nonlocal balance laws - Pennsylvania State UniversityNonlocal vector calculus, volume-constrained...

Archive for Rational Mechanics and Analysis manuscript No.(will be inserted by the editor)

Q. Du · M. D. Gunzburger · R. B.Lehoucq · K. Zhou

A nonlocal vector calculus, nonlocalvolume-constrained problems, andnonlocal balance laws

DRAFT date 15 May 2011Received: date / Accepted: date

Q. Du and K. Zhou were supported in part by the U.S. Department of Energy Officeof Science under grant DE-SC0005346. M. Gunzburger was supported in part by theU.S. Department of Energy Office of Science under grant DE-SC0004970. Sandiais a multiprogram laboratory operated by Sandia Corporation, a Lockheed MartinCompany, for the U.S. Department of Energy under contract DE-AC04-94AL85000.

Q. DuDepartment of MathematicsPenn State UniversityTel.: +1-814-865-3674State College, PA 16802, USAFax: +1-814-865-3735E-mail: [email protected]

M. D. GunzburgerDepartment of Scientific ComputingFlorida State UniversityTallahassee FL 32306-4120, USATel.: +1-850-644-7060Fax: +1-850-644-0098E-mail: [email protected]

R. B. LehoucqSandia National LaboratoriesP.O. Box 5800, MS 1320Albuquerque NM 87185-1320, USATel.: +1-505-845-8929Fax: +1-505-845-7442E-mail: [email protected]

K. ZhouDepartment of MathematicsPenn State UniversityState College, PA 16802, USATel.: +1-814-865-3674Fax: +1-814-865-3735E-mail: [email protected]

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Abstract A vector calculus for nonlocal operators is developed, includingthe definition of nonlocal generalized divergence, gradient, and curl operatorsand the derivation of their adjoint operators. Analogs of several theorems andidentities of the vector calculus for differential operators are also presented.A subsequent incorporation of volume constraints gives rise to well-posedvolume-constrained problems that are analogous to elliptic boundary-valueproblems for differential operators. This is demonstrated via some examples.Relationships between the nonlocal operators and their differential counter-parts are established, first in a distributional sense and then in a weak senseby considering weighted integrals of the nonlocal adjoint operators. An appli-cation of the nonlocal calculus is to pose abstract balance laws with reducedregularity requirements.

Keywords nonlocal · vector calculus · continuum mechanics · peridynamics

1 Introduction

We introduce a vector calculus for nonlocal operators that mimics the classi-cal vector calculus for differential operators. We define, within a Hilbert spacesetting, generalized nonlocal representations of the divergence, gradient, andcurl operators and deduce the corresponding nonlocal adjoint operators. Non-local analogs of the Gauss theorem and the Green’s identities of the vectorcalculus for differential operators are also derived. The nonlocal vector cal-culus can be used to define nonlocal volume-constrained problems that areanalogous to boundary-value problems for partial differential operators. Inaddition, we establish relationships between the nonlocal operators and theirdifferential counterparts.

A balance law postulates that the rate of change of an extensive quantityover any subregion of a body is given by the rate at which that quantityis produced in the subregion minus the flux out of the subregion throughits boundary. Along with kinematics and constitutive relations, balance lawsare a cornerstone of continuum mechanics. Difficulties arise, however, in, forexample, the classical equations of continuum mechanics due to, e.g., shocks,corner singularities, and material failure, all of which are troublesome whendefining an appropriate notion for the “flux through the boundary of thesubregion.” The nonlocal vector calculus we develop has an important ap-plication to balance laws that are nonlocal in the sense that subregions notin direct contact may have a non-zero interaction. This is accomplished bydefining the flux in terms of interactions between possibly disjoint regionsof positive measure possibly sharing no common boundary. An importantfeature of the nonlocal balance laws is that the significant technical detailsassociated with determining normal and tangential traces over the bound-aries of suitable regions is obviated when volume interactions induce flux.Our nonlocal calculus, then, is an alternative to standard approaches for cir-cumventing the technicalities associated with lack of sufficient regularity inlocal balance laws such as measure-theoretic generalizations of the Gauss-Green theorem (see, e.g., [4,18]) and the use of the fractional calculus (see,e.g., [1,24]).

Nonlocal vector calculus, volume-constrained problems and balance laws 3

Preliminary attempts at a nonlocal calculus were the subject of [11,12],which included applications to image processing1 and steady-state diffusion,respectively. However the discussion was limited to scalar problems. In con-trast, this paper extends the ideas in [11,12] to vector and tensor fields andbeyond the consideration of image processing and steady-state diffusion. Forexample, the ideas presented here enable an abstract formulation of the bal-ance laws of momentum and energy and for the peridynamic2 theory for solidmechanics that parallels the vector calculus formulation of the balance lawsof elasticity. The nonlocal vector calculus presented in this paper, however,is sufficiently general that we envisage application to balance laws beyondthose of elasticity, e.g., to the laws of fluid mechanics and electromagnetics.

The paper is organized as follows. The nonlocal vector calculus is de-veloped in Sections 2 and 3. In Section 2, nonlocal generalizations of thedifferential divergence, gradient, and curl operators and of the correspondingadjoint operators are given as are nonlocal generalizations of several theoremsand identities of the vector calculus for differential operators. In Section 3, weintroduce the notion of nonlocal constraint operators that allow us to definenonlocal volume-constrained problems that generalize well-known boundary-value problems for partial differential operators. Amended versions of severalof the theorems and identities of the nonlocal vector calculus are derivedthat account for the constraint operators. Connections between the nonlo-cal operators and their differential counterparts are made in Sections 4 and5. In Section 4, we connect the two classes of operators in a distributionalsense whereas, in Section 5, the connections are made between weighted in-tegrals involving the nonlocal adjoint operators and weak representatives oftheir differential counterparts. The connections made in those two sectionsjustify the use of the terminology “nonlocal divergence, gradient, and curl”operators to refer to the nonlocal operators we define. In Section 6, a briefreview of the conventional notion of a balance law is provided after which anabstract nonlocal balance law is discussed; also, a brief discussion is given ofthe application of our nonlocal vector calculus to the peridynamic theory forcontinuum mechanics.

2 A nonlocal vector calculus

We develop a nonlocal vector calculus that mimics the classical vector calcu-lus for differential operators. The nonlocal vector calculus involves two typesof functions and two types of nonlocal operators. Point functions refer tofunctions defined at points whereas two-point functions refer to functions de-fined for pairs of points. Point operators map two-point functions to pointfunctions whereas two-point operators map point functions to two-point func-tions so that the nomenclature for operators refer to their ranges. Point and

1 The authors of [11] refer to [25] where a discrete nonlocal divergence and gra-dient are introduced within the context of machine learning.

2 Peridynamics was introduced in [20,22]; [23] reviews the peridynamic balancelaws of momentum and energy and provides many citations for the peridynamictheory and its applications. See Section 6.1 for a brief discussion.

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two-point operators are both nonlocal. Point operators involve integrals oftwo-point functions whereas two-point operators explicitly involve pairs ofpoint functions.

We now make more precise the definitions given above. To this end, fora positive integer d, let Ω denote an open subset of Rd. Points in Rd aredenoted by the vectors x, y, or z and the natural Cartesian basis is denoted bye1, . . . , ed. Let n and k denote positive integers. Functions from Ω or subsetsof Ω into Rn×k or Rn or R are referred to as point functions or point mappingsand are denoted by Roman letters, upper-case bold for tensors, lower-casebold for vectors, and plain-face for scalars, respectively, e.g., U(x), u(x), andu(x), respectively. Functions from Ω × Ω or subsets of Ω × Ω to Rn×k, orRn, or R are referred to as two-point functions or two-point mappings andare denoted by Greek letters, upper-case bold for tensors, lower-case boldfor vectors, and plain-face for scalars, respectively, e.g., Ψ(x,y), ψ(x,y),and ψ(x,y), respectively. Symmetric and antisymmetric two-point functionssatisfy, e.g., for the scalar two-point function ψ(x,y), we have ψ(x,y) =ψ(y,x) and ψ(x,y) = −ψ(y,x), respectively.

The dot (or inner) product of two vectors u,v ∈ Rn is denoted by u ·v ∈ R; the dyad (or outer) product is denoted by u ⊗w ∈ Rn×k wheneverw ∈ Rk; given a second-order tensor (matrix) U ∈ Rk×n, the tensor-vector(or matrix-vector) product is denoted by U · v and is given by the vectorwhose components are the dot products of the corresponding rows of U withv.3 For n = 3, the cross product of two vectors u and v is denoted byu × v ∈ R3. The Frobenius product of two second-order tensors A ∈ Rn×kand B ∈ Rn×k, denoted by A : B, is given by the sum of the component-wiseproduct of the two tensors. The trace of B ∈ Rn×n, denoted by tr

(B), is

given by the sum of the diagonal elements of B.Inner products in L2(Ω) and L2(Ω×Ω) are defined in the usual manner.

