displaystyle\widehat{\mathcal{L}^{\delta,\beta}{\bf
u}}=M^{\delta,\beta}\,\widehat{{\bf u}}.$

The following summarizes the results of this subsection.

######  Proposition 1.

The Fourier multipliers $M^{\delta,\beta}$ of the linear peridynamic operator
$\mathcal{L}^{\delta,\beta}$ in (2) are characterized through integral
representations as given by (15), (12) and (14).

We note that the Fourier multipliers of ${\mathcal{N}}$, the Navier operator
given in (6), are similarly defined by

$\widehat{{\mathcal{N}}{\bf u}}=M^{{\mathcal{N}}}\,\widehat{{\bf u}},$

and can be shown to be given explicitly by

$M^{{\mathcal{N}}}(\nu)=-(\lambda^{*}+\mu)\nu\otimes\nu-\mu\|\nu\|^{2}\;I,$
(16)

where $I$ is the identity matrix.

###  3.3 Peridynamic multipliers: Structure and hypergeometric representations

We emphasize that the multipliers of linear peridynamics, given by (12),(14),
and (15), are second-order tensor fields. In this section, we reveal a simple
and explicit structure for the matrix $M^{\delta,\beta}(\nu)$ in terms of
$\nu$ and the derivatives of the scalar multipliers (multipliers of the
nonlocal Laplacian) $m^{\delta,\beta}(\nu)$ given by (9) or, equivalently, by
(10).

####  3.3.1 Hypergeometric formulas

In this section, we derive and present hypergeometric formulas that will be
useful in the subsequent sections. Let $\mathbf{a}=(a_{1},a_{2},\ldots,a_{p})$
and $\mathbf{b}=(b_{1},b_{2},\ldots,b_{q})$ be two vectors of coefficients.
The generalized hypergeometric function ${}_{p}F_{q}$ with parameters
$\mathbf{a}$ and $\mathbf{b}$ is defined as

${}_{p}F_{q}(\mathbf{a};\mathbf{b};z):=\sum_{k=0}^{\infty}\frac{(\mathbf{a})_{k}}{(\mathbf{b})_{k}}\frac{z^{k}}{k!}.$

Here, the notation $(\mathbf{a})_{k}$ represents the product

$(\mathbf{a})_{k}=(a_{1})_{k}(a_{2})_{k}\cdots(a_{p})_{k},$

where $(a)_{k}$ is the Pochhammer symbol

$(a)_{k}=\frac{\Gamma(a+k)}{\Gamma(a)}=a(a+1)(a+2)\cdots(a+k-1).$

We also define the notation

$\prod\mathbf{a}=a_{1}a_{2}\cdots
a_{p}\qquad\text{and}\qquad\mathbf{a}+c=(a_{1}+c,a_{2}+c,\ldots,a_{p}+c),$

and recall the following useful facts about the Pochhammer symbol.

$(a)_{k+1}=a(a+1)\cdots(a+k-1)(a+k)=a(a+1)_{k},$ (17)

and

$\frac{(a+1)_{k}}{(a)_{k}}=\frac{a+k}{a}.$ (18)

In light of (10), we consider the derivatives of a function of the form

$f(z)=z\cdot{}_{p}F_{q}(\mathbf{a};\mathbf{b};z)=z\sum_{k=0}^{\infty}\frac{(\mathbf{a})_{k}}{(\mathbf{b})_{k}}\frac{z^{k}}{k!}=\sum_{k=0}^{\infty}\frac{(\mathbf{a})_{k}}{(\mathbf{b})_{k}}\frac{z^{k+1}}{k!}.$
(19)

######  Lemma 1.

Let $f(x)$ have the form (19). Then

$f^{\prime}(z)={}_{p+1}F_{q+1}(\mathbf{a}^{\prime};\mathbf{b}^{\prime};z)$
(20)

and

$f^{\prime\prime}(z)=\frac{\prod\mathbf{a^{\prime}}}{\prod\mathbf{b^{\prime}}}\;{}_{p+1}F_{q+1}(\mathbf{a}^{\prime}+1;\mathbf{b}^{\prime}+1;z),$
(21)

where

$\mathbf{a^{\prime}}=(2,a_{1},\ldots,a_{p})\qquad\text{and}\qquad\mathbf{b^{\prime}}=(1,b_{1},\ldots,b_{q}).$

###### Proof.

Taking the term-wise first derivative and applying (18) shows that

$f^{\prime}(z)=\sum_{k=0}^{\infty}\frac{(\mathbf{a})_{k}(k+1)}{(\mathbf{b})_{k}}\frac{z^{k}}{k!}=\sum_{k=0}^{\infty}\frac{(\mathbf{a})_{k}(2)_{k}}{(\mathbf{b})_{k}(1)_{k}}\frac{z^{k}}{k!}={}_{p+1}F_{q+1}(\mathbf{a}^{\prime};\mathbf{b}^{\prime};z),$