For example, for vector functions, we have(u,v)Ω =

∫Ω

u · v dx for u(x), v(x) ∈ Rn

(µ,ν)Ω×Ω =

∫Ω

∫Ω

µ · ν dydx for µ(x,y), ν(x,y) ∈ Rn

with analogous expressions involving the Frobenius product and the ordinaryproduct for tensor and scalar functions, respectively.

2.1 Nonlocal point divergence, gradient, and curl operators

The nonlocal point divergence, gradient, and curl operators map two-pointfunctions to point functions and are defined as follows. These operators alongwith their adjoints are the building blocks of our nonlocal calculus.

Definition 1 [Nonlocal point operators] Given the vector two-pointfunction ν : Ω × Ω → Rk and the antisymmetric vector two-point function

3 In matrix notation, the inner, outer, matrix-vector products are given by x ·y =xTy, x⊗ y = xyT , and U · v = UTv.


α : Ω × Ω → Rk, the nonlocal point divergence operator D(ν)

: Ω → R isdefined as

D(ν)(x) :=

∫Ω

(ν(x,y) + ν(y,x)

)·α(y,x) dy for x ∈ Ω. (1a)

Given the scalar two-point function η : Ω × Ω → R and the antisymmet-ric vector two-point function β : Ω × Ω → Rn, the nonlocal point gradientoperator G

(η)

: Ω → Rk is defined as

G(η)(x) :=

∫Ω

(η(x,y) + η(y,x)

)β(y,x) dy for x ∈ Ω. (1b)

Given the vector two-point function µ : Ω ×Ω → R3 and the antisymmetricvector two-point function γ : Ω × Ω → R3, the nonlocal point curl operatorC(µ)

: Ω → R3 is defined as

C(µ)(x) :=

∫Ω

γ(y,x)×(µ(x,y) + µ(y,x)

)dy for x ∈ Ω. � (1c)

The nonlocal point operatorsD, G, and C map vectors to scalars, scalars tovectors, and vectors to vectors, respectively, as is the case for the divergence,gradient, and curl differential operators. Relationships between the nonlocalpoint operators and differential operators are made in Sections 4 and 5 wherewe demonstrate circumstances under which the nonlocal point operators areidentified with the differential operators in the sense of distributions and asweak representations, respectively.

For the sake of simplicity, in much of the rest of the paper, we introducethe following notation:

α := α(x,y) α′ := α(y,x) ψ := ψ(x,y) ψ′ := ψ(y,x)

u := u(x) u′ := u(y) u := u(x) u′ := u(y)

and similarly for other functions.4

2.2 Nonlocal integral theorems

Definition 1 leads to integral theorems for the nonlocal vector calculus thatmimic basic integral theorems of the vector calculus for differential opera-tors;in particular, (2a) can be viewed as nonlocal Gauss’ theorem. The an-tisymmetry, with respect to x and y, of the integrands in the definitions ofthe nonlocal point operators (1) plays a crucial role.

Theorem 1 [Nonlocal integral theorems] Let the functions ν, η, andµ and the operators D, G, and C be defined as in Definition 1. Then, we

4 For example, from (1a), we have D(ν)(x) =

∫Ω

(ν + ν′) ·α dy.

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have the following analogs of the integral theorems of the vector calculus fordifferential operators: ∫

Ω

D(ν)dx = 0 (2a)∫

Ω

G(η)dx = 0 (2b)∫

Ω

C(µ)dx = 0. (2c)

Proof From (1a), we have that∫Ω

D(ν)dx =

∫Ω

∫Ω

(ν + ν′

)·α dy dx.

Because α is antisymmetric, the integrand in the double integral is anti-symmetric, i.e., we have that (ν + ν′) · α = −(ν′ + ν) · α′, from whichit easily follows that that double integral vanishes. Thus, (2a) follows. Thenonlocal integral theorems (2b) and (2c) follow in exactly the same way from(1b) and (1c), respectively.5 �

The nonlocal integral theorems (2a)–(2c) have the simple consequenceslisted in the following corollary that can be viewed as nonlocal integrationby parts formulas.

Corollary 1 [Nonlocal integration by parts formulas] Adopt the func-tions and operators appearing in Definition 1. Then, given the point functionsu(x) : Ω → R, v(x) : Ω → Rn, and w(x) : Ω → R3, we have6∫

Ω

uD(ν)dx +

∫Ω

∫Ω

((u′ − u)α

)· ν dydx = 0 (3a)∫

Ω

v · G(η)dx +

∫Ω

∫Ω

((v′ − v) · β

)η dydx = 0 (3b)∫

Ω

w · C(µ)dx−

∫Ω

∫Ω

(γ × (w′ −w)

)· µ dydx = 0. (3c)

Proof Let ξ = uν; then,

(ξ + ξ′) ·α = (uν + u′ν′) ·α = u(ν + ν′) ·α+ (u′ − u)ν′ ·α

so that, from (1a), we have

D(ξ) =∫Ω

(u(ν + ν′) ·α+ (u′ − u)ν′ ·α

)dy ∀x ∈ Ω.

5 Alternately, (1b) and (1c) can be derived from (1a); one simply chooses ν = ηband ν = b × µ, respectively, in (1a) where b is a constant vector; one also has toassociate β and γ with α.

6 Whenever ν and α are scalar valued functions, the integration by parts formula(3a) appears in [13, Lemma 2.1].


Then, (2a) results in∫Ω

D(ξ) dx =∫Ω

u(∫

Ω

(ν + ν′) ·αdy)dx +

∫Ω

∫Ω

(u′ − u)ν′ ·α dydx = 0

so that, by (1a) and the anti-symmetry of α,∫Ω

uD(ν)dx +

∫Ω

∫Ω

(u− u′)ν′ ·α′ dydx = 0.

A change of variables in the double integral then results in (3a).In a similar manner, (3b) and (3c) can be derived from (2a) along with

(1b) and (1c), respectively. Alternately, (3b) and (3c) easily follow by setting,for an arbitrary constant vector b, ν = ηb and ν = b × µ, respectively, in(3a) and also associating β and γ with α. �

2.3 Nonlocal adjoint operators

The adjoints of the nonlocal point operators are two-point operators and aredefined as follows.

Definition 2 Given a point operator L that maps two-point functions F topoint functions defined over Ω, the adjoint operator L∗ is a two-point operatorthat maps point functions G to two-point functions defined over Ω ×Ω thatsatisfies (

G,L(F ))Ω−(L∗(G), F

)Ω×Ω = 0. (4)

Here, the operators F and G may denote pairs of vector-scalar, scalar-vector,or vector-vector functions. �

Corollary 1 can be used to immediately determine the adjoint operatorscorresponding to the point operators defined in Definition 1.

Corollary 2 [Nonlocal adjoint operators] Given the point function u(x) :Ω → R, the adjoint of D is the two-point operator given by

D∗(u)(x,y) = −(u′ − u)α for x,y ∈ Ω, (5a)

where D∗(u)

: Ω × Ω → Rk. Given the point function v(x) : Ω → Rn, theadjoint of G is the two-point operator given by

G∗(v)(x,y) = −(v′ − v) · β for x,y ∈ Ω, (5b)

where G∗(v)

: Ω × Ω → R. Given the point function w(x) : Ω → R3, theadjoint of C is the two-point operator given by

C∗(w)(x,y) = γ × (w′ −w) for x,y ∈ Ω, (5c)

where C∗(w)

: Ω ×Ω → R3.