Taking a term-wise derivative once again, then reindexing and using (17)
yields

$\begin{split}f^{\prime\prime}(z)&=\sum_{k=1}^{\infty}\frac{(\mathbf{a^{\prime}})_{k}k}{(\mathbf{b^{\prime}})_{k}}\frac{z^{k-1}}{k!}=\sum_{k=0}^{\infty}\frac{(\mathbf{a^{\prime}})_{k+1}(k+1)}{(\mathbf{b^{\prime}})_{k+1}}\frac{z^{k}}{(k+1)!}\\\
&=\frac{\prod\mathbf{a^{\prime}}}{\prod\mathbf{b^{\prime}}}\sum_{k=0}^{\infty}\frac{(\mathbf{a^{\prime}}+1)_{k}}{(\mathbf{b^{\prime}}+1)}\frac{z^{k}}{k!}=\frac{\prod\mathbf{a^{\prime}}}{\prod\mathbf{b^{\prime}}}\;{}_{p+1}F_{q+1}(\mathbf{a}^{\prime}+1;\mathbf{b}^{\prime}+1;z).\end{split}$

∎

Two additional formulas we shall use are found in the following lemmas.

######  Lemma 2.

For any choice of coefficients,

${}_{p}F_{q}(\mathbf{a};\mathbf{b};z)-1=\frac{\prod\mathbf{a}}{\prod\mathbf{b}}z\cdot{}_{p+1}F_{q+1}(1,\mathbf{a}+1;2,\mathbf{b}+1);z).$
(22)

###### Proof.

This is again found by term-wise differentiation, reindexing and applying
(17).

$\begin{split}{}_{p}F_{q}(\mathbf{a};\mathbf{b};z)-1&=\sum_{k=1}^{\infty}\frac{(\mathbf{a})_{k}}{(\mathbf{b})_{k}}\cdot\frac{z^{k}}{k!}=\sum_{k=0}^{\infty}\frac{(\mathbf{a})_{k+1}}{(\mathbf{b})_{k+1}}\cdot\frac{z^{k+1}}{(k+1)!}\\\
&=\frac{\prod\mathbf{a}}{\prod\mathbf{b}}z\
\sum_{k=0}^{\infty}\frac{(\mathbf{a}+1)_{k}(1)_{k}}{(\mathbf{b}+1)_{k}(2)_{k}}\cdot\frac{z^{k}}{k!}.\end{split}$

∎

######  Lemma 3.

For any choice of hypergeometric coefficients and for any numbers $c$ and $d$,

$c\;{}_{p+1}F_{q+1}(1,\mathbf{a};2,\mathbf{b};z)+d\;{}_{p}F_{q}(\mathbf{a};\mathbf{b};z)=(c+d)\;{}_{p+2}F_{q+2}\left(1,\frac{c+2d}{d},\mathbf{a};2,\frac{c+d}{d},\mathbf{b};z\right).$
(23)

###### Proof.

This can be seen by term-wise addition and using (18):

$\begin{split}c\;{}_{p+1}F_{q+1}(1,\mathbf{a};2,\mathbf{b};z)+d\;{}_{p}F_{q}(\mathbf{a};\mathbf{b};z)&=\sum_{k=0}^{\infty}\left(\frac{c(\mathbf{a})_{k}(1)_{k}}{(\mathbf{b})_{k}(2)_{k}}+\frac{d(\mathbf{a})_{k}}{(\mathbf{b})_{k}}\right)\frac{z^{k}}{k!}\\\
&=\sum_{k=0}^{\infty}\left(\frac{c+dk+d}{k+1}\right)\frac{(\mathbf{a})_{k}}{(\mathbf{b})_{k}}\frac{z^{k}}{k!}\\\
&=d\sum_{k=0}^{\infty}\frac{k+\frac{c+d}{d}}{k+1}\frac{(\mathbf{a})_{k}}{(\mathbf{b})_{k}}\frac{z^{k}}{k!}\\\
&=(c+d)\sum_{k=0}^{\infty}\frac{\left(\frac{c+2d}{d}\right)_{k}(1)_{k}}{\left(\frac{c+d}{d}\right)_{k}(2)_{k}}\frac{(\mathbf{a})_{k}}{(\mathbf{b})_{k}}\frac{z^{k}}{k!}.\end{split}$

∎

####  3.3.2 Derivatives of the scalar multipliers

In this section, we show how the tensor multipliers $M^{\delta,\beta}$, and in
particular $M_{b}$ and $M_{s}$, can be recognized in terms of the derivatives
of the scalar multipliers $m^{\delta,\beta}$.

Differentiating $m^{\delta,\beta}$ in (9) with respect to $\nu_{i}$ shows that

$\frac{\partial}{\partial\nu_{i}}m^{\delta,\beta}(\nu)=c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{-w_{i}\sin(\nu\cdot
w)}{\|w\|^{\beta}}\;dw.$ (24)

Substituting this into (14) yields the formula

$\left(M_{s}(\nu)\right)_{ij}=-\frac{\lambda^{*}-\mu}{4}\frac{\partial}{\partial\nu_{i}}m^{\delta,\beta}(\nu)\frac{\partial}{\partial\nu_{j}}m^{\delta,\beta}(\nu).$
(25)

Differentiating a second time in (24) (and replacing $\beta$ by $\beta+2$),
yields