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Proof Let F = ν and G = u in (4). Then, (5a) immediately follows from(3a). Similarly, (5b) and (5c) are immediate consequences of (3b) and (3c),respectively, and (4). �

We can rewrite nonlocal integration by parts formulas in terms of the non-local adjoint operators given by Corollaries 1 and 2, respectively as follows:

∫Ω

uD(ν)dx−

∫Ω

∫Ω

D∗(u)· ν dy dx = 0 (6a)∫

Ω

v · G(η)dx−

∫Ω

∫Ω

G∗(v)η dy dx = 0 (6b)∫

Ω

w · C(µ)dx−

∫Ω

∫Ω

C∗(w)· µ dy dx = 0. (6c)

2.4 Nonlocal Green’s identities

Nonlocal Green’s identities can be derived from (4) by setting F = ΘL∗(H),where L∗(H) may be a scalar or vector or second-order tensor function inwhich cases Θ is a scalar or second-order tensor or fourth-order tensor func-tion, respectively. This leads to the nonlocal Green’s first identity(

G,L(ΘL∗(H)))Ω−(L∗(G), ΘL∗(H)

)Ω×Ω = 0. (7)

Suppose now that Θ is a symmetric tensor. Then, reversing the roles of Gand H and subtracting the result from (7) yields the nonlocal Green’s secondidentity (

G,L(ΘL∗(H)

))Ω−(H,L

(ΘL∗(G)

))Ω

= 0. (8)

Using (7) and (8) immediately yields the following two sets of results.

Corollary 3 [Nonlocal (generalized) Green’s first identities] Giventhe scalar point functions u(x) : Ω → R and v(x) : Ω → R and the two-pointsecond-order tensor function Θ(x,y) : Ω ×Ω → Rn×n, then∫

Ω

uD(Θ · D∗(v)

)dx−

∫Ω

∫Ω

D∗(u) ·Θ · D∗(v) dydx = 0. (9a)

Given the vector point functions u(x) : Ω → Rn and v(x) : Ω → Rn and thetwo-scalar point function θ(x,y) : Ω ×Ω → R, then∫

Ω

v · G(θG∗(u)

)dx−

∫Ω

∫Ω

θG∗(v)G∗(u) dydx = 0. (9b)

Given the vector point functions u(x) : Ω → R3 and w(x) : Ω → R3 and thetwo-point fourth-order tensor function Θ(x,y) : Ω ×Ω → R3×3×3×3, then∫

Ω

w · C(Θ : C∗(u)

)dx−

∫Ω

∫Ω

C∗(w) : Θ : C∗(u) dydx = 0. � (9c)


Corollary 4 [Nonlocal (generalized) Green’s second identities] Weassume the same notation as in Corollary 3 and we also assume that thetensors Θ appearing in (9a) and (9c) are symmetric. Then,∫

Ω

uD(Θ · D∗(v)

)dx−

∫Ω

vD(Θ · D∗(u)

)dx = 0 (10a)

∫Ω

v · G(θG∗(u)

)dx−

∫Ω

u · G(θG∗(v)

)dx = 0 (10b)

∫Ω

w · C(Θ : C∗(u)

)dx−

∫Ω

u · C(Θ : C∗(w)

)dx = 0. � (10c)

2.5 Further observations and results about nonlocal operators

In this subsection, we collect several observations and results that can bededuced from the definitions and results of Sections 2.1–2.4.

2.5.1 A nonlocal divergence operator for tensor functions

A nonlocal divergence operator for tensor functions is defined by applying(1a) to each of the rows of the tensor.

Definition 3 Given the tensor two-point function Ψ : Ω × Ω → Rn×k andthe antisymmetric vector two-point function α : Ω × Ω → Rk the nonlocalpoint divergence operator Dt(Ψ)(x) : Ω → Rn for tensors is defined as

Dt(Ψ)(x) :=∫Ω

(Ψ + Ψ ′

)·α dy for x ∈ Ω. (11a)

A nonlocal Gauss theorem for tensor functions as well as the correspond-ing integration by parts formula and Green’s identities can be derived and theadjoint operator D∗t can defined in the same way as was done above for thenonlocal divergence operator D for vector functions, e.g., see Corollary 2. Inparticular, Definition 2 implies that, given the point function v(x) : Ω → Rn,

D∗t (v)(x,y) = −(v′ − v)⊗α for x,y ∈ Ω, (11b)

where D∗t (v) : Ω × Ω → Rn×k, i.e., D∗t maps a vector point function to atwo-point second-order tensor function.7

7 Despite our notational convention that upper case Greek and Roman fontsdenotes point and two-point tensor points, respectively, we could, in a slight abuseof notation, abbreviate Dt(Ψ) by D(Ψ). In an analogous fashion, D∗t (v) may beabbreviated by D∗(v) when the argument is a vector point function. Note that thisnotational abuse is customary for the differential divergence and gradient operatorsfor which ∇· is used to denote the divergence operator for both vectors and tensorsand ∇ is used to denote the gradient operator on both scalars and vectors.

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2.5.2 Nonlocal vector identities

Compositions of the point operators defined in Sections 2.1 and 2.5.1 with thecorresponding adjoint two-point operators derived in Section 2.3 lead to thefollowing nonlocal vector identities. In this proposition, we set α = β = γand, for the first, second, and fourth results, i.e., those involving nonlocalcurl operators, we set n = k = 3.

Proposition 1 The nonlocal divergence, gradient, and curl operators andthe corresponding adjoint operators satisfy

D(C∗(u)

)= 0 for u : Ω → R3 (12a)

C(D∗(u)

)= 0 for u : Ω → R (12b)

G∗(u) = tr(D∗t (u)

)for u : Ω → Rn (12c)

Dt(D∗t (u)

)− G

(G∗(u)

)= C

(C∗(u)

)for u : Ω → R3. (12d)

Proof We prove (12d); the proofs of (12a)–(12c) are immediate after directsubstitution of the operators involved.

Let u : Ω → R3. Then, by (1c), (5c), (11a), and (11b) and recalling thatα is an antisymmetric function, we have

Dt(D∗t (u)

)− G

(G∗(u)

)= −

∫Ω

((u′ − u)⊗α+ (u− u′)⊗α′

)·α dy

+

∫Ω

((u′ − u) ·α+ (u− u′) ·α′

)α dy

= −2∫Ω

((u′ − u)(α ·α)−

((u′ − u) ·α

)α)dy

= −2∫Ω

(α×

((u′ − u)×α

))dy,

where, for the last equality, we have used the vector identity a × (b × c) =b(c · a) − c(a · b). A simple computation show that the last expression isequal to C

(C∗(u)

)so that (12d) is proved. �

Functions of the form C∗(u) do not entirely comprise the null space ofthe operator D. In fact, for any antisymmetric two-point function ν(x,y), wehave D

(ν)

= 0. However, functions of the form C∗(u) are the only symmetrictwo-point functions belonging to the null space of D. Analogous statementscan be made for the null space of the operator C and two-point functions ofthe form D∗(u).

The four identities in (12) are analogous to the vector identities associatedwith the differential divergence, gradient and curl operator:

∇ · (∇× u) = 0, ∇× (∇u) = 0, ∇ · u = tr(∇u),−∇ · (∇u) +∇(∇ · u) = ∇× (∇× u),


respectively. Because (∇·)∗ = −∇, ∇∗ = −∇·, and (∇×)∗ = (∇×), theseidentities can be written in the form

∇ ·((∇×)∗u)

)= 0, ∇×

((∇·)∗u)

)= 0, ∇∗u = tr

((∇·)∗u

),

∇ ·((∇·)∗u

)−∇

((∇)∗u

)= ∇×

((∇×)∗u

),

which more directly correspond to (12a)–(12d), respectively. In particular,the identities (12a), (12b), and (12d) suggest that D∗, G∗, and C∗ can alsobe viewed as nonlocal analogs of the differential divergence, gradient andcurl operators that, when operating on point functions, result in two-pointfunctions.8

Another set of results for the nonlocal operators that mimic obvious prop-erties of the corresponding differential operators are given in the followingproposition whose proof is a straightforward consequence of (6) and the def-initions of the operators involved.

Proposition 2 Let b and b denote a constant scalar and vector, respectively.Then, the nonlocal adjoint divergence, gradient, and curl operators satisfy

D∗(b) = 0, G∗(b) = 0, and C∗(b) = 0.

Moreover, these three relationships are equivalent to the three nonlocal inte-gral theorems (2a)–(2c). �

These results do not hold for the point divergence, gradient, and curloperators, e.g., D(b), G(b), and C(b) do not necessarily vanish for constantsb and constant vectors b. However, there exist special situation for whichD(b) = 0 holds, and likewise for the other point operators. One such caseoccurs whenever Ω is symmetric with respect to x = y.

2.5.3 Nonlocal curl operators in two and higher dimensions

The nonlocal point and two-point curl operators defined in (1c) and (5c), re-spectively, were defined in three dimensions. These definitions can be gener-alized to arbitrary dimensions by replacing the vector cross product with thewedge product so that, e.g., instead of (5c) we would have C∗(w) : Ω ×Ω →Rr given by

C∗(w)(x,y) := γ ∧ (w′ −w),

where γ(x,y) : Ω ×Ω → Rr with r a positive integer. This suggests a possi-ble exterior calculus-based formalism. However, such a formalism is beyondthe scope of this paper so that only the special cases of r = 3 and lowerdimensions are considered and the vector cross product is retained.

We can also define nonlocal point and two-point curl operators in twospace dimensions, i.e., for Ω ⊂ R2; in fact, we have two types of each kind of

8 Note that G∗(C(µ))6= 0 and C∗

(G(η))6= 0 because of the nonlocality of the

operators.

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curl operator.9 First, we assume, without loss of generality, that µ · e3 = 0,w · e3 = 0, and γ · e3 = 0 in (1c) and (5c). Then, for a vector two-pointfunction µ(x,y) : Ω × Ω → R2, we can view C

(µ)

as the nonlocal scalarpoint function defined by

C(µ)(x) =

∫Ω

((µ2 + µ

′2)γ1 − (µ1 + µ′1)γ2

)dy for x ∈ Ω

and, for a vector point function w : Ω → R2, we can view C∗(w) as thenonlocal scalar two-point function defined by

C∗(w)(x,y) = (w′2 − w2)γ1 − (w′1 − w1)γ2 for x,y ∈ Ω.

Next, assume, again without loss of generality, that µ = µe3, w = we3, andγ ·e3 = 0 in (1c) and (5c). Then, for a scalar two-point function µ(x,y) : Ω×Ω → R, we can view C(µ) as the nonlocal vector point function defined by

C(µ)(x) :=∫Ω

(µ+ µ′)(γ2e1 − γ1e2) dy for x ∈ Ω

and, for a scalar point function w(x) : Ω → R, we can view C∗(w) as thenonlocal vector two-point function defined by

C∗(w)(x,y) = (w′ − w)(γ2e1 − γ1e2) for x,y ∈ Ω.

3 Constraint regions, constraint operators, andvolume-constrained problems

In addition to the divergence, gradient, and curl operators and integralsover a region in Rn, the theorems and identities of the vector calculus fordifferential operators also involve operators acting on functions defined onthe boundary of that region and integrals over that boundary surface.10 The

9 This is analogous to the two types of curl operators in two dimensions, oneoperating on vectors the other on scalars, given by

curlu =∂u1∂x2− ∂u2∂x1

and curlu = − ∂u∂x2

e1 +∂u

∂x1e2.

10 For example, given a region Ω ⊂ Rn having boundary ∂Ω, the divergencetheorem for a vector-valued function u states that∫

Ω

∇ · u dx =∫∂Ω

u · n dx

and the Green’s (generalized) first identity for scalar functions u and v states that,for tensor-valued “constitutive” functions Θ,∫

Ω

u∇ · (Θ∇v) dx +∫Ω

∇u · (Θ∇v) dx =∫∂Ω

u(Θ∇v) · n dx.

The nonlocal divergence theorem (2a) should correspond to the first of these rela-tions and the nonlocal Green’s first identity (9a) should correspond to the second.However, we see that neither (2a) or (9a) contain terms that correspond to theboundary integrals in the above two relations.


nonlocal theorems and identities developed in Section 2 do not contain termsanalogous to the boundary integrals found in the corresponding differentialoperator versions nor do they involve auxiliary boundary operators. However,by viewing boundary operators in the vector calculus for differential operatorsas constraint operators,11 it is a simple matter to rewrite the nonlocal vectortheorems and identities so that they do include such terms. The reason it wasnot necessary to introduce constraint operators and “boundary” integrals inthe theorems and identities of the nonlocal vector calculus is that, in thenonlocal case, “boundary” operators must operate on functions defined overmeasurable volumes, and not on lower-dimensional manifolds [12,20]. As aresult, the actions of these operators are, in a real sense, indistinguishablefrom those of the nonlocal operators we have already defined, except for theresulting domains.

3.1 Constraint regions

We divide the region Ω into disjoint open subsets Ωs and Ωc, i.e., we haveΩ = (Ωs∪Ωc)∪(Ωs∩Ωc), where ( · ) denotes the closure. We refer to Ωs as thesolution domain12 and Ωc as the constraint domain. Note that no relation isassumed between Ωs and Ωc so that, e.g., the four configurations of Figure 1are possible. The bottom-left sketch in that figure depicts a constraint regionthat is a strip following the boundary of Ωs. Not surprisingly, this particularsituation is most closely related to boundary surfaces, but, in the nonlocalcase, constraint regions may take other guises, as depicted in Figure 1. This isanother reason why we first introduced the nonlocal vector calculus withoutsuch regions or without constraint operators.

3.2 Nonlocal point constraint operators

The constraint operators corresponding to each of the nonlocal point op-erators defined in Definition 1 are defined as follows, where the two-pointfunctions α, β, Ψ , η, etc. are defined as in Definition 1.13 Note that eachdefinition also involves redefining, from Ω to Ωs, the domains resulting fromthe actions of the nonlocal point operators D, G, and C.

Definition 4 [Nonlocal point constraint operators] Let Ω = (Ωs ∪Ωc)∪(Ωs∩Ωc). Corresponding to the point divergence operator D

(ν)

: Ωs →

11 In addition to trying to mimic more closely the theorems and identities of thevector calculus for differential operators, we introduce constraint regions and con-straint operators because they are needed to describe nonlocal volume-constrainedproblems and showing their well posedness; see Section 3.5 for examples of nonlocalvolume-constrained problems.12 Why we use the terminology “solution domain” and “constraint domain” willbecome clear when, in Section 3.5, we introduce examples of nonlocal volume-constrained problems that are analogous to differential boundary-value problems.13 A constraint operator corresponding to the nonlocal divergence operator fortensors defined in (11a) can be defined in an entirely analogous manner.

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Fig. 1 Four of the possible configurations for Ω = (Ωs ∪ Ωc) ∪ (Ωs ∩ Ωc), whereΩs and Ωc denote the solution and constraint domains, respectively.

R defined as in (1a) but for x restricted to Ωs, we have the point constraintoperator N (ν) : Ωc → R defined as

N (ν)(x) := −∫Ω

(ν + ν′) ·α dy for x ∈ Ωc. (13a)

Corresponding to the point gradient operator G(η)

: Ωs → Rk defined asin (1b) but for x restricted to Ωs, we have the point constraint operatorS(η) : Ωc → Rk defined as

S(η)(x) := −∫Ω

(η + η′)β dy for x ∈ Ωc. (13b)

Corresponding to the point curl operator C(µ)

: Ωs → R3 defined as in (1c)but for x restricted to Ωs, we have the point constraint operator T (µ) : Ωc →R3 defined as

T (µ)(x) := −∫Ω

γ × (µ+ µ′) dy for x ∈ Ωc. � (13c)

Note that the definitions of the point operators and the correspondingpoint constraint operators are defined using the same integral formulas butthe former is defined for x ∈ Ωs and the latter for x ∈ Ωc.

3.3 Nonlocal integral theorems with constraint operators

It is now a trivial matter to rewrite the nonlocal integral theorems given inTheorem 1 and the nonlocal integration by parts formulas given in Corollary1 in terms of the nonlocal point operators and point constraint operators.We simply take advantage of the fact that the definitions of the two types ofoperators have similar definitions.

Theorem 2 [Nonlocal integral theorems with constraint operators]Assuming the notations and definitions found in Definition 4, we have the


following analogs of the integral theorems of the vector calculus for differentialoperators: ∫

Ωs

D(ν)dx =

∫Ωc

N (ν) dx (14a)∫Ωs

G(η)dx =

∫Ωc

S(η) dx (14b)∫Ωs

C(µ)dx =

∫Ωc

T (µ) dx. � (14c)

Corollary 5 [Nonlocal integration by parts formulas with constraintoperators] Adopt the hypotheses of Theorem 2. Then,∫

Ωs

uD(ν) dx +∫Ω

∫Ω

((u′ − u)ξ

)· ν dydx =

∫Ωc

uN (ν) dx (15a)

∫Ωs

v · G(η) dx +∫Ω

∫Ω

((v′ − v) · β

)η dydx =

∫Ωc

v · S(η) dx (15b)

∫Ωs

w · C(µ) dx−∫Ω

∫Ω

γ ×((w′ −w)

)· µ dydx

=

∫Ωc

w · T (µ) dx. �(15c)

3.4 Nonlocal adjoint operators and Green’s identities with constraintoperators

With the definition of the point constrained operators and the redefinitionof the point operators, the definition of adjoint operators given in (4) can berewritten as (

G,L(F ))Ωs−(L∗(G), F

)Ω×Ω =

(G,B(F )