$\frac{\partial^{2}}{\partial\nu_{i}\partial\nu_{j}}m^{\delta,\beta+2}(\nu)=c^{\delta,\beta+2}\int_{B_{\delta}(0)}\frac{-w_{i}w_{j}\cos(\nu\cdot
w)}{\|w\|^{\beta+2}}\;dw,$ (26)

which implies that

$\int_{B_{\delta}(0)}\frac{w_{i}w_{j}\cos(\nu\cdot
w)}{\|w\|^{\beta+2}}\;dw=-(c^{\delta,\beta+2})^{-1}\frac{\partial^{2}}{\partial\nu_{i}\partial\nu_{j}}m^{\delta,\beta+2}(\nu).$

Moreover,

$\int_{B_{\delta}(0)}\frac{w_{i}w_{j}}{\|w\|^{\beta+2}}\;dw=\delta_{ij}\int_{B_{\delta}(0)}\frac{w_{i}^{2}}{\|w\|^{\beta+2}}\;dw=2(c^{\delta,\beta+2})^{-1}\delta_{ij}.$
(27)

Substituting these last two formulas into (12) shows that

$\left(M_{b}(\nu)\right)_{ij}=-\frac{(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left(\frac{\partial^{2}}{\partial\nu_{i}\partial\nu_{j}}m^{\delta,\beta+2}(\nu)+2\delta_{ij}\right).$
(28)

The scalar multipliers $m^{\delta,\beta}$ can be written as

$\begin{split}m^{\delta,\beta}(\nu)&=-\|\nu\|^{2}{}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&=\frac{4}{\delta^{2}}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right){}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&=\frac{4}{\delta^{2}}f\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\end{split}$
(29)

where $f$ has the form (19) with $p=2$, $q=3$ and coefficients $\mathbf{a}$
and $\mathbf{b}$ defined to match (29). Differentiating once shows that

$\frac{\partial}{\partial\nu_{i}}m^{\delta,\beta}(\nu)=\frac{4}{\delta^{2}}f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\cdot\left(-\frac{1}{2}\delta^{2}\nu_{i}\right)=-2f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\nu_{i}.$
(30)

Differentiating a second time shows that

$\begin{split}\frac{\partial^{2}}{\partial\nu_{i}\partial\nu_{j}}m^{\delta,\beta}(\nu)&=-2f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\delta_{ij}-2f^{\prime\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\nu_{i}\cdot\left(-\frac{1}{2}\delta^{2}\nu_{j}\right)\\\
&=-2f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\delta_{ij}+\delta^{2}f^{\prime\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\nu_{i}\nu_{j}.\end{split}$
(31)

The results of this subsection are summarized by the following.

######  Proposition 2.

The tensor multipliers $M_{b}$ and $M_{s}$ can be represented in terms of the
gradients of the scalar multipliers $m^{\delta,\beta}$ as

$\begin{split}M_{b}(\nu)&=-\frac{(n+2)\;\mu\;c^{\delta,\beta}}{c^{\delta,\beta+2}}\left(\nabla\nabla
m^{\delta,\beta+2}(\nu)+2I\right),\\\ &=-\frac{(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left(\delta^{2}f^{\prime\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\nu\otimes\nu+\left(2-2f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\right)I\right),\end{split}$

where $f$ has the form (19) with $p=2$, $q=3$ and coefficients
$\mathbf{a}=\left(1,\frac{n-\beta}{2}\right)$ and
$\mathbf{b}=\left(2,\frac{n+2}{2},\frac{n+2-\beta}{2}\right)$, and,

$\begin{split}M_{s}(\nu)&=-\frac{\lambda^{*}-\mu}{4}\;\nabla
m^{\delta,\beta}(\nu)\otimes\nabla m^{\delta,\beta}(\nu),\\\
&=-(\lambda^{*}-\mu)\left(f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\right)^{2}\;\nu\otimes\nu.\end{split}$

where $f$ has the form (19) with $p=2$, $q=3$ and coefficients
$\mathbf{a}=\left(1,\frac{n+2-\beta}{2}\right)$ and
$\mathbf{b}=\left(2,\frac{n+2}{2},\frac{n+4-\beta}{2}\right)$.

This result together with the formulas derived in Section 3.3.1 allow us to
express the tensor multipliers as hypergeometric functions.

####  3.3.3 The tensor multipliers: Hypergeometric representations

In this section, we provide a simple and explicit form for the tensor
multipliers $M_{b}(\nu)$ and $M_{s}(\nu)$.

Equations (25), (30) and Lemma 1 show that $M_{s}(\nu)$ is a rank-one
symmetric matrix of the form

$M_{s}(\nu)=\alpha_{s}(\nu)\nu\otimes\nu,$ (32)

with

$\begin{split}\alpha_{s}(\nu)&=-\frac{\lambda^{*}-\mu}{4}\;\left(-2f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\right)^{2}\\\
&=-(\lambda^{*}-\mu)\left(f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\right)^{2}\\\
&=-(\lambda^{*}-\mu){}_{3}F_{4}\left(1,2,\frac{n+2-\beta}{2};1,2,\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)^{2}\\\
&=-(\lambda^{*}-\mu){}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)^{2}.\end{split}$
(33)

Equations (28), (31) and Lemma 1 show that $M_{b}(\nu)$ is a symmetric matrix
of the form

$M_{b}(\nu)=\alpha_{b1}(\nu)I+\alpha_{b2}(\nu)\nu\otimes\nu.$ (34)

The coefficient of the identity matrix is (keeping in mind that this time we
are differentiating $m^{\delta,\beta+2}$),

$\begin{split}\alpha_{b1}(\nu)&=-\frac{(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left(2-2f^{\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\right)\\\
&=\frac{2(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left({}_{3}F_{4}\left(1,2,\frac{n-\beta}{2};1,2,\frac{n+2}{2},\frac{n+2-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)-1\right)\\\
&=\frac{2(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left({}_{1}F_{2}\left(\frac{n-\beta}{2};\frac{n+2}{2},\frac{n+2-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)-1\right).\end{split}$
(35)

Applying Lemma 2 simplifies this expression to show that

$\begin{split}\alpha_{b1}(\nu)&=\frac{2(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left(\frac{2(n-\beta)}{(n+2)(n+2-\beta)}\right)\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&\qquad\times{}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&=-\mu\|\nu\|^{2}{}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right).\end{split}$
(36)

The other coefficient can be computed as

$\begin{split}\alpha_{b2}(\nu)&=-\frac{(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\delta^{2}f^{\prime\prime}\left(-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&=-\frac{(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left(\frac{2(n-\beta)}{(n+2)(n+2-\beta)}\right)\delta^{2}\\\
&\qquad\times{}_{3}F_{4}\left(2,3,\frac{n+2-\beta}{2};2,3,\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&=-2\mu\;{}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right).\end{split}$
(37)

It is interesting to see how these formulas combine to provide a formula for
the trace of the tensor $M_{b}$. Since we know all eigenvalues of $M_{b}$,
using (32)–(37), we can compute the trace as

$\begin{split}\operatorname{trace}M_{b}(\nu)&=n\alpha_{b1}+\|\nu\|^{2}\alpha_{b2}\\\
&=-n\mu\|\nu\|^{2}\;{}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&\qquad-2\mu\|\nu\|^{2}\;{}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right).\end{split}$

Applying Lemma 3, cancelling the repeated $(n+4)/2$ term from the
hypergeometric series coefficients, and then applying (10) shows that

$\begin{split}\operatorname{trace}M_{b}(\nu)&=-(n+2)\mu\|\nu\|^{2}\;{}_{3}F_{4}\left(1,\frac{n+4}{2},\frac{n+2-\beta}{2};2,\frac{n+2}{2},\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&=-(n+2)\mu\|\nu\|^{2}\;{}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&=(n+2)\mu m^{\delta,\beta}(\nu).\end{split}$

This same formula can also be derived directly from (12), since

$\frac{\operatorname{trace}\left(w\otimes
w\right)}{\|w\|^{\beta+2}}\left(\cos(\nu\cdot w)-1\right)=\frac{\cos(\nu\cdot
w)-1}{\|w\|^{\beta}},$

yielding the integrand in (10).

The main results of this subsection are summarized as follows.

######  Proposition 3.

The tensor multipliers $M_{b}$ and $M_{s}$ have the following hypergeometric
representations

$\displaystyle M_{b}(\nu)$
$\displaystyle=-\mu\|\nu\|^{2}\;{}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\;I$
$\displaystyle\quad\;\;-2\mu\;{}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\nu\otimes\nu,$
and, $\displaystyle M_{s}(\nu)$
$\displaystyle=-(\lambda^{*}-\mu)\,\,{}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)^{2}\;\;\nu\otimes\nu.$

An immediate consequence of this result is the convergence of the multipliers
of $\mathcal{L}^{\delta,\beta}$ to the multipliers of ${\mathcal{N}}$ in the
limits as $\delta\rightarrow 0$ or as $\beta\rightarrow n+2$.

######  Proposition 4.

Let $\beta\leq n+2$. Then

$\displaystyle\lim_{\delta\rightarrow 0^{+}}M_{b}(\nu)$ $\displaystyle=$
$\displaystyle-\mu\|\nu\|^{2}\;I-2\mu\;\nu\otimes\nu,$
$\displaystyle\lim_{\delta\rightarrow 0^{+}}M_{s}(\nu)$ $\displaystyle=$
$\displaystyle-(\lambda^{*}-\mu)\;\nu\otimes\nu,$
$\displaystyle\lim_{\delta\rightarrow 0^{+}}M^{\delta,\beta}(\nu)$
$\displaystyle=$
$\displaystyle-(\lambda^{*}+\mu)\nu\otimes\nu-\mu\|\nu\|^{2}\;I=M^{{\mathcal{N}}}(\nu).$

Moreover, let $\delta>0$. Then

$\displaystyle\lim_{\beta\rightarrow n+2^{-}}M_{b}(\nu)$ $\displaystyle=$
$\displaystyle-\mu\|\nu\|^{2}\;I-2\mu\;\nu\otimes\nu,$
$\displaystyle\lim_{\beta\rightarrow n+2^{-}}M_{s}(\nu)$ $\displaystyle=$
$\displaystyle-(\lambda^{*}-\mu)\;\nu\otimes\nu,$
$\displaystyle\lim_{\beta\rightarrow n+2^{-}}M^{\delta,\beta}(\nu)$
$\displaystyle=$
$\displaystyle-(\lambda^{*}+\mu)\nu\otimes\nu-\mu\|\nu\|^{2}\;I=M^{{\mathcal{N}}}(\nu).$

###### Proof.

This result follows from the fact the hypergeometric functions in Proposition
3 are equal to $1$ under the considered limits.