)Ωc, (16)

where B is an operator that maps two-point functions F to point functionsdefined over Ωc. As a result, the adjoint operators D∗, G∗, and C∗ given inCorollary 2 remain unchanged. We can then rewrite nonlocal integration byparts formulas with constraint operators equations given by Corollary 5 asfollows:∫

Ωs

uD(ν)dx−

∫Ω

∫Ω

D∗(u)· ν dydx =

∫Ωc

uN (ν) dx (17a)∫Ωs

v · G(η)dx−

∫Ω

∫Ω

G∗(v)η dydx =

∫Ωc

v · S(η) dx (17b)∫Ωs

w · C(µ)dx−

∫Ω

∫Ω

C∗(w)· µ dydx =

∫Ωc

w · T (µ) dx. (17c)

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The abstract Green’s first and second identities (7) and (8) now take theform(

G,L(ΘL∗(H)))Ωs−(L∗(G), ΘL∗(H)

)Ω×Ω =

(G,B(ΘL∗(H))

)Ωc

(18)

and (G,L

(ΘL∗(H)

))Ωs−(H,L

(ΘL∗(G)

))Ωs

=(G,B

(ΘL∗(H)

))Ωc−(H,B

(ΘL∗(G)

))Ωc,

(19)

respectively. Likewise, the nonlocal Green’s first identities (9a)–(9c) now re-spectively take the form∫

Ωs

uD(Θ · D∗(v)

)dx−

∫Ω

∫Ω

D∗(u) ·Θ · D∗(v)ν dydx

=

∫Ωc

uN(Θ · D∗(v)

)dx

(20a)

∫Ωs

v · G(θG∗(u)

)dx−

∫Ω

∫Ω

θG∗(v)G∗(u) dydx

=

∫Ωc

v · S(θG∗(u)

)dx

(20b)

∫Ωs

w · C(Θ : C∗(u)

)dx−

∫Ω

∫Ω

C∗(w) : Θ : C∗(u) dydx

=

∫Ωc

w · T(Θ : C∗(u)

)dx

(20c)

and the nonlocal Green’s second identities (10a)–(10c) now respectively takethe form∫

Ωs

uD(Θ · D∗(v)

)dx−

∫Ωs

vD(Θ · D∗(u)

)dx

=

∫Ωc

uN(Θ · D∗(v)

)dx−

∫Ωc

vN(Θ · D∗(u)

)dx

(21a)

∫Ωs

v · G(θG∗(u)

)dx−

∫Ωs

u · G(θG∗(v)

)dx

=

∫Ωc

v · S(θG∗(u)

)dx−

∫Ωc

u · S(θG∗(v)

)dx

(21b)

∫Ωs

w · C(Θ : C∗(u)

)dx−

∫Ωs

u · C(Θ : C∗(w)

)dx

=

∫Ωc

w · T(Θ : C∗(u)

)dx−

∫Ωc

u · T(Θ : C∗(v)

)dx.

(21c)


3.5 Examples of nonlocal volume-constrained problems

The nonlocal point operators, the corresponding nonlocal adjoint operators,and the corresponding nonlocal constraint operators given in (1), (5), and(13), respectively, can be used to define nonlocal volume-constrained prob-lems. Here, we merely state problems involving scalar and vector “second-order” operators.

Let Ωc = Ωc1 ∪ Ωc2, where14 Ωc1 ∩ Ωc2 = ∅, and let Θ4, Θ2, and θ de-note fourth-order tensor, second-order tensor, and scalar two-point functions,respectively. The nonlocal volume-constrained problems

D(Θ2 · D∗(u)

)= f in Ω

u = g in Ωc1

N(Θ2 · D∗(u)

)= h in Ωc2

(22a)

Dt(Θ4 : D∗t (u)

)= f in Ω

u = g in Ωc1

Nt(Θ4 : D∗t (u)

)= h in Ωc2

(22b)

C(Θ4 : C∗(u)

)+ G

(θG∗(u)

)= f in Ω

u = g in Ωc1

T(Θ4 : C∗(u)

)+ S

(θG∗(u)

)= h in Ωc2

(22c)

are analogous to the second-order differential boundary-value problems−∇ ·

(K2 · ∇u

)= f in Ω

u = g on ∂Ω1(K2 · ∇u

)· n = h on ∂Ω2

(23a)

−∇ ·

(K4 : ∇u

)= f in Ω

u = g on ∂Ω1(K4 : ∇u

)· n = h on ∂Ω2

(23b)

∇×

(K4 : ∇× u

)−∇

(k∇ · u

)= f in Ω

u = g on ∂Ω1

n×(K4 : ∇× u

)= h1

k∇ · u = h2

}on ∂Ω2,

(23c)

respectively, where ∂Ω = ∂Ω1 ∪ ∂Ω2 denotes the boundary of Ω with ∂Ω1 ∩∂Ω2 = ∅ and where K4, K2, and k denote fourth-order tensor, second-ordertensor, and scalar point functions, respectively.

A special form of the nonlocal volume-constrained problem (22a) corre-sponding to a scalar-valued solution is studied in [12] by appealing to a vari-ational formulation. Well-posedness results are provided there for the case in

14 Either Ωc1 or Ωc2 may be empty.

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which the natural energy space (used to define the variational problem) isequivalent to L2(Ω), the space of square integrable functions. A rigorous con-nection to (23a) is demonstrated as well for such a case. Although the notionof a nonlocal vector calculus is not introduced, see the recent book [2] wherethe authors and their collaborators collect their substantial recent work onnonlocal diffusion and its relationship to (23a); in that work, the nonlocaloperator in (22a) represents the infinitesimal generator. The two papers [3,10] also consider various properties of the nonlocal operator in (22a).

As shown in [7], more generally, the natural energy space associated withthe nonlocal operators used in (22a) may be a strict subspace of L2(Ω),for instance a fractional Sobolev space. For the latter cases, the nonlocalvariational problems possess smoothing properties akin to that for ellipticpartial differential equations but with possibly reduced order [7].

4 Identification of nonlocal operators with differential operatorsin a distributional sense

In this section, we identify, in a distributional sense, nonlocal operators withtheir differential counterparts. For the sake of brevity, we mostly consider thenonlocal point divergence operator D and its adjoint operator D∗. However,analogous results also hold for the nonlocal gradient and curl operators Gand C, respectively, and their adjoints. Further connections between the non-local and differential operators are made in Section 5 where we demonstratecircumstances under which the nonlocal point operators are weak represen-tations of the divergence, gradient, and curl differential operators.

We begin with the identification of the nonlocal point divergence operatorD with the divergence differential operator ∇·. This identification is subjectto the understanding that the former operates on two-point functions whilethe latter operates on point functions.

Proposition 3 Let ν ∈ C∞0 (Ω×Ω), i.e., ν belongs to the space of compactlysupported infinitely differentiable functions over the domain Ω×Ω. Select the(antisymmetric) distribution

α(y,x) = −∇yδ(y − x), (24a)

where ∇ and δ(y−x) denote the differential gradient and Dirac delta distri-bution, respectively. Then,

D(ν)(x) = ∇x · ν(x,x), (24b)

where ∇· denotes the differential divergence operator. In particular, we have

D(u(x) + u(y)

)(x) = ∇x · u(x). (24c)


Proof We have

∇x · ν(x,x) =(∇x · ν(x,y)

)|y=x +

(∇y · ν(x,y)

)|y=x

=(∇y · ν(y,x)

)|y=x +

(∇y · ν(x,y)

)|y=x

=(∇y ·

(ν(y,x) + ν(x,y)

))|y=x

=

∫Ω

∇y ·(ν(y,x) + ν(x,y)

)δ(y − x) dy

= −∫Ω

(ν(y,x) + ν(x,y)

)∇yδ(y − x) dy

= D(ν)(x)

so that the relations (24b) and (24c) now follow. �

Next, we identify the nonlocal Gauss theorem (2a) with the classicalGauss theorem for functions that are compactly supported within Ω.

Corollary 6 The nonlocal Gauss theorem (2a) reduces to∫∂Ω

ν(x,x) · nx = 0.

Proof The result is immediate from (24b) and (2a). �

The following proposition shows that the average of the action of D∗ on apoint function can be related, in a distributional sense, to the action of −∇on that function.

Proposition 4 Let u ∈ C∞0 (Ω) and select α as in (24b). Then,∫Ω

D∗(u) dy = −∇xu.