######  Remark 2.

The same results hold true for the limit from above $\beta\rightarrow
n+2^{+}$.

∎

####  3.3.4 Eigenvalues of the tensor multipliers

The form of the multiplier $M^{\delta,\beta}(\nu)$ is found through (32) and
(34),

$M^{\delta,\beta}(\nu)=\alpha_{b1}(\nu)I+(\alpha_{b2}(\nu)+\alpha_{s}(\nu))\nu\otimes\nu,$
(38)

where $\alpha_{b1}$, $\alpha_{b2}$ and $\alpha_{s}$ are given by (36), (37),
and (33), repectively. This implies that $M^{\delta,\beta}$ is a real
symmetric matrix. Moreover, $\nu$ is an eigenvector of
$M^{\delta,\beta}(\nu)$,

$M^{\delta,\beta}(\nu)\nu=\lambda_{1}(\nu)\nu,$ (39)

where

$\lambda_{1}(\nu)=\alpha_{b1}(\nu)+(\alpha_{b2}(\nu)+\alpha_{s}(\nu))\|\nu\|^{2}.$

Using (33), (36), and (37), this eigenvalue, which is associated with the
direction of $\nu$, has the following hypergeometric representation

$\begin{split}\lambda_{1}(\nu)=-\|\nu\|^{2}\Bigg{(}&\mu\;{}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&+2\mu\;{}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&+(\lambda^{*}-\mu)\;{}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)^{2}\Bigg{)}.\end{split}$
(40)

An alternative expression for this eigenvalue can be obtained by using Lemma 3
to combine the first two hypergeometric functions, yielding

$\begin{split}\lambda_{1}(\nu)=-\|\nu\|^{2}\Bigg{(}&3\mu\;{}_{3}F_{4}\left(1,\frac{5}{2},\frac{n+2-\beta}{2};2,\frac{3}{2},\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\\\
&+(\lambda^{*}-\mu)\;{}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)^{2}\Bigg{)}.\end{split}$
(41)

The other $n-1$ eigenvectors are orthogonal to $\nu$. Denote by $\nu^{\perp}$
a vector in $\mathbb{R}^{n}$ orthogonal to $\nu$. Then

$M^{\delta,\beta}(\nu)\nu^{\perp}=\lambda_{2}(\nu)\nu^{\perp},$ (42)

where

$\lambda_{2}(\nu)=\alpha_{b1}(\nu).$ (43)

Using (36), this eigenvalue, which is associated with orthogonal directions to
$\nu$, has the following hypergeometric representation

$\lambda_{2}(\nu)=-\mu\|\nu\|^{2}{}_{2}F_{3}\left(1,\frac{n+2-\beta}{2};2,\frac{n+4}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right).$
(44)

The results in this subsection are summarized in the following.

######  Theorem 1.

For $\nu\in\mathbb{R}^{n}$, the eigenvalue $\lambda_{1}(\nu)$ of
$M^{\delta,\beta}(\nu)$, associated with the direction of $\nu$, is given by
the hypergeometric representation (41) and the eigenvalue $\lambda_{2}(\nu)$
of $M^{\delta,\beta}(\nu)$, associated with orthogonal directions to $\nu$, is
given by the hypergeometric representation (44).

######  Corollary 1.

Let $\nu\in\mathbb{R}^{n}$. Then, the tensor multipliers
$M^{\delta,\beta}(\nu)$ and $M^{{\mathcal{N}}}(\nu)$ have the same set of
eigenvectors: $\nu$ and $n-1$ eigenvectors orthogonal to $\nu$. Moreover, the
eigenvalues of $M^{\delta,\beta}(\nu)$ converge to the eigenvalues of
$M^{{\mathcal{N}}}(\nu)$ in the local limits as follows: for $\beta\leq n+2$,

$\displaystyle\lim_{\delta\rightarrow 0^{+}}\lambda_{1}(\nu)$
$\displaystyle=-(\lambda^{*}+2\mu)\|\nu\|^{2},$
$\displaystyle\lim_{\delta\rightarrow 0^{+}}\lambda_{2}(\nu)$
$\displaystyle=-\mu\|\nu\|^{2},$ and for $\delta>0,$
$\displaystyle\lim_{\beta\rightarrow n+2^{-}}\lambda_{1}(\nu)$
$\displaystyle=-(\lambda^{*}+2\mu)\|\nu\|^{2},$
$\displaystyle\lim_{\beta\rightarrow n+2^{-}}\lambda_{2}(\nu)$
$\displaystyle=-\mu\|\nu\|^{2}.$

####  3.3.5 Integral representations for the eigenvalues of the peridynamic
multipliers

In this section, we provide integral representations for the eigenvalues
$\lambda_{1}$ and $\lambda_{2}$, given by (40) and (44), respectively.

######  Theorem 2.

The eigenvalue of $M^{\delta,\beta}(\nu)$ associated with the direction of
$\nu$ is given by

$\displaystyle\lambda_{1}(\nu)$ $\displaystyle=$ $\displaystyle(n+2)\mu
c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{(\nu\cdot
w)^{2}}{\|\nu\|^{2}\|w\|^{\beta+2}}(\cos(\nu\cdot w)-1)\;dw$ (45)
$\displaystyle\hskip
108.405pt-(\lambda^{*}-\mu)\left(\frac{c^{\delta,\beta}}{2}\int_{B_{\delta}(0)}\frac{\nu\cdot
w}{\|\nu\|\|w\|^{\beta}}\sin(\nu\cdot w)\;dw\right)^{2}.$

The eigenvalue of $M^{\delta,\beta}(\nu)$ associated with orthogonal
directions to $\nu$ is given by

$\displaystyle\lambda_{2}(\nu)$ $\displaystyle=$ $\displaystyle(n+2)\mu
c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{\nu\cdot
w}{\|\nu\|^{2}\|w\|^{\beta+2}}(\sin(\nu\cdot w)-\nu\cdot w)\;dw.$ (46)

###### Proof.

Solving (39) for $\lambda_{1}(\nu)$ we obtain

$\lambda_{1}(\nu)=M(\nu)\nu\cdot\frac{\nu}{\|\nu\|^{2}}.$ (47)

The result (45) follows from (47) combined with the fact that $M=M_{b}+M_{s}$
and the integral representations of the multipliers given in (12) and (14).

To derive (46), we use (43) and (35) to write

$\lambda_{2}(\nu)=\frac{(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left(2\;{}_{1}F_{2}\left(\frac{n-\beta}{2};\frac{n+2}{2},\frac{n+2-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)-2\right).$
(48)

From (24), (30), and (33), we obtain

$c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}\sin(\nu\cdot
w)\;dw=2\;{}_{1}F_{2}\left(\frac{n+2-\beta}{2};\frac{n+2}{2},\frac{n+4-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right)\nu.$

Thus, by replacing $\beta$ by $\beta+2$, then multiplying both sides of the
above equation by $\cdot\frac{\nu}{\|\nu\|^{2}}$, we find

$c^{\delta,\beta+2}\int_{B_{\delta}(0)}\frac{\nu\cdot
w}{\|w\|^{\beta+2}\|\nu\|^{2}}\sin(\nu\cdot
w)\;dw=2\;{}_{1}F_{2}\left(\frac{n-\beta}{2};\frac{n+2}{2},\frac{n+2-\beta}{2};-\frac{1}{4}\|\nu\|^{2}\delta^{2}\right).$
(49)

Using (27), we have the following identities

$\displaystyle 2$ $\displaystyle=$
$\displaystyle(2I)\nu\cdot\frac{\nu}{\|\nu\|^{2}}$ (50) $\displaystyle=$
$\displaystyle\left(c^{\delta,\beta+2}\int_{B_{\delta}(0)}\frac{w\otimes
w}{\|w\|^{\beta+2}}\;dw\right)\nu\cdot\frac{\nu}{\|\nu\|^{2}}$
$\displaystyle=$ $\displaystyle
c^{\delta,\beta+2}\int_{B_{\delta}(0)}\frac{(\nu\cdot
w)^{2}}{\|w\|^{\beta+2}\|\nu\|^{2}}\;dw.$

Using (48), (49) and (50), we obtain

$\lambda_{2}(\nu)=\frac{(n+2)\mu
c^{\delta,\beta}}{c^{\delta,\beta+2}}\left(c^{\delta,\beta+2}\int_{B_{\delta}(0)}\frac{\nu\cdot
w}{\|w\|^{\beta+2}\|\nu\|^{2}}(\sin(\nu\cdot w)-\nu\cdot w)\;dw\right),$

from which the result follows. ∎

##  4 Eigenvalues of the linear peridynamic operator

Consider $\mathcal{L}^{\delta,\beta}$ as an operator on the periodic torus

${\bf T}^{n}=\prod_{i=1}^{n}[0,\ell_{i}],\qquad\text{with }\ell_{i}>0,\quad
i=1,2,\ldots,n.$

In this section, we use the multiplier approach developed in the previous
sections to identify the eigenvalues and the eigenvector fields of the
operator $\mathcal{L}^{\delta,\beta}$,

$\mathcal{L}^{\delta,\beta}\boldsymbol{\phi}=\lambda\boldsymbol{\phi}.$

Let $\gamma$ be a fixed vector in $\mathbb{R}^{n}$. For any
$k\in\mathbb{Z}^{n}$, define

$\displaystyle\nu_{k}$ $\displaystyle=$ $\displaystyle(2\pi
k_{1}/\ell_{1},2\pi k_{2}/\ell_{2},\ldots,2\pi k_{n}/\ell_{n})^{T},$
$\displaystyle\boldsymbol{\psi}_{k}(x)$ $\displaystyle=$ $\displaystyle
e^{i\nu_{k}\cdot x}\gamma.$