Proof Using (5a) we have that∫Ω

D∗(u) dy =∫Ω

(u(y)− u(x)

)∇yδ(y − x) dy

= −∫Ω

δ(y − x)∇y(u(y)− u(x)

)dy

= −∫Ω

δ(y − x)∇yu(y) dy = −∇xu(x) = (∇x·)∗u(x). �

Now consider the composition of the nonlocal divergence operator andits adjoint, i.e., D

(D∗), which, according to the next proposition, can be

identified, in the sense of distributions, with the Laplace operator ∆ = ∇·∇.

Proposition 5 Let u ∈ C∞0 (Ω) and select |α(x,y)|2 = 12∆yδ(y − x), where∆ denotes the differential Laplace operator. Then,

D(D∗u(x)

)= −∆xu(x).

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Proof Using (1a) and (5a), we have

D(D∗u

)(x) = −2

∫Ω

(u(y)− u(x)

)|α(x)|2 dy

= −∫Ω

(u(y)− u(x)

)∆yδ(y − x) dy

= −∫Ω

δ(y − x)∆y(u(y)− u(x)

)dy

= −∫Ω

δ(y − x)∆yu(y) dy

= −∆xu(x) = −∇x · ∇xu(x) = (∇x·)(∇x·)∗u(x),

where the differential Green’s second identity is used for the third equality.�

An immediate application is to relate the average of the product of operatorsD∗(·) · D∗(·) to the weak Laplacian ∇x · ∇x in a distributional sense.

Corollary 7 Green’s first identity (9a) with Θ = I reduces to∫Ω

∇xu · ∇xv dx =∫Ω

∫Ω

D∗(u) · D∗(v) dydx.

5 Relations between weighted nonlocal operators and weakrepresentations of differential operators

The conventional differential calculus only involves operators mapping pointfunctions to point functions; on the other hand, the nonlocal operators de-fined earlier map two-point functions to point functions or, in case of thenonlocal adjoint operators, point functions to two-point functions. To furtherdemonstrate that the nonlocal operators correspond to nonlocal versions ofthe conventional divergence, gradient, and curl differential operators, we usethe nonlocal operators D, G, and C to define, in Section 5.1, correspond-ing weighted operators that map point functions to point functions. We alsoshow that the adjoint operators corresponding to the weighted operators areweighted integrals of the nonlocal adjoint operators D∗, G∗, and C∗. Then,in Section 5.2, the weighted operators are shown to be nonlocal versions ofthe corresponding differential operators.

5.1 Nonlocal weighted operators

The nonlocal point operators defined in (1) induce new operators, referredto as weighted operators.

Definition 5 Let ω(x,y) : Ω ×Ω → R denote a non-negative scalar-valuedtwo-point function. Let the operators D, G, and C be defined as in (1). Then,


given the point function u(x) : Ω → Rn, the weighted nonlocal divergenceoperator Dω(u) : Ω → R is defined by

Dω(u)(x) := D(ω(x,y)u(x)

)(x) for x ∈ Ω. (25a)

Given the point function u(x) : Ω → R, the weighted nonlocal gradient oper-ator Gω(u) : Ω → Rk is defined by

Gω(u)(x) := G(ω(x,y)u(x)

)(x) for x ∈ Ω. (25b)

Given the point function w(x) : Ω → R3, the weighted nonlocal curl operatorCω(u) : Ω → R3 is defined by

Cω(w)(x) := C(ω(x,y)w(x)

)(x) for x ∈ Ω. � (25c)

The following result shows that the adjoint operators corresponding tothe weighted nonlocal operators are determined as weighted integrals of thecorresponding nonlocal two-point adjoint operators (5).

Proposition 6 Let ω(x,y) : Ω × Ω → R denote a non-negative scalar two-point function and let D∗, G∗, and C∗ denote the nonlocal adjoint operatorsgiven in (5). The adjoint operator D∗ω(u)(x) : Ω → Rk corresponding to theweighted nonlocal divergence operator Dω is given by

D∗ω(u)(x) =∫Ω

D∗(u)(x,y)ω(x,y) dy for x ∈ Ω (26a)

for scalar point functions u(x) : Ω → R. The adjoint operator G∗ω(v)(x) : Ω →R corresponding to the weighted nonlocal gradient operator Gω is given by

G∗ω(v)(x) =∫Ω

G∗(v)(x,y)ω(x,y) dy for x ∈ Ω (26b)

for vector point functions v(x) : Ω → Rn. The adjoint operator C∗ω(w)(x) : Ω→ R3 corresponding to the weighted nonlocal curl operator Cω is given by

C∗ω(w)(x) =∫Ω

C∗(w)(x,y)ω(x,y) dy for x ∈ Ω (26c)

for vector point functions w(x) : Ω → R3.

Proof We have that(Dω(u), u

)Ω

=

∫Ω

u(x)D(ω(x,y)u(x)

)dx

=

∫Ω

∫Ω

D∗(u)(x,y) ·(ω(x,y)u(x)

)dydx

=

∫Ω

(∫Ω

ω(x,y)D∗(u)(x,y) dy)· u(x) dx,

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where (25a) is used for the first equality and (4) and (5a) for the second.But, by definition, the adjoint operator D∗ω(·) corresponding to the operatorDω(·) satisfies (

u,Dω(u))Ω

=(D∗ω(u),u

)Ω.

Comparing the last two results yields (25a). The conclusions (25b) and (25c)are derived in a similar fashion. �

In direct analogy to the results of Section 2.5.1, we extend 25a and 26ato tensors and vectors, respectively.

Definition 6 Let ω(x,y) : Ω × Ω → R denote a non-negative scalar two-point function and let Dt be given as in (11b). Given the tensor point functionU(x) : Ω → Rn×k, the weighted nonlocal divergence operator Dt,ω(U) : Ω →Rn for tensors is defined by

Dt,ω(U)(x) := Dt

(ω(x,y)U(x)

)(x) for x ∈ Ω. � (27)

Corollary 8 The adjoint operator D∗t,ω(u)(x) : Ω → Rn×k corresponding tothe operator Dt,ω is given by

D∗t,ω(u)(x) =

∫Ω

D∗t (u)(x,y)ω(x,y) dy for x ∈ Ω (28)

for vector point functions u : Ω → Rn. �

5.2 Relationships between weighted operators and differential operators

We have seen that the nonlocal operators satisfy many properties that mimictheir differential counterparts. In particular, we established, in Section 4,that the nonlocal point operators can be identified with the correspondingdifferential operators in a distributional sense. In this section, we establishrigorous connections between the weighted operators and their differentialcounterparts by first introducing a cut-off or horizon parameter ε > 0 andthen analyzing what occurs as ε→ 0, that is, in the local limit. To this end,we define

Bε(x) := {y ∈ Rd : |y − x| < ε}.

Let Ω = Rd and let φ denote a positive radial function; selectα(x,y) =

y − x|y − x|

for x 6= y

ω(|x− y|) =

{|y − x|φ(|y − x|) y ∈ Bε(x)0 otherwise

(29a)

with the function φ satisfying a normalization condition∫Bε(x)

|y − x|2φ(|y − x|) dy = d, (29b)


where d denotes the space dimension.Note that α is an antisymmetric function whereas ω is a symmetric func-

tion. We then have, for a scalar function u, that the components of Gω(u)

and

D∗ω(u)

given by (25b) and (26a), respectively, are given by, for j = 1, . . . , d,

dju(x) :=

∫Bε(0)

(u(x + z) + u(x)

)zj φ(|z|) dz (30a)

d∗ju(x) := −∫Bε(0)

(u(x + z)− u(x)

)zj φ(|z|) dz, (30b)

where zj denotes the j-th component of z.The following result ensues.

Lemma 1 Assume that ω and α are defined as in (29a). Let u : Rd → Rand let dju(x) and d

∗ju(x) denote the j-th components of Gω(u) and D∗ω(u),

respectively. Then,dju = −d∗ju. (31)

Proof From (30) we have

dju(x) + d∗ju(x) = −2u(x)

∫Bε(0)

yjφ(|y|)dy = 0

from which the result directly follows. �

Based on this lemma and the definition of the weighted operators, wehave the following results.

Corollary 9 Under the same conditions as in Lemma 1, we have

Dω = −G∗ω, Gω = −D∗ω, and Cω = C∗ω. � (32)

Thus, under the hypotheses of this corollary, the identities in (32) mimictheir counterparts in the vector calculus for differential operators. Identitiessuch as those in Proposition 1 do not generally hold for the weighted operatorsdue to their nonlocal properties.15

Synergistic with the results of Section 4, we have the following.