Then, by applying $\mathcal{L}_{b}$ in (4), and using (12), we obtain

$\displaystyle\mathcal{L}_{b}\boldsymbol{\psi}_{k}(x)$ $\displaystyle=$
$\displaystyle(n+2)\,\mu\,c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{w\otimes
w}{\|w\|^{\beta+2}}\big{(}e^{i\nu_{k}\cdot(x+w)}\,\gamma-e^{i\nu_{k}\cdot
x}\,\gamma\big{)}\,dw,$ (51) $\displaystyle=$
$\displaystyle\left((n+2)\,\mu\,c^{\delta,\beta}\int_{B_{\delta}(0)}\frac{w\otimes
w}{\|w\|^{\beta+2}}\big{(}e^{i\nu_{k}\cdot
w}-1\big{)}\,dw\right)\,\,e^{i\nu_{k}\cdot x}\,\gamma,$ $\displaystyle=$
$\displaystyle M_{b}(\nu_{k})\boldsymbol{\psi}_{k}(x).$

Similarily, by applying $\mathcal{L}_{s}$ in (5), and using (14), we obtain

$\displaystyle\mathcal{L}_{s}\boldsymbol{\psi}_{k}(x)$ $\displaystyle=$
$\displaystyle(\lambda^{*}-\mu)\,\frac{(c^{\delta,\beta})^{2}}{4}\int_{B_{\delta}(0)}\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}\otimes\frac{q}{\|q\|^{\beta}}\,e^{i\nu_{k}\cdot(x+q+w)}\,\gamma\,dqdw,$
(52) $\displaystyle=$
$\displaystyle(\lambda^{*}-\mu)\,\frac{(c^{\delta,\beta})^{2}}{4}\left(\int_{B_{\delta}(0)}\frac{w}{\|w\|^{\beta}}e^{i\nu_{k}\cdot
w}\,dw\right)\otimes\left(\int_{B_{\delta}(0)}\frac{q}{\|q\|^{\beta}}e^{i\nu_{k}\cdot
q}\,dq\right)\,\,e^{i\nu_{k}\cdot x}\,\gamma,$ $\displaystyle=$ $\displaystyle
M_{s}(\nu_{k})\boldsymbol{\psi}_{k}(x).$

Equations (51), (52), and (15) yield

$\displaystyle\mathcal{L}^{\delta,\beta}\boldsymbol{\psi}_{k}$
$\displaystyle=$
$\displaystyle\mathcal{L}_{b}\boldsymbol{\psi}_{k}+\mathcal{L}_{s}\boldsymbol{\psi}_{k},$
(53) $\displaystyle=$ $\displaystyle
M_{b}(\nu_{k})\boldsymbol{\psi}_{k}+M_{s}(\nu_{k})\boldsymbol{\psi}_{k},$
$\displaystyle=$ $\displaystyle
M^{\delta,\beta}(\nu_{k})\boldsymbol{\psi}_{k}.$

Denote by $M_{k}=M^{\delta,\beta}(\nu_{k})$, the tensor multiplier evaluated
at $\nu_{k}$. From (38), (39), and (39), the matrix $M_{k}$ is real symmetric
and has $n$ orthogonal eigenvectors: one in the direction of $\nu_{k}$, and
$n-1$ orthogonal to $\nu_{k}$, denoted by
$\zeta_{k}^{2},\ldots,\zeta_{k}^{n}$. The associated eigenvalues are denoted
by $\lambda_{1}(k):=\lambda_{1}(\nu_{k})$ and
$\lambda_{2}(k):=\lambda_{2}(\nu_{k})$, respectively. Explicitly,

$\displaystyle M_{k}\nu_{k}$ $\displaystyle=$
$\displaystyle\lambda_{1}(k)\nu_{k},$ $\displaystyle M_{k}\zeta_{k}^{j}$
$\displaystyle=$ $\displaystyle\lambda_{2}(k)\zeta_{k}^{j},\quad j=2,\dots,n.$

Define

$\displaystyle\boldsymbol{\phi}_{k}^{1}(x)$ $\displaystyle=$ $\displaystyle
e^{i\nu_{k}\cdot x}\nu_{k},$ (54) $\displaystyle\boldsymbol{\phi}_{k}^{j}(x)$
$\displaystyle=$ $\displaystyle e^{i\nu_{k}\cdot x}\zeta_{k}^{j},\quad
j=2,\dots,n.$ (55)

Then,

$\displaystyle\mathcal{L}^{\delta,\beta}\boldsymbol{\phi}_{k}^{1}$
$\displaystyle=$ $\displaystyle
M^{\delta,\beta}(\nu_{k})\boldsymbol{\phi}_{k}^{1},$ (56) $\displaystyle=$
$\displaystyle e^{i\nu_{k}\cdot x}M_{k}\nu_{k},$ $\displaystyle=$
$\displaystyle e^{i\nu_{k}\cdot x}\lambda_{1}(k)\nu_{k},$ $\displaystyle=$
$\displaystyle\lambda_{1}(k)\boldsymbol{\phi}_{k}^{1},$

and, for $j=2,\ldots,n$,

$\displaystyle\mathcal{L}^{\delta,\beta}\boldsymbol{\phi}_{k}^{j}$
$\displaystyle=$ $\displaystyle
M^{\delta,\beta}(\nu_{k})\boldsymbol{\phi}_{k}^{j},$ (57) $\displaystyle=$
$\displaystyle e^{i\nu_{k}\cdot x}M_{k}\zeta_{k}^{j},$ $\displaystyle=$
$\displaystyle e^{i\nu_{k}\cdot x}\lambda_{2}(k)\zeta_{k}^{j},$
$\displaystyle=$ $\displaystyle\lambda_{2}(k)\boldsymbol{\phi}_{k}^{j}.$

This shows that the eigenvalues $\lambda_{1}(k)$ and $\lambda_{2}(k)$ of the
peridynamic operator $\mathcal{L}^{\delta,\beta}$, defined on the periodic
torus, are the eigenvalues of the tensor multipliers $M_{k}$. The summary is
given in the following result.