Corollary 10 Select the singular distribution

zjφ(|z|) = −∂jδ(z),

where ∂ju denotes the weak derivative of u with respect to xj. Then,

dj = ∂j and d∗j = −∂j . �

The following lemma demonstrates that the components of the weightedgradient operator Gω or, by (32),D∗ω, converge, as ε→ 0, to the correspondingspatial derivatives.

15 The identities in Lemma 1 and Corollary 9 also hold for more general choicesof α and ω, namely, those such that α is anti-symmetric and ω is a radial functionwith support in Bε(x).

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Proposition 7 Let ω be defined as in (29a) and satisfy (29b). Then, forj = 1, . . . , d, the weighted operators dj and d

∗j defined by (30) are bounded

linear operators from H1(Rd) to L2(Rd). Moreover, if u ∈ H1(Rd), then asε→ 0

‖dju− ∂ju‖L2(Rd) → 0 (33a)‖d∗ju+ ∂ju‖L2(Rd) → 0, (33b)

where ∂ju denotes the weak derivative of u with respect to xj. Moreover, if∫Bε(0)

|z|1+sφ(|z|) dz


If u ∈ H1(Rd), we have that

|d̂ju− ∂̂ju| ≤ |d̂ju|+ |∂̂ju|

≤∫Bε(0)

∣∣ sin(yjξj) cos(∑k 6=j

ykξk) û(ξ) yjφ(|y|)∣∣ dy + |ξj û|

≤ |ξj û|∫Bε(0)

y2jφ(|y|) dy + |ξj û|

≤ 2|ξj û| ∈ L2(Rd),

where the third inequality holds because | sin(x)| ≤ |x| and | cos(x)| ≤ 1. ByTaylor’s theorem, we have

d̂ju = i

∫Bε(0)

(yjξj + y3j ξ

3j cos(θ1)/6)(1 + (

∑k 6=j

ykξk)2 cos(θ2)/2) û yjφ(|y|) dy

= iξj û

∫Bε(0)

y2jφ(|y|) dy + Iε(ξ) = iξj û+ Iε(ξ),

where, for some θ1 and θ2,

Iε(ξ) :=i

6

∫Bε(0)

y4j cos(θ1)φ(|y|) dy ξ3j û

+i

2

∫Bε(0)

y2j (∑k 6=j

yiξi)2 cos(θ2)φ(|y|) dy ξj û

+i

12

∫Bε(0)

y4j (∑k 6=j

ykξk)2 cos(θ1) cos(θ2)φ(|y|) dy ξ3j û.

Because û is bounded a.e., we see that, for any ξ,

|Iε(ξ)| ≤ ε2|ξ3j û|+ ε2|(∑k 6=j

ξk)2ξj û|+ ε4|(

∑k 6=j

ξk)2ξ3j û| → 0 as ε→ 0.

Hence, we have

d̂ju→ ∂̂ju a.e. for ξ ∈ Rd as ε→ 0.

Then, by the dominated convergence theorem, we obtain

‖dju− ∂ju‖L2(Rd) → 0 as ε→ 0.

Lemma 1 implies the same results hold for the operator d∗j . �

The above lemma implies that if ω satisfies (33c) for s = 0, then theweighted operators are bounded operators from L2(Rd) to L2(Rd). Moregenerally, for φ satisfying (33c) with positive s, the operators dj and d

∗j

actually map a subspace of L2(Rd), for instance the fractional Sobolev spaceHs(Rd), to L2(Rd), or even map L2(Rd) to H−s(Rd). We refer to [7] forrelated work.

A direct consequence of Lemma 7 is the following result.

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Proposition 8 Under the condition of Lemma 7, the weighted operators Dω,Gω, and Cω and their adjoint operators D∗ω, G∗ω, and C∗ω are bounded linearoperators from Ht(Rd) to Ht−s(Rd) for 0 ≤ s ≤ 1, where d = 3 for theweighted curl operators. Moreover, if u ∈ H1(Rd) and u ∈ [H1(Rd)]d,

Dω(u)→ ∇ · u D∗ω(u)→ −∇u

Gω(u)→ ∇u G∗ω(u)→ −∇ · u

Cω(u)→ ∇× u C∗ω(u)→ ∇× u,

(34)

where the convergence as ε→ 0 is with respect to L2(Rd).

Proof We prove the convergence of the operator Dω(u) to the divergencedifferential operators. First, note that

Dω(u) =d∑i=1

diui and ∇ · u =d∑i=1

∂iui

so that

‖Dω(u)−∇ · u‖L2(Rd) ≤d∑i=1

‖diui − ∂iui‖L2(Rd) → 0 as ε→ 0.

The remaining results can be proved in a similar fashion. �

For the purpose of discussing nonlocal equations, we also need to considercombinations of the weighted operators such as Dω(C1D∗ω(u)), where C1(x)is a tensor point function that, for example, describes the point property ofa material. Note that the two-point property is involved in the definition ofthe weighted operators. In the next corollary, we illustrate that whenever thehorizon ε goes to zero, the combinations of the weighted operators convergeto their local counterparts.

Corollary 11 Let u ∈ H1(Rd), u ∈ [H1(Rd)]d, C1 : Rd → Rd × Rd inL∞(Rd × Rd), and c2 : Rd → R in L∞(Rd). Then,

Dω(C1 · D∗ω(u)

)→ −∇ · (C1 · ∇u)

Gω(c2G∗ω(u)

)→ −∇(c2∇ · u)

Cω(C1 · C∗ω(u)

)→ ∇×

(C1 · (∇× u)

),

(35)

where the convergence as ε→ 0 is with respect to H−1(Rd).16

Proof Using Proposition 8 and a similar method of proof as for Lemma 7,the results are obtained. �16 Similar results can be obtained for the nonlocal divergence operator on tensors;in particular, we have that Dt,ω(C3 : D∗t,ω(u)) → −∇ · (C3 : ∇u) with C3 : Rd →Rd × Rd × Rd × Rd in L∞(Rd × Rd × Rd × Rd).


Let L denote a linear operator that commutes with the differential andnonlocal operators. We then have the following result.

Lemma 2 Let L denote a linear operator that commutes with the differentialand nonlocal operators. Then, if Lu ∈ H1(Rd),

Dω(Lu)→ ∇ · Lu D∗ω(Lu)→ −∇LuGω(Lu)→ ∇Lu G∗ω(Lu)→ −∇ · LuCω(Lu)→ ∇×Lu C∗ω(Lu)→ ∇×Lu,

(36)

where the convergence as ε→ 0 is with respect to L2(Rd).

If L is selected as a differential operator with constant coefficients, orits formal inverse, we can observe convergence in either stronger or weakernorms.

6 Nonlocal balance laws

Sections 2 and 3 introduced a nonlocal vector calculus. Combined with theanalytical theory in sections 4 and 5, a potentially powerful formalism forinstantiating the nonlocal balance laws of continuum mechanics is now athand. We start by discussing the notion of an abstract balance law as givenby

P(Ω) = Q(∂Ω), Ω ⊂ Rd (37)

postulating that the production of an extensive quantity over the open sub-region Ω is equal to the flux of the quantity through the boundary ∂Ω ofthat subregion. The additive quantities P and Q are expressed in terms oflinear functionals over Ω and ∂Ω, respectively.

The purpose of this section is to demonstrate that the flux is given bythe volume interaction between Ω and Rn \Ω. Subsection 6.1 demonstratesthat the flux induced by the peridynamic nonlocal continuum theory is givenby a volume interaction and is elegantly phrased using the nonlocal calcu-lus. However, assuming sufficient regularity, the volume interaction may berelegated to that on the surface ∂Ω. In particular, suppose that

Q(∂Ω) =∫∂Ω

q̆ dHd−1, (38)

where q̆ and Hd−1 are the flux density and a Hausdorff measure over the sur-face ∂Ω, respectively; see [5, Chap. II] for a discussion. Then, under generalconditions, ∫

∂Ω

q̆ dHd−1 =∫∂Ω

q · n dHd−1 =∫Ω

divq dx (39)

for some vector function q, where n denotes the outward unit normal vectorfor Ω. The relations (38) and (39) are abstract versions of Cauchy’s postu-late and theorem, respectively. The relation (39) establishes the fundamental

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result that the flux density q̆ at a point on any orientable surface ∂Ω is linearin the outward normal vector n. An important consequence is that the fluxrepresents an interaction between the subregions external and internal to Ω.This is made explicit by defining the bi-additive form

I(Ω,Rd \Ω) := Q(∂Ω) ∀Ω ⊂ Rd. (40a)

The balance law (37) implies that P(Rd) = 0 so that the addivity of P andbi-additivity of I grants the important relation