######  Theorem 3.

Let $k\in\mathbb{Z}^{n}$. The eigenvalues of the linear peridynamic operator
$\mathcal{L}^{\delta,\beta}$ are $\lambda_{1}(k):=\lambda_{1}(\nu_{k})$ with
associated eigenvector fields $\boldsymbol{\phi}_{k}^{1}$ and
$\lambda_{2}(k):=\lambda_{2}(\nu_{k})$ with associated eigenvector fields
$\boldsymbol{\phi}_{k}^{j}$, for $j=2,\ldots,n$, where $\lambda_{1}$ and
$\lambda_{2}$ are given in Theorem 1, or equivalently, in Theorem 2. The
eigenvector fields $\\{\boldsymbol{\phi}_{k}^{j}\\}_{k\in\mathbb{Z}^{n}}$,
with $j=1,\ldots,n$, are defined in (54) and (55).

Convergence of the eigenvalues of the peridynamic operator to the eigenvalues
of the Navier operator follows immediately from Corollary 1 and Theorem 3. The
eigenvalues of the Navier operator in (6) are given by

$\displaystyle\lambda^{{\mathcal{N}}}_{1}(k)$ $\displaystyle=$
$\displaystyle-(\lambda^{*}+2\mu)\|k\|^{2},\quad\mbox{ and
}\quad\lambda^{{\mathcal{N}}}_{2}(k)=-\mu\;\|k\|^{2}.$ (58)

Figure 1: The eigenvalues (vertical axis) $\lambda_{1}(\nu)$ and
$\lambda_{2}(\nu)$, as given by (45) and (46), for the 3D case. Here $\|\nu\|$
(horizontal axis) is sampled at $1000$ equispaced points in the interval
$[0,15]$ and $\delta$ and $\beta$ are as given in the titles. The shear
modulus and the second Lamé parameter are given by $\mu=1$ and
$\lambda^{*}=-1.9,-1,0,1,2$. For each plot, the dashed line shows
$\lambda_{2}(\nu)$ and the solid lines show $\lambda_{1}(\nu)$ corresponding
to the different values of $\lambda^{*}$ in a decreasing order, i.e., the top
solid curve corresponds to $\lambda^{*}=-1.9$, the second corresponds to
$\lambda^{*}=-1$, etc.

##  5 Discussion

The hypergeometric representations of the eigenvalues, given in (40) and (44),
are utilized to compute the eigenvalues $\lambda_{1}$ and $\lambda_{2}$ as
shown in Figure 1. It easily follows from (58) that for the Navier operator,
the eigenvalues are non-positive and $\lambda^{{\mathcal{N}}}_{1}(\nu)$ is
decreasing in $\lambda^{*}$ for a fixed value of $\mu$, which additionally can
be seen from the eigenvalues’ curves for the Navier operator (top-left) in
Figure 1. The non-positivity of the eigenvalues and the monotonicity of
$\lambda_{1}(\nu)$ as a function of $\lambda^{*}$ hold true as well for the
peridynamic operator. These observations follow from (45) and (46) for any
$\delta>0$ and $\beta<n+2$ and can also be observed in Figure 1. In addition,
in this figure, we note that in the first row (which corresponds to $\delta$
being close to $0$) and the first column (which corresponds to $\beta$ being
close to $n+2$) the eigenvalues satisfy
$\lambda_{1}(\nu)\approx\lambda_{1}^{\mathcal{N}}(\nu)$ and
$\lambda_{2}(\nu)\approx\lambda_{2}^{\mathcal{N}}(\nu)$, which is consistent
with Corollary 1 and the fact that the hypergeometric functions in (41) and
(44) are continuous. Moreover, in the second row of Figure 1, which
corresponds to $\beta=n+1$, we observe that the curves of the eigenvalues
$\lambda_{1}(\nu)$ and $\lambda_{2}(\nu)$ are linear, of order $~{}\|\nu\|$,
for large values of $\|\nu\|$. The asymptotic behavior of the eigenvalues in
the third row of this figure, which corresponds to $\beta=n$, can be shown to
be logarithmic in $\|\nu\|$. Furthermore, for integrable kernels (when
$\beta<n$) it can be seen from the fourth row of Figure 1 that the eigenvalues
are bounded. These observations can be rigorously proved, similar to the
approach followed in [2], using the hypergeometric representations (41) and
(44), and can be used to derive regularity results for peridynamic equations.
Lastly, we notice in the figure that the curves of $\lambda_{1}(\nu)$ for
different values of $\lambda^{*}$ converge to a single curve for large values
of $\|\nu\|$, in the case that $\beta<n$. This can be shown using the integral
representation (45) of $\lambda_{1}$ and the Riemann–Lebesgue Lemma, which
implies that $\sin(\nu\cdot w)$ weakly converges to $0$ in the limit as
$\|\nu\|\rightarrow\infty$.

## References

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