I(Ω,Rd \Ω) + I(Rd \Ω,Ω) = 0 ∀Ω ⊂ Rd. (40b)

Because Rd \Ω2 = Rd \ (Ω1 ∪Ω2) ∪Ω1 and Rd \Ω1 = Rd \ (Ω1 ∪Ω2) ∪Ω2for all disjoint subregions Ω1 and Ω2, we have

I(Ω1, Ω2) + I(Ω2, Ω1) = I(Ω1,Rd \Ω1) + I(Ω2,Rd \Ω2)− I

(Ω1 ∪Ω2,Rd \ (Ω1 ∪Ω2)

)= 0. (40c)

Then, because the disjoint regions Ω1 and Ω2 are arbitrary,

I(Ω,Ω) = 0 ∀Ω ⊂ Rd. (40d)

We have established that the balance law (37) implies that the flux Q inducesthe alternating (or antisymmetric) bilinear form I. The relations (40b)–(40d)correspond to the physical understanding of a flux. The first relation ex-plains that the interaction of the subregion external to Ω upon Ω is equaland opposed to the interaction of Ω upon Rd \ Ω. The latter two relationsexemplify the action-reaction principle and that there is no self-interactionamong subregions, respectively. For example, whenever (37) denotes the bal-ance of linear momentum, the principle of action-reaction is Newton’s thirdlaw. Additivity of linear momentum implies Newton’s third law and need notbe assumed.

The above discussion explains why the balance law (37) can be equiva-lently replaced by

P(Ω) = I(Ω,Rd \Ω), Ω ⊂ Rd (41)

for the alternating bilinear form I. The crucial observation is that Cauchy’stheorem (39) explains that the flux Q has the algebraic structure of an al-ternating (or antisymmetric) bilinear form encapsulating an interaction. Inparticular, the unit outward normal provides an orientation leading to (40b).Interaction is now determined by subregions and not by contact along theirrespective boundaries.

Introduce the bilinear form

J (Ω1, Ω2) :=∫Ω1

∫Ω2

f(x,y) dy dx ∀Ω1, Ω2 ⊂ Rd, (42a)


where f : Rd × Rd → R denotes an interaction density.17 The bilinear formis alternating (or antisymmetric) if and only if

f(x,y) = −f(y,x) ∀x,y ∈ Rd. (42b)

In stark contrast to I, the flux induced by J dispenses the need for deter-mining a unit normal to an orientable surface and instead considers volumeinteractions among regions. Such a relation and (42a) can also be interpretedmore broadly. Indeed, by prescribing certain continuity assumptions on thebilinear form J with respect to its argument in a suitable topology, the exis-tence of the interaction density f follows from the Schwartz Kernel theorem[19].

We now derive the field equation for the balance law

P̃(Ω) = J (Ω,Rd \Ω), Ω ⊂ Rd. (43a)

A result due to Fuglede [9] as used in [4, p. 298] supplies a production densityρ̃ : Rd → R that satisfies∫

Ω

ρ̃(x) dx =

∫Ω

∫Rd\Ω

f(x,y) dy dx.

The antisymmetry of f grants∫Ω

∫Rd\Ω

f(x,y) dy dx =

∫Ω

∫Rdf(x,y) dy dx (43b)

and the field equation

ρ̃(x) =

∫Rdf(x,y) dy (43c)

follows. In an analogous fashion, the field equation for the balance (41)

ρ = divq

ensues and is understood in the sense of distributions. Because the productionof density in the volume Ω is unique, the relation

divq(x) =

∫Rdf(x,y) dy (43d)

results. Evidently the right-hand side of (43d) is identified with the distribu-tion divq and the flux through ∂Ω is given by∫

∂Ω

q · n dHd−1 =∫Ω

∫Rd\Ω

f(x,y) dy dx

so that the relation (43b) is identified as the divergence theorem. We concludethat the balances (41) and (43a) are equivalent in the sense of distributions.

An important distinction between the two interactions I and J is that,by (39), the former vanishes when the intersection of the closures of the

17 See also [23, p. 85] and [17, p. 29].

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subregion has nonzero intersection, e.g., Ω1∩Ω2 = ∅, whereas the interactionJ may not. The balance (43a) is defined to be nonlocal when the distributiondivq is replaced by the right-hand side of (43d) because points y 6= x areinvolved with respect to the interaction density. On the other hand, when thedistribution divq is such that div can be identified with a linear differentialoperator, say ∇·, and q is suitably defined, the balance (43a) is local.18

We remark that under suitable regularity conditions, a closed form ex-pression for an operator A may be derived such that

∇ · A(f(x,y)

)=

∫Rdf(x,y) dy

is satisfied; see [16,17]. For a variational characterization of the previousrelation within the peridynamic theory, see [14].

6.1 Peridynamic continuum theory

The relation (43d) explains that the role of the function q for the local balanceis now assumed by the interaction density f = f(x,y) which is characterizedas a two-point function given its dependence on x and y. To complete thedescription of a balance law for a physical system, we assume further that theinteraction density results from a mapping of an underlying field given by apoint function. This mapping represents the constitutive relation describingthe specific system.

We now demonstrate that the nonlocal calculus developed in the earliersections is synergistic with the peridynamic nonlocal theory of continuummechanics. Suppose that the scalar interaction density f = f(x,y) is replacedby the vector interaction density f = f(x,y). Then a generalization of thelinearized bond-based peridynamic operator introduced by Silling [20] is givenby

−12Dt((D∗tu

)T )=

∫Ω

(α⊗α) · (u′ − u) dy, (44)

where u represents the displacement field, Dtu = Dt(u)

and D∗tu = D∗t(u)

are given by (11a) and (11b), respectively, and the integrand is an interaction

density f = (α⊗α)·(u′−u). A mechanical perspective indicates that(D∗tu

)Tdescribes the deformation of u and that a constitutive relation maps thedeformation to the force density given by the integral operator. Becausethe integrand is antisymmetric with respect to the arguments x and y, theoperator (44) induces an interaction – that of force between subregions.

If we require that the integral operator in (44) only contain only rigidmotion in its null space, e.g.,

Dt((D∗t (Ax + c))T

)= 0, A = −AT , c := constant vector, (45)

18 See [6, pp. 149–150] for a discussion.


then a necessary and sufficient condition for (45) to hold is that α(x,y) =(y − x

)ζ(|y − x|) where ζ : R+ → R. The integral operator is then written

as

−12Dt((D∗tu(x)

)T )=

∫Ω

(y − x

)⊗(y − x

)σ(|y − x|)

·(u(y)− u(x)

)dy, (46)

where σ := ζ−2 and so results in the linearized peridynamic bond-basedforce density. When Ω ≡ Rd, the paper [7] provides analytical conditions onσ describing the amount of smoothing associated with the integral operatorand the well posedness of the balance of linear momentum and associatedequilibrium equation. That paper also demonstrates that as ε→ 0,

−12Dt((D∗tu)T

)→ −µ∇ · (∇u)− 2µ∇(∇ · u),

the Navier operator of linear elasticity with Poisson ratio one-quarter; seealso the paper [8]. The subsequent paper [26] considers volume-constrainedproblems on bounded domains in R and squares in R2.

The state-based peridynamic theory [22] requires the consideration ofboth volumetric and shear deformations. Similar to the above discussion onthe bond-based models, we may also formulate the state-based peridynamicmodel in terms of the nonlocal operators. Let α and ω be given by (29a).Then, the linear state-based peridynamic integral operator [21] is given by

−Dt,ω(η(D∗t,ωu

)T+ (λG∗ωu) I

), (47)

where η and λ are materials constants, G∗ωu = G∗ω(u), Dt,w and D∗t,w are

given by (26b), (27) and (28), respectively. The scalar G∗ωu measures thevolumetric change, or dilatation, in the material so that G∗ωu I is a diago-nal tensor representing volumetric stress. This allows us to readily apply thenonlocal calculus to study the well-posedness of both free-space and volume-constrained linear peridynamic state-based balance laws, and also suggestswhy, in the limit as ε → 0, the operator given by (47) leads to the linearNavier operator of elasticity for linear isotropic materials with general Pois-son ratios, e.g., not relegated to a value of one-quarter. This is the subjectof forthcoming research.

Acknowledgements

The third author thanks Petronela Radu of the University of Nebraska forpointing out the paper [4].

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IntroductionA nonlocal vector calculusConstraint regions, constraint operators, and volume-constrained problemsIdentification of nonlocal operators with differential operators in a distributional senseRelations between weighted nonlocal operators and weak representations of differential operatorsNonlocal balance laws

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